Numerical Studies on Correlations inDynamics and Localization ofTwo Interacting Particles in LatticesbyTirthaprasad ChattarajM. Sc., INDIAN INSTITUTE OF TECHNOLOGY KANPUR, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(CHEMISTRY)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2018c© Tirthaprasad Chattaraj, 2018The following individuals certify that they have read, and recommended to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, a dissertation entitled:Numerical Studies on Correlations inDynamics and Localization of Two Interacting Particles in Latticessubmitted in partial fulfillment of the requirement forby Tirthaprasad Chattarajthedegree of Doctor of Philosophyin ChemistryExamining Committee:Professor Roman V. Krems, ChemistrySupervisorProfessor Grenfell N. Patey, ChemistrySupervisory Committee MemberProfessor Mark Thachuk, ChemistrySupervisory Committee MemberProfessor Yan Alexander Wang, ChemistryUniversity ExaminerProfessor Andrew MacFarlane, ChemistryUniversity ExamineriiAbstractTwo interacting particles in lattices, in the absence of dissipation, can not distin-guish between attractive or repulsive interaction when the range of their tunnellingis limited to nearest neighbor sites. However, we find that, in the case of long-rangetunnelling, the particles exhibit different dynamics for different types of interactionsof the same strength. The nature of dynamical correlations between particles alsobecomes significantly different. For weak interactions, particles develop a characterin correlation which is in between that of antiwalking and cowalking when the tun-nelling is long-range. For strong interactions, particles cowalk independently of theirstatistics. A few recent experiments have demonstrated such effects of interactionson quantum walk of photons, atoms and spin excitations on various lattice platforms.In disordered lattices the effect of coherent backscattering makes particles localizeto their initial position. We find that a weak repulsive interaction reduces localizationand a strong interaction enhances localization. We also calculate the correlationsbetween the particles in the disordered 1D and 2D systems. The effect of long-rangetunnelling on localization of particles in disordered 1D systems has been explored.For large ordered or disordered lattices, computation of localization parametersbecomes difficult. In these cases, an efficient recursive algorithm is used to calcu-late Green’s functions exactly. We extend such algorithm to disordered systems inboth one and two dimensions. We also illustrate that this recursive algorithm mapsdirectly to some graph structures like binary trees. We perform calculations forquantum walk of interacting particles on such graphs. The method is also used tocalculate the properties of interacting particles on lattices with gauge fields. Fordisordered 2D lattices, we introduce and test approximations which produce accu-rate results and make the calculations more efficient. We examine the localizationparameters for a broad range of interaction and disorder strengths and try to finddifferences among parameters within the range.iiiLay SummaryWhile many of us tend to think of owning a rule after finding one, it is the naturethat rules. From mathematics to sociology, some of us only are fortunate or workedhard to see these rules first. - folkloreIn nature there are two types of particles: bosons and fermions. The bosonshave unique behaviour of togetherness while the fermions want to keep a distancebetween them. Such particles in lattices can be described as hopping from one siteto other sites and when more than one particles occupy a site then they interactwith their hopping modified. Depending on different interaction strengths, theirbehaviour might be totally different in both ordered and disordered lattices. For twosuch interacting particles, which is the focus of this thesis, range of hopping affectstheir dynamics differently for attractive or repulsive interactions. In finite disorderedlattices, both of one and two dimensions, these particles get localized. This thesisillustrates how the correlations between the particles in disordered systems changedepending on the interaction strength.ivPrefaceThe work described in this thesis has been published or in preparation for publi-cation as mentioned in the following.Publications on thesis work:Chattaraj, T. and Krems, R. V. (August 1, 2016), Effects of long-range hopping andinteractions on quantum walks in ordered and disordered lattices, Phys. Rev. A 94:023601arxiv.org/pdf/1605.04349.pdfChattaraj, T. (2018), Recursive computation of Greens functions for interactingparticles in disordered lattices and binary trees.arxiv.org/pdf/1808.04898.pdfChattaraj, T. (2018), Localization parameters for two interacting particles in disor-dered two-dimensional lattices.arxiv.org/pdf/1808.06141.pdfChattaraj, T. (2018), Spectral weights of doublon in interacting Hofstadter model.arxiv.org/pdf/1808.10112.pdfvTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Engineering range of coupling in lattices . . . . . . . . . . . . . . . . 82 Correlations in Dynamics of interacting particles . . . . . . . . . . 122.1 Two particle systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Two particle states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Role of interaction and bound state . . . . . . . . . . . . . . . . . . . 182.4 Recent experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Effects of long-range hopping and interaction . . . . . . . . . . . . . . 222.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Green’s Functions of Interacting Particles . . . . . . . . . . . . . . 303.1 Method of recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Two interacting particles in 1D . . . . . . . . . . . . . . . . . . . . . 353.3 Two interacting particles in 2D . . . . . . . . . . . . . . . . . . . . . 373.4 Two interacting particles in binary tree . . . . . . . . . . . . . . . . . 483.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 Quantum Localization of Interacting Particles . . . . . . . . . . . . 504.1 Scattering with single impurities in 1D . . . . . . . . . . . . . . . . . 51vi4.2 Localization in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Localization in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A Few Operator Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 82B s-wave scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85C Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90D Interaction via phonons . . . . . . . . . . . . . . . . . . . . . . . . . 97E Coupling two states coherently . . . . . . . . . . . . . . . . . . . . . 102F Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106G Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108H On Numerical Computations . . . . . . . . . . . . . . . . . . . . . . 114viiList of Figures1.1 Mo¨lmer-Sorensen scheme . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Correlations in two (in)distinguishable bosonic quantum walks . . . . 152.2 Density distributions in two (in)distinguishable bosonic quantum walk 162.3 Lattice spectrum for non-interacting tight-binding model . . . . . . . 202.4 Density of states in 1D tight-binding model . . . . . . . . . . . . . . 212.5 Correlation dynamics of two hardcore bosons . . . . . . . . . . . . . . 252.6 Correlation dynamics of two hardcore bosons . . . . . . . . . . . . . . 262.7 Lattice spectrum for non-interacting coulombic hopping . . . . . . . . 272.8 Lattice spectrum for non-interacting dipolar hopping . . . . . . . . . 282.9 Superfluid - Mott insulator phase transition . . . . . . . . . . . . . . 293.1 Spectral weight of bound state in 1D . . . . . . . . . . . . . . . . . . 363.2 Correlation dynamics calculated from Green’s functions . . . . . . . . 373.3 Recursive calculations of 2D Green’s functions . . . . . . . . . . . . . 383.4 Two particle dynamics in 2D . . . . . . . . . . . . . . . . . . . . . . . 393.5 Scaling of size of calculation with approximation in 2D . . . . . . . . 403.6 Size of the vectors with approximation in 2D . . . . . . . . . . . . . . 413.7 Size of largest vector with approximation in 2D . . . . . . . . . . . . 413.8 Scaling of IPR with lattice sizes in 2D . . . . . . . . . . . . . . . . . 433.9 Calculation of IPR with approximations . . . . . . . . . . . . . . . . 443.10 Spectral weight of bound state in 2D . . . . . . . . . . . . . . . . . . 453.11 Terms in Hofstadter model . . . . . . . . . . . . . . . . . . . . . . . . 463.12 Spectral weights of doublon in Hofstadter model . . . . . . . . . . . . 473.13 Recursion scheme for binary tree . . . . . . . . . . . . . . . . . . . . 483.14 Spectral weight of bound state in binary tree . . . . . . . . . . . . . . 494.1 Two particle scattering in 1D with impurity . . . . . . . . . . . . . . 514.2 Tunneling of composite particles through impurity . . . . . . . . . . . 534.3 Localization of interacting particles in 1D . . . . . . . . . . . . . . . . 554.4 Correlations of localized interacting particles in 1D . . . . . . . . . . 56viii4.5 Recursive calculation size for a large 2D lattice . . . . . . . . . . . . . 574.6 Density distribution in disordered 2D lattice . . . . . . . . . . . . . . 594.7 Comparison between densities with approximations . . . . . . . . . . 594.9 3D disorder-interaction diagram of IPR . . . . . . . . . . . . . . . . . 614.8 Disorder-interaction diagram of IPR . . . . . . . . . . . . . . . . . . . 624.10 IPR vs interaction for fixed disorders . . . . . . . . . . . . . . . . . . 634.11 Correlations in disordered 2D lattics . . . . . . . . . . . . . . . . . . 644.12 Density distribution of interacting particles in disordered 2D lattice . 65B.1 s-wave scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89C.1 Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95ixAcknowledgementsI acknowledge the support that I have received from my supervisor since thebeginning of my doctoral study. I acknowledge those persons who have worked tire-lessly for many many years to build the computational facilities that we enjoy today.Without such computational power, this work would be impossible to accomplish.I acknowledge my friends in lab whose presence and active participation made mywork enjoyable and insightful. Finally I acknowledge my family who have alwaysstayed on my side throughout the entire journey.xamar ma kexi1IntroductionThe interference of quantum objects has been found to give rise to many phenomenathat cannot be understood classically. One of such phenomena was discovered duringmiddle of the last century by R. Hanbury-Brown and R. Twiss [1] who observed thatdetection of two photons by two detectors was correlated. In a similar experimentfrom the 1980s performed by C. Hong, Z. Ou and L. Mandel [2] it was found thattwo identical photons, when interfered and guided toward two separate detectors,tend to appear together. Such correlations are now understood to be caused byfundamental statistics of particles. While bosons show bunching behaviour in theircorrelations, fermions anti-bunch. In the presence of interactions between particles,however, these effects are known to become significantly different. Today one canperform similar experiments not only with photons but also with atoms as well aswith electronic or spin excitations. This has been made possible by the experimentson trapping atoms in external fields [3].In the 1990s it was proposed that random walk of quantum particles [4] in lat-tices can be used for quantum information purposes [5, 6], which inspired numerousexperiments [7, 8, 9, 10, 11, 12] studying wavepacket dynamics in various atomic andoptical systems. Algorithms for fast spatial search [13, 14] are particularly promising.However, in these algorithms one has to optimize the search speed and search proba-bility on a graph. Ballistic propagation of wavepackets on lattices, when applied forsearch [15] in databases, has promised a speedup of√N (N = database size), overclassical search algorithms. Further generalization to multiparticle quantum walk[16] has been shown to be effective not only for quantum information transfer butalso for understanding isomorphism of graphs [17, 18, 19]. Quantum walk in the pres-ence of an impurity has been proposed for the preparation of entangled states [20].The effect of quantum interference on transport of excitations has been shown to bepresent even in biological systems such as photosynthetic light harvesting complexes1[21, 22].In the case of more than one particles, quantum statistics affect quantum walkin lattices [23, 24]. For bosonic and fermionic particles, the nature of multiparticlequantum walk is very different. Bosons exhibit bunching correlations while fermionsand hard-core bosons show anti-bunching correlations in the absence of any inter-action. These correlations can be used to determine the character of particles fromthe studies of their quantum walk when no such information is available otherwise.However, to study such phenomena one needs the most advanced atomic and opti-cal systems where not only single particle resolution [25, 26] has been achieved buttwo-particle correlations [27, 28, 29, 30] can also be experimentally measured.Photons have been at the forefront of understanding the effects of statistics andquantum interference for a long time. Lahini et al. [31, 32] performed experimentson both one particle and two particle photonic quantum walk on waveguide lattices.Recently, Greiner et al. have shown that such experiments can also be performedwith atoms [30] in optical lattices. Bloch et al. have implemented such schemes forspin excitations [29] in optical lattices. Quantum walks in disordered systems havealso been of interest since the work of Anderson [33] explaining the role of disorderin low-dimensional systems leading to exponential localization of non-interactingparticles. All these effects can be studied in optical lattice systems within a rangeof experimentally accessible parameters. Anderson localization has been recentlyillustrated experimentally for 87Rb atoms in optical speckle lattices [34]. In photonicwave guides, it was found that even in disordered systems, localized photons stillremain correlated [32].For more than one particle, the interaction between particles affects both thedynamics and localization. A huge amount of study has been done since the 1980sto understand these effects just for two particles [35, 36, 37, 38, 39, 40]. The two-particle interactions were shown to reduce localization in disordered lattices. A fewof them predicted that interactions may enhance localization [37, 38]. These aresome of the fundamentally important investigations which can only now be exploredand understood. Our work makes an effort to understand these effects not only inone dimensional systems but also in two dimensional systems. The effect of range oftunnelling on localization of interacting particles in 1D systems has been calculatedin this thesis. The effects of interactions on two particle correlations in disordered1D and 2D lattices have also been explored in this thesis.In the case of atoms in optical lattices [41], tunnelling beyond nearest neighborsite is not significant because of the length scales of these lattices. However, fordipolar molecules, tunnelling of excitations to sites far apart can be observed. Thesetunnelling parameters can also be tuned by an external field [42]. Preparation of such2molecules in optical lattices has been achieved [43, 44, 45, 46] very recently. However,the experiments to achieve a higher filling fraction still remain to be developed.Dipolar molecules in optical lattices are expected [47] to have various separate phases.Quantum random walk using Rydberg atoms has also been proposed [48] for long-range tunnelling models.In theoretical investigations at the most simple and fundamental level, the Hub-bard model [49] plays a central role. It is very useful for understanding the effectsof interaction between particles and their dynamics. The extended Hubbard model,where particles interact and tunnel in lattices beyond their nearest neighbors, hasalso become a highly investigated research topic since a few years ago. In this thesis,I try to understand the interplay of such long-range interactions and tunnelings inboth ordered and disordered lattices. One of the most interesting findings of thisthesis is the effect of such long-range tunnelling on dynamical correlations of twoparticles in 1D lattices.There has also been a lot of interest recently in simulating particles on 2D latticesunder synthetic gauge fields. In the case of atoms, these fields are created by a peri-odic shaking of lattice potentials [50, 51]. We attempt to understand the implicationsof such gauge fields on quantum walk of interacting particles.The methods that have been used for the calculations presented in this thesis aremostly based on full diagonalization of hamiltonians and recursive calculations ofGreen’s functions. The recursion method used in this thesis is an extension of pre-vious work in our group [52]. This method makes the calculations significantly moreefficient compared to full diagonalization and also allows for performing calculationswith a much larger basis size. We introduce new boundary conditions that makethe method exact and approximations that make it more efficient while maintainingaccuracy. The underlined mathematics of the method has been also elucidated incalculations for interacting particles on graphs such as binary trees in this thesis.There are many experimental systems relevant for the research presented here.Our research is most relevant for but not limited to cold atom [53] and trapped-ion [54] systems. (See Appendix F for a brief introduction to optical lattices.) Theresults are applicable wherever the model systems can be mapped to the approximatephysics of the systems under investigation. Two particle correlations, in essence,describe the fundamental physics of many interacting particles. For the cold atomsystems, manipulation of interactions between particles has been pursued from the1980s [55, 56, 57]. See Appendices B and C for the discussion of such controls. Therange of tunnelling in lattices can be controlled within a broad range of tunabilityusing the ideas of Mo¨lmer and Sorensen [58]. Section 1.1 describes how such long-range tunnelling of excitations can be achieved in lattices. More details can be found3in Appendix E. Phonons also play an important part in controlling interactionsbetween electrons and excitations, as described in Appendix D. Excitonic systems(see Appendix G) can also exhibit the interactions, for which the fundamental physicsis expected to be very similar to that which has been described in this thesis.1.1 Thesis overviewThis thesis deals mainly with two aspects of two interacting particles. One is long-range hopping of particles in lattices and the interplay of such hopping with the effectof interactions between the particles. Section 1.3 describes how long-range hoppingcan be engineered in most advanced atomic and optical systems. The other is thebehaviour of interacting particles in the presence of impurities. The model that isused in this thesis to understand the effects of interactions between the particles inboth ordered and disordered lattices is mainly an extension of the Hubbard model.The origin of the terms in the model is explained in Section 1.2.The chapters are organized to describe the main results of the work that has beenperformed over the years. Chapter 2 describes the effects of long-range hopping ofparticles in lattices in presence of the interplay with interactions between the par-ticles. Chapter 3 presents a numerical approach to extend the size of calculationsfor bigger lattices. Chapter 4 contains results that have been calculated from thedynamics of two interacting particles in disordered systems. The effects of the in-teraction on localization of particles in finite disordered lattices is the focus of thatchapter. Finally Chapter 5 summarizes the conclusions that can be drawn from thework of the whole thesis.41.2 Hubbard modelIn this section we introduce the notation that will be used extensively throughoutthe thesis. For particles in lattices, one can describe them as hopping from site tosite in the lattice and interacting with each other. This simple physical model is notonly intuitive but also provides the basic understanding for other models.The Hubbard model makes the notations more simplified than in 1st quantizedform. Although it is very easy to write down, it is very difficult to solve for morethan one particle. The model has a nearest neighbor hopping term and onsite ener-gies. In the presence of interactions, most effective models add the onsite two-bodyinteraction term. Terms that describe tunnelling and interactions beyond nearestneighbors are also added in many models.The starting hamiltonian for particles on lattices consists of the kinetic energyterm and various potential energy terms deriving from electron-electron interaction,electron-ion interaction and ion-ion interactionsH = Hkin + Ve−ion + Ve−e + Vion−ion (1.1)where the first two terms are the one-particle termsH0 = Hkin + Ve−ion =Ne∑i=1[p2i2m+N∑n=1V (ri,Rn)], (1.2)where i is the index for the electrons and n for the nuclei.For a basis one can start with the atomic wavefunctions localized on each siteHatom|ψ〉 = Eψ|ψ〉, where 〈ψ′|ψ〉 = δψψ′ . (1.3)One can take these solutions as those for the electron belonging to the nucleus atsite nHnatom|ψn〉 = Eψn|ψn〉, where 〈ψ′n′|ψn〉 = δψψ′δn′n, (1.4)where we have applied the tight binding approximation for the overlap integral.One can also conveniently write these terms using creation (annihilation) opera-tors|ψn〉 = a†ψn|0〉 (1.5)5where the operators follow (anti)commutation relations as described in Appendix Afor (fermions)bosons [aψ′n′ , a†ψn]±= δψψ′δn′n[a†ψ′n′ , a†ψn]±= 0[aψ′n′ , aψn]± = 0 (1.6)The one-particle hamiltonian term then becomesH0 =∑ψ′n′,ψnEψ′n′,ψna†ψ′n′aψn (1.7)whereEψ′n′,ψn =∫dr 〈ψ′n′|r〉〈r|H0|r〉〈r|ψn〉=∫dr 〈ψ′n′|r〉[p22m+N∑n′′=1V (r,Rn′′)]〈r|ψn〉=∫dr φψ′n′(r)[Eψn +∑n′′=nV (r,Rn′′)]φψn(r)=[Eψδψψ′δn,n′ +∫dr φψ(r)∑n′′=nV (r,Rn′′) φψ(r)]+∫dr φψ′(r)∑n′′=nV (r,Rn′′ −Rn) φψ(r)Eψ′n′,ψn =[E˜ψδψψ′δn,n′]+Wψ′,ψδn,n′ (1.8)The one-particle hamiltonian can then be written asH0 =∑ψE˜ψa†ψaψ +∑ψ 6=ψ′Wψ′,ψa†ψ′aψ, (1.9)Here the site indices are removed for simplicity. The terms on the right hand sideof Eq. 1.9 are the energies of the states and the excitation gap between the stateson each site. For ideal two level systems one can disregard these terms as theycontribute to a constant term. It also turns out that the divergent terms (for thecase of Coulomb interactions) in Vion−ion and Ve−ion cancel each other so that we get6a stable system. The two body part of Ve−e can now be written as an addition ofmultiple partsVe−e = V00 + V11 + V ′01 + V′′01 ψ ∈ {0, 1}, (1.10)whereV00 =12∑n6=n′V {}a†0na†0n′a0n′a0nV11 =12∑n6=n′V {}a†1na†1n′a1n′a1nV ′01 =12∑n,n′V {}a†1na†0n′a0n′a1nV ′′01 =12∑n,n′V {}a†1na†0n′a1n′a0n. (1.11)Here V {} contains all the relevant indices for the interaction terms. The first threeterms here are also the interaction energies between the states of the same anddifferent energy levels located at different sites. The last term is responsible fortransfer of the states between sites. Writing an excitation (or quasiparticle) as q†n =a†1na0n, one can findV ′′01 =∑n12V {}q†nqn +∑n6=n′12V {}q†nqn′ . (1.12)We can simplify all these forms by writing the final Hubbard hamiltonian con-sisting of the onsite excitation energy term and the inter-site hopping terms limitedto nearest neighbors only:HHubbard =∑nεnq†nqn +∑〈nn′〉tnn′q†nqn′ . (1.13)The rest of the thesis builds on this form with interaction terms such as V11added and calculates properties of two particles. The simplification of the physicalsystem to such a model after the elimination of many details makes the calculationssignificantly easier to implement.71.3 Engineering range of coupling in latticesIn most physical systems of interest either nearest neighbor hopping or hoppingextended to few nearest neighbors are observed. However, using modern opticalmethods, it is possible to engineer the hopping ranges. In this section we discussa method where phonons can be used effectively to control the coupling betweenparticles at different sites of a chain.Following from appendix E, the effective hamiltonian after including the spatialvariance of the optical field leads to the following equation:H = −Ω2[eı(δt−k·r)σ− + e−ı(δt−k·r)σ+]. (1.14)In the presence of phonon modes at some frequency ωp, we can write the position ofthe mode with the time dependency of field operators included as following:X(t) = X0(ae−ıωpt + a†eıωpt), (1.15)where X0 = 1/√2Mωp (M is the mass of atoms). One can make δ equal to thatof the phonon frequency ωp. When δ > 0, it can excite the phonon mode by onequantum.Writing the spatial dependence of the laser in phonon modes of certain frequencyωp, we find the following, assuming the momentum of the laser mode along themotional mode:H = −Ω2[eı(δt−kX)σ− + e−ı(δt−kX)σ+]= −Ω2[eı(ωpt−kX0(ae−ıωpt+a†eıωpt))σ− + e−ı(ωpt−kX0(ae−ıωpt+a†eıωpt))σ+]= −Ω2[[1− ıkX0(ae−ıωpt + a†eıωpt)]eıωptσ− + [1− ıkX0(aeıωpt + a†e−ıωpt)]e−ıωptσ+],H+ ' ıkX0Ω2[aσ− + a†σ+]+ · · · (1.16)where in the last step the rotating wave approximation has been applied. Similarly,for the negative detuning (δ < 0), one can findH− ' ıkX0Ω2[a†σ− + aσ+]. (1.17)8These methods are very effective in cooling down the vibrational modes to itsground states. One can raise the electronic states higher while going down in phononnumbers using pi−pulses, then decouple from phonon states while returning to theground electronic state and repeat the processes.In the case of two particles at sites i and j in a lattice, phonon modes can beused to effectively couple (Jij) them irrespective of the range of distance between theparticles.|ggn >|een >|egn− 1 >|egn >|egn+ 1 >|gen− 1 >|gen >|gen+ 1 >ω1ω1ω2ω2|ggn− 1 >|ggn >|ggn+ 1 >|een+ 1 >|een >|een− 1 >|ggn > |een >ω1ω1ω2ω2Figure 1.1: The Mo¨lmer-Sorensen scheme. Left: both spins excited. Right: spinsexchange excitations with effectively the same coupling parameter as in the left.Adapted from Reference [58].The scheme is known after Mo¨lmer-Sorensen [58]. The laser frequency can bedetuned at the Mo¨lmer-Sorensen detuning µ = δ − ωp. Now, both types of laserdetuning can be applied to a system of two particles (δ < 0 and δ > 0). When bothparticles are in the ground state with n phonons in state |g, g, n〉, any of them canabsorb the negatively detuned photon and undergo the transition to the excited statewhile the phonon number goes to n − 1. Thus the state |g, e, n − 1〉 or |e, g, n − 1〉is reached in this process. Alternatively, any state can absorb a positively detunedphoton and go to the |g, e, n+1〉 or |e, g, n+1〉 state. Both of these states can absorb9just the oppositely detuned photon than the first time to reach the |e, e, n〉 state.The intermediate states can be made negligible in the whole transition by similarprocedures as described in Appendix E, where we virtually made the excited statecontributing little in the dynamics of two states but now with the Mo¨lmer-Sorensendetuning (δ < ωp) and the coupling g replaced by ηΩ√n. Here η = kX0, and the√n(√n+ 1) factor comes from the phonon annihilation (creation) operation. Theamplitude of the transition |g, g, n〉 → |g, e, n− 1〉 → |e, e, n〉 isΩ21−+ = −ηΩ√nηΩ√n4µ. (1.18)The amplitude of the transition |g, g, n〉 → |g, e, n+ 1〉 → |e, e, n〉 isΩ21+− =ηΩ√n+ 1ηΩ√n+ 14µ. (1.19)So the amplitude for the transition through exciting the first particle without anysignificant transition into intermediate states with different phonon numbers is givenbyΩ21 = Ω21−+ + Ω21+− =(ηΩ)24µ. (1.20)The contribution from the other two paths, where the first particle changes thestate first |g, g, n〉 → |e, g, n − 1〉 → |e, e, n〉 and |g, g, n〉 → |e, g, n + 1〉 → |e, e, n〉,adds to total amplitudeΩ12 =(ηΩ)24µ, (1.21)Ωtot = Ω12 + Ω21 =(ηΩ)22µ. (1.22)Additional detuning (with respect to the phonon frequency) of the photon (withrespect to the energy gap of two states of the particles) µ = δ − ωp and µ = ωp + δgives the full coupling between the two particles10Jij =(ηΩ)22(δ − ωp) −(ηΩ)22(δ + ωp)=(ηΩ)2ωpδ2 − ω2p. (1.23)This method has recently been used (tuning δ at different sites) to engineercoupling between spins in a 1D chain with variable ranges, where one can achievea regular power law form of the coupling with respect to the distance between thespins [59].Jij =J|i− j|α with 0 ≤ α ≤ 3. (1.24)In Chapter 2 of this thesis, we will consider hamiltonians for two particles with suchlong-range hopping in lattices.112Correlations in Dynamics ofinteracting particlesIn this chapter we mainly discuss the effects of long-range hopping on correlations oftwo interacting particles. The statistics of particles play a crucial role in determiningcorrelations between the particles. This role of statistics is fundamental and has beendescribed in the introductory quantum mechanics books [60]. The bunching of bosonsand anti-bunching of fermions has been known to be a result of their fundamentalstatistics. However the role of interaction in determining the dynamics has beenstudied only recently [31, 32, 61]. A few recent experiments have explored theseeffects with photonic wave guides, trapped ion and trapped atom systems. Thepresence of repulsively bound pairs has been observed in cold atomic systems in theabsence of dissipation of energy [61]. Such systems for two particles can be modelledeffectively by the Hubbard hamiltonian with a conserved number of particles andtotal energy. The similarity between the attractive and repulsive interactions is alsowell known for these models with nearest neighbor hopping. In the case of long-range hopping, however, an asymmetry in the effect of the attractive and repulsiveinteractions is observed in our study. It is described in the later part of this chapter.In the following section we describe a few important results that were obtainedfrom exact diagonalization of the full hamiltonian of a 1D system of two particles.These studies were motivated by the experimental and theoretical studies of quantumwalk on lattices [31, 32, 61] and the studies on the effect of the long-range hoppingon eigenstates of the particles in 1D lattices [62, 63, 64]. The existence of the boundpairs in the presence of both repulsive and attractive interactions was established bythese studies. However, the effect of long-range nature of hopping on such boundpairs was not fully understood. Our calculations try to elucidate this effect.122.1 Two particle systemsThe case of two distinguishable particles can be described by the composite wave-function of the two particlesψ(x1, x2) = ψ1(x1)ψ2(x2). (2.1)However, if the particles are indistinguishable, the wavefunctions have to be sym-metrized (anti-symmetrized) for the bosonic (fermionic) particlesψ(x1, x2)± =1√2[ψ1(x1)ψ2(x2)± ψ2(x1)ψ1(x2)] . (2.2)The effect of this symmetrization (anti-symmetrization) can be observed in the ex-pectation value of the square of the relative distance∆ = 〈(x1 − x2)2〉 = 〈x21〉+ 〈x22〉 − 2〈x1x2〉, (2.3)which for the distinguishable particles is∆ = 〈x21〉+ 〈x22〉 − 2〈x1〉〈x2〉, (2.4)and for the indistinguishable particles is∆ = 〈x21〉+ 〈x22〉 − 2〈x1〉〈x2〉 ∓ 2|〈x12〉|2, (2.5)where x12 is an interference term [60]. This interference effect (due to the fundamen-tal statistics) makes two bosons bunch together, while it results in anti-bunching fortwo fermions.This effect of distinguishability can be easily seen when two particle dynamics issimulated in an ideal 1D lattice with nearest neighbor hopping. One can simulatesuch dynamics under the effect of the Hubbard hamiltonian with onsite interactionsfor two bosons. There have been such studies of distinguishability with photons [65].The hamiltonian of two bosonic particles in ideal lattices can be written as thefollowing simplified formH =∑〈nm〉ta†nam +∑nUa†na†nanan (2.6)where n and m are the lattice site indices, t is the hopping amplitude between twosites and U is the onsite interaction energy.13The joint density distribution (%n,m) then can be calculated from eigenfunctions(|λ〉) and eigenenergies (Eλ) of the hamiltonian%n,m(τ) = |〈n,m|∑λe−iEλτ |λ〉〈λ|n′,m′〉|2. (2.7)The initially occupied sites are denoted as n′,m′ and the evolution time as τ . Thisjoint density describes the correlation between the two particlesCn,m(τ) = %n,m(τ) = 〈a†na†maman〉(τ). (2.8)One can also define the correlations as in Eq. 2.9Cn,m = 〈a†na†maman〉 − 〈a†nan〉〈a†mam〉. (2.9)However, our interest is in comparing the effects of interactions and the last termin previous equation is independent of any interactions. This term only act as someconstant additive which can be neglected for further simplification. The total densitydistribution can be calculated from the joint density distribution asρ(n, τ) =12∑m6=n%(n,m, τ) = 〈a†nan〉(τ). (2.10)Figure 2.1 shows a simulation for quantum walk of two distinguishable and in-distinguishable bosons on an ideal lattice in the presence of the interaction (U = 2).The effect can be clearly seen in terms of the correlation elements which include fourcreation and annihilation operators and also in the density terms which include onlytwo creation or annihilation operators. The correlations which describe the jointprobablities of finding two particles on the same site or nearest neighbor sites canbe termed as cowalking correlations which describe the effect of bunching. The jointdensities which describe the particles moving in the opposite direction are termed asantiwalking correlations and describe anti-bunching.As can be seen from the correlations of two particles in Fig. 2.1, the bosonicparticles tend to bunch together and cowalk. However, fermions and hardcore bosonstend to anti-walk as we will see in later sections. When the particles start thequantum walk from adjacent sites, the correlation dynamics is different as for theindistinguishable particles the correlations are symmetric. However, when they startfrom the same lattice site, the distinguishability has no effect and no difference inquantum walk can be observed.14Figure 2.1: Joint densities (Eq. 2.7) in quantum walk for distinguishable bosons(rows 1 and 3)) and indistinguishable bosons (rows 2 and 4). U/t = 2 for all cases.Time = 1/t, 2/t and 4/t respectively for columns 1, 2 and 3. Particles start fromadjacent sites in cases of rows 1 and 2, while from same site in cases of rows 3 and4. Two axes are site indices for two particles. Color scheme- red, yellow, green, blueshow lower joint density in that order. 15This difference due to the distinguishability can be observed also from the simu-lated density terms on a lattice. From Fig. 2.2 it can be seen that the bunching ofindistinguishable bosons (dashed lines) tends to interfere constructively in betweenthe dynamical wavepacket peaks compared to the case of the distinguishable ones(solid lines). For strongly interacting particles, this difference in densities is expectedto become small as they form a bound state which will behave very similar to a singlecomposite particle for both cases.5 10 15 20 25 n00.10.2ρnτ = 1τ = 2τ = 3τ = 4Figure 2.2: Density distributions (Eq. 4.6) in quantum walk of two distinguishableand indistinguishable bosonic particles. The solid lines correspond to distinguishablebosons while dashed lines to indistinguishable ones. Time τ is measured in the unitsof the inverse of the hopping integral t. The interaction U = 2.2.2 Two particle statesFor a Hamiltonian 2.11 with onsite interaction term, the two particle states and en-ergies can be analytically derived following the description of Valiente and Petrosyan[66]. The theoretical description can also be followed from the discussion of Hecker16Denschlag and Daley [67] or Piil and Molmer [68].H =∑mt(a†m+1am + a†mam+1)+∑mU2a†ma†mamam. (2.11)In absence of interaction, this Hamiltonian moves a single particle from positionstate |xm〉 to |xm±1〉 and the wavefunction and energy can be obtained from thefollowing single particle Schrodinger equation 2.12t [ψ(xm−1 + ψ(xm+1)] = E(1)ψ(xm), (2.12)where ψ(xm) is coefficient for position state |xm〉 in the full wavefunction. Taking aplane wave solution ψq(xm) = exp(ıqm), provides the energyE(1)q = 2t cos q. (2.13)In presence of interaction, the two particle Schrodinger equation takes the follow-ing formt [ψ(xm−1, ym′) + ψ(xm+1, ym′) + ψ(xm, ym′+1) + ψ(xm, ym′−1)]+Uψ(xm, ym′)δm,m′ = E(2)ψ(xm, ym′), (2.14)where xm and ym′ are the coordinates of two particles at sites m and m′ respectively.This equation can be simplified in terms of centre of mass R = 12(x+ y) and relativer = (x− y) coordinates. The wavefunction in momentum basis then becomeψ(x, y) = eıKRψK(r), (2.15)and Eq. 2.14 simplifies totK [ψK(m− 1) + ψK(m+ 1)] + UψK(m)δr,0 = E(2)K ψ(m) (2.16)with tK = 2t cos(K2) yieldsE(2),U=0K,k = 4t cos(K2) cos(k) (2.17)upon plane wave basis ψK,k(m) = exp(±ıkm).A solution to the interacting problem can be approached from substitution ofE(2),U=0K,k into Eq. 2.16 with ψK,k(0) = C. Given the symmetry ψK,k(r) = ψK,k(−r)for bosonic particles, this yields17ψK,k(r) = C[cos(kr) +U2tKcsc(k) sin(k|r|)]. (2.18)For |K| = pi (tK = 0), Eq. 2.16 is simply U = E(2)pi and ψpi = δr0. ForK ∈ (−pi, pi),an ansatz ψK(r) = Cα|r|K can be used. With this ansatz, Eq. 2.16 provides2tKαK + U = E(2)K , (2.19)tKα|m+1|K + α(m−1)Kα|m|K= E(2)K , (2.20)Following which the solution for αK is foundαK =U2tK±√1 +(U2tK)2, (2.21)with the wavefunction (normalized) and energy taking the following formE(2)K =√U2 + 4t2K , (2.22)ψK(m) =√| U2tK|4√1 +(U2tK)2√1 + ( U2tK)2− | U2tK| . (2.23)2.3 Role of interaction and bound stateIn the presence of strong interaction of both attractive and repulsive type, two parti-cles co-walk irrespective of their statistics. The presence of bound states is responsi-ble for such behaviour. It is best explained in the momentum space for ideal lattices.The real space hamiltonian can be written asH = T + V =∑mnta†nam +∑mnVmna†ma†nanam. (2.24)Both in the absence or presence of the interaction between two particles, the mo-mentum dependent eigenenergies of the hamiltonian can be obtained by the Fourier18transform. For the case of 1D lattices, one obtains the following expressions whichcan then be numerically diagonalized to find the eigenenergies:〈k′1, k′2|H|k1, k2〉 =1N2∑m′,n′,m,n〈m′, n′|T + V|m,n〉e−ı(−k′1m′−k′2n′+k1m+k2n). (2.25)The hopping part can be simplified further〈k′1, k′2|T |k1, k2〉 =1N2∑m′,n′,m,n〈m′, n′|tm′′n′′q†m′′qn′′ |m,n〉e−ı(−k′1m′−k′2n′+k1m+k2n)=1N2∑m′,n′,m,n[tmm′δnn′ + tmn′δnm′ + tm′nδmn′ + tnn′δmm′ ]e−ı(−k′1m′−k′2n′+k1m+k2n)= 2∑m−ntm−n[eık1(m−n) + eık2(m−n)]δk1k′1δk2k′2 . (2.26)Similarly,〈k′1, k′2|V|k1, k2〉 =1N2∑m′,n′,m,n〈m′, n′|∑m”n”Vm”n”q†m”q†n”qn”qm”|m,n〉e−ı(−k′1m′−k′2n′+k1m+k2n)=1N∑m−nVm−n[eı(k1−k′1)(m−n) + eı(k2−k′2)(m−n)]δk1+k2,k′1+k′2 (2.27)These simplified equations can be used in diagonalization to obtain the eigenen-ergies for the two particles in momentum basis or K-space. Figure 2.3 shows theeigenenergies in K-space for the non-interacting particles with only nearest neighborhopping, which is the case in the tight binding modelH =∑〈mn〉ta†nam +∑〈mn〉V a†ma†nanam. (2.28)For interacting particles, the bound states separate from the continuum beyonda critical interaction strength between the particles. The wavefunction and energyof the bound states have been derived analytically before [69, 70, 52, 61, 66, 71, 72].The energy of the bound states (Eb) for sufficiently strong interactions can be solved19in any dimension. It takes the following form in 1D. For very strong interaction thedispersion of the bound states become flat. The two bound particles, for very stronginteraction, can be represented as a single composite particle with modified hopping,which is of the order t2/VFigure 2.3: Lattice spectrum for non-interacting tight-binding model. The spectrumis calculated for 50 lattice sites. Energy is calculated in the units of the hoppingintegral. Each dot for a fixed K denote separate (k1, k2) combinations with K =k1 + k2 where −pi < k1, k2 ≤ pi.Eb(K) ' V + 4t2 cos2(K)V. (2.29)As shown in Fig. 2.4, the bound states in 1D separate from the continuum atthe interaction strength V = 4. The states responsible for co-walking, in the non-interacting case, lie around middle of the continuum. With interaction strengthincreased, these states move away from the centre within the continuum band. Atthe critical interaction strength V = ±4, these states separate from the continuumas the bound states. This will be illustrated better in next chapter. At very stronginteractions, the energy of the bound state becomes that of the interaction strength.The continuum states, however, remain very much unaffected by the interactionbetween the two particles.20-4 -2 0 2 4 6 8 E 050100150200DOS(E )V = 0V = 2V = 4V = 6V = 8Figure 2.4: Density of states in 1D tight-binding model. The exact diagonalization isdone on a lattice of 86 sites. The bound state separates from the continuum beyondthe critical interaction strength V = 4.2.4 Recent experimentsTo elucidate this effect of binding on the dynamics and the correlations, severalexperiments have been performed on various lattice systems. Peruzzo et al [10]studied quantum walk of two identical photons in an array of 21 coupled waveguideson a silicon oxynitride quantum photonic platform. In their setup, the photons weremade distinguishable by a temporal delay larger than the coherence time when theyarrived on the photonic lattice. When the photons arrived at the same time onthe lattices side by side, a correlation of cowalking was measured. Lahini et al [31]performed similar experiments on waveguide lattices [73] but with photons arriving21on two adjacent channels of waveguides in random phases. When averaged over manymeasurements, they also found cowalking photon correlations. In waveguide lattices,there have also been experiments [74] to understand the tunnelling properties ofbound particles. Recent developments [75, 76, 77] have made it possible to controlthe interaction between two photons in such systems.In another study from Bloch et al [29], such two particle correlations were mea-sured between two spin excitations in a magnetic spin chain of 87Rb atoms. From atwo dimensional quantum degenerate gas of 87Rb atoms, multiple one dimensionalchains/tubes were first formed. Two spins in adjacent sites at the centre of thesechains were then excited before freezing the dynamics by increasing the confiningpotential. Measurements were made with single site resolution [26], after removingexcess atoms except from the desired state. These measurements accounted only forthe tubes or lattices with two spin atoms left. Joint measurements of the two spinsthen revealed the bosonic cowalking character of the quantum dynamics of two cor-related spins. In such studies the onsite interaction energy has been modified withremarkable control.The most recent study on the two-particle quantum walk in cold-atom systems hasbeen performed by Greiner et al [30]. In their study, the 87Rb atoms themselves aremeasured as quantum walkers. In the experiment, a similar prescription is followed.Preparation of 1D chains from a 2D degenerate gas by confining the gas in onedirection with an optical lattice beam followed by narrow confining beams to retainonly two atoms side by side when lattice depth was decreased to remove all otheratoms. After the initial state preparation, the lattice depth was then again increasedfor the dynamics to take place under controlled parameters. A joint measurement isthen performed with the single site resolution [25].These studies have experimentally verified the effect of interaction on correlationsof two-particle quantum walk. However, in all such studies, only nearest neighborhopping was predominant. The case of long-range hopping has so far not been studiedexperimentally for two particle quantum walk. An interplay between the long-rangehopping with the long-range interaction is now predicted by our study to make thedynamics different for different types of interactions.2.5 Effects of long-range hopping and interactionThe dynamics of the interacting particles in the case of nearest neighbor hoppingand interaction is independent of the sign of the interaction [78]. Both attractiveand repulsive interactions have the same effect on the correlations and dynamical22behaviour in the quantum walk. However, this dynamical symmetry with respectto the sign of the interaction is no longer the case when the particles can hop tosites at long-range beyond nearest neighbors in the lattice. For a single particle,the distribution of eigenenergies on both sides of the zero energy line in K-spaceis symmetric (cosine) for the case of nearest neighbor hopping. For two particles,this symmetry remains in the absence of interaction as shown in Fig. 2.3. Forlong-range hopping, this symmetry breaks even for a single particle. In addition tothis asymmetry, the presence of interaction produces the bound state which can becontrolled by tuning the strength of interaction.We simulate the dynamics of the correlations for the simplest long-range hoppingcase, which is taken as the isotropic power law decay of the hopping integral withrespect to the distance between the sites. We find that not only the dynamics becomedifferent for the different signs of the interactions, but the nature of the correlationsalso becomes significantly different from that of the nearest neighbor models.For the simulations we consider two hardcore bosons intended to map Frenkelexcitons (composite electron-hole pairs, see Appendix B), which do not change signwhen exchanged, but cannot occupy same sites under the effect of following hamil-tonian:H =∑nmtnma†nam +∑nmVnma†na†maman. (2.30)The hardcore bosons follow mixed statisticsana†m = δnm + (1− 2δnm) a†man, (2.31)where a(†)n anihiliates(creates) a particle at site n. Calculations are done for both thenearest-neighbor and long-range interaction and tunneling, which decay isotropicallyas an inverse power of distance. Both short and long range of the tunneling andinteraction are considered:tnm =t|n−m|α , (α = 1, 3) (2.32)Vnm =V|n−m|β . (β = 1, 3) (2.33)We define the interaction as attractive (t/V<0) or repulsive (t/V>0) by the sign offor the ratio of the interaction and hopping.The initial state is indexed to one of such vectors, which is symmetrizedΨ(0) = |n′m′〉 ≡ 1√2(n′1m′2 + n′2m′1) (2.34)23and the time evolution of this state is calculated using the eigenenergies Eλ andeigenstates |λ〉 of the full hamiltonian,Ψ(t) =∑λexp(− iEλt~)|λ〉〈λ|Ψ(0)〉. (2.35)The wavefunctions and energies in can be analytically derived following Eq. 2.17,E(2),V=0K (k) =∑d>0td cos(Kd2)cos(kd) (2.36)where d = |n − m|. However, the non-local character of the hopping may rendermean field analysis inaccurate. The states can be analytically derived following Eqs.2.19 and 2.20, with αK and E(2)K as unknowns.The pair correlations (or joint probabilities) are calculated directly from the co-efficients of the two particle basis vectorsCnm = 〈a†na†maman〉. (2.37)For different combinations of α and β, we observe a few features of the quantumwalk for hardcore bosons. Two fermions would also have similar features but it wasfound [23] that the correlations in momentum space would be different between twohardcore bosons and two fermions. In real space, both hardcore bosons and fermionshave been observed to have the same correlations.We find that, when the hopping is long-range, the dynamics for repulsive interac-tions are faster and for attractive interactions they are slower for the same magnitudeof the interaction strength, as displayed in Figs. 2.5 and 2.6. The dispersion of thecontinuum states below zero energy becomes flatter for the case of long-range hop-ping. These states contribute to make the dynamics slower for the attractive case.The dispersion of continuum states above zero energy becomes steep, which con-tributes to the faster dynamics in the repulsive case in the presence of long-rangehopping.For short-range nearest neighbor hopping there is no asymmetry in dynamicswith respect to the sign of the interaction. The expected anti-walking character isobserved without any interaction. For sufficiently strong interactions the particlesbecome bound and show cowalking character in the dynamics. However, in thecase of the long-range hopping, the correlations are no longer only of cowalking orantiwalking types. The correlations develop a character in between that of cowalkingand antiwalking, where one particle stays at the initial position, while the otherparticle extends to the boundaries.24Figure 2.5: Correlation dynamics (pair correlations at time 2pi/t, Eq. 2.37) of twohardcore bosons with different range of hopping with long-range interaction (β = 1).For hopping limited to only nearest neigbors (NN), the dynamics is symmetric withrespect to the sign of the interaction. For long-range hopping (α = 3 and α = 1),the particles spread faster for repulsive interactions.25Figure 2.6: Correlation dynamics ((pair correlations at time 2pi/t, Eq. 2.37) ) of twohardcore bosons with different range of hopping with short-range interaction (β = 3).For hopping limited to only nearest neigbors (NN), the dynamics is symmetric withrespect to the sign of the interaction. For long-range hopping (α = 3 and α = 1),the particles spread faster for repulsive interactions.26These effects can be understood when the lattice spectrum similar to the tightbinding model is calculated. As can be seen from Figs. 2.7 and 2.8, the dispersionsbecome asymmetric in the case of the long-range hopping. In presence of interaction,one state moves out of the continuum states, which is termed as bound state. Thisbound state separates from the continuum with lesser strength of interaction for theattractive case and requires higher strength of interaction to make it move out of thecontinuum in the repulsive case for the long-range hoping cases, as the dispersionsbecome asymmetric. One can utilize this phenomenon by simply changing the sign ofthe interaction between the two particles to control their quantum walk on a lattice.How this can be done is explained in Appendix C.When Figs. 2.5 and 2.6 are compared, one can observe that the effect of long-range hopping is much more dominant than that of long-range interaction. In thelimit of infinitely large lattices, the upper and lower bounds for the lattice dispersioncan be calculated from the values of Riemann zeta functions and are (in the unitsof t) equal to (+4.80,−3.60), (+6.58,−3.30), (+∞,−2.80), respectively, for α = 3, 2,and 1.Figure 2.7: Lattice spectrum for non-interacting particles with Coulombic hopping.The spectrum is calculated for 50 lattice sites. Energy is calculated in the units ofhopping integral. Each dot for a fixed K denotes separate (k1, k2) combinations withK = k1 + k2 where −pi < k1, k2 ≤ pi.27Figure 2.8: Lattice spectrum for non-interacting particles with dipolar hopping. Thespectrum is calculated for 50 lattice sites. Energy is calculated in the units of hoppingintegral. Each dot for a fixed K denotes separate (k1, k2) combinations with K =k1 + k2 where −pi < k1, k2 ≤ pi.All calculations mentioned in this chapter were performed by the method of fulldiagonalization which limits the size of the lattice that can be considered. In thenext chapter we discuss how similar calculations can be performed for far largerlattice systems. This can be performed by exploiting the properties of the modelhamiltonians. These hamiltonian matrices are generally sparse. This sparsity canbe used to develop a method based on recursion (similar in essence to the famousLanczos method) which will allow one to calculate desired properties from thesematrices in an efficient and accurate manner. However, as we will see, one will thenrequire to perform the same iterative calculations many times for each selection ofenergy within the full band of the dispersion.A qualitative argument can also be made on the effect of the asymmetry inspectrum with respect to the sign of the interaction in presence of long-range hoppingon the superfluid to Mott insulator (MI) phase diagram. For a Hamiltonian in Eq.2.38,H =∑mµa†mam +∑mt(a†mam+1 + a†m+1am)+∑mUa†ma†mamam (2.38)the diagram [79] shows transition to Mott insulator state when the particles getbound. For repulsively interacting particles in presence of long-range hopping, theMott insulator phase is expected to be smaller as transition to the bound state now28requires higher energy and smaller t/U . The Mott insulator state can even be absentfor the α = 1 in 1D, when interaction is repulsive. For the attractive cases, the Mottinsulator region is expected to grow larger, as binding becomes easier in presence oflong-range hopping.0123t/Uµ/UMIMIMISuperfluidFigure 2.9: Qualitative phase diagram for transition between Mott insulator andsuperfluid state when hopping is long range and interaction is repulsive (dottedlines) or attractive (dashed lines).2.6 ConclusionIn this chapter, fundamental physics of the most simple model systems has beenfound to be very rich in structure. Such simple model systems consisting of onlytwo particles show different types of correlations in dynamics in the presence of in-teraction when their tunnelling in lattices is long-range. However, method of fulldiagonalization, which was applied to obtain results of this chapter, limits the size ofthe lattices that can be considered. A recursive algorithm to compute Green’s func-tions can be used as described in next Chapter to obtain values related to propertiesof interest of larger lattice systems.293Green’s Functions of InteractingParticlesSolving coupled differential equations for each lattice site in quantum random walksbecomes extremely difficult for a large system size. Full diagonalization has so farbeen employed in most studies to understand the properties of random walk withboth short-range and long-range hopping cases. However there exists a method ofrecursion which can be used to calculate Green’s functions of interacting particles infairly large lattice systems [52, 80, 81, 82]. Both the one-particle and two-particleGreen’s functions can be calculated exactly by this method for any ordered or dis-ordered systems, as will be described in this chapter. In the case of the tight bind-ing model, the hamiltonian is readily solved by a continued fraction method. Thiscontinued fraction method was applied by Haydock et al [83] and Morita [84] whodeveloped such algorithms for calculations of the density of states [85, 86] in ideal 3Dlattices of various kinds (fcc, bcc, sc) for non-interacting particles. In the case of thedisordered 1D systems, Thouless et al. [87] computed Green’s functions iterativelyto find the effect of different onsite energy distributions on conductivity. In our case,we adapt the recursive formulation to real space for finite lattices, both ordered anddisordered, of both one and two dimensions.The two-particle correlations are known to play an important role in the prop-erties of many lattice systems [88, 89, 90, 91, 92]. One must account for the two-particle correlations to understand such systems. Recently, an efficient formulationin momentum space for ideal lattices was developed to calculate few-particle Green’sfunctions, which also elucidated the effect of the interaction on few-particle boundcomplexes [80, 81, 82].A method, where such few-particle Green’s functions can be efficiently calculatedin disordered systems, was under development in our group [52]. In this thesis, it is30extended to 2D systems and shown to be exactly mappable to some arbitrary graphs(e.g. binary trees). We here illustrate the method with the use of recursive Green’sfunctions. We limit ourselves to discussing the method for the two-particle Green’sfunctions in lattices and trees with nearest neighbor hopping. However, the methodcan be easily generalized to the cases of longer-range hopping and a larger numberof particles.For fairly large lattices, this recursive method is very useful. For calculationsof properties related to two-particle correlations in 1D lattices, one can go beyondone thousand lattice sites thus eliminating finite size effects. For two particles in 2Dlattices, around two thousand lattice sites can be considered. Using this recursion, weperform calculations for two particles in binary trees consisting of up to 9 generations.There is also a possibility to improve upon this and make the calculations even moreefficient.The calculation of the density of states of various systems from the real spaceGreen’s functions and the spectral profile of the two-particle bound state is alsoefficient irrespective of the strength of the interaction between the particles. Thedynamics of the interacting particles and their correlations are also shown to be cal-culated efficiently once the important Green’s elements are found. However, to docalculations for large 2D lattices, approximations have to be applied, as the basis sizebecomes very large even for systems with as few as twenty sites per dimension. Weintroduce such approximation and their usefulness in the later part of this chapter,which is mostly relevant to disordered systems. We also perform some preliminarycalculations for the two-particle Green’s functions in 2D lattices with complex hop-ping parameters, intended to simulate the effects of gauge fields.The algorithm is explained in the next section. Later sections will present a fewof the calculations of dynamics and properties such as the density of states and thespectral weight of the two interacting particles in 1D and 2D lattices and in binarytrees.313.1 Method of recursionWe start with the most simple and extensively studied case of two particles in aone dimensional lattice. The lattice can be perfect or disordered. Each case can besimulated very efficiently using the recursive Green’s function method in real space.The Green’s function for some hamiltonian H is defined as following:G(ω) =1ω −H (3.1)where ω = E+ ıη is a complex number with η a very small positive real number andG(m,n, ω) = 〈mn|G(ω)|m′n′〉 is a time-independent propagator from two particlesoccupying sites m′, n′ to sites m, n in the 1D lattice. We omit the indices m′, n′wherever unnecessary for brevity from now on.For a hamiltonian of the form of Eq. 3.2, where m is the onsite energy, tmn is thehopping element moving the particle from site m to site n and Vmn is the interactionbetween particles at sites m and n,H =∑mma†mam +∑〈mn〉tmna†man +∑〈mn〉Vmna†ma†nanam, (3.2)the following type of recurrence relations will emerge. Here, the vectors 〈mn| fromleft and |m′n′〉 from right are applied to the identity (ω−H)G(ω) = 1 from Eq. 3.1to find the relations for functions like G(m,n, ω) sorted on the left hand side of Eq.3.3 and their related Green’s functions on the right hand side... = ..(ω − m−1 − n+1 − Vm−1n+1)G(m− 1, n+ 1, ω) = δm−1,m′δn+1,n′ + δm−1,n′δn+1,m′− tm−2,m−1G(m− 2, n+ 1, ω)− tm,m−1G(m,n+ 1, ω)− tn,n+1G(m− 1, n, ω)− tn+2,nG(m− 1,m+ 2, ω)32(ω − m − n − Vmn)G(m,n, ω) = δm,m′δn,n′ + δm,n′δn,m′− tm−1,mG(m− 1, n, ω)− tm+1,mG(m+ 1, n, ω)− tn−1,nG(m,n− 1, ω)− tn+1,nG(m,n+ 1, ω)(ω − m+1 − n−1 − Vm+1n−1)G(m+ 1, n− 1, ω) = δm+1,m′δn−1,n′ + δm+1,n′δn−1,m′− tm,m+1G(m,n− 1, ω)− tm+2,m+1G(m+ 2, n− 1, ω)− tn−2,n−1G(m+ 1, n− 2, ω)− tn,n−1G(m+ 1, n, ω).. = ..(3.3)Here, only nearest neighbor hopping and interaction is considered. Once allG(m,n, ω)are found, the dynamics can be easily computed by the Fourier transformation ofthe Green’s function amplitudes from the energy domain to the time domainG(m,n, t) =∑ωe−ıωtG(m,n,E + ıη). (3.4)The spectral weights of eigenstates for any initial state or wave packet of the twoparticles at sites m′ and n′, can be computed from a single Green’s elementA(m′, n′, E) =−1piIm[G(m′, n′, E + ıη)]. (3.5)The density of states (DOS) of the lattice systems up to a scaling factor can alsobe computed from all such single Green’s elementsDOS(E) =∑m′,n′A(m′, n′, E). (3.6)If there is translational symmetry present in the system, then only a few initial stateswith increasing relative distance (|m′ − n′|) might prove sufficient for convergence.Transport properties calculated from Green’s elements such as G(m′ ± 1, n′, ω) orG(m′ ± 1, n′ ± 1, ω)) might also be of key interest.Now, the recursive functions are formulated in the form of Eq. 3.3 consisting ofvectors in a chain. One needs to first find some good quantum numbers and groupGreen’s elements according to such numbers. We find R=m + n for the Green’sfunctions of the form G(m,n, ω) in real space is such a number, as the hamiltonian33does not connect functions with same R directly, as can be checked from Eq. 3.3.We sort all such functions in a single vector GR, as in Eq. 3.7. One can also noticethat GR is only connected to GR−1 and GR+1 by the hamiltonian, as in the Eq. 3.8GR=m+n(ω) =..G(m− 1, n+ 1, ω)G(m,n, ω)G(m+ 1, n− 1, ω)..(3.7)GR = αRGR−1 + βRGR+1 + C (3.8)where C = 0 ( or 6= 0) when R 6= m′ + n′ (or = m′ + n′ = R′).These vectors form a one dimensional chain in terms of their connection to onlythe nearest neighbor vectors and each of their elements can be solved exactly bythe following prescription. This particular form also appears in many other areas ofquantum physics and therefore a similar method in principle can be constructed.On the left and right boundary of the chain, the following equations hold forsystems with open boundary conditionG0 = β0G1 and GL = αLGL−1, (3.9)where 0 and L are the minimum and maximum index possible for R.If we can simplify Eq. 3.8 as in Eq. 3.9, then all the calculations will become arecursion of vectors:GR = ARGR−1 and GR = BRGR+1, if R 6= R′. (3.10)We find A0 = β0 and BL = αL. These are our open boundary conditions. Now,substituting Eq. 3.10 to Eq. 3.8 for R < R′ and R > R′, we find the followingequations respectively:GR = αRGR−1 + βRGR+1BRGR+1 = αRBR−1GR + βRGR+1BRGR+1 = αRBR−1BRGR+1 + βRGR+1[1− αRBR−1]BR = βRBR = [1− αRBR−1]−1βR(3.11)34GR = αRGR−1 + βRGR+1ARGR−1 = αRGR−1 + βRAR+1GRARGR−1 = αRGR−1 + βRAR+1ARGR−1[1− βRAR+1]AR = αRAR = [1− βRAR+1]−1αR(3.12)One can compute these AR and BR matrices recursively starting from Eq. 3.9before one reaches R = R′ from both sides of the chain. At R = R′, applying AR′+1and BR′−1 to Eq. 3.8, one finds the following:GR′ = αR′GR′−1 + βR′GR′+1 + C,GR′ = αR′BR′−1GR′ + βR′AR′+1GR′ + C,GR′ = [1− αR′BR′−1 − βR′AR′+1]−1C.(3.13)Once GR′ is found, all other GR can be found by Eq. 3.10, hence solving theproblem of finding all the Green’s elements for a given m′ and n′ for a single ωwithout diagonalization. To find all the eigenenergies, one has to scan over a rangeof ω that can be used in Eq. 3.16 to compute dynamics. The calculations for each ωare distinct from each other and can be parallelized. The value of η has to be chosenarbitrarily. This choice can be benchmarked by comparing a few sample calculationswith full diagonalization.3.2 Two interacting particles in 1DAs has been remarked in the previous section, this recursive algorithm enables usto do calculations for much larger system sizes than what can be performed usingfull diagonalization procedures. These calculations are also very efficient. We cancalculate dynamics in ideal or disordered lattices. We can calculate the density ofstates and spectral weight for any initial wave packet preparation. In Fig. 3.1 wecalculate G(m′, n′, E) for |m′ − n′| = 1 for a lattice size of 500 sites. The particleswere taken as hard-core bosons and the hamiltonian as in Eq. 3.1435-4 -2 0 2 4 6E050100150200DOS (E )01234A (E )V = 0V = 2V = 4V = 6-4 -2 0 2 4 6EFigure 3.1: Density of states (DOS) calculated for a 1D ideal lattice of two interactingpartcles from full diagonalization are plotted as dotted lines. The spectral weightsfor two particles initially prepared at one lattice site apart calculated using Eq. 3.5is shown in solid lines. The inset shows the calculation of density of states using Eq.3.6.H =∑〈mn〉ta†man +∑〈mn〉V a†ma†nanam (3.14)These calculations reveal the weight of the bound state even inside the continuumas the interaction strength is increased. We find the spectral profile for two particlesinitially located at nearest neighbor sites in a 1D lattice. Figure 3.1 shows thechanges to the profile when the interaction strength is increased. At zero interactionstrength, the profile is symmetric and has a sharp peak at E = 0 with linear dropto the end of the band edge. For V = 2, the profile has a sharp rise at E = 2 and36a sharp drop at the band edge. For stronger interactions, the profile matches withthe distribution of bound states. These spectral weights also appear in the contextof many-particle systems with different filling fractions [93].The correlation dynamics also shows excellent agreement with that obtained fromfull diagonalization as shown in Fig. 3.2. The effectiveness of the method can berealized even by searching less than 500 points (for each ω needed in Eq. 3.16) withinthe full the energy band (−4 ≤ E ≤ +4). As shown in Fig. 3.2, searching less than 50points per unit of the energy width turns out to start producing errors in the rangeof a unit percentage in such calculations. We prefer the approach of first findingthe most important bandwidth from the calculated spectral weight which minimizesthe number of search points and enhances the efficiency of the computation in suchcalculations.Figure 3.2: Comparison between recursive calculation using only 10(left) and 50(mid-dle) search points within unit energy bandwidth with that of full diagonaliza-tion(right) for correlation dynamics of two particles in a 1D ideal lattice for V = 1and at time = 10 (in the unit of hopping). Color scheme- red, yellow, green, blueshow lower joint density in that order.3.3 Two interacting particles in 2DIn the two dimensional systems, the difficulty of doing full diagonalization for twoparticles grows approximately as N12, where N is the number of sites per dimen-sion. As in the case of 2D lattices, one needs to consider a large number of sites toavoid significant finite size effects in the calculations of transport and localizationproperties, these numbers soon become not viable for doing any reasonable calcu-lations. The recursive calculations, as described earlier, break the total calculationinto multiple parts, which can be solved recursively as described before.37-8 -4 0 4 8E-0.0500.05Im[G(n,m; n′,m′; E)]| n - m | = 1| n - m | = 10Figure 3.3: Comparison between recursive calculation using only with that of fulldiagonalization for two particles in a 2D disordered lattice for W = 1, V = 1 (inthe unit of hopping) as in Eq. 3.2 with chosen randomly from box distribution[−W2, W2]. r = |n−m| is the distance between two particles in number of minimumsteps in a lattice. Full symbols are for full diagonalization, smaller filled symbolsinside are for recursive calculations.The recursive calculations are exact when the recursion includes the boundaries.The recursions can also be done locally limited to only a small part of the lattice.Figure 3.3 shows the imaginary part of a few randomly selected Green’s functions fora fixed initial state and final occupations at different distance (minimum number ofsteps between two particles) in a 2D lattice. As it shows, the recursive calculationsmatch exactly with that of full diagonalization irrespective of the distance betweenthe two particles. However, there can be some numerical errors depending on theimplementation of the algorithm.For a 2D system as large as consisting of more than two thousand sites, thisrecursive method also becomes very difficult to implement. In the case of disorderedcases, where one needs to average over many realizations of disorder to account for38any robust results, implementation of the full recursive method becomes challenging.In these circumstances, one finds it necessary to employ some approximations whichhelps in reducing the size of the calculations significantly while producing accurateresults. We propose and test such approximations. These approximations are veryuseful in the cases of the disordered systems.We had found in our previous study [78] that the presence of the disorder enhancescorrelations for smaller distances between particles, effectively enhancing cowalkingand binding. This effect allows us to make some approximation on the maximumdistance between two particles. Green’s functions with larger distances can be ap-proximated to make no contribution to the calculations and hence neglected. Thisselection of Green’s elements depending on the distance between two particles canalso be done dynamically. One can select elements of importance differently at dif-ferent times of propagation.Figure 3.4: Two particle dynamics in 2D ideal lattice (Green’s probability, Eq. 3.16),calculated recursively using approximation of maximum allowed distance betweentwo particles r = 5 (left) and r = 10 (right) at time= 5 for V = 1. Color scheme-red, yellow, green, blue show lower density in that order.In Fig. 3.4, the approximation was applied for the case of a 2D ideal latticewith 20 sites in each dimension. While the overall spread shows similarity for twolimiting distance approximations, the exact distributions are different as expected.This shows that in the case of ideal lattices applying this approximation will not beaccurate at large times.For large systems, employment of such approximations are inevitable. As shownin Fig. 3.5, the total number of elements for a full calculation in 2D becomes close totens of millions for size with 100 sites per dimension. The largest vector involved in39the recursive calculation also includes close to a few million Green’s elements withoutapproximation. One simple approximation to make is to neglect the propagators withlarger relative distance between the two particles and set a maximum allowed relativedistanceG(n,m;n′,m′;ω) ' 0 for |n−m| > r (3.15)where r is a limiting (Hamming) distance in the number of minimum steps betweentwo particles.0 20 40 60 80 100r01e+052e+053e+05size [R′]0 20 40 60 80 100r01e+072e+073e+074e+075e+07total sizeFigure 3.5: Size of largest vector (on the left) and total number of elements involvedin recursive calculations for given maximum allowed relative minimum step distancer (Hamming distance) between two particles in a 2D lattice of 100x100 sites.This approximation allows one to perform calculations considering the wholelattice but with much fewer number of elements compared to the full recursion.This approximation does not constrain the particles over the lattice but neglectsthe elements that can contribute to larger distances between the particles than amaximum chosen distance. As can be seen from Figs. 3.6 and 3.7, doing calculationsfor a lattice of 100 sites per dimension with the maximum relative distance > 10 willbe very difficult.400 100 200 300 400R0500010000size[R]r = 4r = 5r = 6r = 7r = 8r = 9r = 10Figure 3.6: Size of vectors (Eq. 3.7) in recursive calculations for maximum allowedr (Hamming distance) in a 2D lattice of 100x100 sites. The difficulty of calculationsare determined by the size of the vectors with maximum number of elements.10 20 30 40 50 60 70 80 90 100Nx = Ny0200040006000800010000size [R′(Ν)]r = 4r = 5r = 6r = 7r = 8r = 9r = 10Figure 3.7: Size of largest vector (most often the R′ in Eq. 3.7) in recursive calcu-lation for maximum allowed r in a 2D lattice of Nx,Ny sites. For larger r, the sizeincreases faster with the size of the lattices. N = Nx +Ny.41Using this method, we calculate all Green’s elements for a given onsite energydisorder, chosen randomly from a uniform distribution of width W ([−W2, W2]), forparticles initially at adjacent sites at very large times (τ = 1000)G(m,n, τ) =∑ωe−ıωτG(m,n, ω). (3.16)Once we find all such Green’s elements, that is for every pair of site indices (i, j),populated by two particles, we calculate the joint density distribution (%), densitydistribution (ρ) and inverse participation ratio (I) for each realization of disorder:%(m,n, τ) = |G(m,n, τ)|2, (3.17)ρ(m, τ) =12∑n6=m%(m,n, τ), (3.18)I =∑m ρ(m, τ)2∑m ρ(m, τ). (3.19)We average them over many realizations. From the scaling of IPR calculated for2D disordered systems in the range of W = 4, V = 4, as shown in Fig. 3.8, wefind that a minimum lattice size of 30x30 sites should be considered for results thatwould be close to results in larger system sizes. This scaling even for the case underconsideration, where the most delocalized behaviour is expected, hints at localizationfor two weakly interacting particles in disordered 2D lattices. The curve appears toa constancy for larger system sizes which is a signature of localized state rather thanan exponential decay, characteristic of delocalization.In disordered systems, the calculations of macroscopic properties such as theinverse participation ratio calculated from density distributions averaged at timemuch larger than that required to hit the boundaries, do not produce large errorseven when the maximum allowed distance is kept as small as r = 5. As shown in Fig.3.9, with r = 5, the results are within the range of 10% errors. However for specificGreen’s elements, these errors might be large. Specifically the elements describingtransport from center to boundaries are expected to have large finite size effects forsmall or medium sized systems. Localization lengths calculated from such elementscan have errors that are not negligible.4214 16 18 20 22 24 26 28 30 32Nx = Ny0.0070.0080.009ΙFigure 3.8: IPR for increasing lattice sizes in 2D for the case of W = 1, V = 4averaged over 50 realizations of disorder.435 10r80100120140ΙW = 1W = 5W = 95 10r100150200ΙW = 1W = 4W = 9Figure 3.9: Relative IPR with maximum allowed relative distance r for a small 2Dlattice of 12x13 sites. V = 0 (left), V = 4 (right). The IPR is not normalized asη makes the calculated densities not normalized. Averaged over 50 realizations ofdisorder.The spectral weight (Eq. 3.5) for the two particles located at adjacent sites ona 2D lattice with nearest neighbor interaction can be calculated from the imaginarypart of a single Green’s propagator. Figure 3.10 shows the spectral weight for suchtwo particles, that is the bound state for different interaction strength. The spectralweight shows a sharp peak at E = 0 for the non-interacting case with an exponentialdrop to the band edges. The spectral weight for the interacting case shows a sharprise following a linear drop to the band edges.Spectral weights of doublon in Hofstadter modelIn recent years, there has been a lot of interest in implemeting the Hofstadter model[94] in optical lattices for neutral atoms [95, 50, 51, 96]. The Hofstadter model takesaccount the effect of external magnetic fields on electrons in lattices by making thehopping amplitude complex. The model is mimicked for neutral atoms by periodicmodulation of lattice potentials, which averages to zero force, but produces a complexphase factor on momentum dependent hopping or tunneling amplitudes of atomsin lattices [96, 97]. This opens the possibility of simulating integer and fractionalquantum Hall [98] systems and topological insulators [99] in disordered 2D opticallattice systems [100].The model can be derived by Peierls substitution [101] from the tight bindingHubbard model and accounts for a phase for hopping44-4 -2 0 2 4E00.10.2A(E )V = 04 6E00.20.4A(E )V = 412 13E00.511.52A(E )V = 12Figure 3.10: The spectral weights (3.5) for two particles located side by side in a 2Dlattice of 20 sites in each direction. For the non-interacting case, the spectral weightshow a peak at E = 0 with broad wings on both sides. For the interacting case, thespectra becomes narrow with a long tail as the interaction strength is increased.45VJ JJ JJeiφJe−iφ′XYFigure 3.11: The hopping and interaction terms in 2D Hofstadter model for hardcorebosons. The phases φ, φ′ for hopping terms on X axis depends on lattice site indicesof Y axis.H =∑〈ij〉e−ıφijJija†iaj +∑iUia†ia†jajai. (3.20)Experimentally, a periodically modulated potential after averaging over the fulltime period [102] can effectively add the directional phases to the hopping terms andhave the same dispersion as after performing the Peierls substitution [96].In the 2D lattices, we implement the same hamiltonian for two interacting hard-core bosonsHpq =∑〈ij〉[e−ı2pipqiyJix,ix+1a†ixaix+1 + Jiy ,iy+1a†iyaiy+1 + h.c.]+∑〈ij〉V a†ia†jajai, (3.21)where i, j are the site indices of two particles and the axes dependency is removedfrom the interaction term for simplicity. The terms are elaborated in Fig. 3.11.For this hamiltonian we calculate two-particle Green’s functions and spectralweight for two particles located at adjacent sites. We find each spectrum splitsinto several bands depending on the value of q as shown in Fig. 3.12. The non-interacting particles show a sharp peak at E = 0 with the q − 1 number of broad46peaks on both sides. Each of these broad peaks has more peaks inside them. Forincreasing interactions, these broads peaks seem to be merging with each other whilethe overall shape for q 6=∞ appearing totally different from the q =∞ (3.10) case.One observation can be made from the calculated results, which is, the weight ofthe spectra shifts toward lower side of the energy bandwidth for higher interactionstrength (V ) and higher ratio of qpuntil qp< 12. For qp> 12, this trend is expected toreverse as qpand q−pphas same spectra.-4 -2 0 2 400.10.20.3V = 0 q = 2-4 -2 0 2 400.20.4V = 0 q = 3-4 -2 0 2 400.20.4V = 0 q = 44 600.20.4V = 4 q = 24 600.20.4V = 4 q = 34 600.51V = 4 q = 412 13E00.511.5V = 12 q = 212 13E00.511.5V = 12 q = 312 13E00.511.5V = 12 q = 4Figure 3.12: Spectral weight of two interacting particles in 2D Hofstadter model.The number of broad peaks show clear dependence of q while stronger interactionseems to be merging these peaks. p = 1.473.4 Two interacting particles in binary treeThe structure of the recursive calculations maps directly to binary trees when eachlevel of the branches of the tree is taken as a full vector involved in recursive calcula-tions. These tree structures are also known as the Bethe lattice (with a boundary).The root node (L = 0) of the tree splits into two branches of same level (L = 1).Each node on these branches splits into two different nodes. Any node within thevectors does not connect to each other by the hamiltonian and each such vector isconnected to nearest neighbor vectors only. The two boundary conditions necessaryfor the computation of the Green’s functions correspond to the vectors at the highestlevel on the left and right branches as shown in Fig. 3.13. Systems such as binarytrees not only act as a model system interesting for its mathematical form but similarforms can be found in biological systems where transport of excitations may proveto be relevant.L = 0L = 4GRFigure 3.13: Binary tree of 4 levels. Each level separated between left and rightbranches. Each level within each branch can be considered as a vector involved inrecursive calculation as they are not connected to each other by hamiltonian.The spectral weight of interacting particles on a binary tree can be calculatedfrom Eq. 3.5. For two particles placed on the same site of L = 0 on this graph(with maximum L = 8), Fig. 3.14 describes the spectra. The spectrum showsdiscontinuous peaks as opposed to continuous spectra in 1D and 2D lattices. With48stronger interactions, these peaks tend to merge together and a single continuousspectrum enveloping multiple peaks seems to be emerging.-5 0 5E00.40.8A(E )V = 0, recursiveV = 5, recursiveV = 0, fullV = 5, fullFigure 3.14: Spectral weight for two particles located on same site calculated for abinary tree of L=8. For non-interacting particles, the spectrum has multiple peakswith the highest peak at E = 0. For strongly interacting particles, the individualpeaks are compacted into one peak.3.5 ConclusionIn this chapter two-particle Green’s functions have been calculated efficiently usinga recursive algorithm. These calculations provide insights into the problem of in-teracting particles. Possible extensions for calculations of response properties fromtwo-particle correlations can be avenues of further research. In the next chapterwe attempt to understand the behaviour of two particles and their correlations indisordered one- and two-dimensional systems.494Quantum Localization of InteractingParticlesAfter a few experimental observations from 1990s [103, 104, 105, 106, 107], there hasbeen renewed interest in understanding the effect of interactions on the localizationof particles in 1D and 2D systems. These experiments had reported observationsof persistent currents in 1D wires [103, 104, 105] and a localization-delocalizationtransition in 2D lattices [106, 107]. This is of high interest as the scaling theory [108]predicts an absence of such transition in 1D and 2D systems. Since then there hasbeen a plethora of studies. Investigations on whether the inter-particle interaction isresponsible for such phenomena were started immediately. To understand the effectof interparticle interaction on localization, understanding the case of two particleswas necessary. However, while some studies [35, 36, 109] predicted the effect ofinteraction in delocalizing the particles in disordered lattices, some numerically foundthat the interaction-induced delocalization effect is limited to weak interaction cases[40, 110] and for strong interactions the two particles become more localized [37, 38].The differences arose from the calculations using random matrix theory [111]. Somestudies have also noted a universal sub-diffusive behaviour after transient localizationinduced by the interaction [112]. The localization-delocalization transition was alsosupported by some numerical studies in 1D [113, 114, 115] and in 2D [39], althoughthe latter were based on significant approximations.In this chapter we perform numerical calculations to understand not only theeffect of interaction on localization in 1D systems, but also the effect of the rangeof both tunnelling and interaction. In the case of 2D, where calculations are verydifficult to perform, we apply the recursive method described in Section 3.1 and findlocalization parameters for the short-range tunnelling and interaction case.504.1 Scattering with single impurities in 1DHere we study the case of two interacting hardcore bosons in 1D systems by exactdiagonalization. We first study the particles interacting with impurities. In particularthe case of two particles initially placed in adjacent sites, in the middle of a 1Dlattice with two impurities, each placed towards the edges of the lattice as in Fig.4.1. We let the wavepackets tunnel out of the impurities for a certain time andexamine the dependency of that dynamics on inter-particle interactions. We notethat weak interaction increases the tunnelling of the particles through the impuritieswhile strong interactions reduce the tunnelling. The long-range nature of tunnellingpermits tunnelling through the impurities. We observe that the particles with stronginteractions get bound hence heavier as their dispersion also becomes flatter, resultingin the slow tunnelling through the impurities. The impurities were modelled by δ-function potentials and particles outside the impurities were assumed to not scatterback inside the impurities again.Figure 4.1: Initial preparation of two particles (red) with two impurities placedtowards the edges (green) of the 1D lattice.The hamiltonian is as given in Eq. 2.30,H =∑mεma†mam +∑mntmna†man +∑mnVmna†ma†nanam (4.1)whereεm = 0 ∀ m 6= m1,m2 and εm =∞ for i = m1,m2 (4.2)andtmn =t|m− n|α , Vmn =V|m− n|β for α, β ∈ {1, 3,∞} (4.3)Figure 4.2 shows the wavepacket density remaining inside the impurities aftera certain time allowing multiple scattering with the impurities. The effect of the51interaction increases the tunnelling through the impurities in the weak interactioncases and increases trapping of the particles in the strong interaction cases. For long-range hopping, the particles tunnel out faster as expected, however, the long-rangenature of interaction has very minimal effect on controlling the scattering throughthe impurities compared to the effect of long-range nature of tunnelling. As Fig. 4.2illustrates, the tunnelling probability goes through a maximum for an interactionstrength V/t > 0 for both dipolar and Coulombic isotropic hopping. The asymmetryin tunnelling with respect to the sign of the interaction in the case of long-rangehopping can also be noted. In these calculations the particles that tunnel throughthe impurities were dynamically removed from the calculations, with very short timesteps in the unit of the inverse of the hopping parameter t (typically 1000/t). Thelength between the two impurities is chosen to be 10 sites to allow a few scatteringevents to take place. However, the particles with strong attractive interaction andvery strong repulsive interaction behave as very slow particles. For very stronglybound particles the number of scattering events is less compared to weakly attractiveparticles which exhibit faster dynamics. At larger times the interference between thescattered part of the wavepackets within the impurities makes the character of thepropagating wavepacket different from that of purely bound wavepacket projectedtoward impurities, which further modifies the scattering with the impurities. A timeof 20/t was chosen for the results plotted in Fig. 4.2. For the weakly interactingparticles, when a sufficient number of collisions with the impurities is allowed, itis found that the weak repulsively interacting particles tunnel more through theimpurities than the non-interacting ones.Longer tunnelling range leads to more tunnelling through the impurities. Forthe long-range interactions, the effect is not very different from the short-range in-teractions between the particles. However, for weakly repulsive interactions, theshort-range interaction leads to more tunnelling than in the long-range interactioncases. This can be seen prominently present in the case of long-range tunnelling inFig. 4.2. For strong repulsive interaction, the short-range interactions lead to lesstunnelling compared to the long-range interaction cases.52Figure 4.2: Tunneling out of interacting particles through impurities in case of long-range hopping and interaction Eq. 4.3. The particle density is the density remainingafter time 20/t within a length of 10 sites between two impurities.534.2 Localization in 1DAfter gaining some insight into the scattering with isolated impurities, a distributionof impurities is placed (Eq. 4.2) in the lattice to understand the localization proper-ties of disordered 1D systems. The disorders implemented here are both onsite andoffsite in nature. The sites that cannot be occupied become disconnected from therest of the lattice. The long-range character of the hopping makes it possible forthe particles to hop over the impurities. After the dynamics is frozen at a very longtime compared to the hopping parameter (τ t−1), the joint densities (%mn) anddensities (ρm) were calculated from the exact eigenfunctions and eigenenergies in thetwo-particle basis. The density-density correlations (Cmn) are nothing but the jointdensities calculated from the two-particle basis%m,n(τ) = |〈m,n|∑λe−ıEλτ |λ〉〈λ|m′, n′〉|2 (4.4)Cmn(τ) = %m,n(τ) (4.5)ρm(τ) =12∑n 6=m%(m,n, τ). (4.6)Alternatively, one can calculate the localization length as suggested by Oppen et al[110] from Green’s functions and gain insight into the localization behaviour.The inverse participation ratio (IPR) of second rank is calculated from the densitydistribution as in the following equationI = limτt∑mρm(τ)2. (4.7)The participation ratio (Π = I−1) is the parameter which gives the number of sitesparticipating in the distribution and hence is larger for the delocalized systems. Onthe other hand, a higher inverse participation ratio refers to more localized states.Figure 4.3 presents the calculations performed using the method of full diagonaliza-tion for a lattice of 50 sites. The lattice was disordered by 10% of vacancies andthe results were averaged over 5000 such disorders. It can be clearly observed fromour calculations that the particles become more localized for the strong interactioncases compared to the non-interacting ones. The weak repulsive interaction, however,reduces the localization of the particles.54Figure 4.3: Localization parameter, participation ratio (Π) calculated from dynamicsof two interacting particles for interaction strength V initially occupying two sitesside by side in disordered 1D lattices for the cases of long and short-range hoppingin presence of short-range interaction. Averaged over 5000 realizations.The correlations (Cmn) between particles in disordered lattices show an enhance-ment of cowalking between the particles. Figure 4.4 clearly illustrates disorder in-duced enhancement of cowalking correlations even for the weakly interacting par-ticles. For non-interacting particles, emergence of correlations in between that ofcowalking and antiwalking is observed. It can also be seen that the cowalking cor-relations extend toward the edges more prominently than any other correlations. Itcan be inferred that, if the correlations in disordered systems are measured, therewill be a high probability of finding the particles close together. Figure 4.4 alsoshows, that in disordered cases, even in the weak interaction limits, there are veryfew correlations that are important. The particles might be spread over a large partof the lattice depending on the localization length but only a few correlations, mainlythat of the cowalking type, should be taken into account in any such calculations fordisordered systems.55Figure 4.4: Correlations (Cmn, shown in the legends) calculated from dynamics oftwo interacting particles initially occupying adjacent sites in disordered 1D latticesfor the cases of long and short-range hopping (α = 1, 3) in presence of short-rangeinteraction (β = 3) as in Eq. 4.3. Averaged over 5000 realizations.4.3 Localization in 2DTo gain understanding on localization properties of 2D systems the same Hamiltonianas in Eq. 4.1 is simulated with onsite energies (εm) selected randomly from a uniformdistribution of fixed width (W )εm ∈[−W2,W2]. (4.8)56The calculations of localization properties for two interacting particles in two dimen-sional disordered systems cannot be done by the method of full diagonalization asthe basis size grows beyond what can be accounted for, even in the case of small 2Dsystems. To perform such calculations, we use the recursive Green’s function methoddescribed in Section 3.1. The recursive method breaks down the full problem intomultiple smaller size matrix-vector multiplications which make the calculations moreefficient while maintaining accuracy.For a fairly large 2D lattice of 50 sites per dimension, a full diagonalization fortwo particles would entail a total basis size of around three million Green’s functions.This scale is impossible to fully diagonalize even with the help of most sophisticatedcomputers.0 50 100 150 200R010000200003000040000size [VR] fullr = 10r = 50 50 100 150 200R01000200030004000size [R]r = 5r = 6r = 7r = 8r = 9Figure 4.5: Sizes of vectors involved in recursive calculations for a 2D lattice with 50sites per dimension. The Gaussian shaped full distribution reduces to a triangulardistribution in presence of approximations.As shown in Fig. 4.5, the total number of elements to be considered in thecalculations even after applying the approximations as referred in Eq. 3.15, canbecome as large as a few hundred thousand (apply the triangle area law to get thetotal number from the figures). The recursive algorithm can break the calculationto those with vectors having a few thousand of Green’s functions as shown in thefigure. As the calculations involve inversion of matrices, this reduction makes thecalculations significantly more efficient compared to full diagonalization. However,one now has to perform calculations over many search points effectively doing thesame iterations many times to understand the dynamics and correlations of theinteracting particles.57The recursive method allows the exact calculations of these Green’s functions bytaking advantage of the sparsity of the whole matrix. With the recursion method,the full calculation is split into a Gaussian-shaped distribution of vectors as shownin Fig. 4.5. The vectors are coupled as explained in Chapter 3 through Eq. 3.8.However as can be observed from Fig. 4.5, a full calculation for a fairly large 2Dlattice of 50 sites per dimension, even with the help of recursion, remains difficult asit involves tens of matrices with dimensions of the order of tens of thousands, to beconsidered a few hundred times for every energy point within the band. This impliesenormous computational time and resource requirements.The approximation of the maximum relative distance that has been applied forthe calculations of the time-dependent densities in Fig. 4.6, for a disordered 2Dlattice of 50 sites per dimension, makes the calculations significantly faster. Theapproximations can be used to reduce the total number of Green’s functions involvedin the calculation from a few millions to a few hundred thousands. These totalnumber of elements, in the case of a small maximum relative distance to the totallattice size, has a linear distribution of the elements.The method lets us perform the calculations which would be impossible by fulldiagonalization, and is highly accurate and efficient as described in the previouschapter. With the method, we can proceed to take on the challenge of calculatingthe dynamics of a few interacting particles in disordered 2D lattices. We can cal-culate the localization parameters such as the inverse participation ratio (IPR) orany Green’s function of interest from such calculations. As shown in Fig. 4.6, thedensity distribution of two weakly interacting particles in a weakly disordered 2Dlattice appears to be localized. Comparisons between different degrees of approxi-mations (increasing r) with same disorder show that the density distributions areconverging, as can be seen from Fig. 4.7. The density distributions are calculatedfor a single realization of fixed disorder. The differences between the approximationsare not significantly large, even in the absence of averaging over many realizations ofdisorders, which indicates that such approximations, limiting the relative distance,can be used to calculate the properties of disordered lattices.Alternatively, only a few Green’s functions of interest are needed to gain insightinto the localization properties, as suggested by von Oppen et al [110]. However,a medium-size lattice that can be considered for the calculations by the recursionmethod will produce significant finite size effect, and render the calculations of local-ization lengths from Green’s functions involving edges of the lattice highly inaccurate.Thus, the macroscopic properties such as the IPR were employed to understand thelocalization behaviours.As described in the previous chapter, for calculations of the localizations prop-58Figure 4.6: The density distribution for a single realization of same disorder in a 2Dlattice with 50 sites per dimension calculated using approximation of Eq. 3.15. Leftpanel show the density for r = 5, middle panel for r = 7 and right panel for r = 9.Figure 4.7: Difference in density distribution for a single realization of same disorderin a 2D lattice with 50 sites per dimension calculated using approximation of Eq.3.15. Left panel show difference between r = 6 and r = 9, middle panel betweenr = 7 and r = 9, and right panel between r = 8 and r = 9.59erties, one requires averaging over many realizations of disorder. The averagingminimizes differences in results between different realizations of disorders and takesaccount of the different degrees of randomness in each different realization of disor-der. The averaging also produces a density distribution that can be expected of anyrealization of disorder.As explained before, even after approximations, a fairly large lattice size wouldbe difficult to consider for the computation of the localization parameters. Thesecalculations have to not only take into account the number of times the recursionhas to be performed for each point of energy within the bandwidth, but also thenumber of times the same calculations have to be performed for averaging for eachrealization of disorder. However, from Fig. 3.8, it can be observed that even in theranges of weakly disordered and weakly interacting cases, a lattice of medium size,such as containing 20 sites per dimension, won’t produce significant errors. Theseerrors are found to be in the range of 10-20%. Thus, the limitations that we confront,force us to make a choice of doing the calculations for a medium sized lattice for thelocalization calculations.We try to find the overall pattern from the localization parameters over a vastrange of disorder in the lattices and the interactions between particles. From thesepatterns, we attempt to infer if any localization-delocalization transition exists fora 2D disordered system of interacting particles. We also calculate the correlationsbetween the particles over these vast ranges of disorder and interaction strenth,which helps us to understand the effect of disorder on correlations of the interactingparticles that cannot be obtained from density distributions.More importantly, these calculations for medium-size lattice indicate the lengthscales involved in localization of interacting particles in regular 2D lattices and howthe interactions and disorder affect these length scales. The exact numbers that wefind from our calculations might vary because of inherent errors due to the size ofthe lattices and the approximation that has been applied. However, a broad initialunderstanding can be achieved from these calculations.As explained in Eq. 2.7, these calculations directly compute the correlationsbetween particles even in the disordered systems. These correlations reveal the un-derlying structures of localized particles in disordered systems. Measurements oftotal correlations for different relative distances reveal how the most probable rela-tive distance between the particles changes when disorder and interaction strengthsare varied.In Fig. 4.8, the IPR is plotted for a broad range of the interaction and disorder.The same values are shown in Fig. 4.9 where a three-dimensional plot with thecontours reveals the difference between the weak interaction - weak disorder limits60and the strong interaction - strong disorder limits. For the case of 2D lattice systems,these limits can be equated to half of the full bandwidth (V = 4, W = 4) , wherethe systems are found to be least localized. The disorder strength is varied fromW = 1 to W = 12. The interaction strength is varied from V = 0 to V = 8. Thework involves every combination of the integer points for both the disorder and theinteraction strength within this broad range.Figure 4.9: 3D disorder-interaction diagram of IPR computed for a broad range ofinteractions (V = 0 to V = 8) between particles and disorder strengths (W = 1 toW = 12) in 2D lattices. Averaged over 320 realizations of disorders.In Fig. 4.9, the difference can be noted between the two contours colored as lightblue and light green. It is an indication of some physical difference between the tworegions. In Fig. 4.8, the squares highlighted by black rectangles signify a change inthe prominent character of the correlations. On these marked squares, the natureof correlations changes from that of predominantly nearest neighbor ones to that ofnext nearest neighbor ones, between the localized particles. Figure 4.11 illustrateshow the nearest neighbor correlations become more prominent than the next nearestneighbor correlations in the range of strong interaction and strong disorder. Thesechanges indicate some change in the character of the localized particles that cannot be61Figure 4.8: Two dimensional disorder-interaction diagram with black squares show-ing the regions where the nature of correlations changes. The inverse participationratios plotted are averaged over 320 realizations of disorder, shown in the color leg-end.62understood from IPR calculations or from density distributions alone. We quantifythese changes by defining a total correlation parameter ζ which depends on theminimum step distance between the two particles.ζ(rs) =∑mnCmn∣∣|m−n|=rs (4.9)To understand the trend in the computed parameters, we plot the iso-disordersurfaces from Fig. 4.9 in Fig. 4.10. The calculations show very similar behaviourof localization within a broad range of the weak interaction and the weak disorder(0 < W ≤ 4 and 0 ≤ V ≤ 4) strengths. Beyond this region, particles start to localizestrongly. The density distribution for one of the most delocalized points (W = 1 andV = 4) from the weakly localized region is shown in Fig. 4.12.0 1 2 3 4 5 6 7 8V0.010.02ΙW = 1W = 2W = 3W = 4W = 5W = 6W = 7W = 8Figure 4.10: IPR vs interaction for disorder strengths ranging from W = 1 to W = 8.Averaged over 320 realizations of disorders.631 2 3rs0.080.10.120.14 ζ (rs)W = 1W = 2W = 3W = 4W = 5W = 6W = 7W = 81 2 3rs0.050.10.150.2 ζ (rs)W = 1W = 2W = 3W = 4W = 5W = 6W = 7W = 8Figure 4.11: Total correlations in disordered 2D lattics between for two interactingparticles with increasing relative distances. Top and bottom panels show the corre-lations between particles for V = 0 and V = 4 respectively for disorder strengthsranging from W = 1 to W = 8. Averaged over 320 realizations of disorders.64As can be observed, no significant difference in the localization parameters in therange of strong disorder - strong interaction limits are observed compared to the weakdisorder - weak interaction regime. The transition between the regions is smooth.This can be further inferred as an absence of delocalization for interacting particlesin disordered 2D lattices. However, this conclusion remains to be confirmed withother methods. As observed before from the contours in Fig. 4.9 and the changeof behaviour in correlations in Fig. 4.11, there appear to be some physical changesthat should be further studied for any conclusion on the localization-delocalizationtransitions. Besides, a detailed scaling analysis on the whole range of the parametersfrom the lattice of size with 30 sites per dimension to a much larger number of sites perdimension is needed to draw any final conclusion on the localization-delocalizationtransition of interacting particles in two dimensional disordered lattices. Such scale ofcalculations can only be considered after further improvements in the computationalalgorithms and the facilities.X15101520Y15101520 Density0.010.020.030.040.050.010.020.030.04Figure 4.12: Density distribution of interacting particles in disordered 2D lattice forW = 1 and V = 4. Averaged over 320 realizations of disorders.654.4 ConclusionIn this chapter a basic understanding on the length scales of interacting particlesin disordered one- and two-dimensional systems has been obtained by numericalmethods. Enhancement of cowalking correlations has been observed in 1D disorderedsystems. The length scales and correlations calculated for a vast range of parametersbetween interacting particles in disordered 2D systems provide an understanding ofthe localization of interacting particles in two dimensional disordered systems.665ConclusionThe thesis has attempted to achieve the following main goals:1. Develop an understanding of the effect of interaction between particles ontheir correlations in lattices. The correlations were calculated between two hard corebosons in 1D ideal and disordered lattices.2. Develop an understanding of the effect of the range of tunnelling on correlationsbetween particles in lattices. The range of tunnelling was modelled as decayingisotropically with some power of distance. The nature of dynamical correlationsfor both dipolar and Coulombic hopping were found to be different from nearestneighbor hopping models.3. Extend the method of recursive computation of Green’s functions to interactingparticles in disordered 2D lattices and introduce approximations to make it moreaccurate and efficient. The method is shown to be helpful in calculating dynamicsand correlations of interacting particles in lattices of larger size than which can becomputed by full diagonalization method.4. Extend the recursive method for computation of Green’s functions of inter-acting particles in some arbitrary graphs. The cases of binary trees were specificallytaken as an example of such graphs.5. Develop an understanding of correlations and localization properties of inter-acting particles in disordered 2D lattices. The localization parameters of interacting67particles were calculated for a broad range of disorder and interaction strengths.More specifically, we have found for 1D lattice systems that two interacting par-ticles can distinguish between the nature of the interactions (whether repulsive orattractive) when the tunnelling of the particles in lattices becomes long-range. Theparticles become bound with lesser strength of interaction for the attractive casecompared to the repulsive case when tunnelling is long-range. The difference inbinding for the different kind of interactions (attractive vs repulsive) makes thedynamics significantly different in the case of long-range hopping. These featuresmay be used to control the dynamics or transport of particles in lattices by tun-ing the sign and strength of the interaction between the particles. In contrast tothe effect of long-range tunnelling, the effects of long-range interaction were foundto be insignificant. Although such effects are studied only since few years before[116, 117, 118, 119, 120, 121], it seems many more observations are yet to be ob-served.The method of recursion has been developed to calculate Green’s functions ofinteracting particles in one and two dimensional disordered lattices and in binarytrees, which was used to extend the size of the calculations significantly comparedto lattice systems which can be fully diagonalized. The spectral weights for the in-teracting particles have also been calculated for various systems of interest, whichprovides significant insights into those systems. The calculated Green’s functions ofinteracting particles in real space can be Fourier transformed to momentum space.Using the recursion method, exact Green’s functions were calculated for disordered2D systems, which provided insights into the behaviour of interacting particles indisordered systems. The approximations which have been introduced, make thecalculations significantly more efficient while maintaining accuracy. The insightsinto correlations of interacting particles obtained here can also be incorporated intoelectronic structure calculations. For example, the spectral weights obtained fromtwo-particle basis (Fig. 3.1) provides the most significant parts of spectral weightscalculated from full many-body calculations using methods as DMRG [93]. This in-dicates, going from single density basis to two density basis can significantly improvepredictions on properties of interacting electrons in various material systems.For two particles on binary trees, the spectral weight for non-interacting particlesis found to be discontinuous. For stronger interactions, the individual peaks of thespectra tend to merge into one profile, which becomes continuous. The calculationsof Green’s functions of interacting particles on such trees is shown to be exact whencompared to full diagonalization.68Some preliminary calculations to model the effects of uniform magnetic fieldson interacting particles show the splitting of spectra as expected. However, morecalculations remain to be performed to gain further insights into the underlyingphysics of the model systems that have been considered. Calculations of two-particlecorrelations in the presence of inhomogeneous magnetic fields can be considered infuture research.Localization properties of one and two dimensional finite disordered systems havebeen calculated for a vast range of parameters which provides important insightsabout the length scales of the spread of interacting particles in disordered systems.The effect of long range tunnelling on localization of interacting particles in disor-dered 1D systems has been calculated. Long-range tunnelling changes the localiza-tion parameters significantly. However, the prominent disorder-induced correlationswere found to be similar for different ranges of tunnelling. While our results indi-cate the absence of a localization-delocalization transition in disordered 2D systems,further scaling analysis is required to reach a non-ambiguous conclusion. The calcu-lations for the scaling analysis need to be performed for larger lattice sizes, which inthe case of 2D lattices with two correlated particles, will require huge computing re-sources. The effects of interactions on dynamical localization or delocalization mustalso be understood. The role of dissipation on the dynamics and localization are alsoyet to be understood.The calculations of two-particle Green’s functions have shown several interestingfeatures. The bound state has been found to play an important role in controlling thedynamics of the particles. This bound state only becomes effective after a criticalinteraction strength that depends on the band-width of the continuous states. Indisordered systems, the particles were found to be correlated differently in comparisonto the case of ordered systems. However, for strong interactions, disorder was foundto enhance the co-walking correlations. These calculations can be used as the basisto understand the behaviour of a larger number of particles in disordered lattices.The recursive method is expected to be useful for calculations of higher order many-body terms such as three-particle Green’s functions. However, in two and threedimensions, for the calculations of Green’s functions for a larger number of particles,one might require more efficient methods. The large basis size in higher dimensionsfor more particles still remains a hindrance for understanding the physics throughnumerical approaches. However, for interacting particles in disordered systems, thisapproach seems to be the only one that can provide meaningful results.69Bibliography[1] R. Hanbury Brown and R. Q. Twiss. 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The bosonic creation (a†)and annihilation (a) operators satisfy their commutation relation as in the followingequation.aa† − a†a = 1 (A.1)Transforming these operators by addition of constants don’t change their com-mutation properties.a˜† = a† + α and a˜ = a+ α∗a˜a˜† − a˜†a˜ = 1 (A.2)For the commutation relations with higher powers of bose operators one can easilyfind the following.a(a†)n − (a†)n a = n (a†)n−1 = ∂ (a†)n∂a†a† (a)n − (a)n a† = n (a)n−1 = −∂ (a)n∂a(A.3)This can be generalized further for functions that can be expanded in power seriesof bose operators.82af(a†)− f (a†) a = ∂f (a†)∂a†a†f (a)− f (a) a† = −∂f (a)∂a(A.4)One can define the exponential function of these operators as in the followingequation.eαa†=∞∑n=0αnn!(a†)n(A.5)From previous equations it is easy to show the following.aeαa† − eαa†a = αeαa†a†eαa − eαaa† = −αeαa (A.6)Multiplying inverse of the respective exponentials in the previous equation thefollowing relations can be derived.e−αa†aeαa†= α + ae−αaa†eαa = −α + a† (A.7)Similarly one can obtain the following relations.e−αa†eβaeαa†= eβ(e−αa†aeαa†)= eβ(α+a)= eβαeβae−αaeβa†eαa = eβ(e−αaa†eαa)= eβ(−α+a†)= e−βαeβa†(A.8)For a similar operator algebra where a and a† appear together, following relationscan be derived.83eαa†aae−αa†a = e−αaeαa†aa†e−αa†a = eαa† (A.9)Note that∂∂αeαa†aae−αa†a = eαa†a(a†a− aa†)ae−αa†a= −eαa†aae−αa†a (A.10)For such functions f ′(α) = −f(α) has solutions f(α) = e−α ·C and the constant(C) can be found from initial condition f(0) = a.Fermi operatorsOperators of creation (c†) and annihilation (c) of fermions satisfy following commu-tation relations.cc† + c†c = 1 (A.11)(c†)2= 0c2 = 0From which it is easy to show thate−αc†ceαc†=(1− αc†) c (1 + αc†)= c− α2c† + α (cc† − c†c)e−αcc†eαc = (1− αc) c† (1 + αc)= c† − α2c− α (cc† − c†c) (A.12)andeαc†cce−αc†c = e−αceαc†cc†e−αc†c = eαc† (A.13).84Bs-wave scatteringIn this section we discuss the scattering of two quantum particles interacting throughmost simple potentials [122]. This discussion helps in understanding interaction ofquantum particles in most simple forms which can also be read from any introductoryquantum mechanics book. The particles in a many body environment ’feel’ each otherwhen their wavelength is in the range of the interaction potential. This wavelengthis of the same order as the thermal de Broglie wavelengthΛ ' [2pi/(mkBT )]1/2 (B.1)where m, kB, T are, respectively, the mass, the Boltzmann constant and the temper-ature. The interaction becomes important when Λ ' n−1/3 (n is the density) that iswhen the de Broglie wavelength reaches the limit of the interatomic distance.Assuming a certain form of the interaction potential, the Schro¨dinger equationneeds to be solved to understand the behaviour of the interacting particles. Assuminga centro-symmetric potential, the Schro¨dinger equation for the radial part is (with~ = 1) [12mr(− d2dr2− 2rddr+l(l + 1)r2)+ V(r)]Rl(r) = ERl(r) (B.2)Here l is the quantum number for angular momentum of the two particles. Multiply-ing by 2mr on both sides and using the substitutions ε = 2mrE and U(r) = 2mrV(r),Eq. B.2 reduces toR′′l +2rR′l +[ε− U(r)− l(l + 1)r2]Rl(r) = 0. (B.3)85With one more substitution χl(r) = rRl(r), it becomes the following simple 1DSchro¨dinger equation with ε = k2 and Ueff (r) taking account of the rotational energyχ′′l +[k2 − Ueff (r)]χl = 0. (B.4)The rotational energy produces an effective barrier in the potential and a classicalturning point rcl =√l(l+1)k. If the typical range of the potential r0 is less than thede Broglie wavelength Λ, then interaction is not important. Associating a typicalwavenumber k (where k = 1Λ) for the relative motion of atoms, the condition emergesas kr0 1 when interaction is negligible. Equating it with the previous equation,we find that the interaction is important when the range of the potential becomescloser to that of rcl.kr0 =√l(l + 1)r0/rcl 1⇐⇒ rcl r0 for l 6= 0.In the low energy limit, when l = 0, this condition is not sufficient. Scattering cantake place when the barrier is absent. In this s-wave regime, the previous inequalitiesare valid at low collision energies.Looking at the case of a free particle for l = 0 we can gain primary understandingof the scattering eventχ′′0 + k2χ0(r) = 0, (B.5)with the general solution χ0(k, r) = c0 sin(kr+ η0). The radial wave function is thenR0(k, r) =c0krsin(kr + η0) (B.6)where η0 = 0 ensures non-singularity at the origin.The general solution for arbitrary l is found in terms of spherical Bessel functionsjl and spherical Neumann functions nlRl(kr) = cl [cos ηljl(kr) + sin ηlnl(kr)] . (B.7)For l = 0 this produces R0(k, r) = c0j0(kr). This solution is also valid whenr r0, where the free particle assumption is valid for short range potentials. Atlarge distances(r 1k= Λ)for short range potentials, the spherical Bessel andNeumann functions take their asymptotic form and we get the following formula withthe asymptotic phase shift ηl(k) even when the scattering is elastic and conservesasymptotic momentum k.86Rl(kr) 'r→∞clkr[cos ηl sin(kr − lpi2) + sin ηl cos(kr − lpi2)]'r→∞clkrsin(kr + ηl − lpi2) (B.8)'r→∞cl2kı[e−ıηle−ı(kr−lpi2)r− eıηl eı(kr− lpi2)r]'r→∞c′l2k[e−ıkrr− eılpie2ıηl eıkrr](B.9)The final form shows that the outgoing wavefunction has a phase factor Sl = e2ıηlcompared to the incoming wave other than the sign. The length parameter a (a =−ηlk) possesses valuable information about the interaction potential. One can inferwhether the interaction is attractive (a < 0) or repulsive (a > 0) from it. It alsoprovides information about the range of interaction.In the presence of an interaction potential as simple as a spherical well with flatbottom as in the following equation, one can find resonant scattering energies wherethis interaction changes from attractive to repulsive or vice versaU(r) ={−κ20 for r ≤ r00 for r > r0. (B.10)Let’s define some parameters for this potential. For energy in continuum range,( = k2c > 0) K+ =√κ20 + k2c and in the well depth ( = k2b < 0) K− =√κ20 − k2b .The 1D s-wave Schro¨dinger equation is thenχ′′0 +[k2c − U(r)]χ0(r) = 0 for l = 0 and > 0 (B.11)which has the following solutions with a phase shift outside the wellχ0(k, r) ={C< sin(K+r) for r ≤ r0C> sin(kcr + η0) for r > r0. (B.12)Applying the boundary condition to both of these solutions at r = r0, the phaseshift can be found.χ′0/χ0|r=r0 = K+ cotK+r0 = kc cot(kcr0 + η0) (B.13)Now, define the scattering length a in the limit kc → 0 (hence independent of kc).87a = limkc→0a(kc) = − limkc→0η0(kc)kc(B.14)In this limit, K+ → κ0 and the previous boundary condition becomesκ0 cotκ0r0 =1r0 − a. (B.15)Following this we findar0= 1− tanκ0r0κ0r0. (B.16)The behaviour of the scattering length as a function of the interaction strength isshown in Figure B.1. The Figure shows that it diverges at half of odd integer valuesof κ0r0.For energies within the range of the well depth, the 1D Schro¨dinger equationtakes the following form.χ′′0 +[−k2b − U(r)]χ0(r) = 0 for l = 0 and < 0, (B.17)which has the following solutions without any phase shiftχ0(k, r) ={C< sin(K−r) for r ≤ r0C>e−kbr for r > r0 and kb > 0.(B.18)Applying the boundary condition to both of these solutions at r = r0, one obtainsthe following relationsχ′0/χ0|r=r0 = K− cotK−r0 = −kb. (B.19)In the limit kb → 0 hence K− → κ0, we findκ0 cotκ0r0 = 0. (B.20)It has the solutions forκ0r0 =(υ +12)pi (B.21)where υ = 0, 1, 2, ...υmax is the vibrational quantum number of the well. For existenceof at least one bound state, it is required that κ0r0|min = pi2 . However, this solution isderived from a 3D spherically symmetric potential. In 2D, it turns out that a boundstate exists for any shallow potential.880 0.5 1 1.5 2 2.5 3κ0r0 /pi-5-2.502.55 a/r0Figure B.1: The s-wave scattering length a normalized to the well width as a functionof the depth of the well for the case of very low energy scattering. The scatteringlength is one unit of the width of the well except in the resonant regions where itdiverges for certain well depth.Understanding the basics of two particle scattering even at low energies show thatthe interaction between the two particles can be controlled if one can find the resonantscattering energies. Not only that, the interaction can also be made attractive orrepulsive around the resonances with very small change in scattering energy. In thenext section we learn how this is achieved experimentally.89CFeshbach resonanceIn Appendix B, scattering of two particles on one single potential was briefly de-scribed. Every atom or molecule has many states that can couple while the scatter-ing takes place. Here we will take a look at the most simple of such situation withonly two states involved. One of them will be an open channel, where particles comeand leave without any binding. The other we will take as a closed channel whereparticles can form a composite bound state in the process. The coupling betweentwo such channels will allow control of the scattering length and hence the effectiveinteraction strength between the particles. In an experiment, the coupling can becontrolled both optically or using external magnetic fields [55, 56]. For the case ofstates coupled via magnetic field, we will take one of the states to be a triplet andthe other state as a singlet. Coupling between these kind of states can be achievedvia spin flip of one of the particles.In the presence of a magnetic field the electron spins exhibit the Zeeman inter-actionHZ = γes1 ·B + γes2 ·B = γeS ·B (C.1)where γe is the gyromagnetic constant of the electron. As can be seen this Hamil-tonian conserves total spin and does not couple singlet and triplet channels by anyspin flip. There is also nuclear Zeeman interaction describing the coupling of nuclearspins to magnetic fieldHZ = γni1 ·B + γni2 ·B = γni ·B. (C.2)This term does not couple the singlet or triplet channels either. The hyperfineeffect couples the electron spins with nuclear spins90Hhf = α1i1 · s1 + α2i2 · s2, (C.3)where α1,2 are the hyperfine interaction constants. This equation can be written intotal spin conserving (H+hf ) and non-conserving terms (H−hf ). The non-conservingterm can couple singlet and triplet channels via change in total spinH±hf =α12(s1 ± s2) · i1 ± α22(s1 ± s2) · i2. (C.4)Hhf = H+hf +H−hf (C.5)When two particles are identical (α1 = α2), these terms can be simplified further.H±hf =α2(s1 ± s2) · (i1 ± i2) (C.6)Now solving the radial part similar to the Eq. B.2 without any magnetic fieldapplied for some Hamiltonian Hr yields the following equation with spin indexesaddedR′′l,S +2rR′l,S +[ε− Ul,S(r)− l(l + 1)r2]Rl,S(r) = 0. (C.7)where Ul,S(r) contains the spin-spin interaction part beside the singlet or triplet chan-nel potential which also conserve total spin. In a basis like {|ν, l,ml〉|s1, s2, S,Ms〉},the energies of the open channel (ε > 0) for wavefunctions Rl,S(k, r) belongs incontinuumεk = k2c (C.8)and for the closed channel (ε < 0) for wavefunctions Rl,S(v, r) is that of boundvibrational states with ro-vibrational energyεS,lv = −κ2v,S + l(l + 1)Rlv,S. (C.9)where Rlv,S = 〈Rl,S(v, r)|r−2|Rl,S(v, r)〉.In the presence of a magnetic field, as both channels get coupled, one needs tosolve the following secular equation|〈i′1, i′2,m′1,m′2|〈s′1, s′2, S ′,M ′s|〈ν ′, l′,m′l|H−E|ν, l,ml〉|s1, s2, S,Ms〉|i1, i2,m1,m2〉| = 0(C.10)91where the total Hamiltonian now contains both the Zeeman and hyperfine partsH = Hr +HZ +H+hf +H−hf . (C.11)Although this Hamiltonian couples different electronic spin states, it doesn’tchange the total spin projection of the nuclear and electronic spin states (MF =m1+m2+Ms). One can see that from rewriting the hyperfine terms with (szj, sxj, syj)and (izj, ixj, iyj) operators where j ∈ {1, 2}.H+hf =2∑j=1α2[Szijz +12{S+ij− + S+ij+}] (C.12)H−hf =2∑j=1(−)j−1α2[(s1z − s2z)ijz + 12{(s1+ − s2+)ij− + (s1− − s2−)ij+}] (C.13)The secular equation now reads as follows.|(εl,Sv + EσB − E)δσ,σ′ + 〈v′, l′, S ′|v, l, S〉〈σ′|H+hf |σ〉+ 〈v′, l′, S ′|v, l, S〉〈σ′|H−hf |σ〉| = 0(C.14)where σ(′) has all the electronic and nuclear spin indices. In most cases the singlet-triplet coupling via the third term in the equation doesn’t dominate as the radialwavefunctions are very different in small distances because the potentials are gen-erally very different and contribute negligible overlap which suppresses the last twoterms. However, a large overlap at long distances for both the singlet and tripletstates, if any, makes this term significant. For cases when the energies of both theasymptotic singlet and triplet states are resonant at long distances, the radial overlapcan be taken approximately as full overlap and a spin flip can happen.|(εl,Sv + EσB − E)δσ,σ′ + 〈σ′|H+hf |σ〉+ 〈σ′|H−hf |σ〉| = 0 (C.15)So, we look for a way to make any two of such potentials resonant with each other.Let’s look at how the energies of the potentials change in response to an appliedperturbation U in general. For s-wave scattering the energy of the continuum statesis given by k2 and those of unperturbed bound states by ευ. This bound state in thepresence of a perturbation mixes with all the other bound and continuum states,H = H + U (C.16)92|υ〉 U→ |φ〉 and ευ U→ −κ2 (C.17)|φ〉 =∑υ′|υ′〉〈υ′|φ〉+∫dk|k〉〈k|φ〉 (C.18)which leads to the very general following form,〈υ|H|φ〉 =∑υ′〈υ|H|υ′〉〈υ′|φ〉+∑υ′〈υ|U |υ′〉〈υ′|φ〉+∫dk〈υ|H|k〉〈k|φ〉+∫dk〈υ|U |k〉〈k|φ〉. (C.19)Applying orthogonality of states results in a simpler form−(κ2 + ευ)〈υ|φ〉 =∫dk〈υ|U |k〉〈k|φ〉. (C.20)From coupling of the continuum states with that of perturbed states we find thefollowing:〈k|H|φ〉 =∑υ′〈k|H|υ′〉〈υ′|φ〉+∑υ′〈k|U |υ′〉〈υ′|φ〉+∫dk〈k|H|k〉〈k|φ〉+∫dk〈k|U |k〉〈k|φ〉. (C.21)This equation can be simplified by assuming 〈k|H|φ〉 = −κ2〈k|φ〉 to lowest order ofperturbation on left hand side and applying the orthogonalities−(κ2 + k2)〈k|φ〉 =∑υ′〈k|U |υ′〉〈υ′|φ〉 (C.22)Substituting this equation into Eq. C.20 we find(κ2 + ευ) =∫dk|〈υ|U |k〉|2κ2 + k2. (C.23)For the case of a general form for the potential U , which is short range andcentro-symmetric, this equation can be approximated as〈υ|U |k〉 =∫dr〈υ|r〉U(r)eık·r =∫dr〈υ|r〉U(r)∣∣∣r<r0= u0 (C.24)93At long range this can be taken as zero for a short range perturbation potential.Eq. C.23 now simplifies to a simple integral(κ2 + ευ) = u20∫ kmax0dk4pik2κ2 + k2= 4piu20[kmax − κ tan−1(kmaxκ)]= ε0 − κ 4piu20 tan−1(kmaxκ)(C.25)where ε0 = 4piu20kmax and ε0∣∣∣κ→0= ευ which is the threshold for the perturbedenergy for unperturbed asymptotic bound state of very small energy. The detuningfrom threshold is given by εres = ευ − ε0 which can be written asεres = −κ2 − κ 4piu20 tan−1(kmaxκ). (C.26)In the range for κ, where tan−1(kmaxκ)= pi2and independent of kmax, a charac-teristic parameter (R∗) can be introducedR∗ =12pi2u20and εresR∗ = −R∗κ2 − κ (C.27)which gives us the relation of the perturbed energy to that of the unperturbed energy−κ2 = − 14R∗2[−1 +√1− 4R∗2(ευ − ε0)]. (C.28)To generalize this form further, one can define a characteristic scattering lengthparameter aresares =1κ+R∗κ2= − 1R∗εres. (C.29)The open channel also contributes to the scattering length which can be termedas the background contribution abg. The general total scattering length then takesthe following form:a = ares + abg = abg − 1R∗εres. (C.30)To bring two potentials at asymptote on resonance, the Zeeman interaction is veryuseful when both have different spin projections. Suppose they reach a resonance94at field B0 at asymptotic energy ε0, then the differential change of energy is linearlyproportional to that of the difference in field strength close to a resonanceευ(B)− ε0 = 2µδµM(B −B0) (C.31)This can be substituted into Eq. C.33 to find the following general form.a = abg − 12µR∗δµM(B −B0) (C.32)a = abg(1− ∆BB −B0)(C.33)where ∆B =1abgR∗2µδµM.0 1 2 3 4 5B (arb. unit) -6-30369 a/abgFigure C.1: s-wave scattering length in Feshbach resonance.95In almost all cases abg 6= 0. This possibility of tuning resonant scattering bymagnetic fields has applications ranging from formation of molecules (polar andnon-polar) to control of interatomic interaction (both attractive and repulsive) inoptical lattices.In this thesis, a broad range of interaction parameter (both attractive and repul-sive) is considered to explore the physics between two interacting particles in bothordered and disordered lattices. The experimental method, described here only inbrief, justifies for undertaking of such consideration where a broad range of interac-tion strength can be realized.96DInteraction via phononsIn lattices, there are always lattice motional states present as phonons. Thesephonons can also play an important role in controlling inter-particle interactions.The general Hamiltonian in a 1D lattice where the motions of lattice particlesare coupled to nearest neighbors only and are harmonic, one can write the collec-tive motional modes in terms of creation (annihilation) operators giving rise to thebosonic quasiparticles well known as phononsH =∑lp2l2m+ V (y1, y2, ....) (D.1)V =∑l12K(yl − yl+1)2 where Vl,l′ =2K if l = l′−K if l = l′ ± 10 else(D.2)Vq =∑|l−l′|=0,±1eıq(l−l′)Vl,l′= 2K −K(eıq + e−ıq)= 4K sin2(q2)(D.3)= Mω2qωq = 2√KMsin(q2)(D.4)The individual site (l, l′) displacement coordinates and momenta can be Fouriertransformed to collective lattice modes97yq =1√N∑le−ıqlyl and pq =1√N∑leıqlpl (D.5)where the periodic boundary condition yl+N = yl would make it quantized withinfirst Brillouin zone (q = 2pinN,−N2< n < N2+ 1, n ∈ Z).H =∑q12Mp†qpq +12Vqy†qyq (D.6)Writing the collective modes in terms of the creation (annihilation) operatorsgives rise to quasiparticles well known as phononsyq =√12Mωq(a†q + a−q)and pq = ı√Mωq2(a†q − a−q)(D.7)H = ωq(a†qaq +12)(D.8)In general, when the range of coupling is taken beyond nearest neighbors and thepotential is not so simple one will require to diagonalize full Hamiltonian in order tofind the uncoupled normal modes and their energies.In 3D lattices one needs to uncouple the coordinate displacements along threeseparate axes, that are coupled via the potential term, into three orthogonal direc-tions of polarizations(s).aqs =1√2Mωq,s(Mωq,syq + ıp†q) · sq (D.9)a†qs =1√2Mωq,s(Mωq,syq − ıp†q) · sq (D.10)H = ωq,s(a†q,saq,s +12)where ωq,s =√V sqM(D.11)In an ideal lattice, without any electron-phonon coupling the unperturbed Hamil-tonian is the sum of the individual electron and phonon energiesH0 =∑kεkc†kck +∑q,sωq,sa†q,saq,s. (D.12)98The interaction of phonons with electrons can be seen as scattering differentelectronic states via the potential due to the displacement of lattice particles fromtheir equilibrium positionsHI =∑k,k′,l〈k|V (r− l− yl)|k′〉c†kck′=∑k,k′,leı(k′−k)·(l+yl)Vk−k′c†kck′'∑k,k′,leı(k′−k)·l (1 + ı(k− k′) · yl)Vk−k′c†kck′=∑k,k′,leı(k′−k)·l(1 + ı1√N(k− k′) ·∑qeıq·lyq)Vk−k′c†kck′= NV0∑kc†kck + ı√N∑k,k′(k− k′) · yk−k′Vk−k′c†kck′= NV0∑kc†kck + ı√N2Mωq,s∑k,k′,s(k− k′) · s (a†q,s + a−q,s)Vk−k′c†kck′ (D.13)where q = k− k′. After some readjustment we are led to the simple form which nowhas the electron-phonon interactionHF = H0 +He−p =∑kεkc†kck +∑qωqa†qaq +∑k,k′Mkk′(a†q + a−q)c†kck′ . (D.14)We can now look for how this electron-phonon interaction can effectively mediateelectron-electron interaction, both attractive and repulsive.The energy of the electrons can be calculated by using theHe−p as a perturbation.To zeroth order the electron and phonon energies contribute separately. The firstorder term vanishes as there are number non-conserving phonon operators in He−p.The effect of interaction can be found in the second order termE2 = 〈Φ|He−p 1E0 −H0He−p|Φ〉, (D.15)where |Φ〉 = |nk, nq〉 has nq number of phonons in mode q and nk number of electronsin state k.With the full form ofHe−p inserted in the previous equation we find the following:99E2 = 〈Φ|∑k,k′Mkk′(a†q + a−q)c†kck′1E0 −H0∑k′′,k′′′Mk′′k′′′(a†q′ + a−q′)c†k′′ck′′′ |Φ〉= 〈Φ|∑k,k′|Mkk′ |2{a†−qc†kck′1E0 −H0a−qc†k′ck + a†qc†kck′1E0 −H0a†qc†k′ck}|Φ〉=∑k,k′|Mkk′ |2〈nk(1− nk′)〉( 〈n−q〉Ek − Ek′ + ω−q +〈nq + 1〉Ek − Ek′ − ωq)(D.16)With ωq = ω−q (in systems with time reversal symmetry), this equation simplifiesby letting nq = n−qE2 =∑k,k′|Mkk′ |2〈nk〉(2(Ek − Ek′)〈nq〉(Ek − Ek′)2 − (ωq)2 +〈1− nk′〉Ek − Ek′ − ωq). (D.17)The first term of Eq. D.17 shows that the electron-phonon interaction is pro-portional to the phonon number. The second term has phonon mediated electron-electron interaction term. As can be observed, this term can be both repulsive orattractiveEpm =∑k,k′|Mkk′|2〈−nknk′〉 ωq(Ek − Ek′)2 − (ωq)2 . (D.18)Another way of arriving at this form of electron-electron interaction from theFrohlich Hamiltonian, is by the powerful technique of canonical transformation. Oneneeds to transformHF = H0 +He−ptoH˜ = e−s HF es (D.19)= (1− s+ s22− · · ·) HF (1 + s+ s22+ · · ·)= H0 +He−p + [H0, s] + [He−p, s] + · · · (D.20)One can eliminate the phononic operators in H˜ by a choice in s up to certainorders of perturbation in s. Since H0 does not include the phononic operators, onechoice is to have phononic operators in s in such a way that He−p gets cancelled by[H0, s] and the final form gives rise to the following form where only the electron-electron interaction terms will be retained.100H˜ = H0 + [He−p, s]Chosing s in the following ways =∑k,k′Mkk′(Aa†−q +Baq)c†kck′ (D.21)one can find A = − 1Ek−Ek′+ω−q and B = −1Ek−Ek′−ωq .The transformed Hamiltonian (H˜) then takes the following form where the electron-electron interaction term without any direct phonon term included can be observed.H˜ = H0 +∑k,k′|Mkk′|2 ωq(Ek − Ek′)2 − (ωq)2 c†k′+qc†k−qckck′ + · · ·In recent times, unprecedented control has been achieved to have a controllednumber of phonons in a system. There are also experiments where one studiestwo phonon correlations[123]. Phonons can not only mediate interaction betweenparticles in a lattice, but been used to make the particles hop to sites at long rangein lattices as described in the introduction chapter.101ECoupling two states coherentlyIn this section we discuss how two states can be coherently coupled using opticaltools of two detuned lasers. Preparation of states at lattice sites requires not onlyspatial control but also temporal control. In this section we discuss the simplest caseof effectively controlling the preparation of a site in a superposition of two states.Let’s consider the system of two closely lying energy levels (|1〉 and |2〉) coupledoptically to a far lying excited state (|e〉) .The Hamiltonian of the system isH0 = ω1|1〉+ ω2|2〉+ ωe|e〉. (E.1)Two lasers with frequencies λ1 = (ωe − ∆) − ω1 (which is proportional to thegap between a detuned level below the excited state from the first state) and λ2 =(ωe−∆)− (ω2 + δ) (which corresponds to the gap between the detuned level slightlyabove the second state with the detuning ∆ and the detuned level below the excitedstate with the detuning δ) can be used to effectively couple two states to the excitedstate.So we have (with ωe1 = ωe − ω1 and ωe2 = ωe − ω2)λ1 = ωe1 −∆λ2 = ωe1 −∆− (ω12 + δ)λ1 = ωe2 −∆ + ω12λ2 = ωe2 −∆− δ (E.2)In the presence of an external field the interaction Hamiltonian becomes thefollowing:102HI = −µ · E(t)= − [µe1|1〉〈e|+ µ1e|e〉〈1|+ µe2|2〉〈e|+ µ2e|e〉〈2|]E0[cos(λ1t) + cos(λ2t)] (E.3)where the spatial variance of the field is ignored.Solving the Schrodinger equationı~∂ψ(t)∂t= (H0 +HI)ψ(t) (E.4)the wavefunctionψ(t) = C1(t)|1〉+ C2(t)|2〉+ Ce(t)|e〉 (E.5)one can find the folowing relations:ıC˙1 = ω1C1 − gCe [cos(λ1t) + cos(λ2t+ φ)]ıC˙2 = ω2C2 − gCe [cos(λ1t) + cos(λ2t+ φ)]ıC˙e = ωeCe − g∗ [C1 + C2] [cos(λ1t) + cos(λ2t+ φ)] (E.6)where explicit time dependence on the coefficients has not been shown and we assumeg = µe1E0 = µe2E0. One can write these coefficients in their own rotating frame tomake the equations look simplerC˜i = Cieıωit for i ∈ 1, 2, e and ı = √−1 (E.7)and transform the previous set of equation into the following form with cosines nowexpanded into exponentialsı ˙˜C1 =g2C˜ee−ıωe1t [eıλ1t + e−ıλ1t + eıλ2t + e−ıλ2t]ı ˙˜C2 =g2C˜ee−ıωe2t [eıλ1t + e−ıλ1t + eıλ2t + e−ıλ2t]ı ˙˜Ce =g∗2[C˜1eıωe1t + C˜2eıωe2t] [eıλ1t + e−ıλ1t + eıλ2t + e−ıλ2t]. (E.8)After change of the variables we find the following:103ı ˙˜C1 =g2C˜e[e−ı∆t + e−ı(∆+δ+ω12)t]ı ˙˜C2 =g2C˜e[e−ı(∆−ω12)t + e−ı(∆+δ)t]ı ˙˜Ce =g∗2[C˜1[eı∆t + eı(∆+δ+ω12)t]+ C˜2[eı(∆−ω12)t + eı(∆+δ)t]](E.9)Solving for the excited state we find the following after integration at time τC˜e(τ) = −g∗2[C˜1(τ)[eı∆τ − 1∆+eı(∆+δ+ω12)τ − 1∆ + δ + ω12]+ C˜2(τ)[eı(∆−ω12)τ − 1∆− ω12 +eı(∆+δ)τ − 1∆ + δ]]' − g∗2∆[C˜1[eı∆τ + eı(∆+δ+ω12)τ − 2]+ C˜2 [eı(∆−ω12)τ + eı(∆+δ)τ − 2]] (E.10)where in the denominators we have applied the assumption that ∆ ω12 > δ.Putting this equation back into the Schro¨dinger equation for C˜1 and C˜2 one canfind the dynamics between the two states:˙˜C1 = ı|g|24∆[C˜1[eı∆τ + eı(∆+δ+ω12)τ − 2]+C˜2[eı(∆−ω12)τ + eı(∆+δ)τ − 2]] [e−ı∆t + e−ı(∆+δ+ω12)t]' ı |g|24∆[C˜1[2 + eı(δ+ω12)τ + e−ı(δ+ω12)τ]+C˜2[2e−ıω12τ + eıδτ + e−ı(δ+2ω12)τ]' ı |g|24∆[2C˜1 + C˜2eıδτ](E.11)where the rotating-wave approximation has been applied to disregard exponentialswhich have large frequencies as ∆ in first step and ω12 in the second step.Similarly˙˜C2 ' ı |g|24∆[2C˜2 + C˜1e−ıδτ](E.12)which results in the following effective Hamiltonian where the two states are coupledwith the Rabi frequency Ω = |g|22∆104Heff = −|g|22∆[|1〉〈1|+ |2〉〈2|]− |g|24∆[eıδτ |1〉〈2|+ e−ıδτ |2〉〈1|] (E.13)' −Ω2[eıδτσ− + e−ıδτσ+](E.14)where σ+(−) are the raising (lowering) operators.The phase between the two laser beams can now be put back into the equationHeff = −Ω2[eı(δτ+φ)σ− + e−ı(δτ+φ)σ+](E.15)Fixing the detuning to δ = 0, makes the dynamics between the states easily control-lable using the phase between two photons. See details in reference [54].Heff = −Ω2[eıφσ− + e−ıφσ+](E.16)In the introductory chapter, this approach has been used to effectively couplesites at long range, whcih has been recently implemented experimentally [59].105FOptical latticesOptical lattices are created as standing electric field waves by constructive interfer-ence of multiple laser beams. The lattice constant of these systems is typically inthe order of the µm. In these lattices, the dynamics of the atoms loaded, can be eas-ily controlled as the inter-site tunnelling parameter can be determined by fixing theintensity of the laser beams. The atoms feel the lattice potential according to theirAC-Stark shift. The geometry, shape and trap depth of these lattice systems can bemodified experimentally. The temperature of the atoms loaded in these lattices canalso be very low (< 10−9K).The atoms interact with the laser field through the dipole moment induced bythe electric fieldHI = d · E where E = E(x) cos(ωt) (F.1)The change in ground state energy due to this perturbation can be calculated asE (2)g =∑e 6=g|〈e|d|g〉|24[1Ee − Eg − ω −1Ee − Eg + ω]E(x)2 (F.2)The first order term cancels as the expectation value of the dipole operator forthe ground state is zero.The lifetime of the excited state(1Γe)can be accounted in the equation as mod-ified excited state energy from uncertainty principle.E (2)g =∑e 6=g|〈e|d|g〉|24[1Ee − ıΓe2 − Eg − ω− 1Ee − ıΓe2 − Eg + ω]E(x)2 (F.3)106Assuming a two level system, after applying the rotating wave approximation,one obtains the spatial potential felt by the particles locallyVg(x) =Ω2δ4δ2 + Γ2e' Ω24δ2δ (F.4)where Ω(x) = 〈e|d|g〉E(x) is the Rabi frequency.For the red detuned photons (δ < 0), this potential draws the particles to theintensity maximum while for the blue detuned (δ > 0) photons to the intensityminimum. Particles in the excited state however will feel the opposite effect.The imaginary part can be understood as the photon absorption rate related tothe lifetime of the ground state and represents the rate of loss of the ground state.Γ(x) =12Ω2Γe4δ2 + Γ2e' Ω28δ2Γe (F.5)The geometry of 1D, 2D and 3D optical lattices can be ideally represented by thefollowing equations for standing wavesV 1Dx = V0 cos2(kx), k =2piλ, a =λ2(F.6)where a is the lattice constant. In two dimensions four beams can be used to createa rectangular optical lattice with many desired types of potentialsV 2Dx,y = V0[cos2(kx) + cos2(ky) + 21 · 2 cos(φ) cos(kx) cos(ky)](F.7)while using three beams one can prepare hexagonal optical lattices.For the case of a 3D optical lattice, one can use three pairs of beamsV 3Dx,y,z = V0[cos2(kx) + cos2(ky) + cos2(kz) +m2(ω2x + ω2y + ω2z)](F.8)The states of particles generally spread over a few lattice sites and can be ap-proximated by Wannier orbitals (see Appendix G). In many cases of interest of suchsystems, it would suffice to consider the overlap of the wavefunctions localized oneach lattice site with that of nearest neighbors sites giving rise to hopping of particleslimited to nearest neighbors.107GExcitonsExcitons are composite quasiparticles of interacting electrons and holes. Its energydepends on electron-electron interactions, hole-hole interactions and electron-hole in-teractions. The theoretical models that were analyzed in this thesis also hold relevantfor few cases of exciton physics, specifically to physics of interacting Frenkel excitons.In this Appendix we discuss the basic understanding of excitonic Hamiltonians.electron-electron interactionElectrons in lattices are understood through the following Hamiltonian which con-sists of kinetic energy of the electrons, electron-electron Coulombic interaction andinteraction with the lattice potential. In absence of any magnetic interaction, theSchro¨dinger equation takes the following form.[∫d3xψ†(x)(−~2∆2m+ VL(x))ψ(x)+12∫d3xd3x′ψ†(x)ψ†(x′)e|x− x′|ψ(x)ψ(x′)]|Φ〉 = E|Φ〉 (G.1)where the interaction is approximated to two body form only and the wavefunctionsare written in terms of fermionic operators ψ†(x) and ψ(x). These operators can beexpanded in terms of eigenfunctions φj(x) and φ∗j(x).108ψ(x) =∑jcjφj(x)ψ†(x) =∑jc†jφ∗j(x) (G.2)The full wavefunction can be taken as that of electrons occupying the eigenstates.|Φ〉 = c†j1c†j2.....c†jN |0〉 (G.3)The Hamiltonian now transforms into the following form.H =∑mnc†mcn∫d3xφ∗(x)(−~2∆2m+ VL(x))φ(x)+12∑mnm′n′c†mc†ncn′cm′∫d3xd3x′φ∗(x)φ∗(x′)e|x− x′|φ(x)φ(x′) (G.4)From the kinetic energy part one can find the following.〈Φ|c†mcn|Φ〉 = 〈0|cj1cj2.....cjN c†mcn c†j1c†j2..c†j′ ..c†jN |0〉= 〈0|cj1cj2.....cjN c†m c†j1c†j2.. ..c†jN |0〉(−1)exchange= 〈0|cj1cj2.. ..cjN c†j1c†j2.. ..c†jN |0〉= δmn for m ∈ j1, j2, ...., jN (G.5)where the number of exchanges are same for application of both c†m and cn hence nominus sign. Similarly in the interaction term the following terms can be found.〈Φ|c†mc†ncn′cm′|Φ〉 = δnn′δmm′ − δnm′δmn′ for m,n ∈ j1, j2, ..., jN and m 6= n,m′ 6= n′(G.6)where in the first term of right hand side, the number of exchanges of fermionicoperators stay even as one can observe from the previous equation first applying c†mand cm′ then c†n and cn′ . The second term will require one extra exchange betweenc†m and c†n hence having a minus sign. For bosonic operators, however, no minus signwill be there for both Coulombic interaction and exchange terms.109Thus we arrive at the Hartree-Fock expression of total energy where the interac-tion term with no minus sign is described as that of Coulombic interaction and theterm with minus sign is described as that of exchange interaction.〈Φ|H|Φ〉 =∑j1∫d3xφ∗j1(x)(−~2∆2m+ VL(x))φj1(x)+12∑j1,j2∫d3xd3x′φ∗j1(x)φ∗j2(x′)e|x− x′|φj2(x′)φj1(x)−12∑j1,j2∫d3xd3x′φ∗j1(x)φ∗j2(x′)e|x− x′|φj1(x′)φj2(x)E = E0 + EC + Eex (G.7)electron-hole interactionEmpty states within a full valence band (V ) can be represented by particles calledhole for simplification.cj,V = d†jc†j,V = dj (G.8)The fermionic operators d here represents holes. The Hamiltonian now becomesH = EV −∑jd†jdjEj,V +12∑j1,j2,j3,j4d†j1d†j2dj3dj4V(j1j2|j3j4) (G.9)where Ej,V now contains both interaction and exchange terms of electrons and Vcontains explicit hole-hole interaction term as can be seen from following equations.c†mc†ncn′cm′ = dmdnd†n′d†m′= δnn′δmm′ − δnm′δnm′− δnn′d†m′dm + δmn′d†m′dn − δmm′d†n′dn + δnm′d†n′dm+ d†n′d†m′dmdn (G.10)110V(j1j2|j3j4) =∫d3xd3x′φ∗j1(x)φ∗j2(x′)e2|x− x′|φj3(x′)φj4(x) (G.11)To deal with holes in valence band and electrons in conduction band we use thefermionic operators with extra notation.cj1,J1c†j2,J2+ c†j2,J2cj1,J1 = δj1,j2δJ1,J2 (G.12)where J symbols are band indexes.Now the total interaction can be written as following.H ′ =12∑j1,j2,j3,j4;J1,J2,J3,J4c†j1,J1c†j2,J2cj3,J3cj4,J4·∫d3xd3x′φ∗j1,J1(x)φ∗j2,J2(x′)e2|x− x′|φj3,J3(x′)φj4,J4(x)(G.13)Using the hole operators in valence band and electron operators in conductionband one can remove the band indexes and write the interaction Hamiltonian interms of electron-electron interaction in conduction band, hole-hole interaction invalence band and electron-hole interaction between conduction and valence band.H ′ = He−e +Hh−h +He−h (G.14)He−e =12∑j1,j2,j3,j4c†j1c†j2cj3c†j4V(j1j2|j3j4) (G.15)Hh−h =12∑j1,j2,j3,j4dj1dj2d†j3d†j4V(j1j2|j3j4) (G.16)The electron-hole interaction is combination of many parts.111He−h =∑j1,j2,j3,j4[c†j1dj2d†j3cj4 + dj1c†j2cj3d†j4+ c†j1dj2cj3d†j4+ dj1c†j2d†j3cj4]V(j1j2|j3j4)(G.17)From each terms in electron-hole interaction Hamiltonian induced electron energyterms can be found. For examplec†j1dj2d†j3cj4 = c†j1cj4dj2d†j3= c†j1cj4[δj2j3 − d†j3dj2](G.18)Wannier excitonsFor electrons and holes with relative motion delocalized over whole lattice, one candescribe these electrons according to their occupation at valence band maximumand holes to their occupation at conduction band minimum where the dispersionis parabolic. For such a single electron-hole pair in 3D lattices, they behave essen-tially as Hydrogen atom and similar spectra is expected beside the effect of latticeconfinement. In such cases the Hamiltonian takes the following form.H =∑k(E0C +~2k22mC)+∑k(−E0V + ~2k22mV)+12∑k1,k2,k3,k4c†k1c†k2ck3ck4V(k1k2|k3k4)+12∑k1,k2,k3,k4d†k1d†k2dk3dk4V(k3k4|k1k2)+∑k1,k2,k3,k4c†k1ck2d†k3dk4V(k1k4|k2k3) + · · · (G.19)where E0C is the minimum of conduction band and E0V is the maximum of valenceband.Writing a single combined quasiparticle operator that can be called exciton replac-ing separate operators for electrons and holes, one can show that these quasiparticleseffectively are boson like.q†j = c†ked†kh (G.20)112q†j1q†j2= q†j2q†j1(G.21)qj1q†j2= dk′hck′ec†ked†kh= dk′h(δk′e,ke − c†keck′e)d†kh= δk′e,ke(δk′h,kh − d†khdk′h)− c†ke(δk′h,kh − d†khdk′h)ck′e (G.22)[q†j1 , q†j2]−= [qj1 , qj2 ]− = 0[qj1 , q†j2]−= δj1,j2 −Dj1,j2 (G.23)where Dj1,j2 is the operator describing deviation from bosonic commutation ruleessentially consisting of scattering components between electron and hole.Dj1,j2 = δk′e,ked†khdk′h + δk′h,khc†keck′e (G.24)Frenkel excitonsFor electron hole pairs in lattices with relative motion limited to single sites only areanother extreme of exciton physics. These excitons are named after Frenkel. Theirstates can be better expanded in terms of atomic states rather than Bloch waves orin terms of Wannier functions (e.g. atoms in optical lattices) or in terms of deltafunctions for further simplification. In such cases the lattice site index replaces thestate index for electrons and holes.H =∑mnEeffe c†mcn +∑mnEeffh d†mdn−∑mnc†mcmd†ndnV(mn|mn) +∑mnc†mcnd†mdnV(mn|mn) + · · · (G.25)where Eeffe and Eeffh consists of not only that of kinetic energies of electrons andholes respectively but also of electron and hole scattering terms.113HOn Numerical ComputationsThe numerical computations involved many intricacies spanning through basis se-lections, basis transformations, algorithm implementations and sorting of computedresults. Each of these steps involves further complications. A few LAPACK andBLAS routines (dgemm, zgemm, zgemv, dsyev, zsyev) were extensively used com-bined with random number generator (Mersenne Twister) and parallelization method(OpenMP). The results can be sensitive to the use of these libraries in proper se-quence. Each calculation had been dynamically optimized to produce numbers withinrange of tested accuracy while using minimum disk space of both running and statictype. Each of the results was benchmarked against either previously existing reportsor some alternatively calculated result. The most computational difficulty involvedhandling of available running memory on computing nodes for optimization of effi-ciency and accuracy. Each calculation had to pass rigorous testing. The scale of thecalculations were also often not easy for implementation, specifically for large systemsizes of higher dimension. The calculations for disordered systems involve averagingover many disorders till the results converges, which is time consuming. The recur-sive calculations add significant multiplicity for each calculation on a separate pointof energy within full bandwidth, which, although, allows for the calculations involv-ing larger system sizes, however, limit the sizes of the calculations from considerationof consumption of time. For large system sizes, the available running memory oncomputing nodes also limits the use of parallel computation method. Splitting ofeach calculation over several computing nodes is often the case. Sorting of elementswithin dynamic multidimensional arrays is also important for the efficiency of thecalculations. While providing prototypes for each code is not very useful here, aschoice of coding platform and style changes from person to person, this appendixonly intends to express to the reader a very brief sense on the detail of numericalimplementations which lies underneath the presented results in this thesis.114
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Numerical studies on correlations in dynamics and localization of two interacting particles in lattices Chattaraj, Tirthaprasad 2018
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Title | Numerical studies on correlations in dynamics and localization of two interacting particles in lattices |
Creator |
Chattaraj, Tirthaprasad |
Publisher | University of British Columbia |
Date Issued | 2018 |
Description | Two interacting particles in lattices, in the absence of dissipation, can not distin- guish between attractive or repulsive interaction when the range of their tunnelling is limited to nearest neighbor sites. However, we find that, in the case of long-range tunnelling, the particles exhibit different dynamics for different types of interactions of the same strength. The nature of dynamical correlations between particles also becomes significantly different. For weak interactions, particles develop a character in correlation which is in between that of antiwalking and cowalking when the tunnelling is long-range. For strong interactions, particles cowalk independently of their statistics. A few recent experiments have demonstrated such effects of interactions on quantum walk of photons, atoms and spin excitations on various lattice platforms. In disordered lattices the effect of coherent backscattering makes particles localize to their initial position. We find that a weak repulsive interaction reduces localization and a strong interaction enhances localization. We also calculate the correlations between the particles in the disordered 1D and 2D systems. The effect of long-range tunnelling on localization of particles in disordered 1D systems has been explored. For large ordered or disordered lattices, computation of localization parameters becomes difficult. In these cases, an efficient recursive algorithm is used to calculate Green’s functions exactly. We extend such algorithm to disordered systems in both one and two dimensions. We also illustrate that this recursive algorithm maps directly to some graph structures like binary trees. We perform calculations for quantum walk of interacting particles on such graphs. The method is also used to calculate the properties of interacting particles on lattices with gauge fields. For disordered 2D lattices, we introduce and test approximations which produce accurate results and make the calculations more efficient. We examine the localization parameters for a broad range of interaction and disorder strengths and try to find differences among parameters within the range. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2018-10-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0372343 |
URI | http://hdl.handle.net/2429/67296 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemistry |
Affiliation |
Science, Faculty of Chemistry, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2018-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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