Peierls Bipolarons andLocalization in Solid-State andMolecular SystemsbyJohn SousM.Sc., University of Waterloo, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2018c© John Sous 2018The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, the dissertationentitled:Peierls Bipolarons and Localization in Solid-State and Molecular Sys-temssubmitted by John Sous in partial fulfilment of the requirements for the degreeof Doctor of Philosophy in PhysicsExamining Committee:Co-Supervisor Mona Berciu, PhysicsCo-Supervisor Roman V. Krems, ChemistrySupervisory Committee Member Kirk W. Madison, PhysicsUniversity Examiner Joel Feldman, MathUniversity Examiner Gren N. Patey, ChemistryAdditional Supervisory Committee Members:Supervisory Committee Member George A. Sawatzky, PhysicsSupervisory Committee Member Gordon W. Semenoff, PhysicsiiAbstractIn this thesis, I investigate the behavior of particles dressed by quantum fieldexcitations and random interactions.First I consider two-carrier states in the Peierls model describing the modu-lation of the particle hopping due to lattice distortions. I compute the spectralresponse using the Momentum Average approximation. Combining accurate nu-merical techniques and analytical arguments, I provide a complete picture of thePeierls bipolarons. It is found that polarons bind into strongly bound yet lightbipolarons in the singlet sector, even at large values of the electron-phonon cou-pling strength. At finite electron fillings, these bipolarons may condense into ahigh-Tc superconductor. On the other hand, phonons mediate a repulsive in-teraction in the triplet sector, or equivalently (in one dimension), between twohard-core particles, in which case the ground-state dimers bound by sufficientlyattractive bare interactions exhibit two sharp transitions, one of which is the firstknown example of a self-trapping transition at the two-carrier level. In both situ-ations, phonons mediate pair-hopping effective interactions between the carriers.I further study some aspects of the excited spectrum for the two hard-core par-ticles, a situation relevant to ultracold quantum simulators. It is found that therepulsive phonon-mediated interaction binds a repulsive bipolaron embedded inthe excited spectrum.I then turn to the study of quenched randomness in an ultracold molecularplasma. I argue that the quenched ultracold plasma presents an experimentalplatform for studying quantum many-body physics of disordered systems in thelong-time and finite energy-density limits. I analyze an experiment that quenchesa plasma of nitric oxide to an ultracold system of Rydberg molecules, ions andelectrons that exhibits a long-lived state of arrested relaxation. The qualitativefeatures of this state fail to conform with classical models. I develop a microscopiciiiAbstractquantum description for the arrested phase based on an effective many-body spinHamiltonian that includes both dipole-dipole and van der Waals interactions.This effective model appears to offer a way to envision the essential quantumdisordered non-equilibrium physics of this system.This thesis thus examines the quantum many-body response in interactingsystems coupled to bosonic fields or in disordered environments.ivLay SummaryRecent years have witnessed a dramatic revolution in the experimental study ofquantum many-body systems. The cross-fertilization of ideas across the domainsof quantum science from solid-state to atomic, molecular and optical (AMO)physics has led to a swarm of questions regarding the dynamics of quasiparticlesin systems coupled to the environment and/or in the presence of disorder. Thisthesis is motivated by these questions, and investigates particle-bath interactionsrelevant to a plethora of experimental systems, and the dynamics of strongly dis-ordered systems. It presents recent results on fundamental phenomena pertainingto high-Tc superconductivity in systems with strong electron-lattice coupling, self-localization in interacting systems without disorder, and quantum localization atfinite temperatures in quenched ultracold plasmas. This thesis can be viewedas a step towards the search for room-temperature quantum technologies suchas, phonon-mediated high-Tc superconductors and quantum applications of dis-ordered plasmas.vPrefaceMost of the chapters of this thesis have appeared in print elsewhere. In inversechronological order, the resulting preprints and my contributions to them are:• Many-body physics with ultracold plasmas: Quenched randomness and localiza-tion,John Sous and Ed Grant, arXiv:1808.07479; to appear in New Journal ofPhysics.I developed a theoretical rationale for a minimal model for the arrest state of thequenched plasma and detailed the basis of these ideas. I wrote the manuscriptwith Ed Grant.• Light Bipolarons Stabilized by Peierls Electron-Phonon Coupling,John Sous, Monodeep Chakraborty, Roman V. Krems, and Mona Berciu, Phys-ical Review Letters 121, 247001 (2018).I performed Momentum average (MA) and analytical calculations, and wrotethe draft with the co-authors. A followup of this work is currently in prepara-tion.• Possible Many-Body Localization in a Long-Lived Finite-Temperature UltracoldQuasineutral Molecular Plasma,John Sous and Edward Grant, Physical Review Letters 120, 110601 (2018)I developed a theory model to explain the experimental results obtained in theGrant group, and wrote the manuscript with Ed Grant.• Bipolarons bound by repulsive phonon-mediated interactions,John Sous, Mona Berciu, and Roman V. Krems, Physical Review A 96, 063619(2017).viPrefaceI performed numerical and analytical calculations, and wrote the manuscriptwith assistance from Roman V. Krems and advice from Mona Berciu.• Arrested relaxation in an isolated molecular ultracold plasma,R. Haenel, M. Schulz-Weiling, J. Sous, H. Sadeghi, M. Aghigh, L. Melo, J. S.Keller, and E. R. Grant, Physical Review A 96, 023613 (2017).I helped interpret the experimental results developed in the Grant group, andcontributed to the writing of the manuscript.• Phonon-mediated repulsion, sharp transitions and (quasi)self-trapping in theextended Peierls-Hubbard model,J. Sous, M. Chakraborty, C. P. J. Adolphs, R. V. Krems, and M. Berciu,Scientific Reports 7, 1169 (2017).I performed MA and analytical calculations, and assisted in the writing of themanuscript.Other research projects were completed during the course of the doctoralstudies and are not included in this thesis. In inverse chronological order, theresulting preprints are:• Large Trimers from Three-Body Interactions in One-Dimensional Lattices,Arthur Christianen and John Sous, arXiv:1806.10647 (2018).• Extrapolating quantum observables with machine learning: Inferring multiplephase transitions from properties of a single phase,Rodrigo A. Vargas-Herna´ndez, John Sous, Mona Berciu, and Roman V. Krems,Physical Review Letters 121, 255702 (2018).In the last year, in visits to Harvard University, I worked on quantum impuri-ties in cold atom systems. I am currently finalizing a draft with my collaborators,Richard Schmidt, Eugene Demler, Hossein R. Sadeghpour and Thomas C. Kil-lian, on the quench dynamics induced by a Rydberg impurity atom in a Fermigas. I choose not to include this work and other ongoing projects on impuritiesin cold gases in this thesis.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Particles coupled to quantum fields: Lattice polarons and bipo-larons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Outline of Part I . . . . . . . . . . . . . . . . . . . . . . . 31.2 Quenched randomness in quantum systems at finite tempera-tures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Outline of Part II . . . . . . . . . . . . . . . . . . . . . . 71.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7I Polarons and Bipolarons in the Peierls Model of Lat-viiiTable of Contentstice Particle-Phonon Coupling 92 Peierls versus Holstein and Fro¨hlich particle-phonon couplingand their corresponding polarons . . . . . . . . . . . . . . . . . . 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Particle-lattice interactions . . . . . . . . . . . . . . . . . . . . . 112.2.1 Interactions that modulate the particle’s potential energy 122.2.2 Interactions that modulate the particle’s kinetic energy . 152.3 Polaronic effects and superconductivity . . . . . . . . . . . . . . 182.4 Relevance to experimental systems . . . . . . . . . . . . . . . . . 202.4.1 Solid-state systems . . . . . . . . . . . . . . . . . . . . . . 212.4.2 Ultracold quantum simulators . . . . . . . . . . . . . . . . 233 Methods for studying polarons and bipolarons . . . . . . . . . . 253.1 Variational exact diagonalization . . . . . . . . . . . . . . . . . . 253.2 Momentum Average approximation. . . . . . . . . . . . . . . . . . 263.2.1 Green’s function of the Holstein polaron . . . . . . . . . . 263.2.2 Green’s function of the Peierls polaron and bipolaron . . . 313.3 Asymptotic expansions in the anti-adiabatic regime . . . . . . . . 343.3.1 One-particle sector . . . . . . . . . . . . . . . . . . . . . . 353.3.2 Two-particle sector . . . . . . . . . . . . . . . . . . . . . . 374 Two electrons in the one-dimensional Peierls-Hubbard model 394.1 Motivation: Bipolaronic superconductivity . . . . . . . . . . . . . 394.2 Peierls bipolarons . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.1 U = 0 bipolarons . . . . . . . . . . . . . . . . . . . . . . . 414.2.2 Analytical theory in the anti-adiabatic limit . . . . . . . . 444.2.3 Stability to finite repulsive Hubbard U . . . . . . . . . . . 484.3 Summary and concluding remarks . . . . . . . . . . . . . . . . . 505 Two hard-core particles in the one-dimensional extended Peierls-Hubbard model: A. Ground-state properties . . . . . . . . . . . 525.1 Interplay of electron-phonon coupling and extended Hubbard in-teractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52ixTable of Contents5.2 Phonon-mediated repulsion, sharp transitions and (quasi)self-trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2.1 Two-polaron phase diagram . . . . . . . . . . . . . . . . . 545.2.2 Analytical anti-adiabatic theory . . . . . . . . . . . . . . 555.2.3 Two sharp transitions and (quasi)self-trapping . . . . . . 585.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 Two hard-core particles in the one-dimensional Peierls model:B. Repulsively bound bipolarons . . . . . . . . . . . . . . . . . . 636.1 Non-conservative forces and bound states . . . . . . . . . . . . . 636.2 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 646.3 Repulsive bipolaron . . . . . . . . . . . . . . . . . . . . . . . . . 666.4 Beyond the anti-adiabatic limit . . . . . . . . . . . . . . . . . . . 716.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 74II Disordered Ultracold Molecular Plasma 767 Localization and absence of thermalization . . . . . . . . . . . . 777.1 Ergodicity and quantum mechanics . . . . . . . . . . . . . . . . . 777.2 Thermalization and the Eigenstate Thermalization Hypothesis . . 787.3 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.3.1 Single-particle localization . . . . . . . . . . . . . . . . . . 797.3.2 Many-body localization . . . . . . . . . . . . . . . . . . . 807.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828 Possible many-body localization in a long-lived finite-temperature ultracold quasi-neutral molecular plasma . . . . . 848.1 Quenched ultracold molecular plasma . . . . . . . . . . . . . . . 848.1.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 858.2 Molecular physics of the arrested phase . . . . . . . . . . . . . . 878.3 Effective many-body Hamiltonian . . . . . . . . . . . . . . . . . . 908.4 Induced Ising interactions . . . . . . . . . . . . . . . . . . . . . . 918.5 Discussion: Localization versus glassy behavior and slow dynamics 92xTable of Contents8.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 94III Thesis Outcome 969 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . 979.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 989.2.1 Ideas related to Part I . . . . . . . . . . . . . . . . . . . . 989.2.2 Ideas related to Part II . . . . . . . . . . . . . . . . . . . 100Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102AppendicesA Details of the methods used for polarons and bipolarons . . . 126A.1 Action of the phonon annihilation operator in the equation-of-motion for the Green’s function of the Holstein polaron . . . . . 126A.2 Momentum Average approach to the Peierls two-carrier states . . 127A.2.1 Details of MA for two hard-core particles in the Peierlsmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128A.3 Bogoliubov-Born-Green-Kirkwood-Yvon equation-of-motion. . . . 130A.3.1 BBGKY EOM for two particles in the singlet sector of thePeierls model in the anti-adiabatic limit . . . . . . . . . . 130A.3.2 BBGKY EOM for two hard-core particles in the t-t2-Vmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132B Overview of the ultracold plasma experiment and theoreticalconsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134B.1 Double-resonant production of a state selected molecular Rydberggas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134B.2 Selective field ionization spectroscopy of electron binding energy 135B.3 Coupled rate-equation simulations of the electron-impact. . . . . . 137B.3.1 The semi-classical evolution of an n0 = 80 Rydberg gas . . 138xiTable of ContentsB.3.2 The semi-classical evolution of a fully ionized ultracoldplasma with Te(0) = 5 K . . . . . . . . . . . . . . . . . . 140B.4 Ambipolar expansion in a plasma with an ellipsoidal density dis-tribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142B.5 Effective many-body Hamiltonian . . . . . . . . . . . . . . . . . . 144B.5.1 L = 2 case . . . . . . . . . . . . . . . . . . . . . . . . . . 144B.5.2 L > 2 cases . . . . . . . . . . . . . . . . . . . . . . . . . . 146B.6 Induced Van der Waals interactions . . . . . . . . . . . . . . . . 147B.6.1 Non-resonant spin-spin interactions . . . . . . . . . . . . . 148B.6.2 Non-resonant on-site interactions . . . . . . . . . . . . . . 148B.7 Resonance counting and. . . . . . . . . . . . . . . . . . . . . . . . 149xiiList of TablesB.1 Distribution of ions in an idealized Gaussian ellipsoid shellmodel of a quenched ultracold plasma of NO as it entersthe arrest state with a peak density of 4× 1010 cm−3, σx =1.0 µm, σy = 0.55 µm and σz = 0.70 µm. At this point, thequasi-neutral plasma contains a total of 1.9 × 108 NO+ ions (NORydberg molecules). Its average density is 1.4×1010 cm−3 and themean distance between ions is 3.32 µm. . . . . . . . . . . . . . . . 150B.2 Resonance counting parameters in the arrest state of thequenched ultracold plasma. The disorder width W , taken di-rectly from the width of the plasma feature in the SIF spectrum,combined with J˜ – derived from a rough upper-limiting estimateof the average dipole-dipole matrix element, 〈tij〉, based on valuescomputed for ∆n = 0 interactions in alkali metals [201], togetherwith the mean distance between NO+ ions in the shell model ellip-soid – determines Nc, a critical number of dipoles required for de-localization. R∗ describes the length scale for delocalization and τ ∗denotes the delocalization time, given a sufficient number of dipolesat the average density of the experiment. Note that the ultracoldplasma quenched experimentally contains an order of magnitudefewer than Nc dipoles. . . . . . . . . . . . . . . . . . . . . . . . . 152xiiiList of Figures1.1 Schematic representation of a lattice polaron. The carrier(blue spherical dot) distorts the lattice (silver spherical dots). Thequantized lattice vibrations dress the carrier in a polaron cloud(pink region). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Schematic representation of interactions in a many-bodylocalized system. The many-body system of particles (spheres)interacts strongly in the disordered potential landscape. Quantuminferences in the MBL phase are depicted by the white lines. . . . 62.1 Schematic representation of the Holstein model. In a molec-ular crystal local vibrations modify the potential energy of the car-rier (red spherical dot) . Therefore, the resulting quasiparticle hasa large effective mass. . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Schematic representation of the Peierls particle-phononcoupling. The lattice deformation depicted by black arrows mod-ify the distance between the ordered atoms. This in turn modulatesthe hopping integral of the bare particle (magenta dot). Thesedistortions occur in most lattice structures, unlike the vibroniccoupling of the Holstein model depicted in Figure 2.1 of specialrelevance to molecular crystals. . . . . . . . . . . . . . . . . . . . 16xivList of Figures3.1 Schematic representation of the Hilbert space in the anti-adiabatic limit Ω >> t, g. The successive nph-phonon sectors areseparated by Ω, the phonon frequency. t is the bare hopping andg is the particle-phonon coupling strength. The electron-phononcoupling couples the zero-phonon subspace to the one-phonon sub-space with a vertex of amplitude g << Ω. In this limit, one obtainsa low-energy theory toO(g2/Ω) by projecting out the higher energyphonon subspaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1 Two-polaron phase diagram for U = 0 and Ω = 3. Thediagram represents the evolution of the low-energy region of thesinglet sector with λ. Energies are in units of t. The dimensionlesseffective coupling is λ = 2g2/(Ωt). The shaded grey area shows thelower part of the two-polaron continuum. The dark red region rep-resents the lowest energy bipolaron band, while the salmon regionrepresents the higher energy bipolaron band. These results wereobtained with MA and are in good agreement with VED results(blue circles) shown for the low-energy bipolaron. . . . . . . . . . 424.2 Dependence of the effective mass of the low-energy bipo-laron on λ, for U = 0 and Ω = 3.0. m0 = 2me is twicethe free electron mass. The bipolaron’s effective mass is definedas m∗ =(∂2EBP (K)∂K2)−1∣∣∣K=KGS. KGS = 0 for the Peierls bipo-larons. The solid (dashed) lines are VED (MA) results. Note thatm∗ ∼ 2me in the strongly coupled regime, λ > 1. . . . . . . . . . 434.3 Dispersion EBP (K) − EBP (0) of the low-energy bipolaron,for various values of λ = 2g2/(Ωt) at U = 0 and Ω = 3. Theinset shows the polaron dispersion EP (k) − EP (0) for the sameparameters. All energies are in units of t. In the main figure, solidlines are VED results and dashed lines are MA results. Results inthe inset were obtained with MA, and are in good agreement withnumerical results [17]. . . . . . . . . . . . . . . . . . . . . . . . . 45xvList of Figures4.4 Dispersion EBP (K) of both bipolaron bands, for U = 0,Ω = 3 and λ = 2, showing an avoided crossing. EBP (K) is inunits of t. These are MA results. . . . . . . . . . . . . . . . . . . 464.5 UC-λ stability diagram for the Peierls/SSH (solid line) andHolstein (dashed line) bipolarons at Ω = 3. UC is in units of t.For the Peierls/SSH coupling, λ = 2g2/(Ωt), while for the Holsteincoupling, λ = g2H/(2Ωt), where gH is the Holstein electron-phononcoupling parameter. These are VED results, and reveal a starkcontrast between the stability of the two types of bipolarons in thestrong-coupling limit, λ > 1. . . . . . . . . . . . . . . . . . . . . 495.1 Two-polaron phase diagram at t = 1,Ω = 3. The solid blackline shows Vc(λ) below which stable bound states form, while thedashed line shows V¯c(λ); the difference between the two is thestrength of the phonon-mediated interaction. Note that Vc < V¯C ,which means that this interaction is repulsive. The red and greenlines mark the sharp transitions of the bound dimer’s GS. The linesare the VED results and the corresponding symbols are the MAresults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Correlation between the two particles C(δ) =〈ΨGS| 1N∑i nˆinˆi+δ|ΨGS〉 for V = Vc(λ) − 0.5, t = 1, Ω = 3and different λ. Red circles are the λ = 0.1 result, greentriangles are λ = 0.7, and black squares are λ = 2.0. . . . . . . . 585.3 Dimer dispersion ED(K) − ED(0) for a) λ = 1 and variousvalues of V ; and b) V = −30 and various values of λ. Inboth cases t = 1,Ω = 3. The lines are the VED results and thesymbols are the MA results. Note the sharp transitions of the GSmomentum from KGS = 0 to KGS > 0 in both cases. . . . . . . . 60xviList of Figures6.1 V -t2 phase diagram. t is the bare NN hopping. The greyshaded region represents unbound polarons, while the light anddark salmon colored regions represent K = 0 bipolarons and sta-ble bipolarons, respectively. The blue solid line corresponds to thePeierls/SSH model for which V = 4t2. The Peierls/SSH modelline is right on the boundary between the region of stable bipo-larons and the K = 0 bipolarons at t2 = t∗2 ≈ 0.8 corresponding toλC ≈ 1.6; the value of λ marking the onset of stable Peierls/SSHbipolarons, where λ = 2g2/~Ωt. The blue dashed line correspondsto the Peierls/SSH model supplemented with one unit of NN re-pulsion between the bare particles. The black diamond symbolsmark the two points λ = 1.6 and λ = 4 on the Peierls/SSH linefor which splitting of the repulsive bipolaron states from the con-tinuum is illustrated in Figure 6.2. . . . . . . . . . . . . . . . . . 676.2 Energy spectrum of two hard-core bare particles in theone-dimensional Peierls/SSH model. The red line repre-sents the repulsive bipolaron dispersion and the blue shaded re-gion shows the two-polaron continuum band for a) λ = 1.6 andb) λ = 4. The inset of b) illustrates the repulsive bipolaronlog probability distribution Γn(K) = log10[Pn(K)] defined forPn(K) ≡ |〈K,BP |K,n|K,BP |K,n〉|2, where |K,BP 〉 is the bipo-laron state and n is the relative separation of the particles forKBP = 0 (salmon) and KBP = pi (indigo). In both cases, the par-ticles are NN with highest probability. Note that for KBP = pi,even n relative separation between particles is forbidden. For moredetails, see Appendix A.3. . . . . . . . . . . . . . . . . . . . . . . 69xviiList of Figures6.3 The repulsive bipolaron dispersion EK ≡ EBP (K) − EBP (0)in units of the bare NN hopping, t, for various values ofλ = 2g2/~Ωt. a is the lattice constant. The dashed line representsthe onset of Peierls/SSH bipolaron formation. The two blue lineswere illustrated in Figure 6.2, while the red lines label strong cou-pling bipolarons. Note that the bipolaron dispersion exhibits bothlarge curvature and bandwidth, which increase with the couplingstrength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.4 The dependence of the inverse effective mass, ta2/m∗, forthe repulsive bipolaron on λ = 2g2/~Ωt. The effective massof the bipolaron m∗ is defined as m∗ = (∂2EBP (K)/∂K2)−1 forK = 0, t is the bare NN hopping and a is the lattice constant.The inset illustrates the dependence of the energy gap, ∆, betweenthe repulsive bipolaron and the edge of the two-polaron continuumband on λ at various values of K. The dark blue and dark redsolid lines label the gaps for K = 0 and K = pi, respectively; thedashed, dash-dotted and dotted lines label the gaps for K = pi/4,K = pi/2 and K = 3pi/4, respectively. Note that ∆ vanishes forK = pi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728.1 Summary of experimental data. a) Double-resonant selectionof the initial quantum state of the n0f(2) Rydberg gas. b) Laser-crossed differentially pumped supersonic molecular beam. c) Selec-tive field ionization spectrum after 500 ns evolution, showing thesignal of weakly bound electrons combined with a residual popu-lation of 49f(2) Rydberg molecules. After 10 µs, this populationsharpens to signal only high-n Rydbergs and plasma electrons. d)Integrated electron signal as a function of evolution time from 0 to160 µs. Note the onset of the arrest phase before 10 µs. Timescalecompressed by a factor of two after 80 µs. e) x, y-integrated im-ages recorded after a flight time of 400 µs with n0 = 40 for initialRydberg gas peak densities varying from 2×1011 to 1×1012 cm−3.All of these images exhibit the same peak density, 1× 107 cm−3. 86xviiiList of Figures8.2 Schematic representation of NO+ core ions, paired withextravalent electrons to form interacting dipoles di anddj, separated by rij = ri−rj, in this case joining a Rydbergmolecule with an electron bound in the charge of morethan one ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88B.1 Selective field ionization spectrum spectrum as a func-tion of initial Rydberg gas density, ρ0, after 500 ns ofevolution, showing the signal of weakly bound electronscombined with a residual population of 49f(2) Rydbergmolecules, (initial principal quantum number, n0 = 49, inthe f Rydberg series converging to NO+ ion rotationalstate, N+ = 2). After 10 µs, this population sharpens to signalhigh-n Rydbergs and plasma electrons, with a residue of the initialRydberg population, shifted slightly to deeper binding energy byl-mixing and perhaps some small relaxation in n. The prominentfeature that appears at the lowest values of the ramp field gaugesthe potential energy of electrons in high Rydberg states boundto single NO+ ions, combined with electrons bound to the spacecharge of more than one ion. Notice the binding effect of a slightlygreater excess positive charge at the highest initial Rydberg gasdensities. The red feature extends approximately to the bindingenergy of n0 = 80 or 500 GHz. . . . . . . . . . . . . . . . . . . . . 136xixList of FiguresB.2 Semi-classical simulations. (lower) Numbers of ions and elec-trons, Rydberg molecules and neutral dissociation products N(4S)and O(3P) as a function of time during the avalanche of an n0 = 80Rydberg gas of NO to form an ultracold plasma, as predicted by ashell-model coupled rate equation simulation. Here, the initial den-sity distribution of the Rydberg gas is represented by a 5σ Gaussianellipsoid with principal axis dimensions, σx = 1.0 mm, σy = 0.55mm, σz = 0.7 mm and peak density of 4× 1010 cm−3, as measuredfor a typical experimental plasma entering the arrest state afteran evolution of 10 µs. The simulation proceeds in 100 concentricshells enclosing set numbers of kinetically coupled particles, linkedby a common electron temperature that evolves to conserve energyglobally. (upper) Global electron temperature as a function of time. 139B.3 Classical evolution of arrest state. (lower) Numbers of ionsand electrons, Rydberg molecules and neutral dissociation prod-ucts N(4S) and O(3P) as a function of time during the evolutionof an ultracold plasma of NO+ ions and electrons, as predicted bya shell-model coupled rate equation simulation. Here, the initialdensity distribution of the plasma is represented by a 5σ Gaussianellipsoid with principal axis dimensions, σx = 1.0 mm, σy = 0.55mm, σz = 0.7 mm, peak density of 4×1010 cm−3 and initial electrontemperature, Te(0) = 5 K, as measured for a typical experimentalplasma entering the arrest state after am evolution of 10 µs. Thesimulation proceeds in 100 concentric shells enclosing set numbersof kinetically coupled particles, linked by a common electron tem-perature that evolves to conserve energy globally. (upper) Globalelectron temperature as a function of time. . . . . . . . . . . . . 141xxList of FiguresB.4 Hydrodynamic expansion of a Gaussian ellipsoid with thedimensions measured at 10 µs for the typical arrestedplasma described above, modeled by a 100-shell simula-tion, assuming an electron temperature that rises to 60K, with curves, reading from the bottom on the left, forσy(t), σz(t) and σx(t). The lower curve with data shows the mea-sured expansion of a typical molecular NO ultracold plasma witha Vlasov fit for Te = 3 K. . . . . . . . . . . . . . . . . . . . . . . 143B.5 Schematic diagram representing two Rydberg molecules,i and j, dipole coupled in the two-level approximation.In every case, the disorder in the environment of each moleculeperturbs the exact energy level positions of |ei〉 and |ej〉. . . . . . 145B.6 Schematic diagram representing two Rydberg molecules, iand j, dipole coupled in the limits of L = 3. In the very highstate density of the quenched ultracold plasma, the displacementof |e2i 〉 and∣∣e2j〉 will lessen the significance of L = 3 interactionscompared with the case of L = 4. . . . . . . . . . . . . . . . . . . 146B.7 Schematic diagram representing two Rydberg molecules,i and j, dipole coupled in the limits of L = 4. The high statedensity and strong disorder in the quenched ultracold plasma givesthis case of L = 4 greater significance than the restrictive limit ofL = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147xxiAcknowledgementsAs I come to the end of an exciting five-year journey to the doctoral degree at theUniversity of British Columbia (UBC), I feel a sense of gratitude to many indi-viduals who helped shape my scientific personality and influenced me in one wayor another. Above all, I am grateful to my Ph.D. supervisors, Professor RomanKrems and Professor Mona Berciu. Roman has extended me the opportunityto work in the engaging field at the interface of atomic, molecular and optical(AMO) physics and condensed matter physics. Mona helped me master Green’sfunction approaches to polaron problems, which aided the development of a largeportion of my Ph.D. work. Together, their broad knowledge of physics and theirphysical insights have helped me a great deal. I am grateful to have learnt fromtheir scientific no-nonsense approach to research combining both numerical andanalytical tools in search for new physics. Their complementing characteristicshelped me develop a broad scientific approach to new problems. I also wish tothank them both for their advice and patience. I hope this marks the beginningof a long-lasting friendship.I have also been blessed with the opportunity to interact with Professor EdGrant. Ed very generously explained to me his experimental work. These inter-actions organically led to several collaborative projects advancing a theoreticalexplanation of the experimental results obtained in his lab. Ed’s insistence on aphysical conception of every equation, coupled with his elegant style of writingmade our papers a very beautiful and engaging experience. I also wish to thankhim for his advice and encouragement with postdoc applications, and going be-yond the call of duty to teach me how to explain physics to a non-specializedaudience.I wish to acknowledge the visitor’s fellowship support to visit the Institutefor Theoretical, Atomic and Molecular Physics (ITAMP) and the DepartmentxxiiAcknowledgementsof Physics at Harvard University for an extended period last year. I would liketo especially thank Dr. Hossein Sadeghpour for his hospitality and kindness inshowing me around Cambridge and helping me settle. I am also very gratefulto Professor Eugene Demler for welcoming me to his research group. With Dr.Richard Schmidt, Eugene, Hossein and Professor Thomas Killian, I had the op-portunity to work on the new emerging field of Rydberg impurities in ultracoldgases. I appreciate the stimulating environment at ITAMP and Lyman, and allthe people who made this a wonderful experience. Particularly, I wish to thankRichard for teaching me about functional determinants, Hossein for introducingme to the field of Rydberg molecules, and Eugene for fun discussions about thephysics of quantum impurity systems, his insights in physics and advice.During the first year of my Ph.D., I had the opportunity to visit the group ofProfessor Guido Pupillo in Strasbourg. I acknowledge discussions about quantumMonte Carlo methods and polaron physics with his research group members:Edoardo Tignone, Adriano Angelone, Fabio Mezzacapo, Luca Lepori and RogelioDı´az-Me´ndez.During these final stages of my Ph.D., through a DAAD fellowship, I had theopportunity to visit the Technical University of Munich (TUM) and interact withthe group of Professor Frank Pollmann. I wish to thank Frank for his hospital-ity and for engaging discussions about spin liquids, tensor-network methods andfractons. I further acknowledge discussions with Johannes Hauschild, Pablo Sala,Tibor Rakovszky, Kevin Hemery, and Professor Michal Knap.I have been fortune to be part of the stimulating research environment at allthese institutions. I would like to thank my primary collaborators for fruitfulinteractions. Especial thanks to Richard Schmidt for collaborations on Rydbergpolarons, Monodeep Chakraborty for collaborating on bipolaron problems, theexperimental group of Ed Grant, particularly Markus Schulz-Weiling and HosseinSadeghi, Rodrigo Vargas-Herna´ndez for teaching me about Machine Learning andfor a wonderful collaboration, Arthur Christianen for his interest in my ideas onfew-body systems and the three-body problem. Especial thanks are due to manywonderful collaborators involved in ongoing projects: Michael Pretko for teachingme about fractons, A´lvaro Go´mez-Leo´n for collaborating on non-perturbativehierarchal methods to interacting many-polaron systems, Yulia Shchadilova andxxiiiAcknowledgementsTao Shi for teaching me about variational Gaussian methods and collaboratingon the problem of two impurities in a Bose-Einstein condensate.I wish to thank the UBC condensed matter theory faculty: George Sawatzky,Ian Affleck, Philip Stamp and Marcel Franz for many fruitful interactions. I ap-preciate Ian’s encouraging words at the start of my Ph.D. I am beyond captivatedby George’s ability to immediately predict the solution of a research problemwithout any calculations. Philip’s broad knowledge of physics and quantum fieldtheory is quite inspiring and I wish I can master a fraction of his skills. Throughcourses taught by Marcel, I learned about several advanced topics in condensedmatter, particularly topological phases of matter. I like to also thank the AMOfaculty: Kirk Madison, Taka Momose and Valery Millner for providing a friendlyenvironment for interactions and discussions.I acknowledge stimulating discussions with the UBC and Harvard colleagues:Joshua Cantin, Tianrui Xu, Tirtha Chattaraj, Daniel Vieira, Fernando Luna,Nathan Chang, Clemens Adolphs, Mirko Mo¨ller, Alfred Cheung, Hadi Ebrahim-nejad, Bihui Zhu, Hannes Pichler, Hossein Jooya, Rivka Bekenstein, Yao Wang,Fabian Grusdt, Dries Sels, Florentin Reiter, Jamir Marino, Wen Wei Ho andMathias Scheurer.I wish to thank the members of my thesis Examination Committee: GeorgeSawatzky, Kirk Madison and Gordon Semenoff, the University Examiners: JoelFeldman and Gren Patey, and the External Examiner: Frank Marsiglio, for theircritical input that helped sharpen this thesis.In visits, at workshops, and through correspondences, I had the pleasant ex-perience to discuss my ideas with many colleagues. I wish to thank Pri Narang,Jenny Hoffman, Ga´bor Hala´sz, Steven Kivelson, Frank Marsiglio, Ilya Esterlis, SriRaghu, Ted Geballe, David Reichman, Andrew Millis, Steven Johnston, BirgittaWhaley, Mike Zaletel, Ehud Altman, Joel Moore, Ryan Babbush, Sergio Bioxo,Patrick Lee, Vladyslav Kozii, Jonathan Ruhman, Rahul Nandkishore, ShivajiSondhi, Vedikha Khemani, Wojciech De Roeck, Franc¸ois Huveneers, AndreasBuchleitner, Michael Walter, Fabian Heidrich-Meisner, Ulrich Schollwo¨ck andKenji Ohmori.I like to thank my Master’s advisor Marcel Nooijen for encouragement andconfidence in my abilities to move to Physics after studying Biology, and myxxivAcknowledgementsfriend Gre´goire Guillon for his deep insights in Math, Physics and Philosophy,and for his friendship. I thank my high school Physics teacher, Dr. Nabil Elias,for first instilling in me a strong interest in Physics.I would like to thank all who have worked to make my doctoral experiencesmooth, Oliva Dela Cruz-Cordero and Sherry Harbour at UBC, Naomi Taririand Alice Kalemkiarian at ITAMP, Elizabeth Alcock at Harvard University, andClaudine Voelcker at TUM.Away from physics, I had the fortune of good friends. I like to thank ShadyShokry, Ramy Raafat, Michael Kamal, Karim Atef, Georges Helmy, AbraamAtalla, Mike Akl, for their friendship despite the long distance, Fadi Azmy, TarekMorgan, Ashraf and Sherif Malek, Gabriele Lanaro, Arsani William and WassimMalak for the fun times that helped maintain my sanity. I am thankful to havethe friendship of Mark and Mirna, Tina and Myles, Megan, Yassir and Elliot,and Karim, Sandra and Farida. I wish to thank abouna Philopateer Younan andabouna Antonious Fekry for their support and wisdom. Their experience andvision helped me persist to this moment.Finally, I wish to thank my family for their love and support. I owe an oceanof gratitude to my fiance´, Nansy, for her love, support, absolute confidence in myabilities, and patience during what must have been a very difficult period withall my travels and erratic work habits. I wish to thank her parents, Esteer andAdlan, her sister, Tina, and her brother, Michael, for being a second family tome. I wish to thank my uncles, Ibrahim and Malak, and my aunt, Mervat, fortheir love, and my cousins, Mena, Mark and Mary, for their friendship. Aboveall, I wish to thank my parents for their love, sacrifice and unwavering confidencein me. I have learned from them to never stop dreaming, to persist and to alwaysbelieve.xxvTo Nansy, mom and dad.xxviChapter 1Overview1.1 Particles coupled to quantum fields:Lattice polarons and bipolaronsUnderstanding the behavior of interacting particles coupled to quantum fields inclosed systems is fundamental to the study of emergent phenomena manifest onthe macroscopic scale. Not only does the bosonic field modify the basic behaviorof the individual particles that make up the many-body system, but also inducesinter-particle interactions mediated by the exchange of excitations. This leads todramatic changes in the collective behavior of the many-body system.Perhaps the earliest studied example of such physics is the motion of an elec-tron in a polarizable medium, depicted in Figure 1.1. There, the electron deformsthe lattice in its vicinity, which in turn modifies the electron mobility. Theoreticalefforts to explain this phenomenon introduced the “polaron” concept as a quasi-particle description of the electron coupled to the lattice vibrations [1, 2]. Moregenerally, the notion of polarons extends to present a framework for the studyof a particle in a quantum field and applies to a variety of systems. Examplesinclude magnetic polarons that describe the interaction of a charge carrier withmagnetic excitations [3], quantum impurity systems with effects such as the cele-brated orthogonality catastrophe [4], and the spin-bath problem that provides auniversal description of a two-level system in presence of dissipation [5].As mentioned earlier, the field excitations also renormalize the particle-particleinteractions in the many-body system. This ultimately led to the discoveryof the Bardeen-Cooper-Schrieffer (BCS) mechanism of superconductivity [6, 7],where electrons bind through exchange of phonons despite the Coulomb repul-sion, to form a condensate of Cooper pairs [8]. At larger particle-phonon cou-11.1. Particles coupled to quantum fields: Lattice polarons and bipolarons. . .Figure 1.1: Schematic representation of a lattice polaron. The carrier (bluespherical dot) distorts the lattice (silver spherical dots). The quantized latticevibrations dress the carrier in a polaron cloud (pink region).plings, strongly bound pairs of polarons, dubbed “bipolarons”, form [9, 10]. Inthis regime, Bose-Einstein condensation of preformed pairs could lead to hightransition-temperature superconductivity. A full theory of bipolaronic excitationsat finite concentrations likely requires the inclusion of strong-coupling effects thatgo beyond simple mean-field descriptions [11].Most studies of polarons and bipolarons focused on model descriptions de-veloped by Fro¨hlich [12] and Holstein [13]. There, lattice distortions (phonons)modify the potential energy of the bare particles and the resulting polarons andbipolarons are considerably heavier than the bare particles. However, phononsalso modulated the bare particle hopping, as described by the Peierls model [14–16]. Only recently Marchand et al. strongly challenged the notion of heavypolarons as a general feature of particle-phonon coupled systems, showing that21.1. Particles coupled to quantum fields: Lattice polarons and bipolarons. . .Peierls polarons are light at strong coupling strengths [17].Another central question in this field considers whether a sharp transition canexist in the polaronic ground-state as a function of the particle-phonon coupling.It is now understood that such transitions do not exist in the “standard” Fro¨hlichand Holstein models [18]. Remarkably, the same work by Marchand et al. pro-vided indisputable evidence for the existence of a sharp transition in the polaronground-state in the Peierls model [17].The properties of bipolarons in the Peierls model, in particular whether theyare light and/or exhibit sharp transitions were not known before the results pre-sented here (also in [19–21]).1.1.1 Outline of Part IPart I of this thesis investigates these questions concerning the two-carrier prob-lem in the one-dimensional Peierls model, relevant to various solid-state and ul-tracold lattice systems.In Chapter 2, I provide an introduction to the physics of polarons and bipo-larons and motivate the research questions investigated here in the context of thefield. I review previous results on the Holstein and Peierls models and place themwithin the realm of experiments in solid-state and atomic, molecular, and optical(AMO) systems.To tackle the two-carrier problem, I present results obtained using three meth-ods: variational exact diagonalization (VED), the Momentum Average (MA) ap-proach, and perturbational analytical techniques. VED is an unbiased numericalmethod, where the Hamiltonian basis set is constructed iteratively. MA is aquasi-analytical Green’s function technique that utilizes a variational approxima-tion to efficiently and accurately study strongly coupled polaron and bipolaronsystems. The analytical arguments derived using perturbational methods helpexplain the VED and MA numerical results in simple yet intuitive terms. Allthese complementary approaches paint a full picture of the physics of polaronicand bipolaronic systems. Chapter 3 takes the reader through a brief introductionto these methods placing emphasis on the general ideas behind their development.The author of this thesis contributed the development of the MA and analytical31.1. Particles coupled to quantum fields: Lattice polarons and bipolarons. . .approaches. The VED results, which I use here for validation of my MA results,were obtained by Monodeep Chakraborty.In Chapter 4 I study the pairing of two electrons in a singlet state into abipolaron in the Peierls model. I find strongly bound yet light bipolarons evenat very strong electron-phonon coupling strengths. These bipolarons also exhibitmany other unconventional properties, e.g. at strong coupling there are two low-energy bipolaron bands that are stable against strong on-site Hubbard repulsion.These properties result from the specific form of the effective phonon-mediatedinteraction, which is of “pair-hopping” instead of regular density-density type.This unusual effective interaction is bound to have non-trivial consequences forthe superconducting state expected to arise at finite carrier concentrations. Forexample, the phonon-mediated effective electron-electron interaction depends notonly on the exchanged momentum, q, but also on the total momentum of theinteracting electron pair, k + k′, see Chapter 4 for more details.Having established the physics in the singlet sector, I then turn to the studyof pairing of triplet electrons or equivalently, two hard-core particles in the Peierlsmodel in Chapters 5 and 6. In infinite one-dimensional systems, the differencebetween two hard-core particles and two triplet electrons is due to vanishingboundary terms [22, 23].In Chapter 5, I study the ground-state properties of the hard-core or tripletproblem. I show that exchange of phonons generates an effective repulsion be-tween particles, unlike that for the singlet in Chapter 4. A sufficiently strong bareattraction binds a dimer, which also experiences phonon-mediated pair-hoppinginteractions that move the pair as a whole. The two-polaron phase diagram ex-hibits two sharp transitions, leading to light dimers at strong coupling and theflattening of the dimer dispersion at some critical values of the parameters. Thisdimer (quasi)self-trapping occurs at coupling strengths where single polarons aremobile. This illustrates that, depending on the strength of the phonon-mediatedinteractions, the coupling to phonons may completely suppress or strongly en-hance quantum transport of correlated particles.In Chapter 6, I discuss some features of the excited spectrum of the problemconsidered in Chapter 5. I show that the repulsive phonon-mediated interactionsgive rise to “repulsive” bipolarons with unique properties. These bipolaron states41.2. Quenched randomness in quantum systems at finite temperatures. . .appear in the gap between phonon excitations, above the two-polaron continuum.While unstable (due to possible decay to the lower-energy continuum), the bipo-laron is protected by energy and momentum conservation and represents a novelquasiparticle with a large dispersion and a negative effective mass near zero mo-mentum. These states are especially relevant to ultracold experiments with polarmolecules and Rydberg atoms in optical lattices.These results raise doubts about some widely held beliefs regarding bipolaronsand many-polaron systems and indicate that many established ideas must berevisited.1.2 Quenched randomness in quantum systemsat finite temperatures: Disorderedultracold molecular plasmaOur understanding of large-scale quantum phenomena such as superconductiv-ity rests on studies of simple ordered structures such as translationally invariantlattices or homogenous gases. While these are incredibly useful, a complete un-derstanding of quantum mechanics at high temperatures also necessitates the in-clusion of the effects of quenched randomness characteristic of realistic laboratorysettings. The randomness manifests as lack of order, i.e. disorder.Anderson studied the problem of an electron in a random lattice proving thatdisorder can impede transport, especially in lower dimensions, an effect widelyknown as “Anderson localization” [24]. More recently, interest in understandingthe interplay of interactions and disorder culminated with the discovery of theso-called Many-Body Localization (MBL) [25], depicted in Figure 1.2. There,randomness in an interacting many-body system leads to a complete breakdownof thermalization, locking the system in a quantum state at infinite ‘temperature’and for infinite times. Here, temperature is defined in a way to reflect the system’senergy density if it were at thermal equilibrium. It has been shown that thisbreakdown of thermalization can occur at infinite temperature, where all statesin the spectrum are equally probable [26].51.2. Quenched randomness in quantum systems at finite temperatures. . .Figure 1.2: Schematic representation of interactions in a many-body lo-calized system. The many-body system of particles (spheres) interacts stronglyin the disordered potential landscape. Quantum inferences in the MBL phase aredepicted by the white lines.MBL has been observed in deliberately engineered experimental systems withultracold atoms in one and two-dimensional optical lattices [27–32]. In such cases,tuning of the lattice parameters allows investigation of the phase diagram ofthe system as a function of disorder strength. However, such ultracold systemssuffer from decoherence, confining localization to short timescales and low energydensities.It is therefore important to determine experimentally whether conditions existunder which MBL can persist for long times at finite temperatures, and to under-stand if such a robust macroscopic quantum many-body state can occur naturallyin an interacting quantum system without deliberate tuning of experimental pa-rameters.61.3. Main results1.2.1 Outline of Part IIPart II of this thesis explores the possibility of MBL in a long-lived finite-temperature self-assembled macroscopic ultracold plasma.Chapter 7 reviews the concepts of Eigenstate Thermalization Hypothesis(ETH) and MBL. In Chapter 8 I consider an experiment performed in the Grantgroup that quenches a plasma of nitric oxide to an ultracold system of Rydbergmolecules, ions and electrons that exhibits a long-lived state of arrested relaxation[33]. The qualitative features of this state fail to conform with classical models.I develop a microscopic quantum description for the arrested phase based on aneffective many-body spin Hamiltonian that includes both dipole-dipole and vander Waals interactions. This effective model appears to offer a way to envision theessential quantum disordered non-equilibrium physics of this system. It is thusargued that the quenched ultracold plasma presents an experimental platform forstudying quantum many-body physics of disordered systems in the long-time andfinite energy-density limits [34].This study paves the way to the realization of exotic quantum effects, such asentangled macroscopic objects and localization-protected quantum order [35] athigh temperatures.1.3 Main resultsThis thesis presents important findings on the response of realistic quantum many-body systems relevant to current experiments. I investigate the quasiparticle be-havior in systems with coupling to the environment, and the response of quenchedmacroscopic strongly interacting ultracold plasmas. The main findings of thiswork are• Light bipolarons (bound pairs of quasiparticles) exist in a wide class ofphysically motivated systems. At finite electron concentrations, such lightbipolarons may condense to form a high-Tc superconductor.• Under certain conditions, a dimer experiences a self-trapping transition inabsence of disorder. Self-trapping refers to the dimer creating a distortion71.3. Main resultsthat in turn effectively localizes the pair, hence the dimer “self-traps”.• Phonon-mediated interactions between hard-core particles can be repulsiveand can bind repulsively bound bipolaron states embedded in the excitedcontinuum. Such states are of special relevance to ultracold experiments.• Quenched molecular ultracold plasma presents an experimental platform forstudying quantum many-body physics of disordered systems. There, theforces of self-assembly naturally drive the system into a state of possiblelocalization.I summarize these research results and discuss future directions for investiga-tion in Part III of the thesis.8Part IPolarons and Bipolarons in thePeierls Model of LatticeParticle-Phonon Coupling9Chapter 2Peierls versus Holstein andFro¨hlich particle-phonon couplingand their corresponding polarons2.1 IntroductionA particle moving through a crystal distorts the ordering of nearby atoms in thelattice. The quantized distortions (phonons) interact with the bare particle givingrise to a quasiparticle dubbed a “polaron”, i.e. a particle dressed by a cloud ofphonons and with a renormalized dispersion and effective mass, which lead tostrong modifications of transport properties.The current understanding of renormalization of quasiparticles by coupling tobosons follows from polaron theories developed by Landau and Pekar [1, 2] andextended by Fro¨hlich [12], Holstein [13] and Feynman [36, 37]. These theoriesdescribe the particle-boson coupling as one that modulates the bare particle’spotential energy. As the interaction strength increases, the particle becomeseffectively dressed with phonons. The resulting polaron describes a bare particledragging its cloud of phonons and therefore exhibits a larger effective mass. Inaddition, in such theories exchange of bosons between polarons can screen thebare particle-particle repulsion and mediate a strong attractive force that bindspolarons into very heavy objects called bipolarons [38, 39]. A question with along history in this field is whether these polarons and bipolarons exhibit sharptransitions in their ground-state properties as a function of the particle-bosoncoupling. In all the above models there is a smooth crossover from weak couplingto strong coupling and it is now understood that transitions do not occur in suchsystems [18].102.2. Particle-lattice interactionsOn the other hand, bosons also modulate the bare particle’s kinetic energy (orhopping). The effect of this particle-boson coupling has been much less studied.This coupling, modeled by the Peierls/Su-Schrieffer-Heeger (SSH) model [14–16, 40, 41], results in polarons with qualitatively different behavior from Fro¨hlichand Holstein polarons. Remarkably, the Peierls polaron is light at strong couplingand exhibits a sharp transition [17] in its ground-state properties.Here, I study polaron-polaron interactions and bipolarons in the Peierls modeland in the presence of electronic Hubbard interactions. I expect the generalqualitative physics, such as the nature of the phonon-mediated interactions, tonot depend strongly on the dimensionality, see Chapter 4 for a discussion. Forthis reason, I focus here only on the one-dimensional version of the Peierls model,for which numerically accurate results are easily accessible (see Chapter 3 formore discussion about possible extensions of the variational Momentum Average(MA) method to higher dimensions). I contrast the results with the “standard”bipolaron behavior observed in, for example, the Holstein model. Throughoutthis thesis, I study electron-phonon models with optical phonons. I leave thestudy of the effects of higher dimensions on the details of the physics to futurework.2.2 Particle-lattice interactionsThe simplest description of a particle moving in a frozen one-dimensional latticetakes the formH =∑iic†ici −∑iti,i+1(c†ici+1 +H.c.). (2.1)Here, ci annihilates a particle at site i. The first term describes an on-site poten-tial, i, experienced by the particle at site i, while the second term describes inter-site tunneling, ti,i+1, in the nearest-neighbor approximation. Particle-particle in-teractions are ignored, but will be addressed in detail in subsequent chapters.Throughout this thesis I use H.c. to refer to the Hermitian conjugate.Now relax the frozen lattice restriction and allow it to vibrate. Lattice112.2. Particle-lattice interactionsdistortions modify the parameters i and ti,i+1. Taylor expanding about theequilibrium positions R0i , one obtains i ≈ 0 + ∇i(Xi−1 − Xi+1) and ti,i+1 ≈t0 + ∇ti,i+1(Xi − Xi+1), where Xi is the displacement operator of site i. Thissimple exercise demonstrates that phonons modulate a) the particle’s potentialenergy and b) its kinetic energy.In what follows and in the rest of this thesis part, I will study one-dimensionalmodels of electron-phonon coupling.2.2.1 Interactions that modulate the particle’s potentialenergyThe current understanding of the phenomenology of polaronic systems in con-densed matter derives mostly from the consideration of interactions that modu-late the particle’s potential energy. Holstein studied the modulation of a carrier’sbehavior in molecular crystals, due to on-site vibronic coupling [13] (see Fig-ure 2.1). Fro¨hlich considered the long-range phonon modulation of the particle’spotential energy [12] in a model that can be formulated both on the lattice and ina continuum approximation. All these models result in qualitatively similar andgeneric behavior of polarons and bipolarons, which led to a wide belief that allparticle-phonon coupling must lead to the same physics. I will show below andusing new results that this is not correct.Phenomenology of the Holstein modelFor illustrative purposes, I review the basic phenomenology of the Holstein model.The Holstein model (Figure 2.1) in one dimension reads122.2. Particle-lattice interactionstFigure 2.1: Schematic representation of the Holstein model. In a molecularcrystal local vibrations modify the potential energy of the carrier (red sphericaldot) . Therefore, the resulting quasiparticle has a large effective mass.HH = Hp +Hph + VHHp = −t∑i,σ(c†ici+1,,σ +H.c.)Hph =∑qΩqb†qbqVH = gH∑i,σc†i,σci,σ(bi + b†i ). (2.2)Here, σ labels the spin state of the particle, and the phonons, with annihilationoperators bi, are modeled as localized Einstein modes with a flat dispersion Ωq ≡Ω, i.e. Hph = Ω∑ib†ibi, and gH is the particle-phonon coupling strength. TheHolstein coupling is characterized by the dimensionless effective particle-phononcouplingλ =gH22Ωt. (2.3)I have set ~ = 1 and the lattice constant, a = 1, in this subsubsection.Single polaron In the single polaron limit, the Holstein coupling leads to quasi-particle mass enhancement [42]. This can be understood as follows. For gH = 0,the bare particle dispersion is −2tcos(k). As the coupling strength increases,the bare particle becomes more dressed with phonons and its potential energy is132.2. Particle-lattice interactionsgreatly enhanced. The corresponding polaron dispersion is ∼ −2t∗cos(k), withthe renormalized polaron hopping t∗ < t. At strong couplings, λ >> 1, perturba-tion theory predicts an exponentially suppressed polaron hopping t∗ = te−2λt/Ωand the resulting polaron mass is(2te−2λt/Ω)−1[43].Moreover, a long debate about the nature of the polaronic ground-state as afunction of λH was finally firmly settled in [18, 42], where it was shown that theground state evolves smoothly from weak coupling to strong coupling with nosharp transitions. This is true in models with the coupling due to the modulationof the on-site energy for gapped optical phonons.Bipolaron In the bipolaron limit, two soft-core particles, e.g. two fermionsin a singlet state, experience an effective phonon-mediated on-site attraction−∆E∑i nˆi↑nˆi↓, where ∆E → 2g2H/Ω as λ → ∞ [38, 39]. This effective on-siteattraction binds the two fermions into an on-site singlet for any finite coupling gHif ∆E is larger than the bare on-site repulsion U . At strong couplings, λ >> 1and U = 0, the bipolaron mass ≈(4√pi(t2/√2λtΩ)e−8λt/Ω)−1is much largerthan that of two single polarons [38]. Note that there is only one such boundstate in one dimension.On the other hand, phonon-mediated interactions are not sufficient to bindtwo hard-core particles or triplet fermions in the Holstein model [39, 44].As in the case of single polarons, semi-classical approximations employed tostudy the bipolaron problem in the adiabatic limit Ω << t observed a transition[45, 46] from on-site to two-site bipolarons (with the two electrons nearest neigh-bours). A full quantum treatment in one dimension has shown that these sharptransitions do not exist, instead a smooth crossover is observed [38, 39].Summary From this emerges a picture of heavy polarons and bipolarons, preva-lent in the condensed matter community. While I agree with the above results inthe Holstein and related models, I strongly question the unjustified widely heldbelief that all polarons in experimental systems must be heavy. I shall demon-strate in this thesis that polarons and bipolarons can be light in a certain classof systems.Before I proceed I summarize the main points discussed here.142.2. Particle-lattice interactions• Holstein polarons and bipolarons are very heavy at strong couplingstrengths.• Holstein coupling to phonons mediates effective attraction between fermionsin the singlet sector.• Polarons and bipolarons in the Holstein model do not exhibit sharp transi-tions in their ground-state properties, such as their effective mass.2.2.2 Interactions that modulate the particle’s kineticenergyThe Holstein Hamiltonian is an adequate starting point to model particle-latticeinteractions in systems with local on-site structural distortions, such as molecularcrystals. On general grounds, the lattice deformation modulates both the on-siteand the hopping terms. Here, I focus on the modulation of hopping ti,i+1 ≈t0 +∇ti,i+1(Xi −Xi+1) known as the Peierls [14–16] or the Su-Schrieffer-Heeger(SSH) [40, 41] model, shown in Figure 2.2.Let me now provide a historic perspective of what I refer to as the Peierlsmodel. This model has been recently called the ‘BLF-SSH’ model by other authors[47] in an attempt to give due credit to Barisic´, Labbe´ and Friedel (BLF) [14–16], who proposed the model about ten years earlier to Su, Schrieffer and Heeger(SSH) to study superconductivity. SSH introduced it later to study properties ofone-dimensional organic compounds with great success [40, 41].In this thesis, I will study the model in which the coupling of the electronsis to optical phonons; the model resembles the BLF-SSH model insofar as theions couple to the electronic motion (as opposed to electron charge density.) Thismodeling with optical phonons seems to have started with Capone, Stephan andGrilli [48] and, more recently, with Marchand, De Filippis, Cataudella, Berciu,Nagaosa, Prokofev, Mishchenko and Stamp [17] and Cataudella, De Filippis andPerroni [49]. Note that the authors of Ref. [47] called this model with couplingto optical phonons the CSG model after Capone, Stephan and Grilli.I refer to this model as the Peierls model in an attempt to primarily highlightthe coupling of electron hopping to phonons.152.2. Particle-lattice interactionst0tt0<t0t>Figure 2.2: Schematic representation of the Peierls particle-phonon cou-pling. The lattice deformation depicted by black arrows modify the distancebetween the ordered atoms. This in turn modulates the hopping integral of thebare particle (magenta dot). These distortions occur in most lattice structures,unlike the vibronic coupling of the Holstein model depicted in Figure 2.1 of specialrelevance to molecular crystals.Note that using optical modes instead of acoustic modes is not a trivial issue,as the differences between the couplings to the two types of phonons are not yetunderstood.Phenomenology of single polarons in the Peierls modelI review previous results on single polarons in the Peierls model [17]. The Peierlsmodel readsHPeierls = Hp +Hph + VPeierlsHp = −t∑i(c†ici+1 +H.c.)Hph = Ω∑qb†qbq = Ω∑ib†ibiVPeierls = gp∑i(c†ici+1 +H.c.)(bi + b†i − bi+1 − b†i+1). (2.4)I have second quantized the displacement operators X ∼ b† + b in the Taylorapproximation of ti,i+1. Since the spin is a spectator for a single particle in themodel, I drop the spin labels. This very simple model, depicted in Figure 2.2,allows to investigate the effects of the lattice distortions on particle hopping.I characterize the Peierls coupling by the dimensionless effective particle-162.2. Particle-lattice interactionsphonon couplingλ =2gp2Ωt, (2.5)and as before I have set ~ = 1 and the lattice constant a = 1 in this subsubsection.Single polaron The single Peierls polaron problem was only recently studiedin Ref. [17], where it was shown that it can be light at strong couplings. Thisis a consequence of a process in which, for example, the particle hops one siteand creates a phonon, then absorbs this phonon and hops one site further. Allsuch processes contribute to an effective next-nearest-neighbor hopping and thepolaron dispersion is ≈ −2t∗ cos(k) + 2t2cos(2k), where t∗ is the renormalizednearest-neighbor hopping. At strong coupling, the t2 term dominates the disper-sion leading to a sharp transition into a regime of lighter quasiparticles.Studies of models with mixed Peierls and other conventional couplings haveshown that transitions always exist for any finite Peierls coupling [17, 50].Summary This remarkable result presented a departure from the “standard”polaron behavior. Namely• Peierls polarons are light at strong couplings unlike Holstein polarons.• Peierls polarons exhibit a sharp transition in their ground-state properties.Bipolaron Before my work appeared in Refs. [20? , 21] and presented here,it was not known if these effects extend to bipolarons. In the following chaptersI consider the two-carrier problem and provide accurate results for bipolaronsand dimers in the Peierls model. I explicitly define these objects in the relevantchapters below.I shall simplify the notation and refer to the Peierls Hamiltonian 2.4 as H ≡HPeierls with V ≡ VPeierls and g ≡ gp in the following chapters, unless otherwisespecified. I made every chapter self-contained for the purpose of the reader whowishes to view a certain set of results without reference to the other.172.3. Polaronic effects and superconductivity2.3 Polaronic effects and superconductivityThe Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity explains con-ventional superconductivity in a formalism that accounts for the screening of theCoulomb repulsion by an effective phonon-mediated electron-electron attraction[6, 7]. Here, phonons induce an instantaneous (at large phonon frequencies) or aretarded (at small phonon frequencies) attraction between electrons in the Cooperchannel describing the binding of a pair with a zero momentum.This phonon-induced interaction was first derived for two electrons above aFermi sea [51, 52]. Cooper then showed that any such non-zero attraction nomatter how small will lead to an instability in the Fermi sea towards the formationof Cooper pairs, bound electron pairs with a spatial extent larger than the averageinter-particle spacing [8]. Ultimately, this led Bardeen, Cooper and Schrieffer toconsider an electronic model incorporating the attraction in the Cooper channelHBCS =∑~k,σE~kc†~k,σc~k,σ +1N∑~k,~k′V~k,~k′c†~k,↑c†−~k,↓c ~−k′,↓c~k′,↑, (2.6)and found its solution in a mean-field approximation for the Cooper channel:〈c†~k,↑c†−~k,↓〉, describing the formation of Cooper pairs.Consider the s-wave pairing induced by an attractive phonon-mediated inter-actions of the form: V~k,~k′ = −V0 within a shell of thickness ~ΩD around the Fermienergy, |E~k|, |E~k′ | < ~ΩD (recall E~k = ~k − µ), where ΩD is the Debye phononfrequency and µ is the chemical potential. The condensation of Cooper pairs forany non-zero V0 is accompanied by opening of a gap in the spectrum and onefindskBTc =2eγEpi~ΩDe− 1V0ρF , (2.7)where the Euler constant is defined as γE ≈ 0.577 and ρF is the density of statesper spin flavor in the Fermi sea and the factor of 2 accounts for the degeneracyof spin-up and spin-down states.One may expect that increasing the electron-lattice coupling strength, V0,increases the binding between electrons and should lead to superconductivityat higher Tc. However, in this limit the electrons become dressed by clouds of182.3. Polaronic effects and superconductivityphonons forming polarons, with an enhanced effective mass [1, 2, 12, 13, 36, 37,42, 53–59], and the weak-coupling BCS theory breaks down as the bound pairsshrink to a size smaller than the inter-particle spacing.In this limit, a different mechanism of superconductivity was suggested toarise. Alexandrov and Ranninger [60] used the Fro¨hlich model to argue thatheavy bipolarons interact repulsively to second order in perturbation theory andderived an effective Hamiltonian describing the superfluidity of the tightly boundbosonic pairs building on similarities with molecular superconductivity of itiner-ant hardcore pairs [61].This mechanism of superconductivity of tightly bound pairs can be most easilyunderstood as triggered by the onset of coherent superfluid flow of the preformedpairs that Bose-Einstein condense in three dimensions at temperatures lower thanor equal tokBTBECc ≈ 3.31~2ρ2/3BmB, (2.8)where ρB is the density of the bosonic pairs with mass mB.But, the effective mass of bipolarons grows with the coupling strength (andis many orders of magnitude heavier than the free electron mass, me), quench-ing the superconducting coherence and resulting in a suppression of Tc [62]. Attemperatures larger than Tc, such systems typically exhibit charge density waveinsulating behavior. In fact, Anderson used the effective attractive interactioninduced by phonons in local electron-phonon models, e.g. the Holstein model, toexplain the behavior of insulating amorphous semiconductors [9].Based on these results, it has become generally believed impossible to formbipolarons that remain light at strong electron-phonon coupling, making high-Tcbipolaronic superconductivity very unlikely [62].These arguments [62] are derived from studies of the Holstein [13] and Fro¨hlich[12, 54] models, where the phonons modulate the potential energy of the electrons.On the other hand, as argued before, the coupling to phonons also modulates thehopping integrals, as described by the Peierls model [14–16] (known as the Su-Schrieffer-Heeger (SSH) model for polyacetylene [40, 41]). Recently, as explainedabove, it was shown that single polarons in this latter class of models can be lightat strong coupling strengths [17]. I will show in Chapter (4) of this thesis that192.4. Relevance to experimental systemsPeierls bipolarons can be very light, O(1) in me, suggesting that high-Tc bipola-ronic superconductivity may be possible in materials with this type of coupling.To gain intuition of the behavior of Tc with the effective mass, m∗, one can usecoherence arguments to relate the Fermi temperature, TF , of a non-interactinggas of spin-1/2 fermions to the Bose-Einstein Condensation (BEC) temperature,TBECc , of the paired fermions. A Fermi gas of spin-1/2 electrons of mass, me,becomes coherent at temperatures less than or equal tokBTF =(3pi2)2/32~2ρ2/3Fme, (2.9)where ρF is the density of the gas of electrons.At temperatures T < TF , the inter-particle spacing is comparable to the de-Broglie wavelength and the gas behaves quantum mechanically. If the fermionsin the gas pair into quasiparticles with an effective mass m∗ = mB = 2me, we canrelate the BEC temperature at which the gas of bosonic pairs becomes coherentto TF . In such a case, ρF = ρB/2, and a simple calculation givesTBECc ≈ 0.2TF . (2.10)A typical metal has a TF ≈ 10000K, which yields a gigantic TBECc > 1000◦C. ThePeierls bipolarons I discuss in Chapter (4) of this thesis are very light, O(1) inme. A dense enough condensate of such strongly bound pairs gives hope for ahigh Tc. To determine how high Tc can be, one needs to study the conditions forthe formation of a condensate of favorable density, and verify that other interac-tions, such as bipolaron-bipolaron interactions, do not severely renormalize thebipolaron mass.2.4 Relevance to experimental systemsBefore I discuss the various results obtained for the Peierls coupling, I review itsrelevance to experimental systems.202.4. Relevance to experimental systems2.4.1 Solid-state systemsIn simple lattices with a single atom in the unit cell, one expects only the Peierlscoupling (to acoustic phonons) to be non-zero (also, the so-called breathing-modecoupling would be non-zero). On the other hand, complex lattices with a struc-tured unit cell may have a mixture of Holstein-type and Peierls-type couplings,albeit likely to different phonon modes. I generally expect the Peierls coupling,Figure 2.2, to play a significant role in many solids.One may be tempted to question this assertion due to the limited experimen-tal data available on the Peierls model in the limit of low-carrier concentration.Perhaps many experimental results on polaronic and bipolaronic systems withstrong Peierls coupling were misinterpreted in the framework of Holstein or otherphenomenological theories. Coupled with previous results on polarons [17], my re-sults show remarkable deviation from the standard polaron and bipolaron physics.In light of these findings, I believe that a careful re-interpretation of experimentalresults may be called for for many of these systems. For example, assuming a lackof, or a small, mass enhancement automatically implies a weak electron-phononcoupling is no longer justified. Mass enhancement could occur as a result of othermechanisms. Ultimately, this necessitates careful modeling of materials beforeany firm conclusion about their physical behavior can be inferred.The theory developed here provides a general qualitative framework withinwhich to understand the generic effects of the Peierls coupling. A quantitativeunderstanding of polarons and bipolarons in experiments requires, however, care-ful consideration of the microscopic details of the underlying physical situation.Such studies can be performed using straightforward generalizations of the meth-ods developed here and are left to future work.The electron-phonon coupling is characterized by the adiabaticity ratio A =Ω/W , whereW is the electron bandwidth; in one dimensionW = 4t, where t is thebare particle hopping. In many materials, A << 1. This so-called adiabatic limit,Ω << W , presents severe challenges to numerical and analytical approaches.There, at strong electron-phonon coupling strengths, the bare particle excites avery large number of phonons and the resulting polaron clouds extend spatiallyover a large number of sites. This impedes convergence of numerical approaches.212.4. Relevance to experimental systemsSome investigators employed a semi-classical approximation treating thephonons classically to study this limit [63]. The polaron or bipolaron groundstate is then searched in the Born-Oppenheimer potentials created by the differ-ent static lattice configurations. This treatment, however, leads to metastablestates that break the lattice translational symmetry, and unphysical effects suchas localization of polarons (see Ref. [64] for more discussion).Here, I solve the problem fully including quantum fluctuations for Ω ≥ 3t.Ω = 3t is the smallest frequency for which the variational exact diagonalization(see Chapter 3, Section 3.1) numerical calculations were converged. The limitΩ < 4t, the bandwidth of the bare particle, characterizes a regime of competingcarrier and phononic excitations. While the strong adiabatic limit is currentlybeyond reach, I expect the variational Momentum Average (see Chapter 3, Section3.2) results to be valid going to around Ω = 0.5t. This is based on the analysisof the single polaron, where the physical behavior does not change qualitativelygoing from Ω = 3t to Ω = 0.5t [17]. I have also analyzed the behavior in the anti-adiabatic limit A >> 1, where I obtain analytical arguments that help explainthe numerical results.An exciting prospect commands the applicability of my theory or its extensionto materials with A > 1 such as SrTiO3 [65], magic-angle Graphene superlattices[66], and layered MoS2 [67]. All these materials exhibit phenomenology thatsuggests a possible connection to bipolaron physics. Specifically, they exhibitsuperconductivity at very low carrier densities, an effect that can be explained bythe Bose-Einstein condensation of preformed pairs, e.g. bipolarons.In all these systems, polaronic signatures may be seen in the spectral func-tion A(ω) measured in angle-resolved photoemission spectroscopy (ARPES) orscanning tunneling microscope (STM) experiments.The Peierls particle-phonon coupling may also be relevant to a variety ofmolecular systems, such as J-aggregates [68], hybrid organic-inorganic perovskitestructured compounds [69], and photosynthetic complexes [70, 71]. A recentstudy focused on the effects of this coupling in photosynthetic systems and founddeparture from previous results of possible relevance to experiments [72].222.4. Relevance to experimental systems2.4.2 Ultracold quantum simulatorsThe Peierls/SSH model can also be realized in experiments with polar moleculesor cold atoms dressed by Rydberg states in optical lattices. In this case, the bareparticles are hard-core rotational excitations (for molecules) or electronic excita-tions (for cold atoms). If the molecules/atoms are trapped in a Mott insulatorphase [73], the excitations can be transferred between different sites of the op-tical lattice through couplings mediated by the dipole-dipole interactions. Theinter-site transfer of excitations has been observed for molecules in Ref. [74] andfor Rydberg atoms in Ref. [75].The bosonic field is provided by the translational motion of the species trappedin the lattice potential. These are nearly dispersionless Einstein optical latticephonons coupled to the internal excitations of molecules/atoms via the radialdependence of the dipolar interactions. This was illustrated for polar moleculesin Ref. [50, 76] and for cold atoms dressed with Rydberg excitations in Ref. [77]. Inboth cases, the hopping term couples to lattice distortions giving rise to a Peierlscoupling gp = −3(t/a)√~/2mΩ, where m is the mass of the molecule/atom anda is the lattice constant. The frequency of lattice phonons in a one-dimensionalarray is Ω = (2/~)√V0ER, where V0 is the lattice depth and ER = ~2pi2/2ma2is the recoil energy of a molecule in its harmonic trap [78]. The dimensionlessPeierls coupling is then λ ≡ 2g2p/(Ωt) = 18ERt/(~piΩ)2. Thus, λ can be varied bytuning t/Ω.To achieve stronger coupling, one can either increase t, which can be achievedfor cold atoms by dressing with Rydberg states, or decrease Ω, which can beachieved for either atoms or molecules by simply decreasing the intensity of theoptical lattice laser. In the latter case, care must be taken to prevent the Mottinsulator phase from melting. The Mott insulator phase can be stabilized byinducing an on-site repulsion between molecules or atoms by tuning the scatteringlength of molecules/atoms by magnetic Feshbach resonances [79] or by applyingweak DC fields to induce strongly repulsive dipolar interactions between moleculesin the same internal state. The presence of a DC field may induce additionalphonon-mediated interactions [50, 76]. Therefore, the field must be weak enoughto ensure that the Peierls interactions remain dominant.232.4. Relevance to experimental systemsThe soft-core version of the Peierls model may be realized with mixtures ofatoms, where the heavier atoms trapped in an optical lattice provide the phonondegrees of freedom, and the lighter two-component atoms, representing the bareparticles, do not see the trap but couple to the heavy atoms by Feshbach reso-nances [80]. This idea requires further analysis.The polaron or bipolaron appears as a sharp peak in the spectral functionA(ω). Extensions of the stimulated Raman spectroscopy scheme of Ref. [50] canact as a measure of A(ω) enabling the study of polarons and bipolarons in theseultracold experiments.24Chapter 3Methods for studying polaronsand bipolaronsThe author of this thesis performed numerical Momentum Average (MA) andanalytical calculations. Numerical VED results were obtained by MonodeepChakraborty and are used for validation of the MA results. For completeness, Ireview the basic ideas behind the VED and MA methods. I also discuss the ana-lytical techniques used to gain insight into the physics of polarons and bipolarons.This chapter aims to introduce the conceptual nous behind these approachesto the reader, emphasizing the formalism. I shall present calculations and discussthe relevant physics in the following chapters and the accompanying Appendices.3.1 Variational exact diagonalizationThis well-established approach, introduced in Ref. [42], utilizes a variationallyconstructed Hilbert space to find the Hamiltonian eigenstates and eigenenergiesnumerically.For a single polaron, the variational space is constructed beginning with aninitial Bloch state with no phonons, and operating Nh times with the off-diagonalparts of the Hamiltonian. At each step, basis states are added to the basis setif they have a nonzero matrix element with a preexisting basis state. A basisstate is included if it can be reached using Ng phonon creation operators andNt electron hops in any order, with Nt + Ng ≤ Nh. The Hamiltonian matrix isthen built in the variational basis determined by Nh and diagonalized using theLanczos algorithm.By construction, the variational space of Bloch states corresponds to periodiclattices and the prescribed algorithm constructs such Hilbert space without bias.253.2. Momentum Average approximation. . .This establishes VED as a valuable numerical method more efficient than standardbasic approaches.The VED reference results presented in Chapter 4 were obtained using a vari-ant of this method for bipolarons, where the variational basis set is generatedsystematically starting from the Bloch state for an on-site singlet fermionic statewith zero accompanying phonons [38, 39, 44]. Chapter 5 includes reference VEDresults obtained using a version for triplet bipolaronic pairs (equivalently, in onedimension, two hard-core polarons), where the variational basis set is expandedsystematically starting from the Bloch state for two adjacent particles and zerophonons.While for Holstein polarons, VED was successfully extended to study theextreme adiabatic limit [81, 82], for Peierls bipolarons, the current implementationof VED was converged for frequencies Ω ≥ 3t, which I present in this thesis.3.2 Momentum Average approximation: Avariational Green’s function approachThe Momentum Average (MA) approach introduced by Berciu [83] utilizes avariational approximation to solve the equations of motion for the few-particleGreen’s functions in models of particle-boson coupling. Berciu and collaboratorshave applied MA to Holstein-type problems [59] and extended it to single Peierlspolarons [17, 50]. MA has been also extended to study other types of polaronproblems [84, 85]. In this thesis, I extend MA to the study of two-carrier problemsin the Peierls model.In the following subsection I provide an introduction to the MA approach asapplied to the Holstein polaron, see also [59, 83].3.2.1 Green’s function of the Holstein polaronRecall the Holstein model Eq. (2.2)HH =∑kkc†kck + Ω∑ib†ibi + gH∑ini(bi + b†i ). (3.1)263.2. Momentum Average approximation. . .c†k =1√N∑i eikric†i creates a particle in a Bloch state with momentum k, i is thesite index, and k = −2t cos(ka) is the bare particle dispersion. The Hamiltonianis conveniently written in a mixed real space and momentum space representationfor reasons that will become obvious in what follows.Hierarchy of equations of motionThe resolvent of the Green’s function is defined asGˆ(ω) =1ω + iη −HH , (3.2)with η → 0+. One generates the equation of motion for the propagatorG(k, ω) = 〈k|Gˆ(ω)|k〉 (3.3)defined for states |k〉 = c†k|0〉 by repeatedly applying the Dyson’s identity Gˆ(ω) =Gˆ0(ω) + Gˆ(ω)Vˆ Gˆ0(ω) to calculate G(k, ω) = 〈0|ck(Gˆ0(ω) + Gˆ(ω)Vˆ Gˆ0(ω))c†k|0〉.H was separated into a non-interacting part H0, composed of the free particleand free bosons parts, and the Holstein particle-boson coupling Vˆ . Gˆ0(ω) =(ω + iη −H0)−1 is the resolvent of the free particle and the corresponding prop-agator is G0(k, ω) = 〈0|ckGˆ0(ω)c†k|0〉 = 1ω−k+iη . This procedure yields an infinitehierarchy of equations. Writing out the expression for G0(k, ω) in G(k, ω). onefindsG(k, ω) =1ω − k + iη(1 + 〈0|ckGˆ(ω)Vˆ c†k|0〉). (3.4)Consider the Vˆ c†k|0〉 term. The Holstein coupling generates higher order prop-agatorsG(k, ω) =1ω − k + iη(1 +gH√N∑ie−ikriF1(k, i, ω))=1ω − k + iη(1 +gH√N∑q1F1(k, q1, ω)), (3.5)where I have defined the generalized Green’s function of a particle in presence of273.2. Momentum Average approximation. . .a single phonon in real space and momentum space, respectivelyF1(k, i, ω) = 〈0|ckGˆ(ω)c†ib†i |0〉F1(k, q1, ω) = 〈0|ckGˆ(ω)c†k−q1b†q1|0〉. (3.6)This result can be interpreted as follows. The propagator takes the particle fromits initial state with no phonons to a state with a single phonon via all possiblepaths weighted with amplitudes given by the free particle propagator.One expands F1(k, i, ω) using the Dyson’s identity: F1(k, i, ω) =〈0|ckGˆ(ω)Vˆ Gˆ0(ω)c†ib†i |0〉. The free particle propagator Gˆ0(ω) cannot connectstates with different phonon number and therefore is dropped. To proceed, onemust evaluate Vˆ Gˆ0(ω)c†ib†i |0〉 to findF1(k, i, ω) =∑jG0(j − i, ω − Ω)〈0|ckGˆ(ω)Vˆ c†jb†i |0〉. (3.7)MA approximation To appreciate the meaning of the MA approximation,consider Vˆ in Eq. (3.7). The Holstein interaction Vˆ in Eq. (3.1) creates ordestroys phonons only where a particle exists; Vˆ c†jb†i |0〉 = gH(δi,jc†i + c†jb†ib†j)|0〉.Thus, upon the action of Vˆ in Eq. (3.7), one obtains two contributions under thesum,∑j: δi,j〈0|ckGˆ(ω)c†i |0〉 and 〈0|ckGˆ(ω)c†jb†ib†j|0〉.Here comes the crucial approximation in MA: one restricts the size of thepolaron cloud. One throws away terms in∑j 6=i. . . 〈0|ckGˆ(ω)c†jb†ib†j|0〉 that corre-spond to clouds larger than a chosen spatial cut-off. In this simple examplederived here, I allow only one-site polaron clouds, i.e. I retain only the term〈0|ckGˆ(ω)c†i(b†i)2|0〉. This is known as the MA0 level of approximation [86].Interest in strongly bound polaron states with a spatially confined cloud justi-fies the MA0 approximation used in this subsection, which suffices to successfullyyield accurate results for the ground state of the Holstein model [59, 83]. General-ization to more extended polaron clouds is straightforward and leads to improvedaccuracy [86].Alternatively, the MA0 approximation can be viewed as a variational approach283.2. Momentum Average approximation. . .with a trial wavefunction ∑i,j,nαi,j,nc†i(b†j)n|0〉, (3.8)where αi,j,n are the wavefunction coefficients. Since the MA approximation re-stricts Vˆ in the equation of motion, but does not influence the action of the freeresolvent Gˆ0(ω), all translations of the bare particle are retained in the variation-ally restricted Hilbert space.Within the MA0 approach, one findsF1(k, i, ω) = gHG0(0, ω − Ω) [G(k, i, ω) + F2(k, i, ω)] , (3.9)withG(k, i, ω) = 〈0|ckGˆ(ω)c†i |0〉, (3.10)F2(k, i, ω) = 〈0|ckGˆ(ω)c†i(b†i)2|0〉 (3.11)andG0(0, ω − Ω) = 1N∑kG0(k, ω − Ω) := g¯0(ω − Ω). (3.12)is the momentum-averaged propagator obtained by a Fourier transform ofG0(k, ω − Ω).I proceed as before and arrive at the general form for the equation of motionFn(k, i, ω) = 〈0|ckGˆ(ω)Vˆ Gˆ0(ω)c†i(b†i)n|0〉=∑jG0(j − i, ω − Ω)〈0|ckGˆ(ω)Vˆ c†j(b†i)n|0〉. (3.13)Following some algebra, the final MA hierarchy of equations readsF0(k, i, ω) = 〈0|ckGˆ(ω)c†i |0〉,Fn(k, i, ω) = gHg¯0(ω − nΩ) [nFn−1(k, i, ω) + Fn+1(k, i, ω)] . (3.14)Here, the phonon destruction operator produces the prefactor n corresponding to293.2. Momentum Average approximation. . .the number of ways a phonon can be annihilated by Vˆ in a given configuration,see Appendix A.1 for more details.In what follows I suppress the functional dependence in the hierarchy andwrite Fn = αnFn−1 + βnFn+1, where αn = ngHg¯0(n), βn = gHg¯0(n) and the ω andΩ dependences in g¯0 are now implicit. On physical grounds, one expects Fn+1 tobecome vanishing small for sufficiently large n. Thus, one can solve iterativelystarting from Fn ≈ αnFn−1, from which one solves for Fn−1, Fn−2, etc. . .Fn−1 =αn−11− βn−1αnFn−2 (3.15)or Fn =αn1−βnαn+1Fn−1, but Fn = αnFn−1 soαn =αn1− βnαn+1 (3.16)orα1 =α11− β1 α21−β3 α31−.... (3.17)Similarly, for F1 I haveF1 =α11− β1 α21−β3 α31−...F0, (3.18)where F0 is precisely G(k, i, ω − nΩ), see Eq. (3.10). Using Eq. (3.5), I findG(k, ω) =1ω − k + iη (1 +gH√N∑ie−ikriα¯G(k, i, ω)) (3.19)The Fourier transform yieldsG(k, ω) =1ω − k + iη (1 + gHα¯G(k, ω)) (3.20)from which I obtain a closed form for G(k, ω):G(k, ω) =1ω − k + iη − gHα¯ (3.21)303.2. Momentum Average approximation. . .and I have found the MA0 polaron self-energy Σ = gHgα¯.Numerical procedure For the recurrence relation of the type (3.17) in MA0,I use the modified Lenz algorithm to compute continued fractions efficiently andobtain accurate results. More generally, for other extensions and variations ofMA with larger clouds, the recurrence relation can only be solved numericallyas a non-homogenous system of linear equations or using advanced numericaltechniques for continued fractions such as recursive algorithms.From this example one sees that MA devises a clever approximation to simplifythe equations of motion and obtain accurate results efficiently.3.2.2 Green’s function of the Peierls polaron andbipolaronThe appropriate choice of the variational space depends on the details of theHamiltonian and state(s) of interest [86]. For the Holstein model, a one-sitephonon cloud suffices to provides accurate results for single polarons [59] and forstrongly bound on-site bipolarons [87] (see Chapter 2 and the discussion in 2.2.1on Holstein bipolarons). Taken together with the local nature of the Holsteincoupling, this explains why a one-site phonon cloud is accurate to describe suchstates. For the Peierls model, the coupling to phonons is non-local and thereforea bigger cloud is required to yield accurate results. A three-site phonon cloud MAhas been shown to be very accurate for single polarons (within the line-widths ofnumerically exact results for the ground-state dispersion) in such systems [17].I have extended the MA approach to study strongly bound bipolarons inthe Peierls model. I utilize the three-site cloud flavor of MA. Below I sketch themethod as applied to two hard-core particles, e.g. spinless fermions, in the Peierlsmodel. More details can be found in Appendix A.2.MA for two hard-core particles in the Peierls modelHere, I derive the MA equations for two hard-core particles with a three-sitephonon cloud and allow the particles to be arbitrarily far from the cloud but at313.2. Momentum Average approximation. . .most two sites apart from each other, if a cloud is present. Terms correspondingto the particles being further than two sites apart are expected to contributesignificantly only to higher-energy states, if a two-particle state is strongly bound,which is the case of primary interest in this work. For weakly bound dimers, thevariational space must be extended to include configurations with the particlesfurther apart when the cloud is present.To highlight the method, I show a few representative equations, see AppendixA.2.1 for more discussion. As before, I derive the equation of motion using Dyson’sidentity Gˆ(ω) = Gˆ0(ω)+Gˆ(ω)Vˆ Gˆ0(ω), where Vˆ is the Peierls bare particle-phononcoupling termV = g∑i(c†ici+1 +H.c.)(bi + b†i − bi+1 − b†i+1). (3.22)Consider the two-particle propagator G(K, 1, n, ω) = 〈K, 1|Gˆ(ω) |K,n〉 de-fined for two-particle states |K,n〉 = ∑i eiK(Ri+na/2)√N c†ic†i+n |0〉 with the two parti-cles n ≥ 1 sites apart, and a is the lattice constant. Its exact equation of motionisG(K, 1, n, ω) = G0(K, 1, n, ω)+∑ηG0(K, η, n, ω)〈K, 1|Gˆ(ω)Vˆ |K, η〉 .G0(K, 1, n, ω) can be calculated exactly analytically in one dimension [88]. Con-sider now Vˆ |K, η〉. It consists of states with one phonon plus the particles η ± 1sites apart. Thus, the right-hand side of the exact equation of motion containsan infinite number of terms. Because within MA one restricts the particles tobe within two sites of each other when phonons are present, this simplifies the323.2. Momentum Average approximation. . .equation of motion toG(K, 1, n, ω) = G0(K, 1, n, ω)− ge−iKaG0(K, 2, n, ω)F1(−2, 1) + gG0(K, 2, n, ω)F1(−1, 1)− gG0(K, 2, n, ω)F1(0, 1) + geiKaG0(K, 2, n, ω)F1(1, 1)− ge−3iKa/2G0(K, 3, n, ω)F1(−3, 2)− g[e−3iKa/2G0(K, 1, n, ω)− e−iKa/2G0(K, 3, n, ω)]F1(−2, 2)− 2ig sin(Ka/2)G0(K, 1, n, ω)F1(−1, 2)+ g[e3iKa/2G0(K, 1, n, ω)− eiKa/2G0(K, 3, n, ω)]F1(0, 2)+ ge3iKa/2G0(K, 3, n, ω)F1(1, 2), (3.23)where F1(m,n) is shorthand for F1(K,m, n, ω) defined asFl(K,m, n, ω) ≡∑ieiKRi√N〈K, 1|Gˆ(ω)c†i+mc†i+m+nb†li |0〉 , (3.24)i.e. a generalized one-site cloud propagator. By introducing other appropriategeneralized propagators:Fl1,l2(K,m, n, ω) ≡∑ieiKRi√N〈K, 1|Gˆ(ω)c†i+mc†i+m+nb†l1i b†l2i+1 |0〉 , (3.25)Fl1,l2,l3(K,m, n, ω) ≡∑ieiKRi√N〈K, 1|Gˆ(ω)c†i+mc†i+m+nb†l1i−1b†l2i b†l3i+1 |0〉 , (3.26)for two-site cloud and three-site cloud configurations respectively, and repeatedlyapplying the Dyson’s identity, one derives the MA equations of motion for thepropagators in Eqs. (3.23)-(3.26).The resulting hierarchy cannot be solved analytically and I resort to numericalsolvers to obtain the propagator of interest G(K, 1, n, ω). The numerical resultsare presented in the following chapters.MA for two soft-core particles in the Peierls modelI have also extended the MA method to the study of two soft-core particles in thePeierls model. In this extension, I have relaxed the restriction on the distancebetween the bare particles in configurations with phonons. This turns out to be333.3. Asymptotic expansions in the anti-adiabatic regimenecessary to accurately describe the bipolaron states, see Chapter 4 for results.Outlook on MABy construction the MA approach developed here is designed to describe stronglybound states accurately. Indeed, the comparison with VED results in the follow-ing chapters verifies the success of MA in this regime. A suitable increase of thevariational space is necessary to improve the accuracy of the MA approximationfor weakly bound states. This can be done in a rather straightforward way andpromises to establish MA as a valuable and efficient method for the study ofbipolarons as it is for polarons.Furthermore, MA can be easily extended to higher dimensions, where exactnumerics suffer severe convergence issues. Thus, MA offers a promising directionforward in the field of polaron and bipolaron physics.3.3 Asymptotic expansions in theanti-adiabatic regimeTo gain insight into the physical mechanisms underlying the polaron and bipo-laron behavior, one analyzes the asymptotic anti-adiabatic limit, Ω >> t, g. Inthis regime, subspaces with different phonon number are separated by large gapsand the energy scales t and g2/Ω corresponding to single-phonon corrections dom-inate the low-energy physics, see Figure 3.1. This reduces the problem to a two-body (bare particle + phonon) one for the polaron and a three-body (two bareparticles + phonon) one for the bipolaron.Such two- and three-body problems can be solved numerically. To attainanalytical results, however, one renormalizes the bare particle operators by theprojected one-phonon corrections. The resulting single polaron one-body problemis diagonalized by Fourier transforms, while the bipolaron two-body problem issolved in some representative limits by the equation-of-motion (EOM) technique[88].This approach provides an effective Hamiltonian description for the dressedparticles. Formally, the polaron operators in the anti-adiabatic Hilbert space are343.3. Asymptotic expansions in the anti-adiabatic regimeΩΩ gtgnph = 0nph = 1nph = 2Figure 3.1: Schematic representation of the Hilbert space in the anti-adiabatic limit Ω >> t, g. The successive nph-phonon sectors are separated byΩ, the phonon frequency. t is the bare hopping and g is the particle-phonon cou-pling strength. The electron-phonon coupling couples the zero-phonon subspaceto the one-phonon subspace with a vertex of amplitude g << Ω. In this limit,one obtains a low-energy theory to O(g2/Ω) by projecting out the higher energyphonon subspaces.mimicked by the action of the bare particle operators in the renormalized low-energy subspace, if the bare Hamiltonian is appropriately modified to include therelevant particle-phonon corrections. Below I outline the procedure.3.3.1 One-particle sectorLet Pˆ be the projector onto the zero-phonon Hilbert subspace, which is spannedby the states c†i |0〉, ∀i. The effective Hamiltonian in this subspace is, to second353.3. Asymptotic expansions in the anti-adiabatic regimeorder,hˆ1 = Tˆ + Pˆ Vˆ1E0 − Hˆ0Vˆ Pˆ ,where the bare one-body operator is the kinetic energy Tˆ = −t∑i(c†ici+1 +H.c.),the interaction Vˆ is the particle-phonon coupling, and Hˆ0 = Hph is the phononterm. E0 = 0 in this zero-phonon Hilbert subspace.Application to single polarons in the Peierls modelFor single polarons in the Peierls model, this projection leads to [17],hˆ1 = −0∑inˆi + Tˆ + Tˆ2, (3.27)In addition to the nearest-neighbor bare particle hopping, hˆ1 contains the polaronformation energy 0 = 4g2/Ω, and a dynamically generated next-nearest-neighborhopping t2 = g2/Ω = λt/2 (note the unusual sign) resulting from virtual emissionand subsequent absorption of a phonon by the particle, as it hops on and off anintermediate site.Setting a = 1, the polaron dispersion is, therefore,EP (k) = −0 − 2t cos(k) + 2t2 cos(2k).For t > 4t2, i.e. if λ <12, the polaron ground-state (GS) momentum is 0. Forλ > 12, the t2 term dominates the dispersion and the polaron GS momentum iskP = arccost4t2, going asymptotically to pi2as λ → ∞. This explains the sharptransition into a lighter polaron at strong coupling strengths.For more discussions about the polaron behavior in the Peierls and Holsteinmodels, see Chapter 2 and discussions in 2.2.1 and 2.2.2.363.3. Asymptotic expansions in the anti-adiabatic regime3.3.2 Two-particle sectorRepeating the projection onto the two-particle–zero-phonon subspace spanned bythe states c†i,σc†i+n,σ′|0〉, ∀i, n ≥ 1, σ ∈ [↑, ↓], one finds:hˆ2 = hˆ1 + Vˆ0 + Pˆ Vˆ1E0 − Hˆ0Vˆ Pˆ , (3.28)where Vˆ0 is the bare-particle two-body Hubbard interaction.For reference to results on Holstein bipolarons, see Chapter 2 and the discus-sion in 2.2.1.In the next chapters I consider different forms of Vˆ0 and derive hˆ2 for twoparticles with soft-core and hard-core statistics in the Peierls particle-phononmodel. I consider two spinful fermions in Chapter 4 and two triplet fermions orequivalently hard-core particles in Chapters 5 and 6.I solve for the spectrum of hˆ2 numerically as explained below.Approach to the reduced two-body problemThe reduced two-body problem defined by hˆ2 is solved in the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) equation-of-motion (EOM) approach [88, 89].One derives the equations of motion for the resolventGˆ(ω) = (ω + iη − hˆ2)−1 (3.29)evaluated in the two-particle Bloch states |K,n〉 with n the distance between theparticles. By applying hˆ2 on the basis states, I derive a hierarchy of recurrencerelations for the desired propagator.Let me demonstrate the method for two hard-core particles in the one-dimensional t-V modelH = −t∑i(c†ici+1 +H.c.)+ V∑inˆinˆi+1. (3.30)The equations of motion for the two-particle propagator defined as G(K,n, ω) =373.3. Asymptotic expansions in the anti-adiabatic regime〈K, 1| Gˆ(ω) |K,n〉 are(ω + iη − V )G(K, 1, ω) = 1− αKG(K, 2, ω) (3.31)and for any n ≥ 2,(ω + iη)G(K,n, ω) = −αK [G(K,n− 1, ω) +G(K,n+ 1, ω)] . (3.32)Here αK = 2t cos(K2). A general solution to this recurrence relation can be foundnumerically. For more details and a detailed discussion, see Appendix A.3.I will use these methods to solve more complicated two-body problems in thefollowing chapters. Below I make reference to two important results that I shallrefer to repeatedly.Two soft-core particles with on-site interaction For two soft-core parti-cles, any finite on-site attraction U binds the pair and results into one bound statein one dimension [90]. This will be relevant to the discussion of singlet Peierlsbipolarons in Chapter 4.Two hard-core particles with nearest-neighbor inter-site interactionOn the other hand, for two hard-core particles, a nearest-neighbor interactionV > 2t is required to bind a dimer in one dimension [88, 91]. Again, there is onlyone bound state. This is relevant to the discussion of two hard-core polarons orequivalently the pairing of two electrons in the triplet sector in the Peierls model,presented in Chapters 5 and 6.38Chapter 4Two electrons in theone-dimensional Peierls-Hubbardmodel4.1 Motivation: Bipolaronic superconductivitySince the discovery of superconductivity in Hg with a critical temperature Tc =4.2 K [92], the quest for materials with high Tc has been a central driver ofresearch in condensed matter physics, leading to the discovery of many othersuperconductors including the unconventional “high”-Tc cuprate [93] and iron-based [94, 95] families, besides many conventional and unconventional low-Tcones.As explained in Chapter 2, conventional low-Tc superconductivity is under-stood to be a consequence of electron-phonon coupling [6, 7]: Exchange of phononsleads to an effective attractive interaction between electrons in the Fermi sea form-ing Cooper pairs [8], which condense into a superfluid. While there is no proventheory of high-Tc superconductivity, it is widely accepted that phonon-mediatedsuperconductivity cannot exhibit high Tc (at ambient pressure). High Tc wouldrequire strong electron-phonon coupling, but in this limit the electrons becomedressed by clouds of phonons forming polarons, with a renormalized effective mass[1, 2, 12, 13, 36, 37, 42, 53–59]. As demonstrated in Chapter 2, the effective massgrows faster than the phonon-mediated binding in Holstein and related models,resulting in a suppression of Tc [62]. Based on these results, it has become gen-erally believed impossible to form bipolarons (polaron pairs bound by phononexchange) that remain light at strong electron-phonon coupling, making high-Tcbipolaronic superconductivity very unlikely [62, 96, 97].394.2. Peierls bipolaronsThese arguments [62] are derived from studies of the Holstein [13] and Fro¨hlich[12, 54] models, where the phonons modulate the potential energy of the electrons.On the other hand, as argued before, the coupling to phonons also modulates thehopping integrals, as described by the Peierls model [14–16] (known as the Su-Schrieffer-Heeger (SSH) model for polyacetylene [40, 41]). Recently, it was shownthat single polarons in this latter class of models can be light at strong couplingstrengths [17]. See Chapter 2 for an exposition of these ideas.Here, I study phonon-mediated binding of electrons into bipolarons in thePeierls model. I show that Peierls electron-phonon coupling leads to strongphonon-mediated attraction between electrons in the singlet sector (the tripletsector is discussed in Chapter 5). This attraction results in the formation ofstrongly bound yet very light bipolarons: Their mass at strong coupling is closeto twice the free electron mass. Such light bipolarons are expected to condenseinto a superfluid at very high temperatures. This work thus points to a newdirection in the search for high-Tc superconductors: Designing materials withelectron-phonon coupling predominantly of the Peierls-type can lead to phonon-mediated superconductivity at high temperatures.4.2 Peierls bipolaronsI study the singlet state of two spin-12fermions in an infinite one-dimensional chaindescribed by the Peierls-Hubbard modelH = He+Hph+Vˆe−ph. For completeness,He = −t∑i,σ(c†i,σci+1,σ +H.c.)+ U∑inˆi,↑nˆi,↓ (4.1)is the Hubbard model of the bare particles with on-site repulsion U , i is the siteindex and nˆi,σ = c†i,σci,σ counts particles with spin σ at site i. Hph = Ω∑i b†ibi(in units of ~) is the phonon Hamiltonian describing a single Einstein mode withfrequency Ω, andVˆe−ph = g∑i,σ(c†i,σci+1,σ +H.c.)(b†i + bi − b†i+1 − bi+1)(4.2)404.2. Peierls bipolaronsis the Peierls/SSH electron-phonon coupling [17]. We characterize the electron-phonon strength using the dimensionless effective couplingλ =2g2Ωt. (4.3)I investigate the singlet eigenstates using the methods of Chapter 3, namely vari-ational exact diagonalization (VED) [38, 39, 44] and the Momentum Average(MA) approximation [59, 83]. The author of this thesis performed MA calcula-tions. The VED results were obtained by Monodeep Chakraborty and are usedfor validation of the MA results.The behavior of the triplet eigenstates (equivalent to two hard-core particlesin one-dimensional infinite chains) will be discussed in Chapters 5 and 6.4.2.1 U = 0 bipolaronsTwo-polaron phase diagramI first set U = 0 and investigate the stability and properties of the resultingbipolarons. I then discuss the role played by a finite bare on-site repulsion U .Figure 4.1 shows the evolution with λ of the low-energy region of the singletsector, for U = 0 and Ω = 3 (all energies are in units of t = 1). The shaded greyarea shows the lower part of the two-polaron continuum: these states describetwo unbound polarons, their energies being the convolution of two single polaronspectra. The dark red region shows the location of the lowest bipolaron band.VED confirms its existence for all λ > 0, although for weak coupling λ . 0.3,the bipolaron ground state lies just below the continuum and cannot be resolvedon this scale. With increasing λ, the bipolaron band moves further below thecontinuum and for λ & 0.57 it becomes fully separated from it. For strongcoupling λ > 1, this band is accompanied by a higher energy bipolaron band(salmon-colored region), whose evolution with λ closely mirrors that of the lowerband, suggesting a common origin. It is important to note that this second bandlies below the bipolaron + one phonon continuum (not shown) that starts at Ωabove the ground state, and therefore it is an infinitely lived bipolaron.414.2. Peierls bipolarons0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00λ141210864E Unbound PolaronsFigure 4.1: Two-polaron phase diagram for U = 0 and Ω = 3. The diagramrepresents the evolution of the low-energy region of the singlet sector with λ.Energies are in units of t. The dimensionless effective coupling is λ = 2g2/(Ωt).The shaded grey area shows the lower part of the two-polaron continuum. Thedark red region represents the lowest energy bipolaron band, while the salmonregion represents the higher energy bipolaron band. These results were obtainedwith MA and are in good agreement with VED results (blue circles) shown forthe low-energy bipolaron.Light bipolaronsClearly the bandwidths of both bipolaron bands are wide even at extremely strongcouplings λ ∼ 2 (this persists for λ > 2 but such values are unphysical), showingthat the bipolarons remain light even when very strongly bound. This is furtherconfirmed in Figure 4.2, where I plot the low-energy bipolaron’s effective massm∗, in units of two free-particle masses, m0 = 2me = ~2/ta2, where a is the latticeconstant. m∗ varies non-monotonically with λ, with a peak at λ ∼ 0.325 wherethe bipolaron ground state energy starts to drop fast below the lower edge of the424.2. Peierls bipolarons0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0λ0.91.11.31.51.7m/m0Figure 4.2: Dependence of the effective mass of the low-energy bipolaronon λ, for U = 0 and Ω = 3.0. m0 = 2me is twice the free electron mass. Thebipolaron’s effective mass is defined as m∗ =(∂2EBP (K)∂K2)−1∣∣∣K=KGS. KGS = 0 forthe Peierls bipolarons. The solid (dashed) lines are VED (MA) results. Note thatm∗ ∼ 2me in the strongly coupled regime, λ > 1.two-polaron continuum (see Figure 4.1), i.e. the bipolaron crosses over into thestrongly bound regime. Importantly, the ratio m∗/m0 stays close to 1 for λ & 1.In other words, the Peierls bipolaron’s effective mass remains comparable to thatof a pair of free fermions even at very strong coupling λ = 2. For comparison,for the same Ω and λ = 2, the Holstein bipolaron’s bandwidth is 0.0135, i.e. itsmass is increased by about two orders of magnitude.The existence of strongly bound yet light Peierls bipolarons at strong couplingis a central result in this thesis, which violates the widely held notion that stronglycoupled bipolarons are always extremely heavy.434.2. Peierls bipolaronsTwo bipolaron bandsI now discuss the second bipolaron band. For reference, note that the one-dimensional Holstein and Fro¨hlich models host only one bipolaron band withinΩ of the ground-state energy (if U = 0) [39]. This is precisely what is generi-cally expected, see Chapter 3, discussion in 3.3.2. For Holstein coupling, recallthe phonon-mediated effective interaction is an on-site attraction −∆E∑i nˆi↑nˆi↓[38, 39], see Chapter 2 and discussion in 2.2.1. In one dimension, such effectiveon-site attraction binds two fermions into a singlet state, but there is only onebound state. The existence of a second bipolaron state is thus very surprising,and points to a new mechanism behind pairing.To understand this new physics, first consider the dispersion of the lowestenergy bipolaron, and its evolution with increasing λ. This is shown in Figure 4.3for the positive half of the Brillouin zone. The curves have been shifted for ease ofcomparison (their absolute positions can be inferred from Figure 4.1). The insetshows the polaron dispersions for the same coupling parameters.At couplings λ ≤ 0.5 where only this bipolaron band exists, the dispersionhas the standard behavior, monotonically increasing with K. For larger λ, thedispersion has a rather unusual shape, strongly peaked near Ka = pi2. Thisshape is highly suggestive of an avoided crossing with a band located above (thesecond bipolaron state that emerges at these couplings). This is confirmed whenI plot both bands, see Figure 4.4 for λ = 2. The gap that opens between thetwo bands varies only weakly with λ, see Figure 4.1. This behavior suggeststhe existence of two bound states with different symmetries, coupled by a λ-independent symmetry-breaking term.4.2.2 Analytical theory in the anti-adiabatic limitTo unravel the pairing mechanism and explain the origin of the two bipolaronstates and their avoided crossing, I consider the anti-adiabatic limit Ω t, g andobtain analytical results by projecting out the high-energy Hilbert subspaces withone or more phonons [98], as explained in Section 3.3. Note that strong-couplingλ 1 is included within the anti-adiabatic regime if t g Ω such thatg2 Ωt.444.2. Peierls bipolarons0.0 0.2 0.4 0.6 0.8 1.0Ka/pi0.00.20.40.60.81.0EBP(K)−EBP(0)λ=0.5λ=0.8λ=1.0λ=2.00.0 0.5 1.0ka/pi10123EP(k)−EP(0)λ=0.5λ=1.0λ=2.0Figure 4.3: Dispersion EBP (K)−EBP (0) of the low-energy bipolaron, forvarious values of λ = 2g2/(Ωt) at U = 0 and Ω = 3. The inset shows thepolaron dispersion EP (k) − EP (0) for the same parameters. All energies are inunits of t. In the main figure, solid lines are VED results and dashed lines are MAresults. Results in the inset were obtained with MA, and are in good agreementwith numerical results [17].As discussed in Chapter 3.3 in Subsubsection 3.3.1, the effective Hamiltonianin the single-particle sector ishˆ1 = −0∑i,σnˆi,σ −∑i,σ(tc†i,σci+1,σ − t2c†i,σci+2,σ +H.c.)Recall the polaron formation energy 0 = 4g2/Ω, and the dynamically gener-ated next-nearest-neighbor hopping t2 = g2/Ω. This term dominates the polarondispersion EP (k) = −0 − 2t cos(ka) + 2t2 cos(2ka) for large λ and explains thechange in its shape observed in the inset of Figure 4.3 (for detailed discussions454.2. Peierls bipolarons0.0 0.2 0.4 0.6 0.8 1.0Ka/pi14.514.013.513.012.512.0EBP(K)λ=2Figure 4.4: Dispersion EBP (K) of both bipolaron bands, for U = 0, Ω = 3and λ = 2, showing an avoided crossing. EBP (K) is in units of t. These areMA results.see Ref. [17]).In the singlet sector, I find that the effective two-particle Hamiltonian is:hˆ2,s = hˆ1 + Uˆ0,2 + Uˆ1. The additional terms describe short-range phonon-mediatedinteractions between the polarons. SpecificallyUˆ0,2 = −T0,0∑i[c†i−1,↑c†i−1,↓ci,↓ci,↑ +H.c.]+T0,2∑i[(c†i+1,↑c†i−1,↓ − c†i+1,↓c†i−1,↑)ci,↓ci,↑ +H.c.]describes nearest-neighbor “pair-hopping” of an on-site singlet with T0,0 =4g2Ω,and transitions between on-site and next-nearest-neighbor singlets with T0,2 =2g2Ω.They arise through emission and absorption of a phonon, e.g. c†i,↑c†i,↓|0〉Vˆe−ph=⇒464.2. Peierls bipolaronsc†i+1,↑c†i,↓b†i+1|0〉Vˆe−ph=⇒ c†i+1,↑c†i+1,↓|0〉 allows one particle to hop by emitting a phonon,then the second particle absorbs the phonon and hops to its partner’s new site.This is one of the processes contributing to T0,0; all relevant processes can besimilarly inferred.The other effective interaction termUˆ1 = +T1,1∑i,σ[c†i+1,σc†i+2,−σci+1,−σci,σ +H.c.]+J∑i,σc†i+1,σc†i,−σci,σci+1,−σacts when the particles are on adjacent sites and describes the pair-hopping of anearest-neighbor singlet with T1,1 =2g2Ω, and an antiferromagnetic xy exchangewith J = 4g2Ω.Note that none of these terms are of the density-density type of interactionthat is assumed to be the functional form for phonon-mediated effective interac-tions (∆E mechanism explained in Chapter 2 in 2.2.1). More specifically, theseterms can be written in the form∑k,k′,q u(k+ k′, q)c†k+q,↑c†k′−q,↓ck′,↓ck,↑ allowed bytranslational invariance. The interaction vertex, u(k+k′, q), depends not only onthe exchanged momentum, q, as is usually assumed to be the case, but also onthe total momentum of the interacting pair, k + k′. It is therefore important tounderstand the consequences of such interactions, for example, how they affectthe properties of BCS- or Bose-Einstein-Condensate(BEC)-type superconductorsin higher dimensions. Such studies are left for future work.The origin of the two different symmetry states leading to the two bipo-laron bands is now clear. First, let me set t = 0. In this case, the low-energyHilbert subspace factorizes into two sectors, with the particles being separatedeither by an even or by an odd number of sites; the remaining terms in theHamiltonian do not mix these subspaces. To solve for bound states, I calcu-late the two-particle propagator [88] (see Subsection 3.3.2) and check for discretepoles appearing below the continuum. I find that Uˆ0,2 and Uˆ1 can lead to theappearance of a bound state in their respective subspace. The former has amonotonically increasing dispersion, Eeven(K) = −20 − 2T0,0 cos(Ka) + µ(K),474.2. Peierls bipolaronswhere µ(K) = Feven(K)2θ(K)− 12√(Feven(K)θ(K))2 + 4ζ(K), with Feven(K) = 2T0,0 cos(Ka),θ(K) = 1 − (f2(K))2α(K), ζ(K) = α(K)θ(K); f2(K) = 2t2 cos(Ka) and α(K) =2(T0,2+f2(K))2. The latter has a monotonically decreasing dispersion, Eodd(K) =−20 − J + 2(t2 + T1,1) cos(Ka) + κ(K), with κ(K) = (f2(K))2f2(K)+Fodd(K) , whereFodd(K) = −J +2T1,1 cos(Ka). For details of the derivation, see Appendix A.3.1.Note that both these energies are controlled by the energy scale g2/Ω, explainingwhy they evolve similarly with increasing λ. When t is turned on, the nearest-neighbor hopping term breaks this symmetry and leads to the avoided crossing,and hence the two bipolaron bands with unusual dispersions shown in Figure 4.4.These analytical arguments are rigorous only in the anti-adiabatic limit. Thepresented numerical results suggest that the physics remains qualitatively simi-lar away from this limit. For small Ω values, the polaron cloud contains morephonons. This may lead to additional effective interaction terms, however, thoseidentified above are definitely present (albeit with renormalized energy scales).It is therefore necessary to develop methods to extract the functional forms andenergy scales of the various interaction terms, for any value of Ω. Effective mod-els could then use such interaction terms to analyze the physics at finite carrierconcentrations. This work proves that these effective interactions must have un-common (not density-density type) functional forms.4.2.3 Stability to finite repulsive Hubbard UFinally, I address the role of the bare Hubbard repulsion U . In Figure 4.5, thecritical value Uc above which bipolarons dissociate into unbound polarons, for thePeierls/SSH (solid line) and Holstein (dashed line) models is displayed. Clearly,Uc is much larger for Peierls bipolarons than for Holstein bipolarons. This is yetanother qualitative difference between the two models.In the Holstein model, U directly competes with the on-site attraction ∆Emediated by phonons. A smooth crossover from an on-site bipolaron to a weaklybound bipolaron with the particles on neighboring sites is observed for U ∼ ∆E,and a somewhat larger U suffices to dissociate the bipolaron [38].For the Peierls/SSH coupling, consider again the anti-adiabatic limit with t =0. Here, a sufficiently large U will destabilize the bound state in the even sector,484.2. Peierls bipolarons0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00λ010203040506070UcFigure 4.5: UC-λ stability diagram for the Peierls/SSH (solid line)and Holstein (dashed line) bipolarons at Ω = 3. UC is in units of t.For the Peierls/SSH coupling, λ = 2g2/(Ωt), while for the Holstein coupling,λ = g2H/(2Ωt), where gH is the Holstein electron-phonon coupling parameter.These are VED results, and reveal a stark contrast between the stability of thetwo types of bipolarons in the strong-coupling limit, λ > 1.but will have much less effect on the bound state of the odd sector. Hybridizationdue to a finite t will then result in a low-energy bipolaron similar to the boundstate of the odd sector. Consequently, one expects a stable bipolaron even for largevalues of U . Moreover, for a sufficiently large U value, one expects a transition toa bipolaron with ground state momentum KGS = pi/a, favored by the odd boundstate. Indeed, this is verified in the anti-adiabatic limit (not shown).Away from the anti-adiabatic limit, see Figure 4.5, one finds that the t-controlled mixing between even and odd bound states suffices to destabilize bothstates, at large enough U > Uc. However, Uc is an order of magnitude larger thanthat for the Holstein model, for strong couplings. The Peierls/SSH bipolarons are494.3. Summary and concluding remarksthus stable in a much wider range of repulsive U than the Holstein bipolarons.This is a direct consequence of the existence of the two bound states with differentsymmetries, one of which is only weakly affected by large U (at t = 0).4.3 Summary and concluding remarksI have demonstrated the existence of strongly bound yet light Peierls bipolarons.Their stability and effective mass are rather insensitive to on-site Hubbard repul-sion, making my main results valid irrespective of the efficiency of screening ofthe short-range electron-electron repulsion. Moreover, I have also shown that thephonon-mediated effective interactions have non-standard vertices that dependon the total momentum of the pair, k + k′. I explained that pairing is mediatedby pair-hopping terms instead of the customary attractive Hubbard-like terms.This unusual attraction binds two low-energy bipolaron states, instead of one. Asa result of an avoided crossing, these Peierls bipolarons have unique dispersions.These results are qualitatively different from those familiar for the standardq-dependent interactions such as in the Holstein and Fro¨hlich models. The studyof u(k + k′, q) effective interactions opens a field of largely unexplored physics.As discussed earlier in Subsection 2.4.1, this light bipolaron mechanism shouldbe applicable to the study of a variety of solid-state systems. Examples includesuperconductivity in conjugated polymers [40, 41, 99], organic semiconductors[100–103] and some oxides [104–106] with significant Peierls-type coupling.Quantum simulators [107] can also explore large areas of the parameter spaceof this model, see Subsection 2.4.2. This fermionic model may be realized withultracold mixtures of atoms. In this scheme, the heavier atoms trapped in anoptical lattice provide the phonon degrees of freedom, and the lighter fermionicatoms, representing the bare fermions, couple to the heavy atoms by Feshbachresonances [80].To validate the proposed new pathway to high-temperature superconductiv-ity, a detailed understanding of Peierls/SSH couplings at finite carrier concen-trations is required. A few recent studies revealed stark qualitative differences[108, 109] compared to models with Holstein coupling, but many aspects remain504.3. Summary and concluding remarksunstudied. The effect of the repulsive Hubbard U on Tc and its interplay withphonon-mediated interactions are also interesting directions to investigate. Whilea large U weakens the bipolaron binding, other mechanisms may help facilitatesuperconductivity, e.g. [38].Furthermore, the binding mechanism unveiled here poses questions about thenature of superconductivity at finite carrier densities in higher dimensions. Theperturbative arguments explaining the phonon-mediated interactions extend tohigher dimensions. Thus, one expects the light tightly bound bipolarons to con-dense into a BEC-type superconductor with high Tc. In fact, recent work claims arecord Tc for superconductivity in doped organic p-terphenyl molecules [110, 111],where the Peierls coupling is important, and attributes it to a bipolaronic mecha-nism [110]. Similarly, this work may be relevant to understanding electron-phonondriven superconductivity in SrTiO3 [65] and in one-dimensional LaAlO3/SrTiO3interfaces [112]. Another exciting prospect is the study of bipolaronic super-conductivity in hole-doped magic-angle Graphene superlattices [113], where thePeierls electron-phonon coupling is substantial [114], and layered MoS2, wherepolarons and bipolarons have been recently measured [67].All these considerations indicate that the issue of phonon-mediated high-temperature superconductivity must be revisited.51Chapter 5Two hard-core particles in theone-dimensional extendedPeierls-Hubbard model: A.Ground-state propertiesIn the previous chapter I studied the singlet sector and found a stable bipolaron. Inow turn to the triplet sector and study the difference this makes on the nature oftwo-carrier states and their effective interactions. These results also describe twohard-core particles in the one-dimensional Peierls model under consideration. Toappreciate the effective inter-site physics at play, I study the system in presenceof extended Hubbard interactions. Below I review the importance of the interplayof such interactions.5.1 Interplay of electron-phonon coupling andextended Hubbard interactionsStrongly correlated quantum materials exhibit rich physics with many featuresyet to be understood. Correlated lattice systems are modeled by the extendedHubbard model, which includes inter-site interactions giving rise to interestingphysics such as superfluid-Mott insulator transitions [115], antiferromagnetism[116, 117], high-Tc superconductivity [118], twisted superfluidity [119], supersolids[120]. However, the extended Hubbard model does not include interactions withphonons, which are essential for quantum materials. In this chapter I show thatthe interplay of the extended Hubbard interactions with Peierls phonon-mediated525.2. Phonon-mediated repulsion, sharp transitions and (quasi)self-trappingcouplings leads to new unique features, such as self-trapping of correlated pairsand the formation of light (mobile) dimers in the regime of strong interactions,both between the particles and with phonons. For reference, see Chapter 2 for adiscussion on absence of sharp transitions in all previously studied polaronic andbipolaronic systems, with the exception of single polarons in the Peierls model.The interplay of the SSH/Peierls couplings and the extended Hubbard inter-actions alters the behavior of strongly correlated quantum systems. For exam-ple, in the limit of half-filling, an interplay of phonon-mediated attraction withrepulsive Hubbard interactions is known to lead to a competition between theMott-insulator and Peierls-insulator phases [121]. I consider two hard-core par-ticles (or fermions in the triplet state) in a one-dimensional extended Hubbardmodel coupled to phonons through the SSH/Peierls couplings. This is criticalfor understanding quantum transport of interacting excitons in devices basedon organic semiconductors (such as low-temperature solar cells) [122, 123] andthe prospects of observing the Mott-insulator/Peierls-insulator competition withhighly controllable ultracold atoms/molecules systems, which require understand-ing of emergent interactions in the few-particle limit.The extended Peierls-Hubbard model can also be realized for hard-core bosonswith ions in rf-traps [124, 125], Rydberg atoms exchanging excitations [77, 126–128], self-assembled ultracold dipolar crystals [129–131], arrays of polar moleculestrapped in optical lattices [50, 76], arrays of superconducting qubits [132–137],and J-aggregates [68]. See Subsection 2.4.2 for more discussion.As explained below, the interplay of particle statistics, particle interactionsand coupling to phonons leads to unique features such as phonon-mediated repul-sion and sharp transitions in the ground-state properties of dimers including onesuggestive of self-trapping.5.2 Phonon-mediated repulsion, sharptransitions and (quasi)self-trappingTwo hard-core bosons and two identical fermions have the same ground stateenergy in an infinite one-dimensional lattice. As discussed in Refs. [22, 23], the535.2. Phonon-mediated repulsion, sharp transitions and (quasi)self-trappingdifference between the ground states is due to system size-dependent boundaryterms, which vanish in the thermodynamic limit.To this end, I consider two identical fermions (fermionic atoms in the sameinternal state or in a triplet state), or equivalently, two hard-core bosons, placedin an infinite chain described by the extended Peierls-Hubbard Hamiltonian H =Hp +Hph + Vˆ , whereHp = −t∑i(c†ici+1 +H.c.)+ V∑inˆinˆi+1 (5.1)is the extended Hubbard model of the bare particles with infinite on-site repulsion,Hph describes Einstein phonons, and Vˆ is the Peierls particle-phonon coupling Eq.(2.4) for particles without spin degrees of freedom.As before, I characterize the particle-phonon effective coupling by the dimen-sionless λ = 2g2/(Ωt). I set ~ = 1 below.I investigate the problem using variational exact diagonalization (VED) (Sec-tion 3.1) and the Momentum Average (MA) approximation (Section 3.2). Theauthor of this thesis performed MA calculations. The VED results were obtainedby Monodeep Chakraborty and are used for validation of the MA results.5.2.1 Two-polaron phase diagramAgain, I begin by setting V = 0 and study whether exchange of phonons sufficesto bind two hard-core polarons into a bipolaron, or equivalently two fermions intoa triplet bipolaron. As shown in the previous chapter, this is the case in the singletsector. For reference, note that equivalent one-dimensional models with long(er)-range on-site energy-modulating couplings, such as the screened and unscreenedFro¨hlich couplings, host stable triplet bipolarons; for on-site Holstein coupling,bipolarons do not form in the triplet sector [39, 44, 138].I find that for V = 0, (triplet) bipolarons do not form for any coupling λ. Tounderstand the implications of this result, recall that two hard-core bare particles(λ = 0) bind only for V ≤ −2t, see discussion in 3.3.2. This attraction is neededto compensate for the loss of kinetic energy [88]. The SSH polaron dispersion –and hence its kinetic energy – remains significant at all particle-phonon couplings.545.2. Phonon-mediated repulsion, sharp transitions and (quasi)self-trappingThese results thus show that the phonon-mediated interaction is insufficient tocompensate for the kinetic energy that would be lost upon binding a (triplet)bipolaron.Below, when unambiguous, I refer to a dimer/bipolaron without explicit men-tion of “triplet” to imply a dimer/bipolaron in the triplet sector or equivalentlyone composed of two hard-core polarons. When needed I will make clear thecontrast with the singlet bipolaron of Chapter 4.Nature of phonon-mediated interactionsTo characterize quantitatively this phonon-mediated interaction, we compute thevalues of V = Vc(λ) corresponding to the onset of stable bound state (I definethe bound dimer to be stable if its ground state energy is below the two-polaroncontinuum). I then compare Vc(λ) with V¯c(λ), defined as the NN attraction neededto bind two hard-core particles with dispersions identical to those of single SSHpolarons. This latter model mimics the renormalization of the dispersion dueto each particle creating and interacting with its own cloud of phonons, butexcludes the effective interactions due to phonon exchange between the clouds.The phonon exchange occurs in the full model, so |V¯c(λ)| − |Vc(λ)| is an estimateof the phonon-mediated NN attraction between polarons. Figure 5.1 shows that|Vc(λ)| > |V¯c(λ)| for all λ. This means that the phonon-mediated interaction is infact strongly repulsive, in stark contrast to what is observed in the singlet sectorof the Peierls model and the triplet sector of other conventional polaron models[39, 44, 138].5.2.2 Analytical anti-adiabatic theoryThis surprising result can be explained by considering the limit Ω t, g withinperturbation theory (Section 3.3).As shown earlier in Subsubsection 3.3.1, the effective Hamiltonian in thesingle-particle sector is characterized by the appearance of next-nearest-neighbor555.2. Phonon-mediated repulsion, sharp transitions and (quasi)self-trapping0.0 0.5 1.0 1.5 2.0λ20151050VKGS=piKGS 0KGS=0VcV¯cUnbound polaronsFigure 5.1: Two-polaron phase diagram at t = 1,Ω = 3. The solid black lineshows Vc(λ) below which stable bound states form, while the dashed line showsV¯c(λ); the difference between the two is the strength of the phonon-mediatedinteraction. Note that Vc < V¯C , which means that this interaction is repulsive.The red and green lines mark the sharp transitions of the bound dimer’s GS. Thelines are the VED results and the corresponding symbols are the MA results.hoppinghˆ1 = −0∑inˆi −∑i(tc†ici+1 − t2c†i,ci+2 +H.c.).Note the four processes c†i |0〉 Vˆ=⇒ c†i±1b†i±1|0〉 Vˆ=⇒ c†i |0〉 and c†i |0〉 Vˆ=⇒ c†i±1b†i |0〉 Vˆ=⇒c†i |0〉 explain the polaron formation energy 0 = 4g2/Ω [17].Repeating the calculation for two particles, I find the corresponding effective565.2. Phonon-mediated repulsion, sharp transitions and (quasi)self-trappingHamiltonian to behˆ2 = hˆ1 + 0∑inˆinˆi+1,illustrating the appearance of phonon-mediated NN repulsion. Its origin canbe explained as follows: if the polarons are δ ≥ 2 sites apart, each lowers itsenergy by 0 through hops to its adjacent sites and back, accompanied by virtualphonon emission and absorption, as explained above. However, if the polaronsare on adjacent sites, then Fermi statistics blocks half of these processes, i.e. eachparticle can only lower its energy by 0/2. The energy cost for polarons to beadjacent is, thus, 0 = 2λt.Importantly, note that hˆ2 also includes terms such as c†ic†i+1|0〉 hˆ2=⇒ c†i+1c†i+2|0〉.NNN hopping of one particle past the other is forbidden by statistics (the particleat i cannot emit a phonon and move to i+1 because that site is occupied). Instead,these terms describe both particles moving through c†ic†i+1|0〉 Vˆ=⇒ c†ib†i+1c†i+2|0〉 Vˆ=⇒c†i+1c†i+2|0〉. In other words, instead of one particle hopping over the other, whichis forbidden, each particle moves by one site and a phonon is exchanged in theprocess. Thus, this term is also a phonon-mediated effective interaction andwould be absent if phonons could not be exchanged between particles. In thelarge Ω limit it happens to precisely compensate for the NNN hopping forbiddenby the particles’ statistics, but that is not likely to be the case throughout theparameter space. This shows that the functional form of the effective phonon-mediated interaction must also contain such “pair-hopping” terms in addition tothe NN repulsion. Such terms do not appear in models where phonons modulatethe on-site particle energy (e.g., Holstein and Fro¨hlich models). See also discussionin Subsection 4.2.2 on pair-hopping in the singlet sector.For smaller values of Ω, the phonon clouds have more phonons and are moreextended spatially, and thus can mediate longer-range effective interactions andhopping. Indeed, as shown in Figure 5.2 for Ω = 3 and V = Vc(λ)− 0.5, i.e. justinside the dimer stability region, the bound particles favor adjacent locations onlyfor λ → 0. At moderate and strong couplings they are found with highest prob-ability to be 2 or even 3 sites apart, even though the bare attraction is NN only.575.2. Phonon-mediated repulsion, sharp transitions and (quasi)self-trapping0 1 2 3 4 5 6 7δ0.00.10.20.30.40.5C(δ)Figure 5.2: Correlation between the two particles C(δ) =〈ΨGS| 1N∑i nˆinˆi+δ|ΨGS〉 for V = Vc(λ) − 0.5, t = 1, Ω = 3 and differ-ent λ. Red circles are the λ = 0.1 result, green triangles are λ = 0.7, and blacksquares are λ = 2.0.This suggests that the strong phonon-mediated NN repulsion is supplementedby longer-range effective attraction, and/or that binding is due to kinetic energygained through phonon-mediated pair-hopping terms such as the one discussedabove.5.2.3 Two sharp transitions and (quasi)self-trappingI now examine the properties of dimers formed when V is sufficiently large tobalance the phonon-mediated repulsion and the loss of kinetic energy. Figure 5.3shows the dimer dispersion, ED(K), as a function of V and λ, illustrating twounique features of dimers arising from the SSH coupling. At low |V | and/or λ,the dimer ground state has momentum KGS = 0. As λ and/or |V | increases,585.2. Phonon-mediated repulsion, sharp transitions and (quasi)self-trappingthere is a sharp transition to a GS momentum K > 0. Figure 5.3 a) showsthat with increasing |V |, the dimer dispersion develops a rather unusual shapewith a second local minimum appearing at a finite momentum. At V ≈ −4.62tthis minimum becomes degenerate with that at K = 0, and the dimer ground-state momentum jumps discontinuously to KGS ≈ 0.6pi and then continues toincrease with increasing |V |. There is a second sharp transition to KGS = pi atV = −6.12t. These curves are at a fixed λ so the polaron dispersion is unchanged.The change in the dimer dispersion (and in KGS) is therefore due to forcing thebound polarons closer, as |V | increases.In Figure 5.3 b) I follow the evolution of ED(K) with λ for a fixed V = −30t.At small λ one can see a rather heavy dimer with KGS = 0, as expected becausein this limit the non-interacting polarons are quite heavy with EP (k) increasingmonotonically with k [17]. With increasing λ the effective dimer mass increasesfast and the dispersion becomes flat. At a value λ∗ ≈ 0.7 the minimum jumpsdiscontinuously to KGS = pi. It stays there with further increase in λ, but thebandwidth increases dramatically as the phonon-mediated pair-hopping termsbecome dominant, thus making the dimers light at strong coupling.Figure 5.1 illustrates the locations of these sharp transitions for the dimers inthe extended Peierls-Hubbard model on the V -λ phase diagram. To the best ofmy knowledge, this is the first observation of such sharp transitions of the two–polaron ground state. They never occur in Holstein or Fro¨hlich models, wherethe bipolarons always have KGS = 0 [39, 44, 138].The second unique feature illustrated in Figure 5.3 is the flattening of thedimer dispersion at λ = λ∗ ≈ 0.7, suggestive of self-trapping: here the dimers areessentially localized even though the single polarons have finite bandwidth. Thisbehavior can be understood qualitatively as follows. For small λ, the polaron dis-persion is dominated by its NN hopping. A large |V | can bind the polarons onlywhen they are on adjacent sites. When acting on such a configuration, NN hop-ping moves the particles two sites apart to an energetically expensive configura-tion. As a result, the effective dimer dispersion acquires a term ∼ −t2/|V | cos(K),which favors KGS = 0. On the other hand, at finite λ, the pair-hopping processmoving the NN pair as a whole also becomes active and contributes a term of or-der 2t2 cos(K) to the dimer dispersion; this term favors KGS = pi. At λ = λ∗ the595.2. Phonon-mediated repulsion, sharp transitions and (quasi)self-trapping0 0.2 0.4 0.6 0.8 1 K/pi-0.10.00.10.20.3E D(K)-ED(0)V = -3.0V = -4.0V = -4.62V = -5.0V = -6.12U = -7.0V = -9.0a)0 0.2 0.4 0.6 0.8 1K/pi-0.5-0.4-0.3-0.2-0.10.00.1E D(K)-ED(0)λ = 0.25λ = 0.7λ = 1.0λ =3.0b)Figure 5.3: Dimer dispersion ED(K) − ED(0) for a) λ = 1 and variousvalues of V ; and b) V = −30 and various values of λ. In both casest = 1,Ω = 3. The lines are the VED results and the symbols are the MA results.Note the sharp transitions of the GS momentum from KGS = 0 to KGS > 0 inboth cases.605.3. Discussiontwo terms cancel and the bandwidth collapses. However, numerical simulationscannot guarantee that the bandwidth is precisely zero, and I do not have theo-retical arguments why the longer-range hopping should also vanish at λ∗. This iswhy the more conservative term of (quasi)self-trapping is prefered.5.3 DiscussionTo summarize, I have shown here and in the previous chapter that dressing inter-acting particles by phonons through SSH/Peierls couplings leads to very rich two-polaron physics, qualitatively different from that in conventional polaron models.Here, I explained that for bare particles with the statistics of identical fermionsor of hard-core bosons, the phonon-mediated interactions are repulsive, contra-dicting the conventional view that phonons act as “glue” for quasiparticles. I alsoshowed that the pair-hopping terms, which are also mediated by phonon-exchangeand can only arise in models with phonons modulating the particle hopping, playa major role, leading to sharp transitions of the bound dimer’s ground state. Fur-thermore, I observe the collapse of the dimer’s dispersion at coupling strength λ∗,where the single polarons are mobile, suggestive of a self-trapping transition.As discussed in the previous section, all these new observations rest on theinterplay of two generic features: hard-core statistics of bare particles and off-diagonal hopping-dependent particle-phonon couplings. As such, these resultsapply to a wide range of systems and have far-reaching implications for complexquantum systems of interacting dressed particles. The hopping-dependent inter-actions with phonons of the Peierls model are generally present in all materials.They may not always be dominant, but because they lead to qualitatively distinctbehavior of the resulting dressed particles, this raises an important question: Howthe interplay of the Peierls coupling with conventional phonon-induced interac-tions (such as those in the Holstein model) changes the dynamics of polarons?Many of these features presented here are expected to apply to systemswith more particles. The pair-hopping terms must be equally important forfew-polaron ensembles. This suggests that the ground state of hard-core few-polaron states must also exhibit sharp transitions and, perhaps localization (self-615.3. Discussiontrapping).62Chapter 6Two hard-core particles in theone-dimensional Peierls model:B. Repulsively bound bipolaronsIn this chapter, I extend the analysis of the last chapter to the excited spectrumof the two hard-core particles in the Peierls model. I find an exotic bipolaronstate embedded in the excited spectrum bound by the repulsive phonon-mediatedinteractions. This is especially relevant to ultracold experiments, as shall becomeclear below.6.1 Non-conservative forces and bound statesMost composite objects in nature are bound by conservative forces, which areattractive. Dissipative forces act to reduce the stability of bound objects by al-lowing tunneling out of bound states, except in specific cases when dissipationtraps the system in a local minimum of a conservative potential [139]. For par-ticles in a lattice, the dispersion is finite and the two- or few-particle continuumis bounded from both below and above. As a consequence, such particles canbe bound by conservative repulsive interactions, which push the bound state toan energy above the continuum [140]. The influence of such “repulsively bound”states on quantum walks of interacting bosons was recently demonstrated in ex-periments probing the dynamics of ultracold atoms in an optical lattice [141].Repulsively bound states are fundamentally important as a) they are expected torestrict certain quantum phases of many-body quantum systems to a finite rangeof the Hamiltonian parameters [142]; b) they can induce correlations betweenparticles in the high energy part of the continuum through virtual excitations;636.2. Effective Hamiltonianand c) they are thermodynamically unstable states, which can nevertheless trapquantum systems thus impeding thermalization.In this chapter, I demonstrate the binding of hard-core particles by repulsivephonon-mediated interactions in the Peierls/Su-Schrieffer-Heeger (SSH) model[14–16, 40, 41]. This problem is unique for two reasons. Firstly, and as discussedearlier, phonon-mediated interactions in the celebrated Holstein and Fro¨hlichmodels are generally attractive [38, 39, 51, 143]. However, as explained in the pre-vious chapter, the interactions between hard-core particles induced by couplingto quantum phonons described by the less studied Peierls/Su-Schrieffer-Heeger(SSH) model are repulsive [20]. Secondly, the spectrum of phonons is unboundedfrom above. Any bound state embedded in the continuum of phonons should beexpected to decay through coupling to phonons. However, if the phonons aregapped and the dispersion of the bound state is smaller than the gap, this decaymay be prohibited by conservation of energy. Under such conditions, the con-tinuum of two particles + phonon states separates into bands and the repulsivephonon-induced interactions lead to the formation of a stable bound state of twobare particles dressed by phonons, i.e. a bipolaron [144] pushed to an energybetween the bands. The repulsive bipolaron is particularly relevant to ultracoldquantum simulators for which tunable gapped phonons can be readily engineered.In most of the literature on polarons and bipolarons, the focus is on ground-state properties. Excited states are, however, a part of a rich and complex spec-trum, where interactions between dressed particles are much less studied andunderstood. To this end, stable bipolarons in the spectral region above the two-polaron continuum represent a new paradigm for binding of dressed particles.6.2 Effective HamiltonianAs in the previous chapter, I consider the Peierls/Su-Schrieffer-Heeger (SSH)model for hard-core particles (e.g., spinless fermions, hard-core bosons) on a one-dimensional lattice, H = Hp +Hph + Vˆ , whereHp = −t∑i(c†ici+1 +H.c.)(6.1)646.2. Effective Hamiltonianis the tight-binding model of the bare particles with nearest-neighbour (NN) hop-ping. Recall Hph = Ω∑i b†ibi describes an infinite ladder of states separated byΩ in units of ~, andVˆ = g∑i(c†ici+1 +H.c.)(b†i + bi − b†i+1 − bi+1)(6.2)is the Peierls/SSH particle-phonon coupling [14–16, 40, 41]. As before, I charac-terize the particle-phonon coupling by the dimensionless parameter λ = 2g2/ (Ωt).I set the lattice constant a to unity in all the following equations.I focus on the anti-adiabatic limit t, g Ω (as follows from the discussionbelow, the regime of particular interest here is t g Ω) and investigatethe conditions for the formation of repulsively bound bipolarons. The repulsivebipolaron is separated from the higher bipolaron + phonon states by energy gapsthat are proportional to Ω. Thus, the repulsive bipolaron is expected to be stablein this limit.In the two-bare-particle sector, the SSH interaction (6.2) induces both repul-sive phonon-mediated density-density interaction and “pair-hopping” interactionsfor particles with hard-core statistics [20], see Chapter 5. In the anti-adiabaticlimit of interest here, the contribution of N -phonon excitations decays as (g/Ω)N .This allows to derive an effective theory to order (g/Ω) by projecting out higherenergy multi-phonon states. For more details, see Section 3.3 and Subsection5.2.2.Two Peierls/SSH polarons in this limit g/Ω 1 are thus described by thelimiting case of the t-t2-V modelHeff = −0∑inˆi − t∑i(c†ici+1 +H.c.)+t2∑i(c†ici+2 +H.c.)+ V∑inˆinˆi+1, (6.3)with V = 0 = 2λt and t2 = λt/2.This limiting case is characterized by NNN hopping and “pair-hopping” ofNN bound pairs both with a sign opposite to that of the NN hopping. Thus, this656.3. Repulsive bipolaronmodel cannot be taken as a limit to a long-range power law hopping model. Thisis a unique feature of this model which explains why the repulsive bipolarons aredifferent from the t2 = 0 repulsively bound pairs, as I shall demonstrate below.To understand phonon-mediated pairing, I first study the full phase dia-gram of the model (6.3) in the entire range of V and t2. I derive the exactBogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) equation of motion for thetwo-particle propagator G(K,n, ω) = 〈K, 1| Gˆ(ω) |K,n〉 defined for two-particlestates |K,n〉 = ∑i eiK(Ri+n/2)√N c†ic†i+n |0〉 with n ≥ 1 [88, 89] in Appendix A.3.2. Formore discussion about the method, see Subsection 3.3.2. I extract bound stateproperties from the pole of the propagator that appears above the zero-phononcontinuum band. The continuum is the convolution of two single polaron bandssatisfying conservation of momentum.6.3 Repulsive bipolaronBelow I set ~ = 1 and analyze the phase diagram of the model (6.3). Figure 6.1shows the regions in parameter space characterizing the appearance of stablerepulsive bipolarons, metastable repulsive bipolarons, and unbound polarons. Idefine stable bipolarons as those with energy separated from the continuum atall values of the bipolaron momentum K. By contrast, metastable bipolarons aredefined as bipolarons with energy dispersion split from the continuum at K = 0,while merging with the continuum at other values of K. Such bipolarons maydissociate by momentum-changing collisions.To understand the binding mechanism, I first consider the effect of the t2 termon the continuum band. As λ increases, the NNN hopping dominates leading toa transition of the single polaron dispersion from one with a minimum at k = 0to one with a doubly degenerate dispersion with two minima at finite k = ±pi/2.This leads to an asymmetry in the two-polaron continuum with upward shiftingnear the center of the Brillouin zone above the zero of energy. Note that theasymmetry is reversed for t2 → −t2 as the single polaron band would then havetwo maxima instead of two minima. This emphasizes that the NNN hoppingcannot be considered as a cut-off to a longer-range hopping, as eluded to before.666.3. Repulsive bipolaron0.0 0.5 1.0 1.5 2.0 2.5 3.0t2/t024681012V/tStable bipolaronsK=0 bipolaronsUnbound polaronsFigure 6.1: V -t2 phase diagram. t is the bare NN hopping. The grey shaded re-gion represents unbound polarons, while the light and dark salmon colored regionsrepresent K = 0 bipolarons and stable bipolarons, respectively. The blue solidline corresponds to the Peierls/SSH model for which V = 4t2. The Peierls/SSHmodel line is right on the boundary between the region of stable bipolarons andthe K = 0 bipolarons at t2 = t∗2 ≈ 0.8 corresponding to λC ≈ 1.6; the value of λmarking the onset of stable Peierls/SSH bipolarons, where λ = 2g2/~Ωt. The bluedashed line corresponds to the Peierls/SSH model supplemented with one unit ofNN repulsion between the bare particles. The black diamond symbols mark thetwo points λ = 1.6 and λ = 4 on the Peierls/SSH line for which splitting of therepulsive bipolaron states from the continuum is illustrated in Figure 6.2.676.3. Repulsive bipolaronAs t2 increases, the continuum band broadens near the bottom and narrows atthe top and near the edges (not shown) till |t2| > |t|, after which this asymmetrydecreases (see Figure 6.2) as the NNN hopping dominates and the NN hoppingbecomes a perturbative term.To bind polarons, the interaction must compensate for the kinetic energy lostin binding. For t2 = 0, binding happens for V > 2 t [88, 91], see discussionin 3.3.2. However, t2 enhances the kinetic energy of individual polarons shiftingthe continuum band center upwards as explained above. Thus, a higher V isrequired to bind polarons. This is what I observe in numerical calculations shownin Figure 6.1. The K = 0 bound states first appear at a critical value t∗2 ≈ 0.8 t.For t2 < t∗2, a bound state is split everywhere in the Brillouin zone, once it is splitat K = 0. As t2 surpasses t∗2, the asymmetry is lost and the continuum edges shiftupwards and become closer in energy to the continuum center. Thus, a greater Vis required to split the entire spectrum at higher K. This implies that metastableK = 0 bipolarons can decay into the continuum near the Brillouin zone edges bycollisions with other bipolarons or through other K-changing mechanisms.For the Peierls/SSH model, the ratio V/t2 = 4 is fixed, as shown by the blueline in Figure 6.1. When t2 > t∗2, the Peierls/SSH model line is right on the bound-ary between the region of stable bipolarons and the K = 0 bipolarons. Figure 6.2demonstrates the splitting of repulsively bound states from the continuum for twopoints on this line marked by symbols in Figure 6.1. Figure 6.2 a) correspondsto λ = 1.6 (t2 = t∗2), illustrating the onset a bound state at K = 0. Figure 6.2b) corresponds to λ = 4, illustrating the onset of a bound state split from thecontinuum at all values of K. This state merges with the continuum near the edgeof the Brillouin zone. The V/t2 = 4 line at t2 > t∗2 thus marks the onset of stablebipolarons. Note that any repulsion between bare particles contributes directly,pushing the energy of the bound state from the continuum. This is illustratedin Figure 6.1 by the dashed line, which corresponds to the Peierls/SSH modelsupplemented with one unit of NN repulsion between the bare particles.The bipolarons illustrated here have interesting properties. As can be seenfrom the dispersion of the bound state in Figure 6.2, the bipolaron has negativeeffective mass at the band center and positive effective mass at the band minimum.Therefore, coupling the K = 0 bipolarons to photons may result in negative686.3. Repulsive bipolaron3 2 1 0 1 2 3K (in units of a)1210864202E (in units of t)a) λ=1.63 2 1 0 1 2 3K (in units of a)2520151050E (in units of t)b) λ=4.00 1 2 3 4 5 6n10-810-610-410-2100Γn(K)Figure 6.2: Energy spectrum of two hard-core bare particles in theone-dimensional Peierls/SSH model. The red line represents the repulsivebipolaron dispersion and the blue shaded region shows the two-polaron continuumband for a) λ = 1.6 and b) λ = 4. The inset of b) illustrates the repulsivebipolaron log probability distribution Γn(K) = log10[Pn(K)] defined for Pn(K) ≡|〈K,BP |K,n|K,BP |K,n〉|2, where |K,BP 〉 is the bipolaron state and n is therelative separation of the particles for KBP = 0 (salmon) and KBP = pi (indigo).In both cases, the particles are NN with highest probability. Note that for KBP =pi, even n relative separation between particles is forbidden. For more details, seeAppendix A.3.696.3. Repulsive bipolaronrefraction [145–148]. Even more interestingly, the dispersion of the bipolaronexhibits large curvature, which increases with the coupling strength, as illustratedin Figure 6.3.0.0 0.5 1.0 1.5 2.0 2.5 3.0K (in units of a)14121086420EK (in units of t) λ=1.6λ=4.0λ=6.0λ=8.0Figure 6.3: The repulsive bipolaron dispersion EK ≡ EBP (K) − EBP (0)in units of the bare NN hopping, t, for various values of λ = 2g2/~Ωt.a is the lattice constant. The dashed line represents the onset of Peierls/SSHbipolaron formation. The two blue lines were illustrated in Figure 6.2, while thered lines label strong coupling bipolarons. Note that the bipolaron dispersionexhibits both large curvature and bandwidth, which increase with the couplingstrength.The unique shape of the bipolaron band in Figure 6.3 can be explained by con-sidering the limit V >> t, t2. The dispersion is then ∼ [(2t2/V ) + 2t2] cos(K) +(2t22/V ) cos(2K). The first term represents NN hopping of a bound pair com-prised of two NN hops of the constituent polarons in the same direction withamplitudes t2/V and an additional pair-hopping t2. The second term representsNNN of a bound pair where one particle moves two sites away from the its bound706.4. Beyond the anti-adiabatic limitpartner by NNN hopping and then the partner follows, costing an energy V in theprocess. In the Peierls limit, V = 4t2, the analytical expansion acquires higherorder corrections. However, this simple analytical form provides a way to explainthe unusual form of the bipolaron band. In particular, the second term ∼ cos(2K)is responsible for the unusual curvature of the dispersion with a minimum at finiteK < pi. This analysis makes clear that the pair-hopping term is responsible forthe bipolaron’s enhanced band width and reduced effective mass.To illustrate this quantitatively, I plot in Figure 6.4 the inverse of the bipo-laron’s effective mass m∗ as a function of λ. I find that |m∗| decreases rapidlywith λ, and consequently, with V , as V = 2λt for the Peierls/SSH polarons. Forcontrast, a repulsively bound pair in the bare t-V model with t2 = 0 has an ef-fective mass m0 = −V/(2t2) [91], which leads to linear growth of |m0| with V .Furthermore, the energy gap between the K = 0 bipolaron and the two-polaroncontinuum, ∆, grows with λ, as seen in the inset of Figure 6.4. This suggeststhat the K = 0 bipolaron is sufficiently stable at strong coupling. This radicallydistinct behavior highlights the unique nature of the bipolaron which is expectedto be highly mobile near K = 0.6.4 Beyond the anti-adiabatic limitIn the previous section, I considered the anti-adiabatic limit. At lower values ofphonon frequency, the bipolaron acquires higher order vertex corrections. Thisleads to quantitative changes in the results. However, previous studies suggestthat the qualitative physics in the anti-adiabatic limit persists to lower phononfrequencies [17, 20]. In particular, Ref. [17] shows that the sharp transition of theground state of a single SSH polaron from k = 0 to finite k occurs both in theadiabatic and anti-adiabatic regimes. The critical value of λ corresponding to thetransition varies within a small range between 0.5 and 1.25 as the phonon fre-quency is decreased from the anti-adiabatic limit to within the adiabatic regime.The polaron dispersion varies smoothly as the phonon frequency is decreasedbeyond the anti-adiabatic limit and exhibits no irregularities in both the anti-adiabatic and adiabatic limits. Since here I consider the same model, I expect a716.4. Beyond the anti-adiabatic limit2 4 6 8 10 12 14 16λ20151051/m (in units of ta2)2 3 4 5 6 7 8λ02468101214∆/tK=0K=piK=pi/4K=pi/2K=3pi/4Figure 6.4: The dependence of the inverse effective mass, ta2/m∗, for therepulsive bipolaron on λ = 2g2/~Ωt. The effective mass of the bipolaron m∗is defined as m∗ = (∂2EBP (K)/∂K2)−1 for K = 0, t is the bare NN hopping anda is the lattice constant. The inset illustrates the dependence of the energy gap,∆, between the repulsive bipolaron and the edge of the two-polaron continuumband on λ at various values of K. The dark blue and dark red solid lines labelthe gaps for K = 0 and K = pi, respectively; the dashed, dash-dotted and dottedlines label the gaps for K = pi/4, K = pi/2 and K = 3pi/4, respectively. Notethat ∆ vanishes for K = pi.726.4. Beyond the anti-adiabatic limitsimilarly smooth variation of the bipolaron dispersions beyond the anti-adiabaticlimit. In fact, the novel features of the bipolarons considered here can be tracedback to the interplay of the bare particle statistics and phonon-mediated hoppingfor the single SSH polaron.As discussed above, the repulsive interactions stem from the statistics blockingthe phonon-mediated hopping of bare particles into the same lattice site, therebyeliminating part of the renormalization energy (polaron formation energy) of theindividual polarons. Since the renormalization energy of the single polaron isa smooth function of phonon frequency, the repulsive interactions are expectedto extrapolate smoothly beyond the anti-adiabatic limit. The result displayed inFigure 6.4, showing the decrease of the bipolaron mass with the coupling strength,is a consequence of pair-hopping, which is closely connected to the NNN polaronhopping. The NNN polaron hopping is responsible for the decrease in the singlepolaron mass in the anti-adiabatic regime at strong coupling. As Ref. [17] shows,the single SSH polaron becomes heavier as the phonon frequency is decreased butremains light well beyond the anti-adiabatic limit. The same should be expectedfor the repulsive bipolaron.I do not argue that these results extend to the adiabatic regime. However,I use the above observations to suggest the validity of my qualitative resultsat phonon frequencies beyond the anti-adiabatic approximation, as long as Ω islarger than the bipolaron’s bandwidth, the condition that ensures a gap betweenthe bipolaron state and the higher continuum states.Typically, in these lattice models, there are no ultraviolet (UV) divergences.A UV divergence refers to divergence of observables at higher energies. In the lat-tice model I study, the high-energy modes are phonon modes with an energy scaleproportional to the number of phonons excitations, i.e. Eph = nphΩ, where nphis the number of phonons. The absence of UV divergences implies that the con-vergence with respect to nph is not pathological; observables should not dependon the high-energy scale. Therefore, one can see that in the adiabatic limit theeffective interactions obtained within the anti-adiabatic approximation acquirerenormalized energy scales and possibly small higher order corrections. T-matrixand renormalization group approaches can perhaps capture these arguments pro-viding insight into a solution in the adiabatic limit. I expect that the main result736.5. Concluding remarksthat a repulsive bipolaron forms in the Peierls model should apply to a largerpart of parameter space.6.5 Concluding remarksThe repulsive Peierls/SSH bipolaron represents a novel quasiparticle that can bepotentially realized with ultracold quantum simulators. It does not have a directanalog in quantum materials as the presence of acoustic phonons should embedsuch states into a strongly dissipating environment. The lifetime of this quasi-particle in quantum materials would be finite. However, it can leave a signatureas a resonance in a continuum.The repulsive bipolaron discussed here is different from typical repulsivelybound pairs in t-V models. The repulsive bipolaron is bound by phonon-mediateddensity-density and pair-hopping interactions. Experimental observations of thesefield/bath-mediated interactions is fundamental to understanding complex phe-nomena in coupled field theories. Additionally, the bipolaron experiences bothphonon-mediated NNN and pair hopping absent in systems with typical repul-sively bound pairs. This means that quantum interference effects in quantumwalks of the repulsive bipolaron ought to be different from those of t-V or ex-tended Hubbard repulsively bound states. The repulsive bipolaron has a signifi-cantly smaller effective mass than the Hubbard repulsively bound pair owing tothe pair-hopping kinetic terms.Our proposal represents an interesting mechanism for realizing repulsive in-teractions between pseudospins or excitons. Generally, excitons interact via dy-namical interactions that are attractive owing to the specific tensorial form ofthe resonant dipole-dipole coupling [149]. Under conditions discussed here, exci-tons would interact via a phonon-mediated repulsive interaction giving rise to aFrenkel biexciton [150–152] dressed with phonons. Note that Frenkel biexcitonshave never been observed in quantum materials because the hopping of the exci-tations is determined by the dipolar interactions between molecules, whereas thedynamical interactions between the excitations are determined by higher-order(e.g. quadrupolar) interactions due to the symmetry of the molecular states. The746.5. Concluding remarksmechanism introduced here could be the leading mechanism for pairing of exci-tons in materials such as solid molecular hydrogen [153], possessing high energyphonon modes.This emergent phonon-mediated repulsion can be used as a tunable parameterin quantum simulators. Applications can range from stabilizing pre-associated re-pulsively bound pairs in models with bare repulsive interactions to realization ofspin models with pair-hopping and repulsion. For instance, the extension of theeffective model to frustrated lattices such as triangular lattices would enable thestudy of frustration in spin liquids [154] and supersolids [155]. Studies of a frus-trated model closely related to mine reveals a supersolid phase in one dimension[156] and on the triangular lattice [157].The persistence of phonon-mediated interactions at finite concentrations andin higher dimensions is vital for this research direction. I note that the phonon-mediated NN repulsion is a result of statistics blocking hopping to NN sites. Thus,it is likely that the effect will persist in ensembles of greater number of particles,as the more confinement the particles experience, the more likely they are forcedto be NN and interact via the repulsive mechanism discussed here. Phonon-mediated pair-hopping is also likely to survive in the anti-adiabatic limit. This isbecause a single phonon virtual excitation can allow for hopping of at most a pairof NN particles excluding “cluster-hopping” of ensembles of three, four and highernumber of neighboring particles. A systematic approach must be developed toanalyze all such terms and corrections at finite concentrations.75Part IIDisordered Ultracold MolecularPlasma76Chapter 7Localization and absence ofthermalization7.1 Ergodicity and quantum mechanicsQuantum mechanics serves well to describe the dynamics of isolated microscopicmany-body systems [158]. The macroscopic world conforms with the laws ofthermodynamics and Newtonian mechanics [159]. Quantum statistical mechanics[160] bridges these realms by treating the quantum mechanical properties of anensemble of particles statistically and characterizing the properties of the systemin terms of state properties (temperature, chemical potential, etc.), in an approachthat assumes a complex phase space of trajectories with ergodic dynamics [161].However, this is not always the case, and the macroscopic description of quantummany-body systems that fail to behave as expected statistically remains today asa key unsolved problem [25, 162].Ergodicity, when present in an isolated quantum many-body system, emergesas the system thermalizes in a unitary evolution that spreads information amongall the subspaces of the system. The subspaces act as thermal reservoirs for eachother. Most known many-body systems thermalize in this fashion, obeying theEigenstate Thermalization Hypothesis (ETH) [161, 163–166], which motivatesthe use of statistical ensembles that each consists of a single eigenstate of thecorresponding many-body Hamiltonian.Exceptions include fine-tuned integrable systems [167], and the class of so-called many-body localized (MBL) systems [25, 168], which have attracted intenseinterest in recent years. Such systems do not thermalize at finite energy densitiesand are therefore non-ergodic. In these systems, interactions in the disorderedlandscape preserves memory of the initial local conditions for infinitely long times.777.2. Thermalization and the Eigenstate Thermalization HypothesisBelow I review the concept of thermalization and its breakdown in localizedsystems.7.2 Thermalization and the EigenstateThermalization HypothesisConsider a closed quantum system initialized in an arbitrary state. I will partitionthe system into a subsystem and the remainder of the system. The unitary timeevolution under the Hamiltonian dynamics spreads out the quantum informationencoded in the initial state to all parts of the Hilbert space such that at longtimes this information is essentially hidden and its retrieval becomes inaccessible(as it requires the measurement of global operators). One then finds an adequatedescription in terms of a set of statistical quantities, such as the temperature andchemical potential, characterizing the equilibrium steady state of the subsystemthermalized by the rest of the closed system.More formally, one can partition the system into two subsystems A and B,where B is taken to the thermodynamic limit. The reduced density matrix of sub-system A at time t, after tracing over the subsystem B, reads ρA(t) = TrBρ(t).At equilibrium, the Boltzmann probability operator for the full system at tem-perature T reads ρeq(T ) = exp(−H/KBT )Z, where Z is the partition function; andthat for the subsystem is ρeqA (T ) = TrBρeq(T ). If in the limit of long time andlarge system size ρA(t) = ρeqA (T ), one says the system is thermalized.The Eigenstate Thermalization Hypothesis (ETH) [161, 163–166] is a morestrict form of thermalization. It posits that all initially prepared states of amany-body system time-evolved under the Hamiltonian dynamics reach a termi-nal state of thermal equilibrium described by the microcanonical ensemble. Oneway to understand this is to consider initializing the system in a pure state whichis a many-body eigenstate. In this regime, the time evolution of the system be-comes trivial, ρ(t) = ρ(0), and one can see that ETH is naturally satisfied. Here,a time-evolved observable suffers dephasing, such that its off-diagonal coherencevanishes at long times. In other words, the unitary time evolution randomizesthe phases of the off-diagonal terms and they effectively cancel out in the long-787.3. Localizationtime limit. One should note that ETH is a hypothesis. It has been numericallyshown to apply to a large class of systems for arbitrary initial states, with fewexceptions including many-body localized systems. Within ETH, it follows thatsingle-eigenstate ensembles, each consisting of a single eigenstate of the full sys-tem’s Hamiltonian, encode the system’s quantum statistical description in theform of a microcanonical ensemble with an energy domain restricted to containonly one eigenstate [25].Conventionally one expects that all initial out-of-equilibrium states will ulti-mately thermalize in the infinite-time limit. As will be discussed below, localizedsystems retain memory of their initial states and thus defy this expectation.7.3 LocalizationLocalization refers to confinement of the wavefucntions and is most generallyunderstood to be a consequence of quenched disorder. To develop a qualitativeunderstanding of this phenomenon, let me consider a simple many-body spinmodel in an infinite lattice with the Hamiltonian:H =∑iiSzi +∑i,jJij ~Si.~Sj (7.1)Here, i are random on-site static fields drawn from a distribution of width Wand the spin-spin interactions, Jij, represent spin flip-flop (which can be viewedas particle hopping) and Ising interactions, which are taken here to be uniformand non-zero only for nearest neighbors, i.e. Jij = J 6= 0.7.3.1 Single-particle localizationThe single-particle limit confines the problem to a Hilbert space spanned by basisstates with one spin up (down) and all other spins down (up).In absence of disorder, i.e. i = , the eigenstates are Bloch states ~|k〉 =1√N∑i ei~k.~ri |i〉, linear combinations of states localized on the sites labeled bythe index i. A wave packet initialized at a central site spreads throughout thelattice under the unitary Hamiltonian dynamics. In the limit of infinite time, the797.3. Localizationprobability of finding the particle at this initial site decays to zero. To account forthe influence of disorder, one can perturbatively expand about the zero-disorderextended Bloch states to find diffusive behavior in three dimensions and higher[169].On the other hand, if the single-particle states are localized, one finds thatwave packet retains a finite probability at the central site after infinite time. Todevelop a deeper intuition of this effect, consider the limit, W >> J , whereone starts with an assumed state of complete localization and then asks whetherdelocalizing perturbations can destabilize this phase. This expansion about theinfinite disorder limit, also known as the locator expansion, accounts for all possi-ble pathways for the propagation of excitations. The divergence of the expansionsignals a breakdown of perturbation theory and is commonly taken to signal de-localization.In one and two dimensions, disordered systems exhibit a fully localized spec-trum for arbitrary disorder strength with spatially confined eigenstates∼ e(−~r−~Rζ),where ζ is the localization length. In three dimensions, a ‘mobility edge’ in theenergy spectrum separates localized and extended states for weak disorder, andone finds a critical disorder strength beyond which the entire spectrum becomeslocalized [170].This analysis reveals that under the action of the Hamiltonian, Eq. (7.1), theprobability for a spin state to remain in its initial configuration stays non-zerofor infinite time, if the system is in the localized phase, signalling an obviousbreakdown of thermalization and violation of ETH.7.3.2 Many-body localizationNow consider the many-body problem defined in the Hilbert space of arbitraryspin configurations.In absence of interactions, J = 0, the many-body eigenstates of Eq. (7.1) areproduct states |Szi 〉⊗∣∣Szj 〉⊗. . .. The interactions J 6= 0 mix these states. A typicalapproach that accounts for this mechanism of delocalization expands perturba-tively about the trivially localized product state to construct the wavefunctionsin the strong disorder limit W >> J . Remarkably, Basko et al. [168] verified807.3. Localizationthat such one-dimensional interacting many-body system remains localized atnon-zero temperatures to all orders in perturbation theory for sufficiently strongdisorder. This result was rigorously confirmed by Imbrie [171], who provided afull mathematical proof for the existence of MBL in one dimension taking intoaccount subtle resonant non-perturbative effects missing in perturbation theory,thus establishing MBL as a true non-thermal phase of matter.This discovery represents a milestone in quantum statistical mechanics asit suggests the possibility of a quantum phase transition as a function of thedisorder strength between a thermalized (ETH) phase and an MBL phase withall eigenstates violating ETH, where precisely at the transition point the laws ofstatistical mechanics break down. This transition is a dynamical phase transitionthat manifests at the level of ensembles of eigenstates, but is blind to equilibriumthermodynamics [172, 173].MBL acts in a profound way to constrain transport processes in a quantumsystem. In the MBL phase, any local operator, time-evolved with the Hamiltonianand averaged over time, 〈Oˆ〉, becomes a local integral of motion (LIOM) [174–178]. Following any perturbation in a state of MBL, local integrals of motionassociated with Oˆ survive for long times. Oˆ determines how far a conservedquantity can propagate through a system as it quenches to a state of MBL. It isthose emergent LIOM that explain the failure of thermalization in MBL systems,as I detail below.It has become common to refer to the physical degrees of freedom of the bareHamiltonian as p-bits, where the p stands for “physical”. In the case of Eq. (7.1),those are the N local physical two-level degrees of freedom, {~Si}. In a fermionicsystem, those would be the occupation number of single-particle orbitals. In thefully many-body localized (FMBL) regime characterized by localization of all theeigenstates of the system, the localized system is described by the HamiltonianH = E0 +∑iτ zi +∑i,jJijτzi τzj +∞∑n=1∑i,j,{k}K(n)i{k}jτzi τzk1. . . τ zknτzj , (7.2)where {~τi} represent the set of localized two-level degrees of freedom, commonlynamed l-bits, where the l stands for “localized”. Here, the sums are restricted817.4. Experimentssuch that each term appears only once and the magnitudes of Jijand K(n)i{k}j falloff exponentially with distance.One understands the l-bit Hamiltonian, Eq. (7.2), to describe this set of lo-calized conserved constants of motion. For non-interacting fermions, those wouldbe the occupation number of the localized single-particle orbitals. In the limitof weak disorder, one may view these l-bits as dressed p-bits, with the dressingfalling off exponentially with the distance. One can see that the l-bits mutuallyinteract with exponentially suppressed long-range interactions reflecting the factthat p-bits must have non-zero weight on distant l-bits.Given the Hamiltonian Eq. (7.2), I can now explain a number of results.The Hamiltonian commutes with the {τ zi } and therefore they have simultaneouseigenstates reflecting the conserved constants of motion that survive for infinitetime. A state described by Eq. (7.2) characterizes a configuration where eachl-bit precesses about its z-axis at a rate set by its interactions with all other {τ zi },leading to dephasing. Thus, it is understood that a FMBL dephases but doesnot dissipate; it stores information of local observables for infinite time, but withrandom phases.7.4 ExperimentsObservation of distinct MBL behavior requires isolation from any thermal envi-ronment. Consequently, MBL was first observed in deliberately engineered exper-imental systems with ultracold atoms in one and two-dimensional optical lattices[27–32, 179]. In such cases, tuning of the lattice parameters allows investigation ofthe phase diagram of the system as a function of disorder strength. Concurrently,there have been efforts to study disordered dynamics in solid-state systems withweak coupling to thermal baths searching MBL [180, 181]. To study MBL and theassociated MBL-ETH transition, experiments have mostly resorted to preparinghighly out-of-equilibrium initial states and probing their time evolution, using thenon-thermal response of a single local observable as a signature of localization.The majority of these experimental systems (especially ultracold quantum sim-ulators) suffer from decoherence, confining localization to short timescales and827.4. Experimentsfinite system sizes.It is important to determine experimentally whether conditions exist underwhich MBL can persist for long times at finite temperatures, and to understandif such a robust macroscopic quantum many-body state can occur naturally inan interacting quantum system without deliberate tuning of experimental pa-rameters. Such a realization could pave the way to exotic quantum effects, suchas entangled macroscopic objects and localization-protected quantum order [35],which could have societal and technological implications [182]. Below, I discuss adifferent platform for observing MBL behavior: Quenched ultracold plasma.83Chapter 8Possible many-body localizationin a long-lived finite-temperatureultracold quasi-neutral molecularplasmaMotivated by the question of MBL at long times and in large systems, I explorein this chapter the quenched ultracold molecular plasma of the Grant group asan arena in which to study quantum many-body effects in presence of disorder[33, 183]. Note that the ultracold plasma system offers complexity, as encounteredin quantum materials, but evolves from state-selected initial conditions that allowfor a description in terms of a specific set of atomic and molecular degrees offreedom.Experimental results presented here were obtained by the Grant group. Theauthor of this thesis contributed the development of the theory model that ex-plains the arrest state of the quenched plasma seen in the experimental data.8.1 Quenched ultracold molecular plasmaExperimental work has recently established laboratory conditions under which ahigh-density molecular ultracold plasma evolves from a cold Rydberg gas of nitricoxide, adiabatically sequesters energy in a reservoir of global mass transport, andrelaxes to form a spatially correlated, strongly coupled plasma [33, 184]. Thissystem naturally evolves to form an arrested phase that has a long lifetime withrespect to recombination and neutral dissociation, and a very slow rate of freeexpansion. These volumes exhibit state properties that are independent of initial848.1. Quenched ultracold molecular plasmaquantum state and density, parameters which critically affect the timescale ofrelaxation, suggesting a robust process of self-assembly that reaches an arrestedstate, far from conventional thermal equilibrium.Departure from classical models suggests localization in the disposition of en-ergy [33]. In an effort to explain this state of arrested relaxation, I have developeda quantum mechanical description of the system in terms of a power law interact-ing spin model. My results suggest a new avenue for theoretical and experimentalefforts to understand slow dynamics, glassiness [185] and MBL in the long-timelimit in systems of high energy density.8.1.1 ExperimentDouble-resonant pulsed-laser excitation of nitric oxide entrained in a supersonicmolecular beam forms a characteristic Gaussian ellipsoid volume of state-selectedRydberg gas that propagates in z with a well-defined velocity, longitudinal tem-perature (T|| = 500 mK) transverse temperature (T⊥ < 5 mK) and preciselyknown initial density in a range from ρ0 = 1010 to 1012 cm−3 (See Figure 8.1,Ref. [186] and Appendix B).Rydberg molecules in the leading edge of the nearest-neighbor distance distri-bution interact to produce NO+ ions and free electrons [187]. Electron-Rydbergcollisions trigger an ionization avalanche on a time scale from nanoseconds tomicroseconds depending on initial density and principal quantum number, n0.Inelastic collisions heat electrons and the system proceeds to a quasi-equilibrium of ions, electrons and high-Rydberg molecules of nitric oxide. Thisrelaxation and the transient state it produces entirely parallels the many ob-servations of ultracold plasma evolution observed in atomic systems under theconditions of a magneto-optical trap (MOT) [188].This avalanche unfolds directly in sequences of density-classified selective fieldionization (SFI) spectra measured as a function of delay after initial formation ofthe Rydberg gas [33]. For a moderate ρ0 = 3 × 1011 cm−3, the ramp-field signalof the selected Rydberg state, n0 gives way on a 100 ns timescale to form theSFI spectrum of a system in which electrons bind very weakly to single ions ina narrow distribution of high Rydberg states or in a quasi-free state held by the858.1. Quenched ultracold molecular plasma0"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 20 30 40 time (µs) Integrated electron signal (arb.)0 10 20 30 40 time (µs)Distance to G2 (mm)80 100 120 140 160 1800"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 20 30 40 time (µs) Integrated electron signal (arb.)0 10 20 30 40 time (µs)Distance to G2 (mm)0"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 20 30 40 time (µs) Integrated electron signal (arb.)0 10 20 30 40 time (µs)Distance to G2 (mm)0"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 20 30 40 time (µs) Integrated electron signal (arb.)0 10 20 30 40 time (µs)Distance to G2 (mm)0"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 20 30 40 time (µs) Integrated electron signal (arb.)0 10 20 30 40 time (µs)Distance to G2 (mm)0"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 20 30 40 time (µs) Integrated electron signal (arb.)0 10 20 30 40 time (µs)Distance to G2 (mm)0"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 20 30 40 time (µs) Integrated electron signal (arb.)0 0 0 0 40 time (µs)Distance to G2 (mm)0"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 20 30 40 time (µs) Integrated electron signal (arb.)0 0 0 0 40 time (µs)Distance to G2 (mm)0"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 20 30 40 time (µs) Integrated electron signal (arb.)0 10 20 30 40 time (µs)Distance to G2 (mm)0"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 20 30 40 time (µs) Integrated electron signal (arb.)0 10 20 30 40 time (µs)Distance to G2 (mm)0"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 20 30 40 time (µs) Integrated electron signal (arb.)0 0 2 30 40 time (µs)Distance to G2 (mm)0"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs) 50!40!30!20!10!0Electron signal (arb.)0 10 2 30 40 time (µs) Integrated electron signal (arb.)0 10 2 30 40 time (µs)Distance to G2 (mm)≈ 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 15 0 15 15 0 15 15 0 15 15 0 15 mm mm mm mmmmmmmmmm 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 15 0 15 15 0 15 15 0 15 15 0 15 mm mm mm mmmmmmmmmm 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 15 0 15 15 0 15 15 0 15 15 0 15 mm mm mm mmmmmmmmmm 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 15 0 15 15 0 15 15 0 15 15 0 15 mm mm mm mmmmmmmmmm 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 15 0 15 15 0 15 15 0 15 15 0 15 mm mm mm mmmmmmmmmm 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 30 15 0 15 30 15 0 15 15 0 15 15 0 15 15 0 15 mm mm mm mmmmmmmmmm0 20 40 60 80 100 120 0 2 4 6 8 10 12 Recoil velocity (m s-1) ω1 pulse energy (µJ) Field (V cm-1)Ti e (µs)a) b)d)e)0 50 100 150 200 250 ρ 0 (cm-3)10101011N+ = 0 N+ = 21012c)Figure 8.1: Summary of experimental data. a) Double-resonant selection ofthe initial quantum state of the n0f(2) Rydberg gas. b) Laser-crossed differen-tially pumped supersonic molecular beam. c) Selective field ionization spectrumafter 500 ns evolution, showing the signal of weakly bound electrons combinedwith a residual population of 49f(2) Rydberg molecules. After 10 µs, this popula-tion sharpens to signal only high-n Rydbergs and plasma electrons. d) Integratedelectron signal as a function of evolution time from 0 to 160 µs. Note the onset ofthe arrest phase before 10 µs. Timescale compressed by a factor of two after 80µs. e) x, y-integrated images recorded after a flight time of 400 µs with n0 = 40for initial Rydberg gas peak densities varying from 2× 1011 to 1× 1012 cm−3. Allof these images exhibit the same peak density, 1× 107 cm−3.868.2. Molecular physics of the arrested phaseplasma space charge (Appendix B).The peak density of the plasma decays for as much as 10 µs until it reaches avalue of ∼ 4× 1010 cm−3, independent of the initially selected n0 and ρ0. There-after the number of charged particles remains constant for at least a millisecond.On this hydrodynamic timescale, the plasma bifurcates, disposing substantial en-ergy in the relative velocity of plasma volumes separating in ±x, the cross-beamaxis of laser propagation [184].The avalanche to plasma proceeds at a rate predicted with accuracy by semi-classical coupled rate equations [33] (more details in Appendix B). This picturealso calls for the rapid collisional relaxation of Rydberg molecules, accompaniedby an increase in electron temperature to 60 K or more. Bifurcation accounts for aloss of electron energy. But, the volumes that remain cease to evolve, quenchinginstead to form an arrested phase that expands slowly, at a rate reflecting aninitial electron temperature no higher than a few degrees Kelvin. These volumesshow no sign of loss owing to the fast dissociative recombination of NO+ ions withelectrons predicted classically for low Te [189], or predissociation of NO Rydbergs,well-known to occur with relaxation in n [190].Thus, from the experiment, one learns that 5 µs after avalanche begins, Ry-dberg relaxation ceases. The experiment detects no sign of ion acceleration byhot electrons and the surviving number of ions and electrons remains constantfor the entire remaining observation period, extending to as long as 1 ms. Withthe vast phase space available to energized electrons and neutral nitrogen andoxygen atom fragments, this persistent localization of energy in the electrostaticseparation of cold ions and electrons represents a very significant departure froma thermalized phase. Current experimental evidence thus points strongly to en-ergy localization and absence of thermalization within the accessible time of theexperiment.8.2 Molecular physics of the arrested phaseDirect measurements of its electron binding energy together with its observedexpansion rate establish experimentally that the bifurcated plasma contains only878.2. Molecular physics of the arrested phaseNO +NO+e-e-rijdidjFigure 8.2: Schematic representation of NO+ core ions, paired with ex-travalent electrons to form interacting dipoles di and dj, separated byrij = ri − rj, in this case joining a Rydberg molecule with an electronbound in the charge of more than one ion.high-Rydberg molecules (n > 80) and NO+ ions in combination with cold elec-trons (initial Te < 5 K) bound by the space charge. As noted above, semi-classicalmodels mixing these species in any proportion predict thermal relaxation, elec-tron heating, expansion and dissipation on a rapid timescale with very evidentconsequences completely unobserved in the experiment. Instead, beyond an evo-lution time of 10 µs or less, the experiment finds that the plasma settles in a stateof arrested relaxation of universal density and low internal energy manifested bya slow free expansion.To describe this apparent state of suppressed relaxation, I proceed now todevelop formal representations of the predominant interactions in this arrestedphase. Under the evidently cold, quasi-neutral conditions of the relaxed plasma,ions pair with extravalent electrons to form dipoles which interact as representedschematically in Figure 8.2.Assuming an intermolecular spacing that exceeds the dimensions of individualion-electron separations, one can describe the Coulomb interactions represented888.2. Molecular physics of the arrested phasein Figure 8.2 in terms of a simple Hamiltonian:H =∑i(P2i2m+ hi) +∑i,jVij (8.1)where hi describes the local relationship of each electron with its proximal NO+core. This local representation extends to account for the interactions of a boundextravalent electron with vibrational, rotational and electronic degrees of free-dom of the core, as described, for example, by Multichannel Quantum DefectTheory [191]. Each ion-electron pair has momentum, Pi and Vij ≡ V (ri − rj)describes the potential energy of the interacting multipoles, represented in Fig-ure 8.2 to lowest order as induced dipoles with an interaction defined by V ddij =[di · dj − 3(di · rij)(dj · rij)] /r3ij.The plasma also very likely includes ion-electron pairs of positive total energy.This implies the existence of local Hamiltonians of much greater complexity thatdefine quasi-Rydberg bound states with dipole and higher-order moments formedby the interaction of an extravalent electron with more than one ion.Representing the eigenstates of hi by |ei〉, one can write a reduced Hamiltonianfor the pairwise dipole-dipole interactions [192, 193] in the arrested phase:Hdd =∑iP2i2m+∑i,jV ddij (8.2)where one evaluates V ddij in the |ei〉 basis.Note that this Hamiltonian conventionally refers to the case where a narrowbandwidth laser prepares a Rydberg gas in which a particular set of dipole-dipoleinteractions give rise to a small, specific set of coupled states [194–196]. Bycontrast, the avalanche and relaxation of the molecular ultracold plasma spon-taneously populates a great many different states that evolve spatially withoutreference to a dipole blockade of any kind.This system relaxes to a quenched regime of ultracold temperature, from whichit expands radially at a rate of a few meters per second. Dipolar energy interac-tions proceed on a much faster timescale [75, 126, 197, 198]. Cross sections forclose-coupled collisions are minuscule by comparison [199]. One can thus assume898.3. Effective many-body Hamiltonianthat the coupled states defined by dipole-dipole interactions evolve adiabaticallywith the motion of ion centres.This separation of timescales enables to write an effective Hamiltonian describ-ing pairwise interactions that slowly evolve in an instantaneous frame of slowlymoving ions and Rydberg molecules: Heff = P∑i,j Vddij , where P represents aprojector onto the low-energy degrees of freedom owing to dipole-dipole coupling.8.3 Effective many-body HamiltonianWhile the complexity of the |ei〉 basis makes it difficult to compute the exact formof Heff , the concept of this projection enables to make some general observations.Confining the present consideration to pairwise dipolar interaction between Ryd-berg or quasi-Rydberg molecules, I choose a set of basis states |e1〉, |e2〉, ... ∣∣eL〉that spans the low-energy regime. The superscript with lower (higher) integerlabel refers to the state with larger (smaller) electron binding energy.Quenching gives rise to a vast distribution of pair-wise interactions, creatinga random potential landscape. Resonant dipole-dipole interactions in this densemanifold of basis states cause excitation exchange. In the disorder potential, theseprocesses are dominated by low energy-excitations involving L states in number,where one expects L to be small (from 2 to 4). The most probable interactionsselect L-level systems composed of different basis states from dipole to dipole.Representing excitations by spins, one can write an XY model [200] (alsosee Appendix B) that describes these interactions in terms of their effective spindynamicsHeff =∑iiSˆzi +∑i,jJij(Sˆ+i Sˆ−j +H.c.) (8.3)where Sˆ in each case denotes a spin-L operator defined as Sˆγ = ~σˆγ/2, for whichσγ is the corresponding spin-L Pauli matrix that spans the space of the L activelevels and γ = x, y or z. H.c. refers to Hermitian conjugate.This Hamiltonian reflects both the diagonal and off-diagonal disorder createdby the variation in L-level system from dipole to dipole. The first term in Heff908.4. Induced Ising interactionsdescribes the diagonal disorder arising from random contributions to the on-siteenergy of any particular dipole owing to its random local environment. In spinlanguage,∑i iSˆzi represents a Gaussian-distributed random local field of widthW . The representative SFI spectrum in Figure 8.1 directly gauges a W of ∼ 500GHz for the quenched ultracold plasma.In the second term, Jij = tij/r3ij determines the off-diagonal disordered am-plitudes of the spin flip-flops. To visualize the associated disorder, recognize thatthe second term varies as tij ∝ |di||dj|, where every interaction selects a differentdi and dj. Over the present range of W , a simple pair of dipoles formed by sand p Rydberg states of the same n couple with an average tij of 75 GHz µm3[201]. Note that tij falls exponentially with the difference in principal quantumnumbers, ∆nij [202].8.4 Induced Ising interactionsIn the limit |Jij| << W most appropriate to the experiment, sequences of interac-tions can add an Ising term that describes a van der Waals shift of pairs of dipoles[203]. Consider, for example, three mutually nearest-neighbor spins i, j and kin the L = 2 case. A third-order process couples spins i and j via spin k in thefollowing fashion: |↓i, ↑j, ↑k〉Sˆ+i Sˆ−j−−−→ |↑i, ↓j, ↑k〉Sˆ+j Sˆ−k−−−→ |↑i, ↑j, ↓k〉 Sˆ+k Sˆ−i−−−→ |↓i, ↑j, ↑k〉;defining a self interaction that changes the pairwise energies of i, j.All such processes occur with an amplitude, Uij ≈ J2ijJ˜/W 2, where J˜ estimatesJij, for an average value of tij at an average distance separating spins. J˜ simplyestimates the average nearest-neighbor interaction for the purpose of evaluatingthe third order perturbation. At the point of arrest in the quenched ultracoldplasma, one estimates a J˜ on the order of 2 GHz, as derived in Appendix B.Uij is inherently random owing to the randomness in Jij. It is also important tonote that this limit gives rise to additional perturbative processes that renormalizethe local on-site fields by van der Waals terms and slightly affect the pairwise flip-flop amplitudes [201, 203, 204]. I simply absorb these effects in the definitions ofi and Jij. A detailed derivation is presented in Appendix B.Summarizing the results, I arrive at a general spin model with dipolar and918.5. Discussion: Localization versus glassy behavior and slow dynamicsVan der Waals interactions (the explicit derivation is provided in Appendix B):Heff =∑iiSˆzi +∑i,jJij(Sˆ+i Sˆ−j +H.c.)+∑i,jUijSˆzi Sˆzj (8.4)where Uij = Dij/r6ij and Dij = t2ijJ˜/W2.The appearance of this third term underlines the many-body nature of Eq.(B.9). Even in this extreme limit, its dynamics are non-trivial, clearly involvingmore than spin flip-flops with emergent correlations between spins.8.5 Discussion: Localization versus glassybehavior and slow dynamicsThe complexity of this Hamiltonian places an exact solution of Eq. (8.4) beyondreach for the conditions of the plasma. But, I can gauge some likely properties ofsuch a solution by analogy to published work on simpler systems.In the single-excitation limit, this Hamiltonian reduces to the dipolar XYmodel, which has been studied by locator expansion methods measuring the prob-ability of resonant pairs [24, 205]. When Jij scales by a power law α that equalsthe dimension d, the single-spin model with diagonal disorder displays criticalbehavior characterized by extended states with subdiffusive dynamics [24, 205].Dipolar systems in three dimensions can form extended states but yet exhibitnon-ergodic behavior [206].Off-diagonal disorder in the presence of long-range spin flip-flop interactionsof arbitrary order in one dimension yields algebraic localization as opposed toexponential Anderson localization, challenging the generality of the rule that sayssystems must delocalize for α ≤ d [207].The many-body problem is more involved, because the Ising term has off-diagonal matrix elements in the resonant pair states [208]. This mechanism cou-ples distant resonant pairs, transferring energy from one pair to the other to causedelocalization. A study of power law coupled systems predicts that spin flip-flops928.5. Discussion: Localization versus glassy behavior and slow dynamics(order α) and spin Ising interactions (order β) in an iterated pairs configurationin which β ≤ α localize for β/2 > d [209].A locator expansion approach developed for β > α applied to Eq. (8.4)confined to diagonal disorder predicts a critical dimension, dc = 2 [203]. Forthe case of d > dc, this theory holds that a diverging number of resonancesdrives delocalization whenever the number of dipoles exceeds a critical value Nc.For a system described by Eq. (8.4) under the conditions observed for arrest,Nc = (W/J˜)4 ≈ 3× 109. Experimentally the quenched ultracold plasma containsan order of magnitude fewer dipoles than this number, Nc, required for resonancedelocalization. See Appendix B for more discussion.In any event, Nandkishore and Sondhi [210] point out that locator expansionarguments might not hold generally, and that power law interactions can give riseto an MBL phase in higher dimensions. Their arguments build on the idea that, inmany systems, long-range interactions drive the system to a non-trivial correlatedphase described by emergent short-range interactions, well characterized by aperturbation theory approach for which a locator expansion can be applied. Inthis context, MBL with long-range interactions in higher dimensions becomesquite possible.Rare thermal regions (Griffiths regions) are thought to destabilize MBL sys-tems of higher dimension [211–215], creating a glassy state, characterized by aslow evolution to a delocalized phase. However, other results contradict this no-tion, and support the possibility of localization in all dimensions [216]. A commonfeature of MBL and glassy phases is slow dynamics [217–219].Given the apparent conflict of available theoretical results (for example, see[213, 216, 220, 221]) and infeasibility of reliable numerical simulations, experi-ments stand to play an important role in elucidating localization, and differenti-ating between localization and glassiness.A related study has investigated the behavior of a three-dimensional dipolarsystem of nitrogen-vacancy color centers in diamond in the presence of on-sitedisorder [222]. The experimental results point to slow dynamics consistent withthe observations of the arrest phase.Generally speaking, the foregoing analysis suggests that the model definedby Eq. (8.4) ought to exhibit some form of localization or at least very slow938.6. Concluding remarksdynamics, since all the terms in the Hamiltonian are disordered and the termsresponsible for delocalization (Jij and Uij) are expected to be much smaller thanW . This very slow dynamical behavior provides an explanation to the observedarrested relaxation and the accompanied localization in the energy coordinateseen in the experiment.8.6 Concluding remarksExperiment paints a very clear picture of arrested relaxation in an ultracold neu-tral plasma of NO+ ions and electrons. Formed by electron-impact avalanchein a state-selected Rydberg gas of supersonically cooled nitric oxide, this plasmaspontaneously bifurcates, quenching the energy distributions of ions and electronsto form compressed volumes of canonical density and ultracold temperature. Ex-perimental images detail the time evolution of the electron binding energy andthree-dimensional density distribution. These metrics directly show evidence foran initial redistribution of Rydberg population driven by collision with energeticelectrons. Thereafter, all sign of free electrons disappears, and the relaxation ofthe system ceases. These dynamics fail to conform with conventional models ofplasma coupled-rate processes and NO predissociation.This proceeds from this well-defined base of experimental observations to for-mulate a conceptual rationale for localization as it might occur in an ensembleof strongly interacting random dipoles. I refer to the selective field ionization(SFI) spectrum of electron binding energies to gauge the ensemble disorder in theenergies associated with dipole-dipole resonant couplings, and construct a schemeto describe the resulting web of interactions in terms of a projection to some lownumber, L, of random coupled transitions. The vast array of accessible pathways,and the randomness of the system of states populated in the quenched plasmafar exceeds the complexity typically encountered in few-level quantum optics sys-tems. Nevertheless, I take the simplest possible approach of an XY Hamiltonianfor interacting dipolar spins with emergent Ising interactions. Applying the ideaof a locator expansion, I consider the conditions required for delocalization in anL = 2 limit. I find Nc, a critical number of dipoles required to sustain delocal-948.6. Concluding remarksization, that exceeds the number of dipoles in the plasma by more than an orderof magnitude. I thus argue that the plasma ought to exhibit slow dynamics on along observational timescale at least.I present this quenched ultracold plasma as a class of system in which local-ization arises naturally, in which a quench acts to guide the propagation of thesystem toward a global many-body localized state. This evolution must drawupon a subtle interplay between dissociation and resonant but rare dipole-dipoleinteractions giving rise to novel many-body phenomena. Tracking the state of themolecular ensemble, as it proceeds from the semi-classical conditions of a spectro-scopically defined Rydberg gas, provides a unique opportunity to observe stagesin the development of localization. While the construction of a picture of the fullmany-body Fock space of the random Rydberg/excitonic dipoles will require newtheoretical tools, strong evidence of dipole-dipole interactions remains, providinga logical basis for the projection to a simple leading order of resonance and aminimal effective model for localization. Further work is required to wrestle withthe theoretical complexity of the quenched plasma. Effects such as Griffiths re-gions and dissociation may prove to have singular importance in this molecularsystem. The emergence of discerning experimental tools promise both greaterfundamental understanding and perhaps important new applications.95Part IIIThesis Outcome96Chapter 9Conclusions9.1 Summary of resultsIn this thesis, I have studied the pairing of polarons into bipolarons in the Peierlsmodel of particle-phonon coupling, and the possibility of many-body localiza-tion (MBL) in the arrested state of a quenched ultracold molecular plasma. SeeChapter 1 for an overview.Part I of the thesis investigates two-particle states of the Peierls model. InChapter 4, I show that two electrons in the Peierls model pair into a light singletbipolaron. I further explain that these light bipolarons should condense intoa superfluid at high temperatures [19]. In Chapter 5, I study the triplet sector(also relevant to hard-core particles) and show that phonon-mediated interactionsare repulsive. In this case, the ground-state dimer bound by bare attractionexhibits two sharp transitions, one of which is self-trapping [20]. To the bestof my knowledge, this is the first example of a self-trapping transition at thetwo-particle level. In Chapter 6, I study the excited spectrum of two hard-coreparticles in the Peierls model and show that a stable exotic repulsively boundbipolaron appears in the spectrum [21]. This repulsive bipolaron state is relevantto currently ongoing ultracold experiments.These results indicate that many of the widely accepted ideas about polaronsand bipolarons must be questioned and perhaps revisited.Part II of the thesis turns to the issue of the absence of thermalization in thearrested state of the quenched ultracold molecular plasma. Chapter 7 introducesthe general concepts of ergodicity in quantum many-body systems, reviews thetopics of Eigenstate Thermalization Hypothesis (ETH) and quantum localizationin single-particle and many-body systems. Classical models predict dissociationof the plasma on timescales much shorter than the observation lifetime [33]. In979.2. Future directionsChapter 8, I develop a theory to explain this behavior in terms of an effectivedisordered spin model with dipolar interactions [34]. This offers a way to explainthe slow relaxation of the plasma owing to localization of transport degrees offreedom in a possible MBL phase. The arrested relaxation of the plasma presentsan exotic state where quantum behavior in presence of disorder preserves thesystem at high temperatures.Our results establish the quenched plasma as a platform to study quantumphysics of self-assembled large macroscopic systems without fine experimentalcontrol.This thesis aims to develop a deep understanding of physically motivatedmany-body systems. As revealed here, particles dressed by interactions, bathsexcitations, and disorder can exhibit surprising behavior even at high tempera-tures. The results presented here represent a start, and future work will hopefullyexplore the consequences of these ideas, which I hope would revolutionize thelandscape of future commercial quantum technologies.9.2 Future directionsThe research findings of this thesis motivate several future directions of investi-gation. Here I highlight a few.9.2.1 Ideas related to Part IThe discovery of light bipolarons and self-trapping of dimers of hard-core particlesin the Peierls model motivates the investigation of a variety of questions relevantto finite concentrations, with a desire to make direct contact with experimentalsystems. Below are some interesting directions I plan to investigate in the nearfuture:Extreme adiabatic limitThe adiabatic limit of phonon frequencies, Ω, much smaller than the electron hop-ping, t, characterizes a regime of phonon-induced interactions with a large number989.2. Future directionsof phonon-generated effective terms, rendering this limit particularly challengingto numerical methods. One approach treats the phonons classically and exploresthe behavior of electrons in the potential landscape of localized ions in the lattice.However, this treatment breaks the lattice symmetry. It is, therefore, of theoret-ical interest to develop new methods, such as that of Refs. [81, 82], to study thislimit, most relevant to materials.Experimental systemsSome exceptional materials possess phonon frequencies, Ω, much larger than theelectron hopping, t, such as SrTiO3 [65], magic-angle graphene superlattices [66],and layered MoS2 [67]. Out theoretical techniques are capable of addressingthis limit. There, one can explore the nature of phonon-mediated interactionsand their relevance to the observed phenomenology in experiments. For exam-ple, SrTiO3 exhibits superconductivity at very low carrier densities attributedto electron-phonon coupling. Magic-angle graphene superlattices exhibit similarphenomenology, but are believed to posses interactions mediated by both elec-tronic correlations and phonons.To study these systems, I imagine an approach that builds effective latticemodels representative of the experimental situation taking information from abinitio calculations and symmetry considerations. These models differ from ours inthat they incorporate details of the experimental situation. I expect the generalphysics, such as the phonon-mediated pair-hopping, to carry over, but the adaptedcalculation will allow direct comparison to experiments. I plan to devise modelsthat tackle the limit of single and few electrons and pave the way for extensionsto finite doping.Finite concentrationsThe biggest hurdle to high-temperature bipolaronic superconductivity has beenthe characteristically heavy masses of bipolarons at strong electron-phonon cou-plings. I have established that the Peierls coupling stabilizes strongly bound yetvery light bipolarons. To understand the phases of the Peierls model in the ther-modynamics limit, future studies must focus on finite concentrations of electrons.999.2. Future directionsA general phase diagram at zero temperature can be quantified by three pa-rameters, the dimensionless electron-phonon coupling, λ, the phonon frequency,Ω, and the electron filling factor, n¯. Moreover, finite temperature phase dia-grams could shed light on some very interesting effects. One generally expectsregimes of charge density wave insulating behavior and superconductivity. Identi-fying the nature of these phases and their stability conditions is of major interest.Will density-wave behavior exhibit bond order? Do electron-phonon interactionsinduce magnetic correlations? What are the effects of the long-range Coulombrepulsion on superconductivity? How high is Tc for superconductivity? Theseare samples of different questions relevant to the understanding of the nature ofphonon-mediated interactions.The exploration of these problems in higher dimensions demands the devel-opment of new theoretical tools. Most numerical tools suffer severe limitationsin higher-dimensional systems and I envision semi-analytical tools motivated byphysical principles to lead the search for superconductivity in the near term. Tothis end, identifying the regimes of potential interest can help guide the develop-ment of effective approaches. For example, numerical knowledge of the nature ofinteractions can direct mean-field approximations in the relevant coupling chan-nels and provide a phenomenological base to a physical theory.9.2.2 Ideas related to Part IIHaving established MBL as a possible explanation for the arrest state of thequenched plasma, one might ask: What are the definitive measures of localizationor its lack thereof in the ultracold plasma experiment? To answer this question,some possible directions are:Spectroscopic signatures of MBL in the ultracold plasmaSeveral authors [25, 223, 224] have identified the presence of ‘soft gap’ at zerofrequency in the spectral response of a system as a robust signature of MBLthat survives coupling to the environment. This gap reflects the energy-levelrepulsion between eigenstates, which means that a transition out of a localizedstate becomes suppressed as the frequency approaches zero.1009.2. Future directionsThe complexity of interaction pathways afforded by the large number of dipo-lar states of varying principal quantum number, n, presents the question ofwhether this generic feature carries over to the ultracold plasma. To providea definitive answer to this line of inquiry, I shall explore theoretically and in con-junction with experiments the spectral behavior of a disordered system of randomdipoles, each with more than two molecular states per dipole. Within this ap-proach, I will study the manifestations of many-body localization in the selectivefield ionization (SFI) spectroscopy. This will require adaptation of the theoreticalwork developed in [223] to the case of SFI spectroscopy.Microscopic mechanism of localization in the ultracold plasmaIn my analysis of the dynamics in the plasma, I have identified a minimal effec-tive model describing the low-order leading resonant transitions in the disorderedlandscape characteristic of the arrested phase. I resorted to resonance countingfor the L = 2 case to argue for the likelihood of localization. My approach doesnot identify the local integrals of motion emerging in the presumably localizedlong-range interacting disordered system. I expect that in presence of randomdipolar interactions and random local fields, interactions can only dominantlyhybridize few, perhaps two, dipoles into entangled structures. 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Thereexists n ways to annihilate a phonon in a given configuration by the action of Vˆ .This can be seen from [bi, (b†i )n] = n(b†i )n−1, which is a special case of the identity[bi, f(b†i )] =∂∂b†if(b†i ), where f(b†i ) is any given function of b†i .126A.2. Momentum Average approach to the Peierls two-carrier statesA.2 Momentum Average approach to thePeierls two-carrier statesIn Subsection 3.2.2, I discuss the Momentum Average (MA) approach to Peierlspolarons and bipolarons. Here, I provide more details about MA and its applica-tion to Peierls bipolaron-type problems.The Momentum Average (MA) approximation is a non-perturbative quasi-analytical technique designed to solve the equation of motion for the relevantGreen’s function G(k, ω) = 〈k|(ω −H + iη)−1|k〉 in the Bogoliubov-Born-Green-Kirkwood-Yvon (BBKGY) hierarchy. The hierarchy consists of an infinite set ofcoupled equations that are impossible to solve exactly. By neglecting exponen-tially small contributions in the expansion, one simplifies the equations of motionto a form that is readily solvable numerically. The guide to approximating thehierarchy follows from the variational meaning of MA: essentially one solves theproblem in a variational subspace.The choice of the variational space depends on the details of the Hamiltonianand state(s) of interest [86]. For the Holstein model, a one-site phonon cloudsuffices to provide accurate results for single polarons [83, 86] and for so-called S0bipolarons describing two electrons in an on-site singlet bound state [87]. Takentogether with the local nature of the Holstein coupling, this explains why a one-site phonon cloud is accurate to describe such states. For the Edwards andSSH models, the coupling to phonons is non-local and therefore a bigger cloud isrequired to yield accurate results. A three-site phonon cloud MA has been shownto be very accurate for such models [17, 226].In Part I of this thesis, I generalize MA to study strongly bound two-carrierstates in the (extended) Peierls-Hubbard model. I derive the MA equations for twoparticles with a three-site phonon cloud and allow the particles to be arbitrarilyfar from the cloud.In the case of hard-core bare particles, discussed in Chapter 5, I have re-stricted the bare particles to be at most two sites apart from each other, if acloud is present. Terms corresponding to the particles being further than twosites apart are expected to contribute significantly only to higher-energy states,127A.2. Momentum Average approach to the Peierls two-carrier statesif a two-particle state is strongly bound, which is the case of primary interest tous. For weakly bound dimers, the variational space must be extended to includeconfigurations where the particles are further apart. The variational space canbe increased systematically until convergence is achieved.On the other hand, for the study of two electrons in the singlet sector, dis-cussed in Chapter 4, I have found that it is necessary to relax this restriction.A.2.1 Details of MA for two hard-core particles in thePeierls modelIn Subsubsection 3.2.2, I present few representative equations for the Momen-tum Average (MA) approach used to study two hard-core particles in the Peierlsmodel. Here I provide more details.I derive the equations of motion using Dyson’s identity Gˆ(ω) = Gˆ0(ω) +Gˆ(ω)Vˆ Gˆ0(ω) where Gˆ(ω) = (ω − H + iη)−1, Gˆ0(ω) = (ω − H0 + iη)−1 withH0 = Hp +Hph, and Vˆ is the Peierls particle-phonon coupling term.Consider the two-particle propagator G(K, 1, n, ω) = 〈K, 1|Gˆ(w) |K,n〉 de-fined for two-particle states |K,n〉 = ∑i eiK(Ri+na/2)√N c†ic†i+n |0〉 with the two parti-cles n ≥ 1 sites apart, and a is the lattice constant. Using Dyson’s identity andinserting a resolution of the identity, its exact equation of motion can be writtenas:G(K, 1, n, ω) = G0(K, 1, n, ω)+∑ηG0(K, η, n, ω)〈K, 1|Gˆ(ω)Vˆ |K, η〉 .Consider Vˆ |K, η〉. It consists of states with one phonon plus the particles η ± 1sites apart. Thus, the right-hand side of the exact equation of motion contains aninfinite number of terms. Within MA I restrict the particles to be within two sites128A.2. Momentum Average approach to the Peierls two-carrier statesof each other when phonons are present; this simplifies the equation of motion to:G(K, 1, n, ω) = G0(K, 1, n, ω)− ge−iKaG0(K, 2, n, ω)F1(−2, 1) + gG0(K, 2, n, ω)F1(−1, 1)− gG0(K, 2, n, ω)F1(0, 1) + geiKaG0(K, 2, n, ω)F1(1, 1)− ge−3iKa/2G0(K, 3, n, ω)F1(−3, 2)− g[e−3iKa/2G0(K, 1, n, ω)− e−iKa/2G0(K, 3, n, ω)]F1(−2, 2)− 2ig sin(Ka/2)G0(K, 1, n, ω)F1(−1, 2)+ g[e3iKa/2G0(K, 1, n, ω)− eiKa/2G0(K, 3, n, ω)]F1(0, 2)+ ge3iKa/2G0(K, 3, n, ω)F1(1, 2), (A.2)where F1(m,n) is shorthand for F1(K,m, n, ω) defined asFl(K,m, n, ω) ≡∑ieiKRi√N〈K, 1|Gˆ(ω)c†i+mc†i+m+nb†li |0〉 , (A.3)i.e. a generalized one-site cloud propagator. I then introduce other appropriategeneralized propagators:Fl1,l2(K,m, n, ω) ≡∑ieiKRi√N〈K, 1|Gˆ(ω)c†i+mc†i+m+nb†l1i b†l2i+1 |0〉 , (A.4)Fl1,l2,l3(K,m, n, ω) ≡∑ieiKRi√N〈K, 1|Gˆ(ω)c†i+mc†i+m+nb†l1i−1b†l2i b†l3i+1 |0〉 , (A.5)for two-site cloud and three-site cloud configurations respectively, and repeat-edly apply the Dyson’s identity, to derive the MA equations of motion for thepropagators in Eqs. (A.2)-(A.5).This linear system of coupled equations is solved numerically and the propa-gator of interest G(K, 1, n, ω) is computed. Dimer bound state properties, suchas ED(K) presented in Chapter 5 can be extracted from the propagator.By construction the MA approach developed here is designed to describestrongly bound states accurately. Indeed, the comparison with variational exactdiagonalization (VED) results in Chapter 5 shows that the MA approximation isaccurate in this regime.Similarly, the unrestricted MA used for two electrons in the singlet sector and129A.3. Bogoliubov-Born-Green-Kirkwood-Yvon equation-of-motion. . .presented in Chapter 4 shows excellent agreement with VED.A.3 Bogoliubov-Born-Green-Kirkwood-Yvonequation-of-motion technique for thetwo-body Hamiltonian hˆ2I use the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) equation-of-motion(EOM) approach, outlined in Subsection 3.3.2, to find the discrete bound statedispersion of the projected two-body Hamiltonian hˆ2, defined in Eq. (3.28):hˆ2 = hˆ1 + Vˆ0 + Pˆ Vˆ1E0 − Hˆ0Vˆ Pˆ . (A.6)I apply this to the projected two-body Hamiltonians derived in the anti-adiabaticlimit. The results are presented in Chapters 4, 5 and 6.A.3.1 BBGKY EOM for two particles in the singletsector of the Peierls model in the anti-adiabaticlimitHere I apply the BBGKY EOM method to two particles in the singlet sector of thePeierls model in the anti-adiabatic limit, used to compute the results presentedin Subsection 4.2.2.In absence of bare particle hopping, i.e. t = 0, the Hamiltonian factories intoan even and an odd sector:hˆeven = hˆt=01 + Uˆ0,2 (A.7)hˆodd = hˆt=01 + Uˆ1, (A.8)wherehˆ1 = −0∑i,σnˆi,σ +∑i,σt2(c†i,σci+2,σ +H.c.),130A.3. Bogoliubov-Born-Green-Kirkwood-Yvon equation-of-motion. . .Uˆ0,2 = −T0,0∑i[c†i−1,↑c†i−1,↓ci,↓ci,↑ +H.c.]+T0,2∑i[(c†i+1,↑c†i−1,↓ − c†i+1,↓c†i−1,↑)ci,↓ci,↑ +H.c.]andUˆ1 = +T1,1∑i,σ[c†i+1,σc†i+2,−σci+1,−σci,σ +H.c.]+ J∑i,σc†i+1,σc†i,−σci,σci+1,−σ.I define the two-particle states|K,n〉 =∑ieiK(Ri+n2)√Ns†i,n|0〉,∀n ≥ 0. Here s†i,n :=(c†i,↑c†i+n,↓ − c†i,↓c†i+n,↑)/√2(1 + δn, 0) creates a singlet statewith the two particles n sites apart.I solve for the propagators Gδ(K,n, ω) = 〈K, δ| Gˆ(ω) |K,n〉, where Gˆ(ω) =(ω + iη − hˆeven(odd))−1 is the resolvent in the even (odd) sector. I use the short-hand g(n) for Gδ(K,n, ω), where δ = 0 (δ = 1) labels the propagator of interestin the even (odd) sector. The bound state energy is at the lowest discrete pole ofthese propagators below the two-polaron continuum.Solution in the even sectorI generate the EOM for g(n) ≡ G0(K,n, ω) = 〈K, 0| Gˆ(ω) |K,n〉 using Gˆ(ω)(ω +iη − hˆeven)−1 = 1:(ω + iη + Feven(K))g(0) = 1 +√2 (T0,2 + f2(K)) g(2)(ω + iη)g(2) =√2 (T0,2 + f2(K)) g(0) + f2(K)g(4) (A.9)and for any even n > 3,(ω + iη)g(n) = f2(K) [g(n− 2) + g(n+ 2)] . (A.10)131A.3. Bogoliubov-Born-Green-Kirkwood-Yvon equation-of-motion. . .Here, Feven(K) = 2T0,0 cos(K) and f2(K) = 2t2 cos(K). I solve this recursionrelation using the methods for continued fractions [88]. The results are presentedin Subsection 4.2.2.Solution in the odd sectorHere I generate the EOM for g(n) ≡ G1(K,n, ω) = 〈K, 1| Gˆ(ω) |K,n〉 usingGˆ(ω)(ω + iη − hˆodd)−1 = 1:(ω + iη − f2(K)− Fodd(K))g(1) = 1 + f2(K)g(3) (A.11)and for any odd n ≥ 3,(ω + iη)g(n) = f2(K) [g(n− 2) + g(n+ 2)] , (A.12)where Fodd(K) = −J+2T1,1 cos(K). I solve this recursion relation using the samemethods. The results are presented in Subsection 4.2.2.A.3.2 BBGKY EOM for two hard-core particles in thet-t2-V modelHere I apply the BBGKY EOM method to two hard-core particles in the one-dimensional model t-t2-V model, Eq. (6.3),H = −0∑inˆi − t∑i(c†ici+1 +H.c.)+ t2∑i(c†ici+2 +H.c.)+ V∑inˆinˆi+1, (A.13)relevant to the discussions in Chapters 5 and 6. I define the two-particle states|K,n〉 =∑ieiK(Ri+n2)√Nc†ic†i+n|0〉,∀n ≥ 1, and the propagators G(K,n, ω) = 〈K, 1| Gˆ(ω) |K,n〉, where Gˆ(ω) =(ω+ iη−H)−1 is the resolvent of interest. I use the short-hand notation g(n) for132A.3. Bogoliubov-Born-Green-Kirkwood-Yvon equation-of-motion. . .G(K,n, ω). The repulsive bipolaron bound state energy (once it appears) is atthe highest discrete pole of these propagators above the two-polaron continuum.Using the identity Gˆ(ω)(ω + iη −H)−1 = 1, I generate the EOM:(ω + iη + 20 − βK − V )g(1) = 1− αKg(2) + βKg(3)(ω + iη + 20)g(2) = −αK [g(1) + g(3)] + βKg(4) (A.14)and for any n ≥ 3,(ω + iη + 20)g(n) = −αK [g(n− 1) + g(n+ 1)]+βK [g(n− 2) + g(n+ 2)]. (A.15)Here, αK = 2t cos(K2)and βK = 2t2 cos(K).The physically acceptable analytical solution for recurrence relations of thistype is available in [227], however it is rather complicated and its poles cannot beextracted analytically. A general solution can be found numerically. Results inFigure 6.1 represent solutions obtained solving this system of equations numeri-cally.I note that for K = pi, αK vanishes limiting the EOM to odd n. This explainswhy even n states are forbidden as shown in the inset of Figure 6.2 b).133Appendix BOverview of the ultracold plasmaexperiment and theoreticalconsiderationsHere, I present a detailed discussion of the ultracold plasma experiment and thetheory developed in Part II of this thesis.B.1 Double-resonant production of a stateselected molecular Rydberg gasLaser pulses, ω1 and ω2, cross a molecular beam to define a Gaussian ellipsoidalvolume in which a sequence of resonant electronic transitions transfer populationfrom the X 2Π1/2 ground state of nitric oxide to an intermediate state, A2Σ+with angular momentum neglecting spin, N ′ = 0, and then to a specified level inthe mixed n0f(2) Rydberg series to create a state-selected Rydberg gas of nitricoxide, in which quantities (0) and (2) refer to rotational quantum numbers of theNO+ 1Σ+ cation core.The intensity of ω1 determines the density of Rydberg molecules formed bysaturated absorption of ω2. For a given ω1 intensity, the peak Rydberg gas densityvaries with ω1 − ω2 delay according to the well-known decay rate of the A 2Σ+state. Choosing Iω1 and ∆tω1−ω2 , the experimentalists precisely control the initialpeak density of the Rydberg gas ellipsoid over a two-decade range from ρ0 = 1010to 1012 cm−3 [186].In the core of this ellipsoid, Rydberg molecules, propagating in the molecularbeam have a local longitudinal temperature of T|| = 500 mK and a transverse134B.2. Selective field ionization spectroscopy of electron binding energytemperature, T⊥ < 5 mK. These molecules interact at a density-determined rateto form NO+ ions and free electrons. Initially created electrons collide withRydberg molecules to trigger electron-impact avalanche on a time-scale that varieswith density from nanoseconds to microseconds (see below).B.2 Selective field ionization spectroscopy ofelectron binding energySelective field ionization (SFI) produces an electron signal waveform that varieswith the amplitude of a linearly rising electrostatic field. Electrons in a Ryd-berg state with principal quantum number, n, ionize diabatically when the fieldamplitude reaches the electron binding energy threshold, 1/9n2 [228].For low density Rydberg gases, SFI has served as an exacting probe of the cou-pling of electron orbital angular momentum coupling with core rotation. Studiesof nitric oxide in particular have shown that nf(2) Rydberg states of NO traversethe Stark manifold to form NO+ in rotational states N+ = 2 and 0 [229].Experiments described in the main text operate in a diabatic regime, employ-ing a slew rate of 0.7 V cm−1 ns−1. Under these conditions, SFI features thatappear when the field rises to an amplitude of F V cm−1 measure electrons boundby energy Eb in cm−1, according to Eb = 4√F .Quasi-free electrons, weakly bound in the attractive potential of more thanone cation, ionize at a low field that varies with the number of excess ions in theplasma.The SFI spectrum presented in the text as Figure 8.1 c) and shown here asFigure B.1 maps the electron binding energy as a function of the initial Rydberggas density for a molecular nitric oxide ultracold plasma after 500 ns of evolution.At a glance, the spectrum at higher density (1012 cm−3) shows direct evidence ofeither electrons bound to an increasing space charge or a broader distribution ofhigh-n Rydberg states.This contracts to a narrower distribution of very weakly bound electrons inplasmas of lower density (1010 cm−3). Here, one observes the spectrum of aresidue of molecules with the originally selected principal quantum number of135B.2. Selective field ionization spectroscopy of electron binding energyField (V cm-1)0 50 100 150 200 250 ρ 0 (cm-3)10101011N+ = 0 N+ = 21012Figure B.1: Selective field ionization spectrum spectrum as a function ofinitial Rydberg gas density, ρ0, after 500 ns of evolution, showing thesignal of weakly bound electrons combined with a residual population of49f(2) Rydberg molecules, (initial principal quantum number, n0 = 49,in the f Rydberg series converging to NO+ ion rotational state, N+ = 2).After 10 µs, this population sharpens to signal high-n Rydbergs and plasmaelectrons, with a residue of the initial Rydberg population, shifted slightly todeeper binding energy by l-mixing and perhaps some small relaxation in n. Theprominent feature that appears at the lowest values of the ramp field gauges thepotential energy of electrons in high Rydberg states bound to single NO+ ions,combined with electrons bound to the space charge of more than one ion. Noticethe binding effect of a slightly greater excess positive charge at the highest initialRydberg gas densities. The red feature extends approximately to the bindingenergy of n0 = 80 or 500 GHz.136B.3. Coupled rate-equation simulations of the electron-impact. . .the Rydberg gas, shifted slightly to deeper apparent binding energy by evidentl-mixing or slight relaxation in n.The experimentalists have used SFI measurements like these to characterizethe avalanche and evolution dynamics of a great many Rydberg gases of varyingdensity and initial principal quantum number. Relaxation times vary, but all ofthese spectra evolve to form the same final spectrum of weakly bound electronswith traces of residual Rydberg gas for systems of low initial density.B.3 Coupled rate-equation simulations of theelectron-impact avalanche to ultracoldplasma in a molecular Rydberg gasThe semi-classical mechanics embodied in a system of coupled rate equationsserves well to describe the avalanche of a molecular Rydberg gas to ultracoldplasma. In this picture, Rydberg molecule densities, labeled ρi, evolve over aladder of principal quantum numbers, ni, according to:−dρidt=∑jkijρeρi −∑jkjiρeρj+ki,ionρeρi − ki,tbrρ3e + ki,PDρi (B.1)The free-electron density changes as:dρedt=∑ikionρ2e −∑ikitbrρ3e − kDRρ2e (B.2)A variational reaction rate formalism determines Te-dependant rate coeffi-cients, kij, for electron impact transitions from Rydberg state i to j, kiion, forcollisional ionization from state i and kitbr, for three-body recombination to statei [230, 231]. Unimolecular rate constants, ki,PD, describe the principal quantumnumber dependant rate of Rydberg predissociation [190, 232, 233], averaged overazimuthal quantum number, l [234]. kDR accounts for direct dissociative recom-137B.3. Coupled rate-equation simulations of the electron-impact. . .bination [189]The relaxation of molecules in the manifold of Rydberg states determines thetemperature of electrons released by avalanche. Conservation of total energy perunit volume requires:Etot =32kBTe(t)ρe(t)−R∑iρi(t)n2i+32kBTρDRe −R∑iρPDin2i(B.3)where R is the Rydberg constant for NO, and ρDRe and ρPDi represent the numberof electrons and Rydberg molecules of level i lost to dissociative recombinationand predissociation, respectively [235, 236].To realistically represent the density distribution produced by crossed-beamlaser excitation of the cylindrical distribution of NO ground-state molecules inthe molecular beam, one uses a concentric system of 100 shells of defined densityspanning a Gaussian ellipsoid to 5σ in three dimensions. Avalanche proceedsas determined by the initial Rydberg molecule density of each shell. Each shellconserves the combined density of stationary molecules, ions and neutral frag-mentation products. Electrons satisfy local quasi-neutrality, but are otherwiseassumed mobile, and thermally equilibrated over the entire volume [33].B.3.1 The semi-classical evolution of an n0 = 80 RydberggasFigure B.2 shows the global evolution of particle densities and electron tempera-ture calculated for an n0 = 80 Rydberg gas at an initial density of 4× 1010 cm−3[33], representing one limit of the SFI spectrum obtained as above for an ultracoldplasma in its arrest state after an evolution time of 10 µs. By this point, the realsystem begins a phase of unchanging composition and very slow expansion thatlasts at least a millisecond – as long a period as it can be observed.The semi-classical simulation result shown in Figure B.2 informs that theSFI spectrum shown in Figure B.1 cannot possibly signal a conventional gas of138B.3. Coupled rate-equation simulations of the electron-impact. . .0102030405060700 5 10 15 20Temperature (K)Time (µs) 1 1 2Time (µs)Electron temperature (K)010203040506070051015200 5 10 15 20Number of particlesTime (µs)x 107Number of particles05201015 0 5 10 15 20Time (µs)x 107N (4S) + O (3P)NO+ + e-NO*051015200 5 10 15 20Number of particlesTime (µs)x 107Number of particles 211 0 5 10 15 20Time (µs)x 107N (4S) + O (3P)NO+ + e-NO*0102030405060700 5 10 15 20Temperature (K)Time (µs) Time (µs)Electron temperature (K)12345060700102030405060700 5 10 15 20Temperature (K)Time (µs) 1 1 2Time (µs)Electron temperature (K)010203040506070051015200 5 10 15 20Number of particlesTime (µs)x 107Number of particles05201015 0 5 10 15 20Time (µs)x 107N (4S) + O (3P)NO+ + e-NO*051015200 5 10 15 20Number of particlesTime (µs)x 107Number of particles 211 0 5 10 15 20Time (µs)x 107N (4S) + O (3P)NO+ + e-NO*0102030405060700 5 10 15 20Temperature (K)Time (µs) Time (µs)Electron temperature (K)1234506070Figure B.2: Semi-classical simulations. (lower) Numbers of ions and elec-trons, Rydberg molecules and neutral dissociation products N(4S) and O(3P) asa function of time during the avalanche of an n0 = 80 Rydberg gas of NO toform an ultracold plasma, as predicted by a shell-model coupled rate equationsimulation. Here, the initial density distribution of the Rydberg gas is repre-sented by a 5σ Gaussian ellipsoid with principal axis dimensions, σx = 1.0 mm,σy = 0.55 mm, σz = 0.7 mm and peak density of 4 × 1010 cm−3, as measuredfor a typical experimental plasma entering the arrest state after an evolution of10 µs. The simulation proceeds in 100 concentric shells enclosing set numbersof kinetically coupled particles, linked by a common electron temperature thatevolves to conserve energy globally. (upper) Global electron temperature as afunction of time.139B.3. Coupled rate-equation simulations of the electron-impact. . .long lived high-Rydberg molecules. Instead, a proven semi-classical rate modelconfigured for the density distribution of the experiment, predicts the decay ofsuch a high-Rydberg gas to plasma on the timescale of a microsecond or less.In the model, predissociation consumes residual Rydbergs in all n-levels withina few microseconds and the formation of neutral atomic products quickly slows.This must occur conventionally because the rising electron temperature stabilizesthe classical plasma state by suppressing three-body recombination. The realarrested state, however, shows no sign of an electron temperature higher than afew degrees K.B.3.2 The semi-classical evolution of a fully ionizedultracold plasma with Te(0) = 5 KOne instead tests the kinetic stability of a conventional ultracold plasma composedentirely of ions and electrons. Again, one assumes initial conditions that fit withthe observed properties of the arrest state: NO+ and electrons present at a densityof 4×1010 cm−3 in an ellipsoid with Gaussian dimensions, σx = 1.0 mm, σy = 0.55mm, σz = 0.7 mm, represented by simulations evolving in 100 shells, with electrontemperature equilibration [33]. In keeping with the very slow rate of plasmaexpansion observed in the experiment, one sets the initial electron temperatureto 5 K.Figure B.3 shows how this classical arrest state evolves in time. The formationand rapid decay of NO Rydberg molecules signifies an immediate process of three-body recombination, which decreases the charged particle density of the plasma,Predissociation reduces the steady-state density of Rydberg molecules to a valueof nearly zero, but three-body recombination persists as shown by the risingdensity of neutral atom fragments. Eventually, this process slows as the electrontemperature rises. Could this hot-electron ultracold plasma represent the endstate of arrested relaxation? Absolutely not. As detailed in the next section, aplasma with an electron temperature of 60 K would expand to a volume largerthan the experimental chamber in less than 100 µs.140B.4. Ambipolar expansion in a plasma with an ellipsoidal density distribution0102030405060700 5 10 15 20Temperature (K)Time (µs)1 1 2 2 3Time (µs)Electron temperature (K)010203040506070051015200 5 10 15 20Number of particlesTime (µs)x 107Number of particles0520101510 15 20 25 30Time (µs)x 107N (4S) + O (3P)NO+ + e-NO*051015200 5 10 15 20Number of particlesTime (µs)x 107Number of particles 211 0 5 10 15 20Time (µs)x 107N (4S) + O (3P)NO+ + e-NO*0102030405060700 5 10 15 20Temperature (K)Time (µs) Time (µs)Electron temperature (K)12345060700102030405060700 5 10 15 20Temperature (K)Time (µs)1 1 2 2 3Time (µs)Electron temperature (K)010203040506070051015200 5 10 15 20Number of particlesTime (µs)x 107Number of particles0520101510 15 20 25 30Time (µs)x 107N (4S) + O (3P)NO+ + e-NO*051015200 5 10 15 20Number of particlesTime (µs)x 107Number of particles 211 0 5 10 15 20Time (µs)x 107N (4S) + O (3P)NO+ + e-NO*0102030405060700 5 10 15 20Temperature (K)Time (µs) Time (µs)Electron temperature (K)1234506070Figure B.3: Classical evolution of arrest state. (lower) Numbers of ions andelectrons, Rydberg molecules and neutral dissociation products N(4S) and O(3P)as a function of time during the evolution of an ultracold plasma of NO+ ions andelectrons, as predicted by a shell-model coupled rate equation simulation. Here,the initial density distribution of the plasma is represented by a 5σ Gaussianellipsoid with principal axis dimensions, σx = 1.0 mm, σy = 0.55 mm, σz = 0.7mm, peak density of 4× 1010 cm−3 and initial electron temperature, Te(0) = 5 K,as measured for a typical experimental plasma entering the arrest state after amevolution of 10 µs. The simulation proceeds in 100 concentric shells enclosing setnumbers of kinetically coupled particles, linked by a common electron temperaturethat evolves to conserve energy globally. (upper) Global electron temperature asa function of time. 141B.4. Ambipolar expansion in a plasma with an ellipsoidal density distributionB.4 Ambipolar expansion in a plasma with anellipsoidal density distributionThe self-similar expansion of a spherical Gaussian plasma is well-described by ananalytic solution of the so-called Vlasov equations for electrons and ions with self-consistent electric fields. For a distribution of width σ, in the limit of Te Ti,this solution reduces to [237]:e∇φ = kBTeρ−1∇ρ = −kBTe rσ2(B.4)In essence, the thermal pressure of the electron gas produces an electrostatic forcethat radially accelerates the ion density distribution according to the gradientin the electrostatic potential. In approximate terms, the expanding electronstransfer kinetic energy to the ions, accelerating the distribution to an averageballistic velocity,kBTe ≈ mi〈v2i〉(B.5)The velocity varies linearly with radial distance, ∂tr = γr, where γ falls withtime as the distribution expands, and the electron temperature cools accordingto ∂tTe = −2γTe.To model the ellipsoidal plasma, one represent its charge distribution by a setof concentric shells. In this shell model, the density difference from each shell j toshell j + 1 establishes a potential gradient that determines the local electrostaticforce in each principal axis direction, k [238]:emi∇φk,j(t) =∂uk,j(t)∂t=kBTe(t)miρj(t)ρj(t)− ρj+1(t)rk,j(t)− rk,j+1(t) (B.6)where ρj(t) represents the density of ions in shell j.The radial coordinates of each shell evolve according to its instantaneous ve-locity along each axis, uk,j(t).∂rk,j(t)∂t= uk,j(t) = γk,j(t)rk,j(t) (B.7)142B.4. Ambipolar expansion in a plasma with an ellipsoidal density distributionwhich in turn determines shell volume and thus its density, ρj(t). The electrontemperature supplies the thermal energy that drives this ambipolar expansion.Ions accelerate and Te falls according to:3kB2∂Te(t)∂t= − mi∑j Nj∑k,jNjuk,j(t)∂uk,j(t)∂t(B.8)050010001500200025000 5 10 15 20 25 30 35 40σ(t)(µm)Time (µs)Figure B.4: Hydrodynamic expansion of a Gaussian ellipsoid with thedimensions measured at 10 µs for the typical arrested plasma describedabove, modeled by a 100-shell simulation, assuming an electron tem-perature that rises to 60 K, with curves, reading from the bottom onthe left, for σy(t), σz(t) and σx(t). The lower curve with data shows the mea-sured expansion of a typical molecular NO ultracold plasma with a Vlasov fit forTe = 3 K.Figure B.4 compares the ambipolar expansion of an ellipsoidal plasma, simu-lated for an initial volume with the starting dimensions described above and an143B.5. Effective many-body Hamiltonianinitial electron temperature of 60 K, compared with the time evolution of theGaussian width measured in z by experiment. Note that the choice of a largeinitial volume intrinsically slows the simulated expansion. Yet, nevertheless, theelectron heating that arises inevitably from three-body recombination in a classi-cal ultracold plasma demands a rate of expansion that is completely unsupportedby experimental observation.B.5 Effective many-body HamiltonianExperimental observations inform that the molecular ultracold plasma of nitricoxide evolves to a state of arrested relaxation in which extravalent electrons oc-cupy a narrow distribution of weakly bound states. This distribution of statessupports a vast distribution of pair-wise interactions, creating a random poten-tial landscape. Resonant dipole-dipole interactions in this dense manifold of basisstates cause excitation exchange. In the disorder potential, these processes aredominated by low energy-excitations involving L states in number, where I expectL to be small (from 2 to 4). The most probable interactions select L-level systemscomposed of different basis states from dipole to dipole. Thus, the states |e1〉,|e2〉 ... ∣∣eL〉 vary from one dipole to the next and from time to time.Representing excitations by spins, I can write an XY model [200] that describesthese interactions in terms of their effective spin dynamicsHeff =∑iiSˆzi +∑i,jJij(Sˆ+i Sˆ−j +H.c.), (B.9)where Sˆ in each case denotes a spin-L operator defined as Sˆγ = ~σˆγ/2, for whichσγ is the corresponding spin-L Pauli matrix that spans the space of the L activelevels and γ = x, y or z. H.c. refers to Hermitian conjugate.Let me now consider specific examples of this construction.B.5.1 L = 2 caseFigure B.5 diagrams a case that is uniquely defined for every pair of interactingdipoles. In the limit of isolated pairs, this two-level interaction is exactly resonant.144B.5. Effective many-body HamiltonianConditions described below randomly displace these energy level positions.For each particular dipole i, described by states |e1i 〉 and |e2i 〉, let me define aprojection operator for the higher-energy state (which I will call spin-up) σˆe2i =|e2i 〉 〈e2i | = (1 + σˆzi )/2 and the lower-energy state (spin-down) σˆe1i = |e1i 〉 〈e1i | =(1−σˆzi )/2. Thus, I can represent the two levels of a dipole i, with an energy spacingi, by a one-body operator iSˆzi = (~i/2)σˆzi . This defines an energy ±~i/2depending on which state |e2i 〉 or |e1i 〉 is occupied, respectively, i.e. |e2i 〉 ≡ |↑i〉 and|e1i 〉 ≡ |↓i〉.•• ••Si+Sj-i j i j|ei2> |ej2>|ei1> |ej1>N = 2|ej2>|ej1>|ei2>|ei1>Figure B.5: Schematic diagram representing two Rydberg molecules, iand j, dipole coupled in the two-level approximation. In every case,the disorder in the environment of each molecule perturbs the exact energy levelpositions of |ei〉 and |ej〉.The on-site energy is given by i = E12i + Di [149], where E12i is the en-ergy separation between the two states |e1i 〉 and |e2i 〉 evaluated for the localHamiltonian hi. hi varies with the random potential landscape from one dipoleto the next and thus is responsible for the diagonal disorder in the on-siteterm. Di =∑j 6=i〈e2i , e1j∣∣V ddi,j ∣∣e2i , e1j〉 − 〈e1i , e1j ∣∣V ddi,j ∣∣e1i , e1j〉 represents the shift ina dipole’s energy due to dipole-dipole interactions [149]. This term is identicallyzero for parity-conserving states [239].Lowering and raising operators, σˆ−i = |e1i 〉 〈e2i | and its Hermitian conju-gate σˆ+i = |e2i 〉 〈e1i |, define a resonant spin flip-flop between dipoles i andj: Jij(Sˆ+i Sˆ−j + H.c.) = (~Jij/2)(σˆ+i σˆ−j + H.c.) with amplitude Jij = tij/r3ij;tij =〈e2i , e1j∣∣V ddi,j ∣∣e1i , e2j〉. This refers to the dipole-dipole mediated transferof excitation [149] represented by, for example, Sˆ+i Sˆ−j |↓i〉 |↑j〉 = |↑i〉 |↓j〉 i.e.|e2i 〉∣∣e1j〉 Sˆ+i Sˆ−j−−−→ |e1i 〉 ∣∣e2j〉. One can expect this class of matrix element to be non-zero for many of the local eigenstates of hi and hj, as the dipole-dipole operator145B.5. Effective many-body Hamiltoniancouples states of different parity, limited only by a few selection rules [239].Additionally, note that dipole-dipole interactions lead to a two-body Ising termof the form Sˆzi Sˆzj . This term originates from dipole-dipole induced shifts of pairsof dipoles [149] and has an amplitude〈e2i , e2j∣∣V ddi,j ∣∣e2i , e2j〉 + 〈e1i , e1j ∣∣V ddi,j ∣∣e1i , e1j〉.This term is also identically zero for parity conserving states [239]. Since, thearrested phase includes no external parity-breaking fields and neglecting localfield fluctuations, one assumes Di = 0 → i = E12i and no dipole-dipole inducedIsing interaction.B.5.2 L > 2 casesOne can easily imagine systematic coupling schemes that involve three or fourL-level interactions. Excitation transfer still governs the dynamics via terms likeJij(Sˆ+i Sˆ−j + H.c.), where the Sˆ operators live in the active L-dimensional sub-spaces. Figures B.6 and B.7 schematically detail examples of these interactions.••Si+Sj-i j i j|ei2> |ej2>|ei1> |ej1>N = 3••|ei3> |ej3>|ej2>|ej1>|ej3>|ei2>|ei1>|ei3>Figure B.6: Schematic diagram representing two Rydberg molecules, iand j, dipole coupled in the limits of L = 3. In the very high state densityof the quenched ultracold plasma, the displacement of |e2i 〉 and∣∣e2j〉 will lessenthe significance of L = 3 interactions compared with the case of L = 4.Figure B.6 represents an interaction of overwhelming importance in studies ofRydberg quantum optics. Typically, a narrow bandwidth laser excites a resonantpair state, such as 23P3/2 + 23P3/2 ↔ 23s + 24s in Cs [240]. Excitation transferin this L = 3 case operates for example as:Sˆ+i Sˆ−j |Si = −1〉 |Sj = 1〉 = |Si = 0〉 |Sj = 0〉 , (B.10)146B.6. Induced Van der Waals interactionsi.e. |e1i 〉∣∣e3j〉 Sˆ+i Sˆ−j−−−→ |e2i 〉 ∣∣e2j〉.For a gas of Rydberg molecules occupying a dense manifold of disorderedstates, the case of L = 3 becomes an operationally indistinguishable special caseof the more general L = 4 interaction, which maps onto a spin of 3/2.Si+Sj-i j i j|ei2> |ej2>|ei1> |ej1>N = 4••|ei3> |ej3>••|ei4> |ej4>|ej2>|ej1>|ej3>|ej4>|ei2>|ei1>|ei3>|ei4>Figure B.7: Schematic diagram representing two Rydberg molecules, iand j, dipole coupled in the limits of L = 4. The high state density andstrong disorder in the quenched ultracold plasma gives this case of L = 4 greatersignificance than the restrictive limit of L = 3.Here, I represent the interaction as an excitation transfer that operates as:Sˆ+i Sˆ−j |Si = −3/2〉 |Sj = 3/2〉= |Si = −1/2〉 |Sj = 1/2〉 , (B.11)i.e. |e1i 〉∣∣e4j〉 Sˆ+i Sˆ−j−−−→ |e2i 〉 ∣∣e3j〉.I can extend such sequences to higher L, but low-energy resonant dipole-dipoleexcitation exchange in the dense manifold of basis states will most prominentlyinvolve a small number of L-levels per dipole.B.6 Induced Van der Waals interactionsIn the limit |Jij| W most appropriate to the experiment, the effective Hamil-tonian takes the form of a generalized spin model with dipole-dipole and van derWaals interactions:147B.6. Induced Van der Waals interactionsHeff =∑iiSˆzi +∑i,jJij(Sˆ+i Sˆ−j +H.c.) +∑i,jUijSˆzi Sˆzj , (B.12)where Uij = Dij/r6ij and Dij = t2ijJ˜/W2.B.6.1 Non-resonant spin-spin interactionsThe appearance of the term,∑i,j UijSˆzi Sˆzj , underlines the many-body nature ofthis model. One obtains this term by treating Jij as a perturbation in (B.9)[203]. For the L = 2 case, this occurs at the third order, while for all other L,this term appears at the second order [203]. Thus, such a term arises generallyin the |Jij| W limit in three dimensions.The van der Waals interactions occur with an amplitude, Uij ≈ J2ijJ˜/W 2,where J˜ estimates Jij at the average distance separating spins. I do not expectthese interactions to depend strongly on the off-diagonal disorder, as they arisefrom the off-resonant part of∑i,j Jij(Sˆ+i Sˆ−j + H.c.), which presumably does notcause real transitions [203]. Thus, one can rationalize the use of J˜ here as anaverage weighting term. I leave the task of studying the effect of off-diagonaldisorder to future work.B.6.2 Non-resonant on-site interactionsIt is also important to note that this limit gives rise to additional perturbativeprocesses that renormalize the local on-site fields∑i iSˆzi by van der Waals terms[203].Similar considerations from a completely different atomistic perspective verifythat this term is approximately∑l 6=i hCij6 /r6ij where h is the Planck constantand Cij6 denotes the C6 coefficients for the van der Waals interaction between theoff-resonant dipoles i and j [201, 204].The induced on-site terms will also vary randomly owing to the randomnessin the potential landscape. I simply absorb such terms in the definition of i.148B.7. Resonance counting and. . .B.7 Resonance counting and the number ofdipoles in the quenched ultracold plasmaRef. [203] considers the problem of delocalization via resonance counting argu-ments in the model of Eq. (B.12) for the general case of α < β, under conditionsfor which d > dc. Here α refers to the power law that regulates Jij and β refersto Uij. d and dc stand for dimensionality and critical dimensionality. This workconcludes that delocalization occurs at arbitrary disorder given sufficient systemsize.For local disorder, W , and average spin flip-flop amplitude, J˜ , the resonantpair criterion defines, Nc, a critical number of dipoles above which the system de-localizes. Here, I compare this theoretical estimate with an accurate experimentalmeasure of the number of dipoles present in the arrest state of the quenched ul-tracold plasma.Controlled conditions of supersonic expansion precisely define the cylindricaldensity distribution of nitric oxide in the molecular beam [186]. Co-propagatinglaser beams, Gaussian ω1 and ω2, cross orthogonally in the x, y plane to define aGaussian ellipsoidal excitation volume.When ω2 saturates the second step of double resonance, the intensity of ω1controls the peak density of the Rydberg gas volume up to a maximum of 6 ×1012 cm−3, obtained upon saturation of the first step. Density varies from shotto shot, and experimental work have developed an accurate means of classifyingand binning individual SIF traces according to initial Rydberg gas peak density,as displayed in Figure B.1. Coupled rate simulations describing the kinetics ofthe avalanche of Rydberg gas to plasma confirm these estimates of peak density.Two methods of plasma tomography determine the evolution of plasma sizeand relative density distribution as a function of time. In the SFI apparatus, aperpendicular imaging grid that translates in the molecular beam propagationdirection, z, yields an electron signal waveform that gauges the changing plasmadensity and width as a function of evolution time. This waveform, followed to apoint of evident arrest at about 5 µs, and well beyond, as illustrated by Figure 8.1in the main text, establish a case for arrested relaxation.149B.7. Resonance counting and. . .Table B.1: Distribution of ions in an idealized Gaussian ellipsoid shellmodel of a quenched ultracold plasma of NO as it enters the arreststate with a peak density of 4× 1010 cm−3, σx = 1.0 µm, σy = 0.55 µm andσz = 0.70 µm. At this point, the quasi-neutral plasma contains a total of 1.9×108NO+ ions (NO Rydberg molecules). Its average density is 1.4 × 1010 cm−3 andthe mean distance between ions is 3.32 µm.Shell Density Volume Particle Fraction awsNum cm−3 cm3 Number ×100 µm1 4.0× 1010 1.8× 10−6 7.0× 104 0.04 1.812 3.9× 1010 1.1× 10−5 4.4× 105 0.23 1.833 3.7× 1010 9.0× 10−5 3.3× 106 1.75 1.864 3.3× 1010 2.8× 10−4 9.3× 106 4.87 1.935 2.6× 1010 8.3× 10−4 2.2× 107 11.52 2.086 2.1× 1010 1.2× 10−3 2.6× 107 13.40 2.267 1.5× 1010 1.8× 10−3 2.8× 107 14.46 2.498 1.1× 1010 2.4× 10−3 2.6× 107 13.81 2.789 7.4× 109 3.4× 10−3 2.5× 107 13.28 3.1910 4.3× 109 5.1× 10−3 2.2× 107 11.56 3.8111 2.0× 109 8.3× 10−3 1.7× 107 8.85 4.8912 5.6× 108 1.7× 10−2 9.4× 106 4.94 7.5113 7.9× 107 3.1× 10−2 2.4× 106 1.27 14.4714 4.0× 106 5.6× 10−2 2.3× 105 0.12 38.9615 4.4× 104 1.0× 10−1 4.6× 103 0.00 176.22Images projected in the x, y plane together with waveforms in z, recorded afternearly 0.5 ms of flight, detail a slow ballistic expansion in Cartesian coordinatesthat are extrapolated back to an evolution time of 10 µs to determine the absolutedensity distribution of the arrested ultracold plasma, described by the shell modelpresented in Table B.1. This representation neglects the redistribution of chargedensity associated with the initial stages of bifurcation. The total number of ionsrepresented by this distribution remains constant for as long as it can be measuredin the long flight-path instrument, at least a half millisecond.The ion density averaged over shells determines 〈|rij|〉. This average distancebetween dipoles, combined with a a rough upper-limiting estimate of the averagedipole-dipole matrix element, 〈tij〉, based on values computed for a ∆n = 0 Fo¨sterresonant interaction in Li [201], yields an upper-limiting estimate of J˜ .150B.7. Resonance counting and. . .However, interaction with charged particles in the plasma environment per-turbs the electronic structure of individual Rydberg molecules. This diminishesthe probability of finding resonant target states, decreasing the real value of J˜ ,and giving rise to a rarity and randomness of resonant dipole-dipole interactionsdistributed over a huge state space defined by the measured distribution of elec-tron binding energies, W .As noted in Figure B.1, a simple measure of the width of the plasma featurein the delayed SFI spectrum determines W . Table B.2 summarizes this and otherparameters of the arrest state derived from experiment, including the J˜ for Liunder the experimental conditions as an upper limit.For short range interactions in a one-dimensional spin chain, perturbative ar-guments applied to disordered interacting spin models, such as the one above,predict many-body localization [171]. However, in higher dimensions especially,long-range resonant interactions play an important role in defining the conditionsunder which localization can occur. It is generally accepted that interactionsgoverned by a coupling amplitude, Jij that decreases with distance as 1/rβij de-localizes any system at finite temperature for which the dimension, d exceedsβ/2.However, building on ideas introduced by Anderson [24] and Levitov [205],Burin [203] offers a means by which to test a dimensionally constrained system forconditions that favor the onset of delocalization. He uses a perturbation approachthat defines limits over which localization can occur in a system as modeled abovein which delocalization proceeds by the Ising interaction of extended resonantpairs.In this picture, a system that violates the dimension constraint delocalizes foran arbitrary size of disorder whenever the number of dipoles exceeds a criticalnumber, Nc, which is determined by the disorder width, W and the averagecoupling strength, J˜ . Coupling terms in the Hamiltonian defined by Eq. (B.12)scale in r according to α = 3, β = 6 and d = 3. This sets a critical number ofdipoles defined by the quantity, Nc = (W/J˜)4 [203]. For the arrest state definedby the density distribution described by the elliptical shell model in Table B.1, themeasured W taken with the upper limiting estimate for J˜ , yields Nc = 3.6× 109.Considering this value of Nc in relation to the average density of the system151B.7. Resonance counting and. . .Table B.2: Resonance counting parameters in the arrest state of thequenched ultracold plasma. The disorder width W , taken directly from thewidth of the plasma feature in the SIF spectrum, combined with J˜ – derived froma rough upper-limiting estimate of the average dipole-dipole matrix element, 〈tij〉,based on values computed for ∆n = 0 interactions in alkali metals [201], togetherwith the mean distance between NO+ ions in the shell model ellipsoid – determinesNc, a critical number of dipoles required for delocalization. R∗ describes thelength scale for delocalization and τ ∗ denotes the delocalization time, given asufficient number of dipoles at the average density of the experiment. Note thatthe ultracold plasma quenched experimentally contains an order of magnitudefewer than Nc dipoles.W 〈tij〉 〈|rij|〉 J˜ Nc R∗ τ ∗GHz GHz(µm)3 µm GHz µm s500 75 3.3 2.0 3.6× 109 4000 0.85at arrest defines R∗, an effective distance between resonant dipoles at which pointthis occurs [203]. For the conditions described in Table B.1, this coupling wouldoccur in a system large enough to contain 3.6 × 109 dipoles a distance of 4 mmor more. At this distance, the upper-limiting dipole-dipole matrix element wouldpredict a characteristic irreversible transition time, τ ∗ on the order of one second[208].I note that the quenched ultracold plasma formed experimentally relaxes to avolume that contains an order of magnitude fewer dipoles than Nc, as determinedfor this case by the model of Ref. [203].As I attempt to convey above, the experiment yields plasmas of well defineddensity distribution and total number of dipoles. However, the precise natureof the associated quantum states and their dipole-dipole interaction is much lesswell known. This limits the certainty with which we can determine Nc. What ismore perturbation theory in a locator expansion formulation may not accuratelydefine the limiting conditions for MBL in higher dimensions [210].Nevertheless, the quenched plasma seems consistently able to find the condi-tions necessary for arrested relaxation. A great many different avalanche startingconditions, defined by varying initial Rydberg gas density and initial principalquantum number, all evolve to retain comparable internal energy and yield an152B.7. Resonance counting and. . .arrest state with much the same density distribution as that described by theshell model detailed in Table B.1.153
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Peierls bipolarons and localization in solid-state and molecular systems Sous, John 2018
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Title | Peierls bipolarons and localization in solid-state and molecular systems |
Creator |
Sous, John |
Publisher | University of British Columbia |
Date Issued | 2018 |
Description | In this thesis, I investigate the behavior of particles dressed by quantum field excitations and random interactions. First I consider two-carrier states in the Peierls model describing the modulation of the particle hopping due to lattice distortions. I compute the spectral response using the Momentum Average approximation. Combining accurate numerical techniques and analytical arguments, I provide a complete picture of the Peierls bipolarons. It is found that polarons bind into strongly bound yet light bipolarons in the singlet sector, even at large values of the electron-phonon coupling strength. At finite electron fillings, these bipolarons may condense into a high-Tc superconductor. On the other hand, phonons mediate a repulsive interaction in the triplet sector, or equivalently (in one dimension), between two hard-core particles, in which case the ground-state dimers bound by sufficiently attractive bare interactions exhibit two sharp transitions, one of which is the first known example of a self-trapping transition at the two-carrier level. In both situations, phonons mediate pair-hopping effective interactions between the carriers. I further study some aspects of the excited spectrum for the two hard-core particles, a situation relevant to ultracold quantum simulators. It is found that the repulsive phonon-mediated interaction binds a repulsive bipolaron embedded in the excited spectrum. I then turn to the study of quenched randomness in an ultracold molecular plasma. I argue that the quenched ultracold plasma presents an experimental platform for studying quantum many-body physics of disordered systems in the long-time and finite energy-density limits. I analyze an experiment that quenches a plasma of nitric oxide to an ultracold system of Rydberg molecules, ions and electrons that exhibits a long-lived state of arrested relaxation. The qualitative features of this state fail to conform with classical models. I develop a microscopic quantum description for the arrested phase based on an effective many-body spin Hamiltonian that includes both dipole-dipole and van der Waals interactions. This effective model appears to offer a way to envision the essential quantum disordered non-equilibrium physics of this system. This thesis thus examines the quantum many-body response in interacting systems coupled to bosonic fields or in disordered environments. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2019-01-04 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0375874 |
URI | http://hdl.handle.net/2429/68177 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2019-02 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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