Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Scale symmetry and the non-equilibrium quantum dynamics of ultra-cold atomic gases Maki, Jeff 2019

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata


24-ubc_2019_may_maki_jeff.pdf [ 1.66MB ]
JSON: 24-1.0377773.json
JSON-LD: 24-1.0377773-ld.json
RDF/XML (Pretty): 24-1.0377773-rdf.xml
RDF/JSON: 24-1.0377773-rdf.json
Turtle: 24-1.0377773-turtle.txt
N-Triples: 24-1.0377773-rdf-ntriples.txt
Original Record: 24-1.0377773-source.json
Full Text

Full Text

Scale Symmetry and theNon-Equilibrium Quantum Dynamicsof Ultra-Cold Atomic GasesbyJeff MakiB.Sc., The University of Calgary, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2019c© Jeff Maki 2019 The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:  Scale Symmetry and the Non-Equilibrium Quantum Dynamics of Ultra-Cold Atomic Gases  submitted by Jeff Maki  in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics   Examining Committee: Fei Zhou, Physics Supervisor  Gordon Semenoff, Physics Supervisory Committee Member  Ian Affleck, Physics University Examiner Richard Froese, Mathematics University Examiner   Additional Supervisory Committee Members: Ed Grant, Chemistry Supervisory Committee Member Philip Stamp, Physics Supervisory Committee Member iiAbstractThe study of the quantum dynamics of ultra-cold atomic gases has become aforefront of atomic research. Experiments studying dynamics have becomeroutine in laboratories, and a plethora of phenomena have been studied.Theoretically, however, the situation is often intractable unless one resorts tonumerical or semiclassical calculations. In this thesis we apply the symmetryassociated with scale invariance to study the dynamics of atomic gases, anddiscuss the implications of this symmetry on the full quantum dynamics. Inparticular we study the time evolution of an expanding two-dimensional Bosegas with attractive contact interactions, and the three-dimensional Fermigas at unitarity. To do this we employ a quantum variational approachand exact symmetry arguments. It is shown that the time evolution due toa scale invariant Hamiltonian produces an emergent conformal symmetry.This emergent conformal symmetry has implications on the time evolutionof an expanding quantum gas. In addition, we examine the effects of brokenscale symmetry on the expansion dynamics. To do this, we develop a non-perturbative formalism that classifies the possible dynamics that can occur.This formalism is then applied to two systems, an ensemble of two-bodysystems, and for the compressional and elliptic flow of a unitary Fermi gas,both in three spatial dimensions.iiLay SummaryA theoretical understanding of the motion of ultra-cold atomic gases is quitecomplex. In order to understand this, a thorough knowledge of strongly in-teracting systems is required. Without the aid of special tools, like symme-tries, such a problem would simply be impossible. In this work we investigatehow the symmetry associated with fractals, i.e. scale symmetry, allows oneto make concrete predictions about the dynamics of ultra-cold gases, andhow their density profiles evolve with time, even for strongly interactingsystems.iiiPrefaceAll of the research was done by the author, J. Maki.The results in Chapter 3 are based on two published works. The firstof which was done with the assistance of Mohammadreza Mohammadi: J.Maki, M. Mohammadi, and F. Zhou, Phys. Rev. A, 90,063609 (2014).In this work I acted as the lead investigator, responsible for the primaryanalysis. M. Mohammadi, worked on the early stages of the project. F.Zhou was the supervisory authority on the project, and aided on both theanalysis, and the manuscript composition.Chapter 3 is also based off the work: J. Maki, S.J. Jiang, and F. Zhou,Euro. Phys. Letts. 118, 5 (2016). In this work, I was the lead investigator,responsible for the primary analysis of these works. S.J. Jiang helped developthe analysis, and contributed to the manuscript edits. Again, F. Zhou wasthe supervisory authority on the project, and aided on both the analysis,and the manuscript composition.The results contained in Chapters 4, 5, and 6 were published in the work:J. Maki, L.M. Zhao, and F. Zhou, Phys. Rev. A 98, 013602 (2018). In thiswork, I acted as the lead investigator, responsible for the development of thetheory, the analysis, and the manuscript. L.M. Zhao aided with the analysisand the numerics. F. Zhou was the supervisory authority on the project,and aided on both the analysis, and the manuscript composition.The results of Chapter 7 are unpublished results which will appear in anupcoming article written by myself, and my supervisor, Fei Zhou. In thiswork, I acted as the lead investigator, under the aid of my supervisor F.Zhou. In this work we both worked on the analysis and the manuscript.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Definition of Scale and Conformal Invariance . . . . . . . . 62.1 Scale Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Conformal Symmetry . . . . . . . . . . . . . . . . . . . . . . 93 Dynamics of Two-Dimensional Bose Gases . . . . . . . . . . 113.1 A Formal Solution to Dynamics . . . . . . . . . . . . . . . . 133.2 Semiclassical Solution . . . . . . . . . . . . . . . . . . . . . . 143.3 Quantum Variational Approach . . . . . . . . . . . . . . . . 143.4 Dynamics of an Inhomogeneous Bose Gas . . . . . . . . . . . 173.4.1 Dynamics of the Density Profile . . . . . . . . . . . . 173.4.2 Dynamics of the Moment of Inertia . . . . . . . . . . 223.4.3 Dynamics of a Repulsively Interacting Bose Gas . . . 243.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24vTable of Contents4 Non-Relativistic Dynamics and Scale Symmetry . . . . . . 264.1 The so(2,1) Algebra . . . . . . . . . . . . . . . . . . . . . . . 274.2 The Comoving Reference Frame . . . . . . . . . . . . . . . . 294.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 The Breaking of Scale Invariance . . . . . . . . . . . . . . . . 345.1 Explicit Form of the Deviation From Scale Invariance . . . . 365.2 Classification of Deviations . . . . . . . . . . . . . . . . . . . 385.3 Non-Perturbative Solution for Relevant Deviations . . . . . . 415.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 Application to Two-Body and One-Body Systems . . . . . 446.1 The Two-Body Problem: Schrodinger Equation in the Co-moving Frame . . . . . . . . . . . . . . . . . . . . . . . . . . 446.2 Near Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 456.3 Weakly Interacting . . . . . . . . . . . . . . . . . . . . . . . 506.4 Experimental Application . . . . . . . . . . . . . . . . . . . . 516.5 Application to a Mobile Impurity . . . . . . . . . . . . . . . 526.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 Application to Many-Body Systems . . . . . . . . . . . . . . 607.1 Definitions for the One-Body Density Matrix . . . . . . . . . 617.2 Isotropic Trap and Compressional Flow . . . . . . . . . . . . 637.2.1 Scale Invariance and Compressional Flow . . . . . . . 637.2.2 Broken Scale Invariance and Compressional Flow . . 657.3 Elliptic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.3.1 Scale Invariance and Elliptic Flow . . . . . . . . . . . 667.3.2 Broken Scale Invariance and Elliptic Flow . . . . . . 677.4 Comparison to the Scaling Solution Ansatz . . . . . . . . . . 687.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74AppendicesA The Action of Scale and Conformal Transformations . . . 83A.1 Scale Transformation . . . . . . . . . . . . . . . . . . . . . . 83A.2 Conformal Transformation . . . . . . . . . . . . . . . . . . . 85viTable of ContentsB Derivation of the Quantum Variational Approach . . . . . 88B.1 The Effective Action . . . . . . . . . . . . . . . . . . . . . . . 88B.2 Scale Invariance at the Semiclassical Level . . . . . . . . . . 90B.3 Coarse Grained Dynamics . . . . . . . . . . . . . . . . . . . 91B.4 Need for Quantization . . . . . . . . . . . . . . . . . . . . . . 94C Quantum Anomaly and the Heisenberg Equation of Motion 96D Existence of Conformal Towers . . . . . . . . . . . . . . . . . 99D.1 Application of Conformal Tower Spectrum to a 1D HarmonicOscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101E Comoving Reference Frame and Time Dependent Traps . 103E.1 Quench of The Trapping potential . . . . . . . . . . . . . . . 104E.2 Efimovian Expansion . . . . . . . . . . . . . . . . . . . . . . 105E.2.1 Experimental Detection of Efimovian Expansion . . . 106F Two-Body Solution . . . . . . . . . . . . . . . . . . . . . . . . . 107F.1 Beat Amplitudes for Moment of Inertia and Contact . . . . . 111G The Density Matrix and Conformal Symmetry . . . . . . . 114G.1 The Density Matrix and Scale Invariance . . . . . . . . . . . 114G.1.1 Scale Invariant Dynamics . . . . . . . . . . . . . . . . 115G.1.2 Broken Scale Invariance . . . . . . . . . . . . . . . . . 116G.1.3 Dynamics of Local Observables . . . . . . . . . . . . 117H Hydrodynamic and Heisenberg Equation of Motion for Com-pressional and Elliptic Flow . . . . . . . . . . . . . . . . . . . 119H.1 Scaling Solution to the Hydrodynamic Flow . . . . . . . . . 119H.1.1 Isotropic Expansion . . . . . . . . . . . . . . . . . . . 121H.1.2 Anisotropic Expansion . . . . . . . . . . . . . . . . . 121H.2 Heisenberg Equation of motion . . . . . . . . . . . . . . . . . 122viiList of Tables7.1 The leading and next leading order time dependence for themoment of inertia: 〈r2i 〉(t), for a unitary Fermi gas. Thetime dependence depends on whether the initial harmonicconfinement is isotropic, with frequency ω, or anisotropic,and whether the initial conditions are a diagonal, or genericnon-diagonal ensemble of conformal tower states. The exactcoefficients, v, A, and B, can be found in Eqs. 7.14, 7.16, and7.24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71viiiList of Figures3.1 The numerical solution of the probability density, |ψ(λ, t)|2,and the resulting density profile. Here the semiclassical so-lution is λsc(t) = (m|V |)1/4√t. a) For λ  λsc(t) (onlythe scattering state contribution is shown, see main text)when λ0/σ = 50, m|V | = 50 and t/(mσ2) = 1000. b) Forλ  λ0 when λ0/σ = 10, m|V | = 27.2 and t/(mσ2) = 1000.The linear depletion in the probability density is specificallyshown by the red dashed lines. c) The density profile asr → 0, Eq. 3.24, (blue solid) and the semiclassical solution(red dashed). This figure first appeared in Ref. [1]. . . . . . . 183.2 a) The temporal evolution of the density profile at a fixedposition r  λ0. For this calculation r/σ = 0.1, m|V | = 27.2,λ0/σ = 10. b) The frequency spectrum (see Eq. 3.25, bluesolid line) is shown alongside the semiclassical frequencies (reddashed line). Only two families are shown explicitly withlabels 1) and 2) corresponding to families with ν = 1 andν = 2, respectively. c) The spectra for r/σ = 0.1, m|V | = 32and λ0/σ = 10. This figure first appeared in Ref. [1]. . . . . . 213.3 Comparison between the Heisenberg and Schrodinger calcu-lations for the moment of inertia, 〈λ2(t)〉. Here we use theinitial conditions set in Fig. 3.2 a). Since energy is conserved:d3〈λ2(t)〉/dt3 = d〈AD〉(t)/dt. The quantum beats are dueto the anomalous term, 〈AD〉(t), defined in Eq. 3.32. Whenthis term is included in the Heisenberg equation of motion,Eq. 3.33, one can show the presence of the discrete invariantbeats. This anomaly highlights the non-trivial nature of theHamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23ixList of Figures3.4 Here we show a schematic for the dynamics for a repulsivelyinteracting Bose gas when λ0/σ = 50, m|V |= 50 and t/(mσ2) =1000. a) The time evolved wave function for an initially Gaus-sian wave function centred at λ0 with width σ. b) The result-ing density profile of the Bose gas. . . . . . . . . . . . . . . . 244.1 The conformal tower states in both a) the laboratory frame,and b) in the co-moving frame. A single conformal toweris depicted with ground state energy E0. In the laboratoryframe, the conformal tower states are evenly spaced but con-tract like t−2 in the long time limit, ωt 1, see Eq. 4.16. Inthe co-moving frame the spectrum does not evolve with time.This plot first appeared in Ref. [2]. . . . . . . . . . . . . . . . 315.1 Here we show the conformal tower spectrum and perturba-tion in both a) the laboratory frame, and b) the comovingframe. The blue (dotted) lines correspond to the conformaltower states. The red (solid) and black (dash-dotted) linescorrespond to deviations with scaling, α = 1 and α = −1,respectively. For scaling α = 1, the interaction eventuallycouples more and more states, see Eq. 5.20, with Ncoupled ∝ t.For α = −1, the interaction vanishes with time, i.e. forlong times, fewer and fewer states are coupled together with,Ncoupled ∝ 1/t. We therefore expect a breakdown of time-dependent perturbation theory for α ≥ 1. This figure firstappeared in Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . 406.1 The probability for the particle to remain in the resonantground state in the comoving frame, for large finite scatteringlengths, a λ0, as a function of −λ0/(4pia) ln(pi/2−τ). Herewe note, λ0 = 1/√ω, and pi/2 − τ = 1/(ωt). This result hasbeen obtained by numerically solving Eq. 6.1 with λ0/(4pia) =0.015 and r0/λ0 = 10−3/2. The system is initially preparedin the ground state of the resonant model. Very quickly theprobability satisfies Eq. 6.7, and develops oscillations at thefrequency v/2 = 20.14. This figure first appeared in Ref. [2]. . 47xList of Figures6.2 The time evolution of 〈x2〉(τ) as a function of−λ0/(4pia) ln(pi/2−τ), where λ0 = 1/√ω, and pi/2 − τ = 1/(ωt). This has beenobtained by numerically solving the near resonant wave func-tion, Eq. 6.1. In this calculation, the system was prepared inthe ground state with λ0/(4pia) = 0.015 and r0/λ0 = 10−3/2.The dynamics can be fit to Eq. 6.9, with oscillations at fre-quency v/2 = 20.14. In the inset, the dynamics over theentire range is shown. This figure first appeared in Ref. [2]. . 496.3 Proposed experimental set up for examining broken scale in-variance on two-body dynamics. a) To create a three dimen-sional lattice, with a lattice constant, al, much larger thanthe optical wavelength, three pairs of coplanar beams areneeded. Each pair of beams will have the same wave num-ber, k, but an angle, θ between them. The resulting opticallattice will be due to the difference between the two beams:δkα = sin(θα/2)kα, for α = x, y, z. For large lattice constants,the optical lattice will look like an ensemble of harmonic trapswith harmonic lengths: λ0 = 1/√ω. If each of the differentpairs of beams lie in different orthogonal planes, the resultwill be a square lattice. b) A schematic of the experiment.Here we show a single dimension of the resulting optical lat-tice. At t = 0 the lattice is removed so the two-body systemscan expand in free space. For times, ω−1  t  al/√ω,the dynamics of the whole system will be equivalent to anensemble of independent two-body systems. This figure firstappeared in Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . 536.4 The time evolution of the trapped impurity according to Eq. 6.18in units of ω, for MωIλ20 = 3. a) The probability of being inthe instantaneous ground state of Eq. 6.18. b) The fluctua-tions of the trapped impurity position, Eq. 6.19, in both thelaboratory (dashed line) and co-moving frames (solid line).This figure first appeared in Ref. [2]. . . . . . . . . . . . . . . 566.5 Thermodynamic and dynamic relevancy for a d-dimensionalquantum gas with s-wave interactions near resonance. Thethermodynamic relevancy differs from the dynamic relevancyby one spatial dimension. . . . . . . . . . . . . . . . . . . . . 59xiList of Figures7.1 The solution to the aspect ratio for various shear viscositiesfor a unitary Fermi gas. These results were obtained by nu-merically solving Eq. 7.31. We have used the experimen-tal parameters in Ref. [3], ωx = 2pi ∗ 230 Hz, ωy = 2.7ωx,ωz = 33ωx. For any value of the shear viscosity, the aspectratio saturates. In the inset, the dynamics of the moment ofinertia in the y direction is shown. This solution fits well withEq. 7.24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70E.1 Solution for λ(t) for the dynamics of a quench trapping poten-tial. The ratio of the trapping potentials is given by ωi = 5ωf .For convenience we plot λ2(t) as a function of time. . . . . . . 105E.2 The dynamics of the moment of inertia for a scale invariantquantum gas inside an expanding trap. The time dependenceof the trap is given by Eq. E.9.The dynamics of the momentof inertia are given by Eq. E.11, and are in strong agreementwith the observations in Ref. [4]. . . . . . . . . . . . . . . . . 106xiiAcknowledgementsI would like to thank my supervisor Fei Zhou for his continued support overthe course of my study. In addition, my heartfelt thanks go to Shao-JianJiang and Li-Ming Zhao for the numerous discussions and help.xiiiDedicationI would like to thank my parents, my family, Jarett, Josh, Marcel, Tali, andall my friends back home for their constant love, good cheer, and support.Keep being the lovely people you are.xivChapter 1IntroductionCold atom systems have been praised for the extreme amount of controlavailable to experiments. In experiments, one is able to tune almost everykey feature of a system using lasers or magnetic fields; including the dimen-sionality of the gas [5], the interaction [6], and many other facets. It is notsurprising that people have considered atomic gas systems as a candidatefor new quantum mechanical technologies [7–9], and as quantum simulators[10–13], i.e. a quantum system that one can fine tune to replicate a morerealistic, or complicated, quantum system. Already atomic systems havebeen used to simulate the physics inside an atom’s nucleus [12], and hightemperature superconductors [11, 13].In order to truly create new technologies from atomic gases, a compre-hensive understanding of atomic gases is needed. Not only does one needto have knowledge of the energetic and thermodynamic properties of thesegases, but even their dynamical properties. In recent years, it has becomeroutine for experiments to create non-equilibrium atomic systems and tostudy how they evolve in time. Almost any conceivable non-equilibrium dy-namical experiment can be performed in a laboratory. This has led to studieson expansion dynamics [14–17], breathing modes [18–21], quench dynamics[6, 15, 22–24], time dependent trap dynamics [4, 25], thermalization andlocalization [23, 26–29], periodic driving [29–34], and more.Although dynamical experiments are routine, theoretical and numericalstudies of dynamics can be quite difficult. In principal, to determine thedynamics of a quantum system with N atoms, it is necessary to solve themany body Schrodinger equation:i∂tψ({~ri}, t) = Hψ({~ri}, t), (1.1)where ψ({~ri}, t) is the many body wave function describing a system ofi = 1, 2, ...N particles with positions, {~ri}, and Hamiltonian, H. SolvingEq. 1.1 is an extremely complex problem. In order to study the dynam-ics, it is necessary to obtain the full spectrum of the N -particle many bodysystem. This is usually impossible, as even understanding the ground stateof an arbitrary many body system can be daunting. For this reason, one1Chapter 1. Introductionusually needs to turn to either numerical simulations, or semiclassical ap-proximationsNumerically, the study of dynamics is often limited to one dimensionalmodels, where the Bethe ansatz can be employed [35–38]. An extension ofthis ansatz to higher dimensions simply does not exist. For higher dimen-sions, different schemes, such as time dependent density functional theory[39–41], need to be applied to dynamics. These numerical schemes oftenutilize the scaling properties of the equation of state, and rely on phe-nomenological parameters imported from experiments or other numericalsimulations. For this reason, a complete analytic understanding of dynami-cal phenomena, from microscopic first principles, is not always possible. Interms of theoretical studies, the study of dynamics is often limited to semi-classical approximations [42–45]. These approximations render the problemtractable, but do not describe all the quantum effects present in dynamics.Beyond these treatments, the understanding of non-equilibrium quantumdynamics remains a difficult problem.One important tool for simplifying the complexities of many body physicsis symmetry. If a symmetry is present in a system, certain aspects of thephysics will be governed by the symmetry alone, regardless of the micro-scopic details of the system. That is, many conceivably different systemscan exhibit the same physics if they obey the same symmetry. The notionof symmetry has been paramount in other areas of modern physics, and hasbeen used to understand phenomena such as phase transitions [46, 47], thestandard model of particle physics [48], and crystallography [49].For the study of dynamics, it is possible to employ the same procedure;to look for symmetries to simplify the problem. In this case, it is necessaryto discuss symmetries that simultaneously transform both the spatial andtemporal coordinates. A specific dynamical symmetry of interest is scalesymmetry, i.e. the invariance associated with scale transformations. Scalesymmetry is the invariance of the physics under dilations of spatial andtemporal coordinates. In the context of quantum gases, a system is scaleinvariant if the governing equation of motion, the many body Schrodingerequation, Eq. 1.1, is invariant under the scale transformation:~r′i = e−b~ri i = 1, 2, ..., N t′ = e−2bt, (1.2)where the set of coordinate {~ri} are the position of the N particles in thesystem, and b is a scaling factor.Generally speaking, scale symmetries have had resounding success instudying a wide array of time independent problems, such as the physics at2Chapter 1. Introductionphase transitions [46, 47, 50, 51], as well as complex biological systems [52].For dynamical systems, the dynamical scale symmetry defined in Eq. 1.2can lead to drastic simplifications. In particular, dynamical scale symmetrywill equate the dynamics at two points in space and time, up to an overallscaling factor, if those two points can be equated by a scale transformation.For this reason, as well as the resounding success in thermodynamics, it isfitting to apply scale symmetry to the problem of dynamics.Thankfully, there are a number of experimentally accessible cold atomsystems which are, or nearly are, scale invariant. Beyond the trivial non-interacting quantum gas, systems such as the two dimensional quantum gas[53–55], the high dimensional degenerate Bose gas at unitarity, [24, 56, 57],the three dimensional Fermi gas at unitarity [3, 16, 58], the Tonks-Girardeaugas in one dimension [59–61], the two dimensional Fermi gas at a p-waveresonance [62], also exhibit, or nearly exhibit, scale symmetry. For thesesystems, scale symmetry has been applied to quantities such as the equationof state [24, 56, 57, 62–65], frequencies and damping rates of collective modes[20, 21, 66], and expansion dynamics [16, 42, 45].Apart from numerical methods, there have been two main theoreticalapproaches for studying the effect of scale invariance on dynamics. Thefirst method is to apply a scaling solution to the dynamics [42–45]. Thescaling solution is a variational approach that is often limited to the semi-classical approximation. However, this approximation does not necessarilyrequire scale invariance, as it also provides an approximate description ofthe dynamics for non-scale invariant systems. Although the scaling solutionis generally applied to semiclassical systems, in Refs. [2, 67] we utilized ascaling solution in order to capture the effects of quantum fluctuations onthe N-body bound states of low dimensional Bose gases, and the dynamicsof two dimensional Bose gases.The second major approach for studying scale invariant dynamics is touse the Heisenberg equation of motion. This approach is useful for address-ing the role of scale invariance on the full quantum dynamics of global ob-servables [4, 25, 66, 68]. When the scale symmetry is broken, the Heisenbergequation of motion becomes much more difficult, and potentially intractablewithout any additional knowledge about the microscopic properties of thesystem. As well, the Heisenberg equation of motion is very specialized to agiven observable; it lacks any microscopic, local, information that is carriedin the many body wave function.A microscopic approach which utilizes the scaling property of the Schrodingerequation, Eq. 1.1, was put forward in Refs. [69, 70]. In this approach, oneuses a transformation that studies the dynamics of the N -body wave func-3Chapter 1. Introductiontion in an expanding non-inertial reference frame. Although this microscopicwave function approach is valid for a variety of systems [71], it is quite ben-eficial for scale invariant systems. This approach can still be difficult as onestill needs to determine the full spectrum of an effective N -body Hamilto-nian, but it can give insights into the dynamics of scale invariant systems.A simple application of this approach is for the dynamics of systems ini-tially prepared in the ground state of a harmonic potential. In this case, itwas shown that the expansion dynamics of the many body wave function isequivalent to the scaling solution [71]. As a result, the dynamics of all localobservables are equivalent to a time dependent rescaling. This method wasused to study the dynamics of three dimensional Fermi gases at unitarity[69, 70, 72], Bose gases [71], the Tonks gas in one dimension [73], and forharmonically trapped scale invariant systems [68, 74].In general, a full microscopic categorization of scale invariant dynamics,for arbitrary initial conditions, is lacking. This is especially true for stronglyinteracting scale invariant systems, where the many body wave function isintractable. Even the energetics of these systems are not completely un-derstood, although tremendous progress has been made [57, 62, 64, 75–77].For scale invariant dynamics, a number of outstanding questions remain;what are the signatures of scale invariance in dynamics, are these signaturesindependent of the interactions present in the system in question, how dothe dynamics differ for different scale invariant systems, and is there anyrelationship between the dynamics and the energetics when one breaks thescale invariance?In this thesis we address these issues and categorize the effects of scaleinvariance on the quantum dynamics of cold atom systems, in the presence,or absence of, resonant interactions. To do this, we utilize the fact that scalesymmetry is intimately connected to another dynamical symmetry, confor-mal symmetry [78]. Conformal symmetry corresponds to the invariance ofthe equations of motion under the following transformation:~r′i =~ri1− bt i = 1, 2, ..., N t′ =t1− bt . (1.3)The first application of this symmetry was in the context of non-relativisticfield theory [78, 79]. For these field theories, it was shown that the spectracan be arranged into a series of towers, where the level spacing in a giventower is a constant [79]. Following the success of conformal symmetry in fieldtheory, later this symmetry was applied to understand the dynamics of coldatomic gases. The first application of conformal symmetry in cold gases waspresented in Ref. [66], where the authors showed that the breathing modes4Chapter 1. Introductionof scale invariant gases in harmonic traps form a tower of excitations, andthe frequency of the oscillations occur at twice the trap frequency.Inspired by the previous symmetry arguments, [66, 70, 79], we show thatthe quantum dynamics due to scale invariant Hamiltonians are constrainedby an emergent conformal symmetry. The presence of this new symme-try allows one to make analytical statements about the quantum dynamics,independent of the microscopic details of the system, and the initial condi-tions. In addition, we develop a non-perturbative formalism that categorizesthe breaking of scale invariance, using both exact symmetry arguments, andthe microscopic wave function method [69, 70]. This formalism allows oneto relate the effect of the broken scale invariance on the dynamics to how aperturbation behaves under a scale transformation. This result allows one todelineate the dynamics near different scale invariant points, and to constructa scaling argument for the dynamics of nearly scale invariant systems.The remainder of this thesis is organized as follows. In chapter 2, wedefine scale and conformal transformations. The relationship between thesetwo symmetries and their action on quantum operators is discussed. Inchapter 3 we study the dynamics of a two dimensional Bose gas using anapproach we developed, the quantum variational approach. In this quantumvariational approach, it is possible to elucidate the effects of scale invarianceon the expansion dynamics of an initially inhomogeneous Bose gas. In chap-ter 4 we put forward the exact implications of scale symmetry. This can bedone by exploiting the connection between scale transformations, conformaltransformations, and an overarching symmetry, the SO(2,1) symmetry. Thisoverarching symmetry allows one to to make concrete predictions about thedynamics of any scale invariant system. In chapter 5 this analysis is extendedto the case of nearly scale invariant systems, where scale symmetry is ex-plicitly, but slightly, broken. Here we develop a non-perturbative approachthat categorizes the possible deviations from scale invariant dynamics. Inchapter 6, we apply our non-perturbative formalism to the two-body systeminteracting with s-wave interactions in three spatial dimensions. In Chapter7 we apply this formalism to the expansion dynamics of a resonant Fermigas of N particles in three spatial dimensions. In particular, we examinethe moment of inertia for the gas, and how scale and conformal symmetryaffect the dynamics of compressional and elliptic flows. We then concludeand summarize our results in Chapter 8.5Chapter 2Definition of Scale andConformal InvarianceBefore discussing the effects of scale symmetry, it is necessary to review thenotion of symmetry transformations in quantum mechanics. In particular,how to perform a generic transformation in quantum mechanics and how toshow that a transformation is a symmetry of the system.In general, a given transformation can be associated with a unitary op-erator, U , that enacts the transformation. We will be focused on transfor-mations that can be done continuously. These transformations have unitaryoperators of the form:U(b) = e−ibG, (2.1)where b can be thought of as a generalized angle that parametrizes themagnitude of the transformation, while G is a Hermitian operator that iscalled the generator of the transformation.In this thesis we will be primarily focused on operators and how theytransform under scale and conformal transformations. For this reason wewill work in the Heisenberg representation where operators have time de-pendence, but the quantum state does not:O(~r, t) = eiHtO(~r)e−iHt. (2.2)where O(~r) is an arbitrary operator.In this representation all the quantum dynamics are contained in the op-erator, and the Schrodinger equation is replaced with the equivalent Heisen-berg equation of motion:i∂tO(~r, t) = [H,O(~r, t)] , (2.3)where H is the Hamiltonian for the system.For a transformation to be a symmetry of the system, the Heisenbergequation of motion for the symmetry transformed operator:62.1. Scale SymmetryOb(~r, t) = U†(b)O(~r, t)U(b), (2.4)must be equivalent. To be more explicit, the Heisenberg equation of motionafter the symmetry change:U †(b)i∂tO(~r, t)U(b) = U †(b)[H,O(~r, t)]U(b),i∂tOb(~r, t) =[U †(b)HU(b), Ob(~r, t)], (2.5)must be equivalent to the original equation of motion, Eq. Scale SymmetryWe are now in a position to define scale symmetry. The generator of scaletransformations is the operator:D = −i∫ddrψ†(~r)(d2+ ~r · ∇r)ψ(~r), (2.6)where ψ(†)(~r) is the annihilation (creation) operator which annihilates (cre-ates) a particle at position ~r. For concreteness we consider Bosonic particles,such that the the creation and annihilation operators satisfy:[ψ(~r, t), ψ†(~r′, t)]= δ(~r − ~r′). (2.7)The following discussions are equally valid for Fermions.Let us consider the Heisenberg equation of motion for the field operator,ψ(~r, t). As shown in Appendix A, the action of the scale transformation onthe field operator is:ψb(~r, t) = eiDbψ(~r, t)e−iDb= e−db/2ψ(~re−b, te−2b). (2.8)As one can see, the generator of scale transformations does indeed performthe transformations defined in Eq. 1.2. The overall rescaling of the fieldoperator is present to conserve normalization. The equation of motion forthe field operator then transforms to:e−2bi∂t′ψ(~r′, t′) =[eiDbHe−iDb, ψ(~r′, t′)], (2.9)72.1. Scale Symmetrywhere the coordinates ~r′ and t′ are defined in Eq. 1.2.To see if the equation of motion is left invariant, we need to considerthe action of a scale transformation on the Hamiltonian. Consider a genericHamiltonian with two body interactions V (~r):H =∫dd~rψ†(~r)(−∇2x2)ψ(~r) +12∫dd~rdd~r′ V (~r−~r′)ψ†(~r)ψ†(~r′)ψ(~r′)ψ(~r).(2.10)The commutator between this Hamiltonian and the generator of scale in-variance is found to be:[D,H] = 2iH− i∫dd~rdd~r′[(1 +~r · ∇r + ~r′ · ∇r′2)V (~r − ~r′)]ψ†(~r)ψ†(~r′)ψ(~r′)ψ(~r).(2.11)If a system is said to be scale invariant, we require the commutator betweenthe generator of scale transformations and the Hamiltonian to be propor-tional to the Hamiltonian. Otherwise, new terms in the Hamiltonian willbe generated under a scale transformation. Therefore, we require that thesecond term on the right hand side of Eq. 2.11 vanishes. If this is true, allterms in the Hamiltonian physically scale like the kinetic energy operator.That is, a scale invariant Hamiltonian, Hs, is defined to be:[D,Hs] = 2iHs. (2.12)In general, we note that the scaling dimension, ∆O, of a given operator, O,as:[D,O] = ∆OiO. (2.13)Given Eq. 2.12, one can show that a scale invariant Hamiltonian transformsas:eiDbHse−iDb = e−2bHs. (2.14)For now let us consider the Hamiltonian in Eq. 2.9 to be scale invariant.Upon substituting this result into Eq. 2.9, one can see that the Heisen-berg equation of motion in the new coordinates is equivalent to the originalequation of motion. In this way we define scale invariance.82.2. Conformal Symmetry2.2 Conformal SymmetryIn this section we define a closely related symmetry, conformal symmetry.This transformations is more abstract than scale symmetry, and is generatedby the operator:C =∫ddr r2ψ†(~r)ψ(~r). (2.15)The generator of conformal symmetry, alongside the scale invariant Hamil-tonian, Hs, and the generator of scale transformations, D, form a represen-tation of the group SO(2, 1) [66]:[Hs, C] = −iD,[D,Hs] = 2iHs,[D,C] = −2iC. (2.16)A more detailed discussion of this symmetry will be presented in Chapter 4.For now, consider the action of a conformal transformation on a fieldoperator ψ(~r, t) [78]. A calculation equivalent to the case of a scale trans-formation gives:ψb(~r, t) = eiCbψ(~r, t)e−iCb= (1− bt)−d/2e−i r22b1−btψ(~r1− bt ,t1− bt). (2.17)One can again compute the effect of a conformal transformation on theequation of motion for a field operator. If the Hamiltonian governing thesystem is scale invariant, one can show that the equation of motion is leftinvariant under the transformation:~r′ =~r1− bt t′ =t1− bt . (2.18)For more details see Appendix A.In general the presence of these two symmetries will restrict the dynam-ics of any operator O with scaling dimension ∆O. I.e. the operator willtransform as:Ob(~r, t) = e−∆ObO(~re−b, te−2b), (2.19)92.2. Conformal Symmetryunder scale transformations, and as:Ob(~r, t) = (1− bt)−∆OO(~r1− bt ,t1− bt), (2.20)under conformal transformations.As one can see, scale and conformal transformations affect both thespatial and temporal coordinates. These two symmetries relate the dynamicsat two different points of space and time. That is, with some previousknowledge of the system at one point in space and time, it is possible topredict the dynamics at a different point in space and time 1. It is for thisreason we expect scale invariant (and conformal invariant) dynamics to beanalytically tractable.1Here we ignore the effect of initial conditions, which can add another length scalethat breaks these symmetries. However, as we will see below, the initial conditions willbe irrelevant in the long time dynamics.10Chapter 3Dynamics ofTwo-Dimensional Bose GasesWe begin our study of scale invariant systems by considering a so-calledquintessential example of a scale invariant quantum gas, the two-dimensionalBose gas with short ranged, attractive, interactions. This system can bereadily created in experiments [53, 54], and is a prime candidate for the in-vestigation of scale invariant dynamics. The Hamiltonian for the interactingtwo dimensional Bose gas is given by:H =∫d2rψ†(~r)(−∇2r2)ψ(~r) +g2∫d2r ψ†(~r)ψ†(~r)ψ(~r)ψ(~r), (3.1)Naively one expects this Hamiltonian to be scale invariant as the couplingconstant, g is dimensionless. However, this is not true when one examinesthe system.For attractive interactions, g < 0, the Hamiltonian is not well defined atshort length scales, i.e. in the ultraviolet limit. The process of regularizingthe potential results in the presence of a new length scale associated withthe two-body bound state. The energy of this bound state is given by:Eb = −Λ2e−4pi/g, (3.2)where Λ is the ultraviolet cut-off for the theory [67]. This new bound stateexplicitly breaks the scale invariance, as this bound state energy will changeunder a scale transformation. Equivalently, one can say that the interactionstrength, g, is no longer a constant, but is a function of this new energyscale:g = − 4pilog(Λ2/|Eb|) . (3.3)This is known as a quantum anomaly: the act of quantization breaks theclassical symmetry of the system.11Chapter 3. Dynamics of Two-Dimensional Bose GasesThe effects of the quantum anomaly on the thermodynamics of two di-mensional Bose gases have been studied theoretically [80–83]. A number ofexperiments have probed the magnitude of scale invariance breaking in theenergetics and dynamics. In terms of energetics, experiments determiningthe equation of state have not been able to detect a significant effect fromthe quantum anomaly at weak interactions. As discussed in Refs.[53, 54] anenergetic signature of scale invariance is that the equation of state can beexpressed as a homogeneous function of the chemical potential and temper-ature. As a result, the equation of state will have a scaling form similar toEq. 2.19, but in terms of the chemical potential and temperature. This hasbeen reproduced quite well in experiments and is evidence for the anomalybeing a weak correction to the scale invariance.In terms of dynamics, it was first shown in Ref. [66] that a harmoni-cally trapped two-dimensional Bose gas would exhibit breathing modes, atexactly twice the trap frequency. This is due to the presence of a hiddensymmetry in the system, namely the SO(2,1) symmetry. The presence of thequantum anomaly will naturally break the scale invariance, and the SO(2,1)symmetry, which will result in a correction to the breathing mode frequency.Although a number of theoretical works have examined how the anomalyaffects this frequency [65, 84–87], the experimental evidence has also shownthat the dynamics are only weakly perturbed by the quantum anomaly [55].Since both the experiments and dynamics provide strong evidence for thescale invariance of a two-dimensional Bose gas, we will neglect the effect ofthe quantum anomaly on the dynamics and examine the signatures of scaleinvariance in the expansion dynamics of an inhomogeneous two-dimensionalBose gas.In the following discussions, we will show using an approach we devel-oped, the quantum variational approach, that the continuous scale invari-ance is broken. This breaking is not due to the quantum anomaly, but ratherdue to the fact that it is experimentally impossible to create a true two di-mensional system; there is always a finite confinement radius. Although thecontinuous scale invariance is broken, it is replaced by a discrete scale invari-ance. This discrete scale invariance will affect the expansion dynamics of aninitially inhomogeneous Bose gas. In particular, we show that the scale in-variance manifests in a logarithmic rise in the density profile near the centerof the condensate, and a set of discrete scale invariant beat frequencies.123.1. A Formal Solution to Dynamics3.1 A Formal Solution to DynamicsTo begin, we will write a formal solution to the problem. In order to deter-mine the density, n(~r, t), it is necessary to evaluate:n(~r, t) = 〈ψ0|eiHtnˆ(~r)e−iHt|ψ0〉. (3.4)where the density operator: nˆ(~r) = ψˆ†(~r)ψˆ(~r),2 H is the Hamiltonian de-fined in Eq. 3.1, and |ψ0〉 is the initial many body state. Although thisexpression is completely general, it is beneficial to insert a complete set ofcoherent states |{ψ(~x)}〉:n(~r, t) =∫Dψ(~x)Dψ′(~x) 〈ψ0|eiHt|{ψ(~x)}〉〈{ψ′(~x)}|e−iHt|ψ0〉[ψ∗(~r)ψ′(~r) 〈{ψ(~x)}|{ψ′(~x)}〉] . (3.5)The coherent states are eigenstates of the annihilation operator:ψˆ(~r)|{ψ(~x)}〉 = ψ(~r)|{ψ(~x)}〉, (3.6)while the eigenvalues satisfy the normalization condition:∫d2r|ψ(~r)|2 = N. (3.7)In addition, it is also possible to write the transition amplitude, 〈{ψ(~x)}|e−iHt|ψ0〉,in terms of functional integrals [88]:〈{ψ(~x)}|e−iHt|ψ0〉=∫ ′DψeiS , (3.8)where S is the action for a non-relativistic Bose gas:S =∫d2x∫ t0dt′ψ∗(~x, t)(i∂t +∇22)ψ(~x, t)− g2|ψ(~x, t)|4, (3.9)and∫ ′Dψ denotes the sum over all field configurations ψ(~x, t) which satisfythe following boundary conditions:ψ(~x, T ) = ψ(~x) ψ(~x, 0) = ψ0(~x). (3.10)2In this section, the annihilation operators are labelled by hats. They are not to beconfused with their corresponding eigenvalue. This notation is only present in this chapteras we will exclusively deal with the eigenvalues of the annihilation operator, and not theoperator itself.133.2. Semiclassical Solution3.2 Semiclassical SolutionAlthough everything stated so far is general, it is impossible to evaluate thedensity profile by means of brute force. For this reason the problem is oftentreated semiclassically. To do this, we minimize the action in Eq. 3.9, andexamine only the fields that minimize the action. This results in the wellknown Gross-Pitaevskii equation [89]:i∂tψsc(~r, t) =(−∇2r2− g|ψsc(~r, t)|2)ψsc(~r, t). (3.11)Eq. 3.11 is scale invariant, and possesses a scaling solution of the form:ψsc(~r, t) =√Nλ2(t)f(rλ(t))ei r22λ˙(t)λ(t) , (3.12)where λ(t) is a yet to be determined function. The function, f(x), is chosento match the initial density profile of the Bose gas, while the phase is chosento satisfy conservation of probability. For more details we refer the readerto Appendix B.The function λ(t) can be determined by substituting Eq. 3.12 into Eq. 3.11and integrating out the position dependence. The result is a differentialequation for λ(t):mλ¨(t) =Vλ(t)3(3.13)where m = C1N and V = NC2 + C3gN2, and C1, C2, C3, are constantsthat depend on the specific shape of the Bose gas [67].Eq. 3.13 is a classical, scale invariant, equation of motion for a particlein an inverse square potential: V (λ) ∝ λ−2. The semiclassical variationalapproach reduces a quantum many body problem to a single particle classicalmechanics problem.3.3 Quantum Variational ApproachThe results presented in the previous section neglect quantum fluctuations.However, we show in Refs. [1, 67] that it is possible to treat the quantumfluctuations in low dimensions by means of coarse graining. This can bedone by noting two things; the first is that the inner products between twodifferent field configurations is approximately given by:143.3. Quantum Variational Approach〈{ψ(~r)}|{ψ′(~r)}〉 ≈ δ ({ψ(~r)} − {ψ′(~r)}) . (3.14)This is valid for dense condensates, n(~r, t) 1, as small deviations in many-body coherent states lead to nearly orthogonal states, i.e. the orthogonalitycatastrophe.The second simplification is to express the fields ψ(~r, t) in terms of slowand fast degrees of freedom:ψ(~r, t) = ψλ(~r, t) + δψ(~r, t). (3.15)The fields δψ(~r, t) represent short wavelength, fast, many body fluctuations,such as phonons, and the fields ψλ(~r, t) represent the isotropic, long wave-length, slow, degrees of freedom that parametrize the motion of the gas as awhole. We parametrize the slow degree of freedom using a single parameteransatz, identical to the semiclassical solution:ψλ(~r, t) =√Nλ2(t)f(rλ(t))ei r22λ˙(t)λ(t) (3.16)The separation of the slow-isotropic and fast-anisotropic degrees of freedomleads to a controllable expansion of the many body fluctuations. This isvalid in the limit of dense condensates [1]. A full calculation is shown inAppendix B, here we quote the result; an effective action for the parameterλ(t):n(~r, t) =∫ ∞0dλNλ2f(rλ)|〈ψλ|e−iHt|ψ0〉|2〈ψλ|e−iHt|ψ0〉 =∫ λ(t)=λλ(0)=λ0Dλ(t)ei∫ t0dt′ 12mλ˙2(t)+ V2λ(t)2 .(3.17)where |ψλ〉, is a state of a condensate with width λ, and again m = C1Nand V = C2N + C3gN2. In Eq. 3.17, we assume the initial wave functionis well described as a condensate with size λ0. It is important to note thatif one takes the semiclassical approximation to Eq. 3.17, one can reproduceEq. 3.13.The key thing to note is that Eq. 3.17 is equivalent to a single particlequantum mechanical problem:Hλ =Pˆλ2m+V2λˆ2. (3.18)153.3. Quantum Variational ApproachIn this effective description, |λ〉 is an eigenstate of the operator λˆ: λˆ|λ〉 =λ|λ〉, representing a condensate with size λ, while Pˆλ is the momentumconjugate to λˆ.The calculation of the density profile reduces to:n(~r, t) =∫ ∞0dλNλ2f(rλ)|ψ(λ, t)|2 (3.19)where ψ(λ, t) = 〈λ|e−iHλt|ψ0〉, is the wave function associated with the sizeof the condensate. In this quantum variational approach, we replace themany body dynamics with a single particle quantum mechanical problem,where the single particle represents the macroscopic size of the condensate.Although Eq. 3.17 is equivalent to Eq. 3.19, a natural question to ask iswhether quantum fluctuations in the size of the condensate are important.To examine this, it is necessary to consider the magnitude of the fluctua-tions, δλ, around the semiclassical solution of size λ0: 〈δλ(t)δλ(t)〉/λ20. Ifthe fluctuations are small compared to the semiclassical path, then Eq. 3.13is an accurate description of the dynamics. As shown in Appendix B, forattractive interactions with V < 0, the amplitude of fluctuations is not con-trolled by the size of the semiclassical path, but by the scale independentparameter 1/√m|V | ∝ 1/(N√gN). This states that for mesoscopic conden-sates, quantum fluctuations are important and it is best to consider the fullquantum mechanical problem associated with Eq. 3.18. For the remainderof this chapter we will focus on this limit.We end this section with a brief discussion on the spectrum of Eq. 3.18for attractive interactions. The spectrum consists of a continuous set ofscattering states, ψ(1)s and ψ(2)s , with energies E =k22m , and a discrete setof bound states, ψb, with energies En = − k2n2m , where (up to normalizationfactors):ψ(1)s = Re√kλJa(kλ)ψ(2)s = Re√kλYa(kλ)ψb =√kλKa(knλ)kn = k0 exp(−npi√m|V |). (3.20)The functions Ja(x), Ya(x), and Ka(x) are the Bessel J, Bessel Y andmodified Bessel K functions of order a = i√m|V | − 1/4, respectively, andn = 1, 2, 3... Here we focus on condensates with m|V | > 1/4 and |g|  1, orequivalently, C2/C3N < |g|  1.163.4. Dynamics of an Inhomogeneous Bose GasIt is important to notice that Eq. 3.18 is singular near λ = 0, and isstrictly speaking ill defined. As a result, when the motion of λ is quantized,it is necessary to introduce an ultraviolet scale, k0, which regularizes thesingular potential. This new length scale is not associated with the quantumanomaly, but rather with the confinement radius of a quasi-two-dimensionalgeometry [82]. Practically, this length scale breaks the continuous scaleinvariance. However, a discrete scale invariance can be induced as a result ofthe original, continuous, scale invariance. This piece of physics is reminiscentof Efimov physics, where the three body bound states also satisfy a discretescaling relation [90]. The discrete scale invariance is explicit in the boundstate spectrum which is equally spaced on a logarithmic scale.3.4 Dynamics of an Inhomogeneous Bose GasAt this stage one can consider the dynamics of a condensate which is initiallyprepared with size λ0. The initial amplitude can be represented as:ψ(λ, t = 0) = 〈λ|ψ0〉 = 1(pi)1/4√σe−(λ−λ0)22σ2 , (3.21)where the spreading, σ, is fixed by requiring that the energy of the effectivemodel is identical to the microscopic model: σ = λ0/(√C1N). Below wepresent our numerical solutions of Eqs. 3.19 and 3.18 with the initial stategiven by Eq. 3.21.In order to evaluate the transition amplitude at time t, it is necessaryto examine how the initial state in Eq. 3.21 is projected onto the completeset of eigenstates. The amount of probability projected onto the boundstates depends on the ratio of the potential energy to the kinetic energy:√m|V |/N . The more kinetic (potential) energy the system possesses, themore probability will be concentrated in the scattering (bound) states. Oncethe projection of the initial state is known, the unitary evolution can becarried out to obtain the probability density, |ψ(λ, t)|2. We focus on thelimit when the time t  √m/|V |λ20, when the initial transient dynamicshave disappeared. We present the probability density in Figs. 3.1 a-b).3.4.1 Dynamics of the Density ProfileIt is now possible to evaluate the dynamics of the density profile. We findtwo robust features due to the scale invariance of Eq. 3.18. The first featuredepends on the shape of the density profile. Near the origin, the density173.4. Dynamics of an Inhomogeneous Bose Gasλ/ σ|ψ(λ,t)|2 σλ0/σλ / σ|ψ(λ,t)|2 σ(V/m)1/4√t/σ−3 −2 −1 0 1 2 3log(r / σ)ρ(r,t) σ2r / σρ(r,t) σ2λ0/σb)c)a)Figure 3.1: The numerical solution of the probability density, |ψ(λ, t)|2,and the resulting density profile. Here the semiclassical solution is λsc(t) =(m|V |)1/4√t. a) For λ  λsc(t) (only the scattering state contribution isshown, see main text) when λ0/σ = 50, m|V | = 50 and t/(mσ2) = 1000.b) For λ  λ0 when λ0/σ = 10, m|V | = 27.2 and t/(mσ2) = 1000. Thelinear depletion in the probability density is specifically shown by the reddashed lines. c) The density profile as r → 0, Eq. 3.24, (blue solid) and thesemiclassical solution (red dashed). This figure first appeared in Ref. [1].183.4. Dynamics of an Inhomogeneous Bose Gaswill approximately be n(0, t) ≈ 〈λˆ−2〉(t). First one might consider approx-imating 〈λˆ−2〉(t) with the most probable value of λ. Often this value cor-responds to the semiclassical value, which would be a solution to Eq. 3.13.This methodology is equivalent to the semiclassical and hydrodynamicalmethods studied previously [42–44].However, there are anomalous contributions to 〈λˆ−2〉 due to small λ. Tosee this, note that all the eigenfunctions for this Hamiltonian have a lineardepletion in the probability density:|ψs,b(kλ√mV )|2 ∝ λ. (3.22)The reason for this can be understood semiclassically. A particle fallinginto a potential has a probability density proportional to the inverse of themomentum at that position. For small λ, the momentum is entirely governedby the potential energy:|ψ(λ)|2 ∝ 1p(λ)≈√2λ2|V | . (3.23)The fact that the effective model has an inverse square potential is a sig-nature of the original scale invariance of the problem. It means that theprobability density must deplete linearly with λ. This is true for all eigen-states, both scattering and bound states.The consequence of the linear depletion in the probability density is thatthe dominant contributions to 〈λˆ−2〉 come from the least probable conden-sate sizes. This results to a logarithmic singularity in the density profile:limr/√t→0ρ(~r, t) =1pim|V |λ20t2log(√tr). (3.24)This logarithmic rise in the density is shown in Fig. 3.1 c). This feature isrobust as it only depends on the linear depletion of the probability densityfor all eigenstates.The second major consequence concerns condensates with larger inter-action energies, i.e. when the initial state has a larger projection onto thebound states. The presence of bound states in Eq. 3.18 will undoubtedlylead to oscillations in the density profile, see Fig. 3.2 a). The frequencies ofthese oscillations correspond to the difference of two bound state energies:ωn,ν = En+ν − En, with En given by Eq. 3.20, and n, ν = 1, 2, 3.... Theexact location of these beat frequencies and their spectral weight will specif-ically depend on the ultraviolet parameter, k0, and the initial conditions, λ0.193.4. Dynamics of an Inhomogeneous Bose GasHowever, the effect of the induced discrete scale invariance of the system ismanifest in the organization of these frequencies. From Eq. 3.20, the beatfrequencies can be written as:log(ωn,νmσ2)= log(k20σ22)− 2pi√m|V |n− log(1− e2piν√m|V |). (3.25)Upon observation, one can see that the beat frequencies are arranged intoa series of families. The separation between frequencies in a given familyis a constant universal value: 2pi/√m|V |. However, there are a number offamilies present, and each family is shifted with respect to one another bythe final term in Eq. 3.25. This final term, which depends on the boundstate spacing, ν, denotes the family of beat frequencies. Our numericalsimulations of the oscilations in the density profile, shown in Figs. 3.2 b-c),are completely consistent with this general analysis.The universal scaling found in Eq. 3.25 is to be contrasted with thesemiclassical solution for a Bose gas with an initial size λ0. In AppendixB, the semiclassical solution for a particle oscillating in an inverse squarepotential has been worked out; here we quote the result:n(~r, t) =Nλ2sc(t)f(rλsc(t)),λsc(t) = = λ0√1− (2(t− nT )/T )2, (3.26)for t ∈ [nT − T/2, nT + T/2] with n = 0, 1, 2, ... and period, T :T = 2λ20√m|V | . (3.27)We note that the period depends only on the initial conditions of the prob-lem, which is a consequence of the scale invariance of the system. Since thissolution oscillates with a period T , the frequency spectrum only containsfrequencies ωn = 2pin/T and n = 1, 2, 3, ... These points are shown along-side the quantum frequency spectrum in Figs. 3.2 b-c). For smaller values of√m|V |, there is a major discrepancy between the quantum and semiclassi-cal oscillations. As noted previously, for larger values of√m|V |, the role ofquantum fluctuations diminish. In this limit, the oscillation spectrum willcollapse onto the semiclassical solution.203.4. Dynamics of an Inhomogeneous Bose Gas1000 1100 1200 1300 1400 1500 160000.050.1t / m σ2 ρ(r,t) σ2  −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 00log(ω m σ2 )ρ(r, ω)−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 000.05log (ω m σ2) ρ(r, ω)1)1)2)2)2)2)1)1)1)1)a)b)c)2)2)Figure 3.2: a) The temporal evolution of the density profile at a fixed po-sition r  λ0. For this calculation r/σ = 0.1, m|V | = 27.2, λ0/σ = 10. b)The frequency spectrum (see Eq. 3.25, blue solid line) is shown alongsidethe semiclassical frequencies (red dashed line). Only two families are shownexplicitly with labels 1) and 2) corresponding to families with ν = 1 andν = 2, respectively. c) The spectra for r/σ = 0.1, m|V | = 32 and λ0/σ = 10.This figure first appeared in Ref. [1].213.4. Dynamics of an Inhomogeneous Bose Gas3.4.2 Dynamics of the Moment of InertiaIn this section we consider another important observable, the moment ofinertia:〈r2〉(t) =∫d2r r2n(r, t). (3.28)The moment of inertia effectively describes the size of the expanding Bosegas as a function of time. Using the effective model, one can write:〈r2〉(t) ≈ 〈r2〉(0)λ20∫dλλ2|ψ(λ, t)|2. (3.29)Unlike the average over density, this average will not contain anomalous con-tributions from small condensate sizes. Therefore the dominant contributionto the moment of inertia will be from the semiclassical solution.Although the dominant contribution may be due to the semiclassicalsolution, the dynamics will not be equivalent to the semiclassical solution.The projection of the wave function onto the bound states will lead to thediscrete scale invariant oscillations at frequencies given by Eq. 3.25. In thissection we show an alternative approach, using the Heisenberg equation ofmotion, to examine the difference between the semiclassical and quantumsolutions.For a scale invariant system, the equation of motion for the moment ofinertia is found to be:∂∂t〈r2〉(t) = 2〈D〉(t)∂2∂t2〈r2〉(t) = 4〈Hλ〉. (3.30)The solution to the Heisenberg equations of motion are equivalent to thesemiclassical solution presented in Eq. 3.26. In this case, the condensateoscillates with a period given by Eq. 3.27. The question then arises, wheredo the discrete scale invariant beats come from?The answer is that Eq. 3.30 is incomplete. In particular, one can showthat Hλ is not self adjoint, i.e. Hλ and H†λ act on different vector spaces.If one were to take into account this discrepancy, one would find a modifiedHeisenberg equation of motion [91–93]:i∂tO(t) = i [H,O(t)] + i(H† −H)O(t), (3.31)223.4. Dynamics of an Inhomogeneous Bose Gas2000 4000 6000 8000 10000t/λ02-0.0002-0.00010.00010.0002d3<λ2>/dt3d3<λ2>/dt3d <AD>/dtFigure 3.3: Comparison between the Heisenberg and Schrodinger calcula-tions for the moment of inertia, 〈λ2(t)〉. Here we use the initial conditionsset in Fig. 3.2 a). Since energy is conserved: d3〈λ2(t)〉/dt3 = d〈AD〉(t)/dt.The quantum beats are due to the anomalous term, 〈AD〉(t), defined inEq. 3.32. When this term is included in the Heisenberg equation of motion,Eq. 3.33, one can show the presence of the discrete invariant beats. Thisanomaly highlights the non-trivial nature of the Hamiltonian.where O(t) is some generic operator. The last term we identify as theanomaly associated with the operator O:AO(t) = i(H† −H)O(t). (3.32)In calculating the moment of inertia one can show that:∂2∂t2〈r2〉(t) = 4〈Hλ〉+ 2AD(t), (3.33)where AD is the anomaly associated with the generator of scale transforma-tions. The anomaly term depends on the microscopic details of the system,and will be the source of the discrete scale invariant quantum beats. InFig. 3.3, we have calculated the moment of inertia using the Schrodingerrepresentation as well as by means of Eq. 3.33 for an initially Gaussian wavefunction. As one can see, the two approaches are identical.For a more detailed discussion of the anomalous term to the equation ofmotion, and its application to the quantum anomaly in two dimensions, werefer the reader to Appendix C.233.5. Summary(V/m t)1/2 t/(m ) |(,t)|2(V/m t)1/2rn(r,t)/Na)b)Figure 3.4: Here we show a schematic for the dynamics for a repulsivelyinteracting Bose gas when λ0/σ = 50, m|V | = 50 and t/(mσ2) = 1000.a) The time evolved wave function for an initially Gaussian wave functioncentred at λ0 with width σ. b) The resulting density profile of the Bose gas.3.4.3 Dynamics of a Repulsively Interacting Bose GasIn the previous discussion, we focused primarily on attractive condensatesas quantum fluctuations are important. For repulsively interacting conden-sates, the situation is different. By examining the fluctuations around thesemiclassical solution, 〈δλ(t)δλ(t)〉/λ20, one can show that the strength ofthe fluctuations is inversely proportional to the semiclassical path size, λ0.In the long time limit, the semiclassical path size grows as λ0 ∝ t. There-fore, the dynamics can be treated semiclassically. The density profile willnot exhibit a logarithmic singularity at short distances, since the probabil-ity of small condensate sizes are exponentially suppressed, and there are nobound states to produce discrete scale invariant beats. In Fig. 3.4 we showa schematic for the time evolved wave function and for the resulting densityprofile using the quantum variational approach. This solution is consistentwith the semiclassical description of the dynamics.3.5 SummaryIn this chapter we have studied the expansion dynamics of an initially in-homogeneous Bose gas in two spatial dimensions. Although the system isclassically scale invariant for repulsive interactions, for attractive interac-tions there is a quantum anomaly. In this work we neglected the effects of243.5. Summarythe quantum anomaly, and looked for the signatures of scale invariance onthe expansion dynamics of an initially inhomogeneous Bose gas.For repulsive interactions, it was shown that quantum fluctuations in thedynamics can be neglected. In this case, the dynamics of the Bose gas arewell described by a time dependent scaling ansatz that satisfies the semi-classical and hydrodynamic equations of motion. On the other hand, forattractive interactions, and weak renormalization effects, quantum fluctu-ations in the size of the condensate can not be neglected. We employ aquantum variational approach that replaces the many body quantum dy-namics with an effective single particle quantum mechanical problem of aparticle in an inverse square potential. This model is naively scale invariant,but requires regularization which is due to the confinement radius of the sys-tem [82]. The act of regularization breaks the continuous scale invariance,but replaces it with a discrete scale invariance. The discrete scale invarianceis manifest in a logarithmic singularity of the density at short distances, andin quantum beats that obey a discrete scaling relation.This approach is not only limited to the dynamics of Bose gases, butcan be used to study their energetics as well. In Ref. [67] we applied thisformalism to study N -body solitonic states of low dimensional Bose gases.The results of this formalism are consistent with previous studies on thespectrum of one-dimensional [94] and two-dimensional [95, 96] Bose gases.Although the quantum variational approach works quite well for theenergetics and dynamics of low-dimensional Bose systems, it has two limi-tations. The first is that the extension to Fermionic systems is not straightforward since Fermionic fields are expressed in terms of Grassmann fields.Secondly, this approach is a variational approach. It’s application dependson the specific system of interest, and for this reason it does not tell oneabout the general features of scale invariant dynamics. The validity of sucha method depends on the variational ansatz, and should be compared toexperiment. For this reason, a microscopic approach to the dynamics wouldbe ideal. In the next chapter we will develop a microscopic formalism thatcan elucidate the exact signatures of scale invariance.25Chapter 4Non-Relativistic Dynamicsand Scale SymmetryIn this chapter we answer the question what are the effects of scale invarianceon the dynamics of non-relativistic quantum systems? To do this, we exploita useful symmetry, the SO(2,1) symmetry. This symmetry is important asit incorporates both scale and conformal transformations.The first application of the SO(2,1) symmetry in cold atoms was pre-sented in Ref. [66], where the authors showed that the breathing modesoccur at twice the trap frequency. Although the two dimensional Bose gasis not scale invariant due to the quantum anomaly, their analysis appliesto any scale invariant system placed in an isotropic harmonic trap. As wewill review, the presence of this symmetry means that the spectrum of ascale invariant gas in an isotropic harmonic oscillator potential can alwaysbe decomposed into a series of evenly spaced states called conformal towers[70, 79].To supplement the algebraic approach, we use an exact microscopic for-malism based on a many body wave function that involves a time depen-dent rescaling of the position coordinates with a gauge transformation. Thismethod was put forward in Refs. [69, 70], where they examined the dynamicsand excitation spectrum of the unitary fermi gas in time dependent harmonictraps. This wave function approach is useful for understanding the expan-sion dynamics of scale invariant gases prepared in the ground state of anisotropic harmonic trap. In this case, the dynamics at all times are equiva-lent to a time dependent rescaling. This trivial rescaling dynamics has beenexploited for studying the dynamics of a number of scale invariant atomicsystems prepared in the ground state of an harmonic oscillator [4, 25, 71–74].In this chapter, we extend the previous analyses to arbitrary initial con-ditions. We begin by showing that the conformal tower states are an idealbasis to study dynamics, as their dynamics are equivalent to a trivial timedependent rescaling at all times. We then show that this algebraic approachis equivalent to understanding the dynamics of the many body wave functionin an expanding, non-inertial reference frame. Using these two techniques,264.1. The so(2,1) Algebrawe show that scale invariance implies that the long time dynamics of a localobservable are equivalent to a time dependent rescaling. Although this isnot true for short times, the trivial rescaling dynamics in the long time limitis true for arbitrary initial conditions, and arbitrary observable; it dependsonly on the scale invariance of the Hamiltonian.4.1 The so(2,1) AlgebraIn general, the Schrodinger equation has a number of symmetries consistentwith Galilean invariance [97]. The full list of symmetries is given in Ap-pendix D, but here we focus on three specific symmetries: time translations,scale transformations, Eq. 2.6, and conformal transformations, Eq. 2.15. Asnoted in chapter 2, the generator for these three transformations form a rep-resentation of the SO(2,1) symmetry; i.e. they form a Lie algebra, Eq. 2.16,the non-relativistic conformal algebra, or so(2,1) algebra [66, 70, 79]. In thissection we will study quantum dynamics by exploiting this algebra.We begin by discussing the implications of the so(2,1) algebra, Eq. 2.16,on the Hamiltonian, Hs + ω2C, a scale invariant gas placed in an isotropicharmonic trap with frequency ω. Here we explicitly define Hs as:Hs =∫ddrψ†(~r)(−∇22)ψ(~r)+12∫ddrddr′ ψ†(~r)ψ†(~r′)Vs(~r−~r′)ψ(~r′)ψ(~r),(4.1)while C is defined in Eq. 2.15, and we note Vs(~r) is a scale invariant po-tential satisfying: Vs(~re−b) = e2bVs(~r). Although Hs + ω2C is not scaleinvariant, it does possess SO(2,1) symmetry, which was first pointed out inRef. [66]. Since the conformal algebra includes both scale and conformaltransformations, it restricts both the spatial and temporal coordinates. Forthis reason, it will be beneficial to explore the explicit consequences of theconformal algebra.In terms of energetics, the spectrum of Hs +ω2C is composed of a seriesof conformal towers [66, 70, 79]. Each state in a given conformal tower willhave energy:En,ν = (2n+ Eν)ω. (4.2)The intra-tower level spacing is fixed to be two harmonic units. The numberof towers and their ground state energies will depend on the specific systembeing investigated. The quintessential example of this is the one dimensional274.1. The so(2,1) Algebraharmonic oscillator. For a single particle, there are two towers: one of evenparity with energy En,e = (2n+ 1/2)ω, and one of odd parity, En,o =(2n+ 3/2)ω. For higher dimensions, one can show that each conformaltower can be labelled by the angular momentum quantum numbers. Theproof of this is shown in Appendix D.These conformal towers turn out to be an exceptional basis to studydynamics. Consider the time evolution of a conformal tower state, |n〉, withcollective quantum number n, under the scale invariant Hamiltonian, Hs:|n(t)〉 = e−iHst|n〉En|n(t)〉 = e−iHst(Hs + ω2C)eiHst|n(t)〉. (4.3)From the commutation relations, Eq. 2.16, one can show:e−iHst(Hs + ω2C)eiHst =(1 + ω2t2)Hs − ω2tD + ω2C. (4.4)Although this operator may seem random, it can be rewritten in a moreconvenient form:(1 + ω2t2) ∫ddrψ†(~r)12(−i∇r − ~r ω2t1 + ω2t2)2+ω22(r2(1 + ω2t2)2)ψ(~r)+12∫ddrddr′ ψ†(~r)ψ†(~r′)Vs(~r − ~r′√1 + ω2t2)ψ(~r′)ψ(~r). (4.5)Eq. 4.5 is nothing more than the operator, Hs + ω2C, but defined in newcoordinates:r˜ =r√1 + ω2t2p˜ =√1 + ω2t2(−i∇r − ~r ω2t1 + ω2t2). (4.6)Eqs. 4.5 and 4.6 specifies that the state |n(t)〉 is an eigenstate of the Hamil-tonian, Hs + ω2C, if one uses the coordinates defined in Eq. 4.6.The dynamics of these conformal tower states are equivalent to the dy-namics of a Gaussian wave packet in free space. The probability densityof the wave function will maintain its shape, and all the dyanmics are con-tained in a time dependent rescaling factor: λ(t) =√1 + ω2t2. As a resultthe conformal tower basis is an excellent basis to consider scale invariant dy-namics, as the trivial rescaling dynamics are encapsulated in the rescalingfactor, λ(t).284.2. The Comoving Reference Frame4.2 The Comoving Reference FrameThe previous discussions were based on the conformal algebra, and connectsscale invariant dynamics in free space, to the states in a time dependentisotropic harmonic oscillator potential. Here we relate this connection to achange of reference frame by examining the N -body Schrodinger equationfor particles interacting through a scale invariant two-body potential, in dspatial dimensions:i∂tψ ({~ri, σi}, t) = Hsψ ({~ri, σi}, t) ,Hs =N∑i=1−12∇2i +12N∑i,j=1Vs(~ri − ~rj), (4.7)where {ri, σi} are the positions and spins of the i = 1, 2, ..., N particles.Consider the following many body wave function:ψ({~ri, σi}, t) = 1λdN/2(t)exp[i2N∑i=1r2iλ˙(t)λ(t)]φ({ ~riλ(t), σi}, τ(t)), (4.8)where λ(t) and τ(t) are yet to be determined functions of time, ~ri and σi arethe position and spin of the i = 1, ...N particles. All the many body infor-mation of the wave function is contained in the wave function, φ({~xi, σi}, τ).This ansatz was first put forward in Ref. [69, 70], and it combines a timedependent rescaling with a gauge transformation. The purpose of this wavefunction is to separate the trivial rescaling dynamics from any non-trivialdynamics. Substitution of this many body wave function into Eq. 4.7 yields:i∂τ∂t∂∂τφ ({~xi, σi}, τ) =(∑i[−121λ2(t)∇˜2i +x2i2λ¨(t)λ(t)]+12∑i,jVs(λ(t) (~xi − ~xj))φ ({~xi, σi}, τ) ,(4.9)where we have defined new coordinates:~x =~riλ(t)τ(t) (4.10)294.2. The Comoving Reference FrameAt this stage, λ(t) and τ(t) are undefined functions of time. Althoughthe choice is arbitrary, the most logical choice for expansion dynamics is todefine λ(t) and τ(t) according to:λ¨(t)λ3(t) = ω2 λ(t) =√1 + ω2t2, (4.11)λ(0) = 1 λ˙(0) = 0. (4.12)∂τ∂t=1λ2(t)τ(t) =1ωarctan (ωt) . (4.13)For this choice, the Schrodinger equation reduces to:i∂∂τφ({~xi, σi}, τ) = H˜φ({~xi, σi}, τ),H˜ =N∑i=1[−12∇˜2i +12ω2x2i]+12N∑i,j=1Vs(~xi − ~xj),(4.14)where we have used the fact that the interaction is scale invariant: Vs(e−b~r) =e2bVs(~r). Eq. 4.14 is nothing more than the Schrodinger equation for a scaleinvariant gas in the presence of a Harmonic trap.Eq. 4.14 can be understood as studying the dynamics of a quantum sys-tem in an expanding, non-inertial reference frame, or comoving frame, withcoordinates defined in Eq. 4.10. Since the comoving frame is a non-inertialreference frame, a fictitious force must appear. This fictitious force is noth-ing more than a harmonic restoring force with frequency√λ¨(t)λ3(t) = ω.The resulting Hamiltonian in this comoving reference frame is the physicalinterpretation of the conformal algebra and Eq. 4.5.The eigenstates of Eq. 4.14 are just the conformal tower states whichwere discussed in the previous section. If one prepares a quantum systemin a given conformal tower state, the dynamics of the wave function in thelaboratory frame will be a simple time dependent rescaling:ψn({~ri, σi}, t) = 1λ(t)dN/2exp[i2N∑i=1r2iλ˙(t)λ(t)]φn({ ~riλ(t), σi})e−iEnτ(t),(4.15)where φn({~ri, σi}) = 〈{~ri}|n〉 is a conformal tower state with energy En.The dynamics of scale invariant systems in a given conformal tower statehave been used to study the dynamics of resonant three dimensional gases304.2. The Comoving Reference Frame2+E04+E06+E08+E00 1 2 3 4 5 6 7 8 9 10 tE02+E04+E06+E08+E0a)b)Figure 4.1: The conformal tower states in both a) the laboratory frame,and b) in the co-moving frame. A single conformal tower is depicted withground state energy E0. In the laboratory frame, the conformal tower statesare evenly spaced but contract like t−2 in the long time limit, ωt  1, seeEq. 4.16. In the co-moving frame the spectrum does not evolve with time.This plot first appeared in Ref. [2].[70], the Tonks girardeau gas in one dimension [73], and for scale invariantgases in time dependent harmonic traps [4, 25].The phase factor −iEnτ(t), is the dynamical phase. Although these con-formal tower states are not eigenstates of the Hamiltonian in the laboratoryframe, Hs, they can still be described by a global dynamical phase, as ifthey were eigenstates. Physically, the dynamical phase is equivalent to theadiabatic phase for an eigenstate with quasi-energy:En(t) =Enλ2(t). (4.16)The conformal tower spectrum is shown in Fig. 4.1, in both the comoving andlaboratory frames. The main difference is that in the laboratory frame it isnecessary to deal with the qausi-energies of Eq. 4.16, while in the comovingframe, the spectrum is time independent.It is important to note that this phase factor freezes at large times:τ(ωt 1) ≈ 1ω[pi2− 1ωt+O(1ω2t2)]. (4.17)As a result, the dynamics of an arbitrary superposition of conformal towerstates will eventually be governed only by λ(t). To see this consider an314.2. The Comoving Reference Framearbitrary superposition of conformal tower states with expansion coefficients,Cn:ψ({~ri, σi}, t) =∑nCn1λdN/2(t)exp[i2n∑i=1r2iλ˙(t)λ(t)]φn({ ~riλ(t), σi})e−iEnτ(t).(4.18)In the comoving frame the wave function will freeze in some configurationwhen the dynamical phase saturates after a few initial time scales. At thispoint, the probability density will maintain its shape, and simply rescaleaccording to λ(t), exactly like an individual eigenstate would. The onlydifference is that for an arbitrary superposition, the asymptotic wave func-tion in the comoving frame is not necessarily equivalent to the initial wavefunction.We can use the saturation of the dynamical phase to state a consequenceof scale symmetry on the dynamics of an arbitrary non-relativistic quantumsystem: the dynamics of some local observable, 〈O(~r)〉(t), will be equivalentto a simple time dependent rescaling at long times:limωt,ωt′→∞〈O(r)〉(t) ≈(λ(t′)λ(t))∆O 〈O(λ(t′)λ(t)~r)〉(t′), (4.19)where ∆O is the scaling dimension of the operator, O(~r). In the long timelimit, λ(t) ≈ ωt, and thus 〈O(~r)〉(t) ∝ t−∆0 . As we will prove in Chapter7, this approximate long time rescaling is due to an emergent conformalsymmetry.To end this section, we comment that the use of the comoving frameis not only limited to connecting the dynamics in free space to that in aharmonic trap. The transformation shown in Eq. 4.8 can be applied to anumber of situations including the dynamics of time dependent traps. Forthis situation, we can use the comoving frame to map the system onto atime independent harmonic trap. For this situation, the equation for λ(t) isgiven by:λ¨(t) =ω20λ3(t)− ω2(t)λ(t), (4.20)where ω0 is a reference trap frequency, and ω(t) is the time dependent trapfrequency. This has been applied to the dynamics of scale invariant gasesplaced in the ground state [4, 25, 68, 74]. In Appendix E, we review this ap-proach, and determine the time dependent scaling factor, λ(t), for a suddenquench in the trap, [74], and for a broadening time dependent trap [4].324.3. Summary4.3 SummaryIn summary, the dynamics of scale symmetric systems are intimately tied toconformal symmetry. Here we used the fact that for scale invariant systems,there exists a set of dynamical states which remain eigenstates of a time-dependent harmonic oscillator potential, at all times. The existence of thesestates rely on the non-relativistic conformal, or SO(2,1), symmetry, which isvalid for Galilean invariant systems with scale invariant Hamiltonians [66, 70,79, 97]. Physically the so(2,1) algebra is equivalent to studying the dynamicsin an expanding, non-inertial, comoving, reference frame. The Schrodingerequation in the comoving frame is simply the scale invariant Hamiltonianplaced in an isotropic harmonic potential. This system possesses the SO(2,1)symmetry, and the spectrum is given by a series conformal towers.We examined the dynamics using this conformal tower basis, and showedthat the quantum state necessarily freezes in the comoving frame. Conse-quently, all the dynamics at long times are governed by the time dependentrescaling factor, λ(t). This has immediate implications for the dynamics oflocal observables. The freezing of the dynamical phase implies that the longtime dynamics of any local observable is simply a time dependent rescaling,independent of the initial conditions. These predictions are robust and donot depend on the microscopic details of the system, just the scale invari-ance of the Hamiltonian, and the conformal algebra. In the next chapterwe will study the robustness of these results to the explicit breaking of scaleinvariance.33Chapter 5The Breaking of ScaleInvarianceFor a generic quantum system, scale invariance requires some form of finetuning. In the context of atomic gases, it is possible to tune the s-waveinteractions by means of a Feshbach resonance [6]. The Feshbach resonanceallows one to tune the interactions from zero to positive or negative infinity.For quantum gases with s-wave interactions, we can relate the interactionstrength to a new length scale, the d-dimensional scattering length, a. Forscale invariant systems, this length scale will either be zero or diverge, andas a result, will not be present in the physics. In this situation, the dynamicsof the quantum gas are well described by the results in Chapter 4.However, an important question remains. How do the dynamics for asystem with explicitly broken scale invariance differ from the scale invariantcase. If the scale invariance is only broken slightly, we can imagine twosituations. The first is that the breaking of scale invariance is irrelevant,and the long time dynamics is equivalent to a scale invariant system; andthe second, the interactions destroy the long time scaling behaviour seen inEq. 4.19.In this chapter, we will answer this question by examining a d-dimensionalquantum gas with either nearly resonant, or weak s-wave interactions. Al-though we consider a specific system, our methods will be quite general,and can apply to a number of cold atom systems. Here we show that therelevancy of the interactions in the long time limit depends on the thermo-dynamic relevancy of the deviation. We explicitly put down a condition thatdelineates a relevant from an irrelevant deviation from scale invariance.For relevant interactions, we show that it is possible to find a non-perturbative solution for the long time dynamics. This approach primarilydepends on the universal scaling properties of the operator that breaks thescale invariance, and only marginally on the microscopic details of the sys-tem. In this situation, the time dependent rescaling dynamics is modifiedby a non-trivial time dependence. The form of this time dependence is fixedby the scaling of the deviation from scale invariance.34Chapter 5. The Breaking of Scale InvarianceTo address the effects of the broken scale invariance for nearly reso-nant, and weakly interacting scale invariant quantum gases, we consider theHamiltonian:H =∫ddrψ†(~r)(−∇2r2)ψ(~r) +12∫ddrddr′ψ†(~r)ψ†(~r′)V (~r− ~r′)ψ(~r′)ψ(~r),(5.1)where V (~r) is a nearly scale invariant two-body potential. In order to studythe breaking of scale invariance, it is beneficial to split Eq. 5.1 into a scaleinvariant piece, Hs, and a deviation, δH:Hs =∫ddrψ†(~r)(−∇2r2)ψ(~r) +12∫ddrddr′ψ†(~r)ψ†(~r′)Vs(~r − ~r′)ψ(~r′)ψ(~r)δH =12∫ddrddr′ψ†(~r)ψ†(~r′)(V (~r − ~r′)− Vs(~r − ~r′))ψ(~r′)ψ(~r). (5.2)For this Hamiltonian, the time evolved wave function is given by:|ψ(t)〉 = e−iHt|ψ0〉, (5.3)where |ψ0〉 is the initial state. Here we note that the state |ψ(t)〉 is relatedto the many body wave function through:〈{~ri}|ψ(t)〉 = ψ({~ri, σi}, t)=1λdN/2(t)exp[i2N∑i=1r2iλ˙(t)λ(t)]φ({ ~riλ(t), σi}, τ(t)). (5.4)For nearly scale invariant systems, it is ideal to expand the total Hamiltonianaround the scale invariant Hamiltonian. This can be done by using theinteraction picture with respect to the scale invariant Hamiltonian, Hs. Inthe interaction picture, the time dependent wave function is given by:|ψ(t)〉 =∑nCn(t)e−iEnτ(t)|n(t)〉Cn(t) = 〈n|Te−i∫ t0dt′ δHI(t)|ψ0〉δHI(t) = eiHstδHe−iHst, (5.5)where T is the time ordering operator, and we have used the notation:355.1. Explicit Form of the Deviation From Scale Invariance〈{~ri}|n(t)〉 = 1λdN/2(t)ei2∑Ni=1r2iλ˙(t)λ(t)φn({ ~riλ(t), σi}). (5.6)The presence of the deviation will induce couplings between the differentconformal tower states. As a result, the occupation of a given conformalstate is no longer time independent, as in the scale invariant case. Theremainder of this chapter will investigate this additional time dependence,and determine the modification to the scale invariant dynamics.5.1 Explicit Form of the Deviation From ScaleInvarianceIn order to solve Eq. 5.5, it is necessary to understand the explicit form ofthe deviation operator, Eq. 5.2. The breaking of scale invariance means thephysics depends on an additional length scale. As mentioned previously,for atomic gases, this length scale is the scattering length. The question isthen how to relate the deviation to this new length scale. To study this weconsider a model contact interaction of strength g(Λ):V =12∫r>Λ−1ddrg(Λ)ψ†(~r)ψ†(~r)ψ(~r)ψ(~r), (5.7)where Λ is the ultraviolet momentum cut off for the theory. Although wehave chosen a specific model to study, it is important to note that g(Λ)is chosen such that the low energy physics is independent of the UV cutoff, and this effective model reproduces the low energy scattering of thetrue interatomic potential; i.e. both the true potential and the contactinteraction can be parametrized by the same scattering length, a. The scaleinvariant value of the coupling constant is denoted by, g∗(Λ). When thecoupling constant is fine tuned to this value, the scattering length will eitherdisappear or be infinite.For this interaction, the deviation operator can then be written as:δH =∫r>Λ−1ddr (g(Λ)− g∗(Λ))ψ†(~r)ψ†(~r)ψ(~r)ψ(~r). (5.8)In order to proceed further, it is important to note that g(Λ) − g∗(Λ)can be related to the beta-function [46, 47]. The beta-function describes theeffects of perturbations and their relevancy to thermodynamic properties,as one coarse grains and examines the system on larger and larger lengthscales. For this reason, the beta-function is intimately connected to how the365.1. Explicit Form of the Deviation From Scale Invariancesystem responds to a scale transformation. In particular, if the system isscale invariant, the system should not change under a scale transformation;in terms of the beta-function, a scale invariant system corresponds to a zeroof the beta-function. The definition of the beta-function is:β(g˜(Λ)) =∂∂ log(Λ)g˜(Λ)g˜(Λ) = CdΛd−2g(Λ), (5.9)where Cd is a dimensional dependent constant. For a d-dimensional quantumgas with short range interactions, the beta-function is given by [67]:β(g˜(Λ)) = (d− 2)g˜(Λ) + g˜2(Λ). (5.10)For three spatial dimensions, Eq. 5.10 has two zeros corresponding to thenon-interacting system, g˜∗(Λ) = 0, and the resonantly interacting system,g˜∗(Λ) = −1.In order to relate the deviation to the beta-function, it is convenient tointroduce a physical length scale associated with the interaction, a. In termsof atomic gases, this length scale is simply the s-wave scattering length. Thescattering length is related to the coupling constant via:a = Λ−1f (g˜(Λ)) . (5.11)Since this is a physical length scale, it must be independent of the ultravioletcutoff of the theory:∂a∂ log(Λ)= 0 = −a+ ∂a∂g˜(Λ)β(g˜(Λ)). (5.12)Near a scale invariant fixed point, it is possible to expand the beta-function to linear order:β(g˜(Λ)) ≈ β′(g˜∗(Λ)) (g˜(Λ)− g˜∗(Λ)) , (5.13)where β′(g˜∗(Λ)) is the derivative of the beta-function evaluated at the scaleinvariant point. Substituting Eq. 5.13 into Eq. 5.12 relates the physicallength scale to the deviation of the coupling constant from its scale invariantvalue:g(Λ)− g∗(Λ) ∝ 1Λd−21(Λa)−β′(g˜∗(Λ)) . (5.14)375.2. Classification of DeviationsAt this point we can write down the deviation operator as:δH =1a−β′(g˜∗(Λ))1Λd−2−β′(g˜∗(Λ))∫r>Λ−1ddrψ†(~r)ψ†(~r)ψ(~r)ψ(~r). (5.15)It is instructive to consider how δH transforms under a scale transformation,ψ(~r)→ e−bd/2ψ(~re−b). In this case, by changing the variables ~r → ~re−b andΛ→ Λeb, one can show:δH → e−(2+β′(g˜∗(Λ)))bδH. (5.16)Eq. 5.16 states that the scaling dimension of δH is no longer 2, but 2 +β′(g˜∗(Λ)). The derivative of the beta-function will be an important quantityfor the remainder of these discussions. For this reason we define:α = −β′(g˜∗(Λ)). (5.17)For the three dimensional gas with s-wave contact interactions, one can showthat α = 1 near resonance, and α = −1 for weak interactions.Now that we know the scaling dimension of the deviation, the only thingleft is to show that δH is independent of the ultraviolet physics. This hasbeen done previously by examining the short distance physics of Fermi sys-tems [75–77, 98]. In short we note that δH is regularized with the scalingdimension 2− α.5.2 Classification of DeviationsNow that the explicit form of the deviation, δH, determined, let us considerhow the deviation couples different conformal tower states, and how the de-viation alters the dynamics of a quantum system. To do this, it is necessaryto determine the time dependence of the expansion coefficients defined inEq. 5.5:Cn(t) =∞∑m=0C(m)n (t)C(m)n (t) = (−i)m∫ t0dt1...∫ tm−10dtm〈n|eiHst1δHe−iHst1 ...eiHstmδHe−iHstm |ψ0〉.(5.18)385.2. Classification of DeviationsA fundamental component of Eq. 5.18 is the matrix element of the deviationoperator between two conformal tower states, in the interaction picture.Utilizing the dynamical properties of the conformal tower states, see Eq.4.5,one obtains:〈n|eiHstδHe−iHst|l〉 = 1λ(t)2−αaαei(En−El)τ(t)〈n|δh|l〉, (5.19)where |n〉, and |l〉 are two conformal tower states, λ(t) = √1 + ω2t2, andδH = a−αδh.As an estimate of the relevancy of the deviation, consider the number ofconformal tower states mixed by the interaction at time t. The number ofcoupled conformal tower states can be estimated by calculating the ratio ofthe interaction strength to the quasi-energy spacing of the conformal towerstates, see Eq. 4.16:Ncoupled ≈(1√1 + ω2t2)2−α 1 + ω2t22≈ (1 + ω2t2)α/2. (5.20)In the long time limit, Ncoupled ∝ tα. Therefore the relevancy of the deviationto the dynamics depends on the derivative of the beta-function near a givenfixed point.We note that this same analysis can be performed in the comoving frame.In this frame, the conformal tower states have no time dependence, but thedeviation will scale as: δH ∝ a−α(1 + ω2t2)α/2. This leads to the samecondition as Eq. 5.20. The strength of the deviation as a function of time,compared to the conformal tower spectrum, is shown in Fig. 5.1 for twodeviations with scaling α = 1, and α = −1.In order to obtain an exact condition on whether a deviation is relevantor irrelevant, let us consider first order perturbation theory, in the long timelimit:Cn(t) ≈ Cn(0)− i∫ t0dt′1(ωt′)2−α1aαei(En−El)τ(t)〈n|δh|ψ0〉. (5.21)For 2 − α > 1, or equivalently α < 1, the dominant contribution to theexpansion coefficient is from short times. In the large time limit, the effect395.2. Classification of Deviations2+E04+E06+E08+E00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t2+E04+E06+E08+E0a)b)Figure 5.1: Here we show the conformal tower spectrum and perturbationin both a) the laboratory frame, and b) the comoving frame. The blue(dotted) lines correspond to the conformal tower states. The red (solid) andblack (dash-dotted) lines correspond to deviations with scaling, α = 1 andα = −1, respectively. For scaling α = 1, the interaction eventually couplesmore and more states, see Eq. 5.20, with Ncoupled ∝ t. For α = −1, theinteraction vanishes with time, i.e. for long times, fewer and fewer statesare coupled together with, Ncoupled ∝ 1/t. We therefore expect a breakdownof time-dependent perturbation theory for α ≥ 1. This figure first appearedin Ref. [2].405.3. Non-Perturbative Solution for Relevant Deviationsof the interaction vanishes like t−1+α. Therefore, the effects of the deviationvanish at long times, and the dynamics are equivalent to the scale invari-ant dynamics discussed in Chapter 4. The only difference between the scaleinvariant dynamics and the perturbed dynamics is that the deviation will al-ter the expansion coefficients at short times. However, this time dependencewill quickly saturate, and the resulting dynamics will be governed by thetime dependent rescaling. If the deviation is sufficiently weak, perturbationtheory and the conformal dynamics will provide an accurate description ofthe dynamics.On the other hand, for α ≥ 1, the contribution at long times divergesas:Cn(t) ≈ 1− i(ωt)α−1α− 11(√ωa)α〈n|V˜ |ψ0〉 α > 1≈ 1− i log(ωt) 1√ωa〈n|V˜ |ψ0〉 α = 1, (5.22)where we have defined the matrix, V˜ , by its matrix elements between twoconformal tower states:〈n|V˜ |l〉 = 1ω1−α/2ei(En−El)pi2ω 〈n|δh|l〉. (5.23)Therefore, the effect of the interaction becomes increasingly important in thelong time limit. As a result, perturbation theory is inadequate at describingthe dynamics. At this point, one can posit that the rescaling dynamics atlong times will be broken by a non-trivial time dependence. In order tocapture this non-trivial time dependence, a non-perturbative approach isneeded.5.3 Non-Perturbative Solution for RelevantDeviationsIn this section, we develop a non-perturbative approach for dealing withnearly scale invariant systems with deviations that have scalings: α ≥ 1.We begin by examining the mth order of Cn(t), Eq. 5.18. Inserting mconformal tower states, and utilizing Eq. 5.19 gives:〈n|C(m)n (t)|ψ0〉 =(− iaα)m ∑l1...lm∫ t0dt1...∫ tm−10dtm415.3. Non-Perturbative Solution for Relevant Deviations 1√1 + ω2t212−α ...( 1√1 + ω2t2m)2−αe−i(En−El1 )τ(t1)...e−i(Elm−1−Elm )τ(tm)〈n|δh|l1〉...〈lm−1|δh|lm〉〈lm|ψ0〉, (5.24)Eq. 5.24 can be simplified by noting that in the long time limit, ωt  1,the dominant contribution to these integrals comes from long times. In thislimit the dynamical phase saturates, see Eq. 4.17, and the time dependenceof the deviation comes from the simple rescaling factor, λ(t). In this case,the summation over the intermediate conformal states can be collapsed:∑l1,...,lme−i(En−Elm )pi2ω 〈n|δh|l1〉...〈lm−1|δh|lm〉〈lm|ψ0〉 = ωm(1−α/2)〈n|V˜ m|ψ0〉,(5.25)where V˜ is defined in Eq. 5.23. The remaining integrations can be carriedout to give:∫ t0dt1...∫ tm−10dtm 1√1 + ω2t212−α ...( 1√1 + ω2t2m)2−α≈ 1m!(ωt)α−1α− 11ωm,(5.26)Eqs. 5.25 and 5.26 allow one to non-perturbatively solve for the expansioncoefficients, in the long time limit:Cn(t ω−1) ≈ exp[−i 1(√ωa)α(ωt)α−1α− 1 V˜]α > 1≈ exp[−i 1√ωalog(ωt)V˜]α = 1. (5.27)Eq. 5.27 is nothing more than the unitary evolution operator under aHamiltonian, V˜ , with time coordinate tα−1/aα. As stated in the previ-ous section, the deviation operator, V˜ , is a regularized, dimensionless, anduniversal operator. It only depends on the statistics and the number ofparticles.The scaling of the effective temporal coordinate, tα−1/aα, is dictatedby the derivative of the beta-function evaluated at the nearby scale invari-ant fixed point. It is thus universal, and only depends on basic symmetryproperties, and the dimension of the system.425.4. SummaryThe presence of this non-perturbative time dependence will lead to non-trivial dynamics for a local observable, O. In particular, due to the scalingof the deviation, one can posit that the dynamics of a local observable willhave the form:〈O〉(t >> ω−1) ≈ (ωt)−∆OF(1aα(ωt)α−1α− 1)(5.28)Eq. 5.28 states that the scale invariant dynamics are modified by somefunction of tα−1/aα. Although the function, F (t), depends on the specificobservable, the number of particles, and more importantly, the matrix V˜ ,the scaling of Eq. 5.28 is robust.5.4 SummaryIn this chapter we examined the dynamical consequences of explicitly break-ing the scale invariance. When the scale invariance is explicitly broken, anew length scale appears. For quantum gases with short ranged interac-tions, this length scale is the s-wave scattering length. This length scalecan be used to parametrize the strength of the deviation from scale invari-ance: δH ∝ a−α, where α, defined in Eq 5.17, is given by the negative ofthe derivative of the beta-function evaluated at the nearby scale invariantpoint.The scaling of the deviation, α, dictates whether a deviation is relevant,α ≥ 1, or irrelevant, α < 1, to the study of dynamics. For irrelevantinteractions, the number of conformal states mixed by the deviation vanishesfor long times. Therefore, the long time dynamics are governed by thetime dependent rescaling. For relevant interactions, the system becomesstrongly interacting in the long time limit, and a non-perturbative approachis required. The results of such an approach is contained in Eqs. 5.27. Theseresults are non-perturbative, and are the main theoretical results of thisthesis. This relevant breaking of scale invariance will modify the dynamicsof local observables, as seen in Eq. 5.28.In the remaining sections, we explore the physical implications of theseresults. In particular, we employ this formalism to the expansion of anensemble of two-body systems in Chapter 6, and to the compressional andelliptic flow of resonantly interacting Fermi gases in three dimensions inChapter 7.43Chapter 6Application to Two-Bodyand One-Body SystemsIn this chapter we employ the formalism of Chapter 5 on a toy model; therelative dynamics of a two-body system, or equivalently, a single particlein the presence of a short ranged potential, in three spatial dimensions.The two particles can be either two bosons, or fermions in the spin singletchannel.For both these systems, the two scale invariant points correspond to thenon-interacting, a = 0, and resonantly interacting, a = ∞, limits. The dy-namics at these points can be understood using the formalism in Chapter4. However, in advance it is not known how the dynamics differ for finitescattering lengths. As stated in Chapter 5, the relevancy of the interactioncan be determined by means of the beta-function. For a three dimensionalquantum gas the relevancy of the deviation from resonance was marginallyrelevant, α = 1, while the deviation from scale invariance was irrelevantfor weak interactions, α = −1. Since the beta-function only depends onbasic properties of the system, and not the number of particles, we ex-pect the dynamics near resonance to exhibit a non-perturbative logarithmictime dependence, while the asymptotic dynamics for weak interactions tobe equivalent to a scale invariant system. This can be shown analyticallyfor our two toy models.6.1 The Two-Body Problem: SchrodingerEquation in the Comoving FrameTo start, we consider the two-body problem, and write down the radialSchrodinger equation for the relative coordinates, in the comoving frame:i∂τχl(x, τ) =[−12∂2x +12x2 +l(l + 1)2x2+ λ2(τ)V (xλ(τ))]χl(x, τ)λ(t) =√1 + ω2t2 = sec(ωτ)446.2. Near Resonanceτ(t) =1ωarctan(ωt) (6.1)where we have set the (reduced) mass to unity, and Yl,m(~x) is the spher-ical harmonic with angular quantum number, l, and projection quantumnumber, m. The radial wave function, χl(x, τ), is related to the full wavefunction via: φl,m(~x, τ) = Yl,m(xˆ)χl(x, τ)/x, and is properly normalized:∫ ∞0dx|χl(x, τ)|2 = 1. (6.2)In what follows we will only focus on the zero angular momentum, or s-wave, scattering of this potential, as higher angular momentum scatteringis suppressed by a factor of (√Er0)2l, where r0 is the range of the potential,and E is the relative energy.For specificity, we will consider the potential to be a square well of depth:V0λ2(τ), and range: r0/λ(τ). This potential is consistent with the timedependence of the interaction in the comoving frame: λ2(τ)V (xλ(τ)), andcaptures all the essential physics at low energies. The scattering length forthis potential depends on the depth and range via:a(τ) =aλ(τ)=r0λ(τ)(1− tan(√V0r0)√V0r0). (6.3)As can be seen from Eq. 6.3, at either of the scale invariant points, a(τ),will remain at the scale invariant point. However, if the scattering length isfinite, it will naturally flow to smaller values as time increases. The questionnow is, how do the dynamics for nearly resonant and weakly interacting two-body systems differ from their scale invariant counterparts.6.2 Near ResonanceFor large scattering lengths, we expand Eq. 6.1 around the resonant, or a =∞, solution. The spectrum of the resonant Hamiltonian will be composed ofconformal towers, each characterized by their angular momentum quantumnumber. Since we only consider s-wave scattering, we only need to focuson the matrix element of the deviation operator between s-wave conformaltower states.Naturally, we expect the deviation from scale invariance to be propor-tional to a−1, or equivalently, has scaling α = 1. In appendix F, we showthat this is indeed the case, and derive an analytical expression for thedeviation operator, see Eq. 5.23:456.2. Near Resonance〈n|V˜ |m〉 = fnfm, (6.4)where:fn =√2pi1/4(2n− 1)!!√(2n)!. (6.5)The deviation operator, V˜ , has only a single non-zero eigenvalue:v =nmax∑n=0f2n 〈n|v〉 =fn√v. (6.6)The eigenvalue, v, diverges with the harmonic quantum number as:√nmax.We note that this sum is controlled by the energy scale set by the range ofthe potential: nmax = 1/(ωr20), where r0 is the range of the potential withr0  a.The formal solution for the wave function in the laboratory frame isgiven by:ψ(~r, t) =1λ3/2(t)exp[ir22λ˙(t)λ(t)]∑nCn(t)e−iEnτ(t)φn(~rλ(t))Cn(t) = 〈n| exp[−i 1√ωalog (ωt) V˜]|ψ0〉= 〈n|ψ0〉+ 〈n|v〉〈v|ψ0〉(e− i√ωalog(ωt)v − 1). (6.7)where φn(~r) are the resonant conformal tower states, see Appendix F. Herewe have assumed real initial expansion coefficients to specifically highlightthe effect of the deviation.The first term in Eq. 6.7 is the initial expansion coefficients, and it pro-duces the scale invariant dynamics. The presence of the second term will pro-duce a beat in the probability density of the form: sin2(v/(√ωa) log(ωt)/2),in the laboratory frame. The amplitude of this beat is proportional to v−1.In the limit of zero range interactions, r0 → 0, v diverges and the beatamplitude vanishes. As a result, the scale invariant dynamics of zero rangemodels is robust to deviations. Only models with a finite range will exhibitnon-perturbative corrections to the dynamics. The beat is illustrated inFig. 6.1, where we show the numerical solution for the probability to be inthe resonant ground state, |C0|2, for a wave function initially in the resonant466.2. Near ResonanceFigure 6.1: The probability for the particle to remain in the resonant groundstate in the comoving frame, for large finite scattering lengths, a λ0, as afunction of −λ0/(4pia) ln(pi/2−τ). Here we note, λ0 = 1/√ω, and pi/2−τ =1/(ωt). This result has been obtained by numerically solving Eq. 6.1 withλ0/(4pia) = 0.015 and r0/λ0 = 10−3/2. The system is initially prepared inthe ground state of the resonant model. Very quickly the probability satisfiesEq. 6.7, and develops oscillations at the frequency v/2 = 20.14. This figurefirst appeared in Ref. [2].ground state. The solution is obtained by numerically solving the time de-pendent Schrodinger equation, Eq. 6.1. The oscillations at frequency, v/2,are easily observed and are well described by Eq. 6.7.This logarithmic beat will be present in all the dynamics of the system.As an example of the breaking of scale invariance, consider the time evolutionof the moment of inertia for the two-body system. The moment of inertiais defined as:limt→∞〈r2〉(t) = 〈r21 + r22〉(t) ≈ λ2(t)(〈X2〉(pi/2) + 〈x2〉(τ(t)). (6.8)In Eq. 6.8, we have assumed that the center of mass motion is not entan-gled with the relative motion, allowing for the center of mass and relativemoments of inertia to decouple. The contribution to the center of mass,476.2. Near Resonance〈X2〉(τ), will saturate as it is trivially scale invariant. The relative motionis complicated due to the breaking of scale invariance. In Fig. 6.2, we showthe relative moment of inertia. In the comoving frame, the log-periodic os-cillations are clearly visible. In the laboratory frame, the dynamics of therelative moment of inertia are given by:limt→∞ω〈x2〉(t) ≈ A+B sin(v1√ωalog(ωt))1ωt+D sin2(v21√ωalog(ωt)).(6.9)In Eq. 6.9, the coefficients A, B, and D depend on the initial conditions,and the range of the potential, r0, through the state |v〉. Explicit expres-sions for these coefficients are given in Appendix F. This result is consistentwith Eq. 5.28, and the time dependent perturbation theory, Eq. 5.22, if oneexpands the sine functions to first order in v/(√ωa) log(ωt). The presenceof the beat in the expansion coefficients can lead to significant deviationsfrom the scale invariant dynamics, which can be seen in Fig. 6.2.In addition to these results, we note that the two-body interaction doesnot break translational invariance in the laboratory frame. Thus, it is mean-ingful to study the dynamics of the momentum distribution, n(k, t). Inparticular, we examine the dynamics of the contact. The contact is de-fined by the assymptotic behaviour of the momentum distribution: C(t) =limk→∞ k4n(k, t) [76, 77]. The dynamics of the contact at resonance, andfor weak interactions have been discussed previously in Ref. [72]. Here weextend their analysis to study the dynamics of the contact away from reso-nance.In order to obtain the momentum distribution near resonance, we willagain assume that the initial wave function is a product state between thecenter of mass, and relative motion. One can then integrate out the centerof mass coordinate to unity, and examine the momentum distribution forthe relative coordinate. The relative momentum distribution for this case isrelated to the Fourier transform of the solution for the relative wave function,Eq. 6.7:ψ(k, t) =∑ne−iEnτλ3/2(t)〈n|e− i√ωa log(ωt)V˜ |ψ0〉 ·∫d3rei r22λ˙(t)λ(t)−i~k·~rφn(~rλ(t)).(6.10)For large momenta, the integrand of Eq. 6.10 will be dominated by the con-tribution at short distances, r  k−1. It is possible to obtain an analyticalexpression for the contact:486.2. Near ResonanceFigure 6.2: The time evolution of 〈x2〉(τ) as a function of−λ0/(4pia) ln(pi/2−τ), where λ0 = 1/√ω, and pi/2 − τ = 1/(ωt). This has been obtained bynumerically solving the near resonant wave function, Eq. 6.1. In this calcu-lation, the system was prepared in the ground state with λ0/(4pia) = 0.015and r0/λ0 = 10−3/2. The dynamics can be fit to Eq. 6.9, with oscillationsat frequency v/2 = 20.14. In the inset, the dynamics over the entire rangeis shown. This figure first appeared in Ref. [2].496.3. Weakly InteractingC(t) = limk→∞k4n(k, t) = limk→∞k4 |ψ(k, t)|2 = 1λ(t)C˜(t)C˜(t) =∣∣∣∣∣∑n〈n|e− i√ωa ln(ωt)V˜ |ψ0〉e−iEnτ(t)√pi2fn∣∣∣∣∣2, (6.11)where the matrix elements of V˜ and fn are given in Eqs. 6.4, and 6.5.At resonance, the contact in the comoving frame tends to a constantwhich depends on the initial conditions. The resulting dynamics are givenby the scaling parameter, λ(t); this result was obtained in Ref. [72]. Nearresonance, however, the expansion coefficients are time dependent due tothe log-periodic beat, Eq. 6.7. The beat in the expansion coefficients willtranslate to a beat in the contact:limt→∞C(t) ≈Eλ(t)+Fλ(t)sin2(v21√ωalog (ωt)), (6.12)where E and F are two coefficients, which are given explicitly in AppendixF, and we have neglected terms that vanish as 1/(ωt)2. The first term is theresonant scale invariant result, while the second term is the deviation thatarises from breaking the scale invariance. Again, the presence of the log-periodic beat only depends on the deviation from resonance. The amplitudeof the beat is controlled by the constant F which depends on the finite range,r0, through the state |v〉.6.3 Weakly InteractingLet us compare the dynamics near resonance to the weakly interacting case.In this case, the scattering length is small, and we expect the deviation to beproportional to a. As shown in Appendix F, this is indeed the case, and thescaling is: α = −1. Therefore, the deviation is irrelevant to the dynamics.The system will be well described by first order perturbation theory:Cn(t) = 1− i∫ t0dt′aλ3(t′)〈n|V˜ |ψ0〉. (6.13)As shown in Appendix F, the matrix, V˜ , in the weakly interacting limit isgiven by:506.4. Experimental Application〈m|V˜ |n〉 = gmgngn = (−1)n 1pi1/412n−1√(2n+ 1)!n!, (6.14)where Γ(n) is the gamma function.6.4 Experimental ApplicationIn this section, we propose a future experiment composed of an ensembleof two-body states, tightly confined in micro-traps that are periodically ar-ranged in an optical lattice, see Fig. 6.3. The trap frequency, ω, of eachmicro-trap depends on the laser intensities: ω2 = (1/3m)∇2V (~r)|~r=~r0 (as-suming cubic lattice symmetry), where the derivatives are evaluated at thelattice sites of the optical lattice, which are formed by the minima of theconfining potential energy, V (~r), see Eq. 6.15. To enhance the effect of bro-ken scale invariance and for the convenience of experimental observation,we further propose to use three pairs of coplanar lasers angled at a small θto create a lattice with a controllable and relatively large lattice constant,al(θ).The laser set up is shown in Fig. 6.3 a). For each dimension, we usetwo coplanar beams with wave vectors ~k1, and ~k2. These two beams willthen interfere and create a standing wave with an effective wave vector,δ~kα = (~k1 − ~k2)α = k sin(θ/2)eˆα, where eˆα is the unit vector for directionα = x, y, z. In this experiment we assume that each pair of coplanar beamsare constructed to produce a periodic potential of the form:V (~r) = V0 cos2(k sinθ2x)cos2(k sinθ2y)cos2(k sinθ2z)(6.15)where V0 is proportional to the laser intensity. The lattice constant for thispotential is given by:al =pik sin θ2. (6.16)In practice, the lattice constant can be tuned up to the order of millimetresby decreasing θ.Now it is possible to create an ensemble of two-body systems via anoptical lattice [99, 100] by downloading pre-cooled atoms either a) in the516.5. Application to a Mobile Impuritypresence of Feshbach resonance, or b) in the absence of scattering. Thetwo situations, a) and b), correspond to the resonant and free particle fixedpoints, respectively. As the tunnelling is negligible in the limit of tightconfinement, each approximately harmonic micro-trap will host conformaltower states. If the atoms are at rest in the ground state, the initial statewill be at the bottom of the tower. At t = 0, the lattice is then turned offbut the magnetic field is simultaneously varied so that the systems are nolonger at resonance (a), or zero scattering (b).The expansion of the two-body ensemble, shown in Fig. 6.3 b), can thenbe observed to verify the main conclusions about the relevance or irrelevanceof deviations from a scale invariant fixed point. Near resonance the dynamicsshould be modified by a non-perturbative log-periodic time dependence, seeEqs. 6.7 and 6.9 as well as Figs. 6.1 and 6.2, while for weak interactions thelong time dynamics are equivalent to a rescaling. As far as λ(t) al/2, eachtwo-body system will expand independently in free space. For systems withinitially tight confinement, 1  √ωal, the effect of broken scale invarianceon the dynamics will then be visible for times: ω−1  t al/√ω.6.5 Application to a Mobile ImpurityAs mentioned previously, the relative motion of the two-body problem isequivalent to a particle in an external potential. However, a more realisticsystem is a massive impurity placed in a quantum gas. The effect of theimpurity is to create an external potential for the quantum gas.Consider a quantum gas interacting with an impurity of mass, M , thatis subjected to a harmonic trap of frequency ωI . In the laboratory frame,the Hamiltonian for the system is:H =∑i−12∇2i −12M∇2I +12Mω2IR2I +∑iU(~ri − ~RI), (6.17)where U(~r) is the short range atom-impurity interaction. In this Hamil-tonian, the motion of the trapped impurity atom is included alongside thequantum gas. The operators, RI and −i∇I are the coordinate and momentaoperators for the impurity atom, while ~ri and −i∇i are still the coordinatesand momenta for the quantum gas.One can still examine the dynamics in the expanding co-moving frame.In order to examine the dynamics of the quantum gas, it is ideal to choose:λ(t) =√1 + ω2t2. This choice of λ(t) then gives the modified effectiveHamiltonian:526.5. Application to a Mobile ImpurityFigure 6.3: Proposed experimental set up for examining broken scale in-variance on two-body dynamics. a) To create a three dimensional lattice,with a lattice constant, al, much larger than the optical wavelength, threepairs of coplanar beams are needed. Each pair of beams will have the samewave number, k, but an angle, θ between them. The resulting optical latticewill be due to the difference between the two beams: δkα = sin(θα/2)kα,for α = x, y, z. For large lattice constants, the optical lattice will look likean ensemble of harmonic traps with harmonic lengths: λ0 = 1/√ω. If eachof the different pairs of beams lie in different orthogonal planes, the resultwill be a square lattice. b) A schematic of the experiment. Here we showa single dimension of the resulting optical lattice. At t = 0 the latticeis removed so the two-body systems can expand in free space. For times,ω−1  t  al/√ω, the dynamics of the whole system will be equivalent toan ensemble of independent two-body systems. This figure first appeared inRef. [2]. 536.5. Application to a Mobile ImpurityH˜ =∑i(−12∇˜2i +12x2i + λ2(τ)U(λ(τ)(~xi − ~XI)))− 12M∇˜2I +12MX2I(ω2Iλ4(τ) + 1)(6.18)where ~XI = ~RI/λ(t).Eq. 6.18 states that the impurity is subject to a harmonic trap of fre-quency:√ω2Iλ4(τ) + 1. As τ approaches pi/2, the frequency of the trapwill diverge. In this case, we expect that the motion of the impurity to beadiabatic, and that the impurity will become more and more localized nearthe origin.To test this hypothesis, we show in Fig. 6.4 a) the probability for theimpurity to be in the instantaneous ground state of the trap, when it wasinitially prepared in the ground state. It is easy to see that the adiabaticapproximation works extremely well in the long time limit. The fluctuationsin the position of the trapped impurity,√〈X2I 〉, can then be related to theinstantaneous trap size:√〈X2I 〉 =√1MωIcos(τ), (6.19)which vanishes in the long time limit. The dynamics of this expectationvalue is shown in Fig. 6.4 b).The adiabatic dynamics in the comoving frame is intuitive when oneconsiders the motion in the laboratory frame. For heavy impurities, or largetrapping potentials, the interaction between the impurity and the gas willnot excite the impurity. As a result, when the gas expands further awayfrom the impurity, the impurity will remain near the origin, and it’s exactposition in the trap will become more and more irrelevant to the expandinggas. The adiabaticity of the impurity results in a form of coarse graineddynamics for the quantum gas. In the long time limit, the dynamics of thequantum gas will be insensitive to the initial preparation of the impurity.All the results obtained for the two-body problem, see Eqs. 6.7 and 6.9, aswell as Figs. 6.1 and 6.2, will be equally valid.Although these results were obtained for a single particle, the extensionto N particles is straightforward. For multiple cold atoms, the dynamicscan be understood by examining how N particles occupy the eigenstates ofV˜ . For fermions, the Pauli-exclusion principle prevents multiple fermionswith the same spin to occupy the state, |v〉. This results in the appearance546.6. Summaryof a single frequency v in the dynamics. On the other hand, if the system iscomposed of bosons which have condensed, the dynamics will be identicalto a single particle, and will still have oscillations at the frequency, v.In general, for a N -body non-interacting quantum gas in the presenceof an external potential, the wave function in the comoving frame can bewritten in terms of:φ(x1, ..., xN , τ) =∑{ni}[ψ({ni}, τ)e−i∑Ni=12niτ×1√N !∑P(±1)PφnP1 (x1)φnP2 (x2)...φnPN (xN )](6.20)where the first summation runs over all distinct combinations of singleparticle conformal states, n1, ..., nN , where each individual ni, runs fromni = 0, 1, ..., nmax, and i = 1, 2, ..., N . The second summation is over all thepermutations of the set n1, ..., nN . The factor of (±1)P ensures the correctparticle exchange symmetry for either bosons, or fermions, with P beingthe number of exchanges to reach the given permutation. The expansioncoefficients, ψ({ni}, τ), are normalized to unity.For this general wave function, one can show that the moment of inertiafor a non-interacting quantum gas has the same form as Eq. 6.9. In AppendixF, explicit expressions for the the coefficients, A, B, and D are given for Nparticles.6.6 SummaryIn this chapter we discussed the dynamics of two systems near resonance andfor weak interactions; the two-body system, and a non-interacting quantumgas in the presence of a massive impurity. Both of these problems are analyt-ically tractable. For s-wave interactions, the deviation from scale invarianceis marginally relevant, α = 1, near resonance, and irrelevant, α = −1, forweakly interacting systems. Near resonance, the dynamics are modified bya logarithmic beat, with a frequency given by Eq. 6.6. This can be clearlyseen in the moment of inertia, see Fig. 6.2 and Eq. 6.9.Although these results were obtained for s-wave interactions in threespatial dimensions, it is possible to use the scaling of the deviation to clas-sify the dynamics of a variety of cold atom systems. In particular, one can556.6. SummaryFigure 6.4: The time evolution of the trapped impurity according to Eq. 6.18in units of ω, for MωIλ20 = 3. a) The probability of being in the instanta-neous ground state of Eq. 6.18. b) The fluctuations of the trapped impurityposition, Eq. 6.19, in both the laboratory (dashed line) and co-moving frames(solid line). This figure first appeared in Ref. [2].566.6. Summaryshow that for s-wave interactions, the dynamics are marginally relevant nearresonance in three spatial dimensions, and for weak interactions in one spa-tial dimension. Similarly, one-dimensional systems with p-wave interactionsbehave identically to a three dimensional quantum gas with s-wave inter-actions; near resonance the deviation from scale invariance is marginallyrelevant, and irrelevant for weak interactions. We summarize these resultsin Fig. 6.5, where we show the dynamic and thermodynamic relevancy of ad-dimensional s-wave gas near resonance.The only caveat to this discussion is for systems that possess a quantumanomaly, for example, the two-dimensional quantum gas. For these systems,the quantum anomaly will break the SO(2,1) symmetry, and the structureof the conformal towers. As a result, our formalism can only be applied tothe non-interacting fixed point.It is interesting to note there is a difference between the dynamical andthermodynamic relevancy of a given perturbation. Therefore, it is an im-portant question to ask why is there a difference? The answer to this isbased on the difference between scale transformations and conformal trans-formations on the action. In thermodynamics, the relevancy of a deviationis determined by examining how the action changes under a scale transfor-mation. Let us consider how the deviation operator affects the scaling ofthe action. The deviation term of the action can be written as:δS = −∫r≥Λ−1dd~r∫dt1(aΛ)−β′(g˜∗(Λ))1Λd−2ψ†(~r, t)ψ†(~r, t)ψ(~r, t)ψ(~r, t).(6.21)where we have used the explicit form of the deviation operator, Eq. 5.16.Under a scale transformation, one can show that the deviation of the actionis given by:δS′ = e−bβ′(g˜∗(Λ))δS. (6.22)as one can see, the effect of the deviation is entirely captured by the betafunction, and how the action changes under scale transformations.In terms of dynamics, it is conformal symmetry that one must consider.As discussed in Chapter 4, the long time dynamics are governed by thetrivial time dynamics of the conformal tower states. We will show in thefollowing chapter that this is due to the emergence of conformal symmetry.Therefore, in order to understand the effect of the deviation on the dy-namics, it is necessary to consider how the deviation operator responds to576.6. Summaryconformal transformations. Using Eq. 6.21, one can show under a conformaltransformation:δS′ = (1− bt)−1−β′(g˜∗(Λ))δS. (6.23)Upon comparison, one can see that scale transformations and conformaltransformations have different effects on the deviation term of the action.This difference is the reason why the condition for relevancy in dynamicsdiffers from the thermodynamic relevancy. In the long time limit, one needsto consider conformal symmetry, not scale symmetry.586.6. SummaryFigure 6.5: Thermodynamic and dynamic relevancy for a d-dimensionalquantum gas with s-wave interactions near resonance. The thermodynamicrelevancy differs from the dynamic relevancy by one spatial dimension.59Chapter 7Application to Many-BodySystemsPreviously we have highlighted the effects of scale invariance on the dynam-ics of arbitrary quantum systems in Chapter 4, and how the dynamics aremodified in the presence of an explicit deviation from scale symmetry inChapter 5. These previous discussions are true for arbitrary Galilean in-variant, non-relativistic, quantum systems. Here we apply the formalismsdeveloped in those two chapters to the dynamics of a three dimensionalFermi gas near resonance, with N particles. This system is strongly in-teracting, and has been the focus of a number of both experimental andtheoretical studies [4, 16, 17, 20, 25, 39, 40, 45, 63, 64, 101, 102].One particular phenomenon of interest to experimentalists is the expan-sion dynamics of the unitary Fermi gas. This can be examined by consideringthe motion of the moment of inertia in a given direction:〈r2i 〉(t) =∫d3r r2i 〈ψ0|ψ†(~r, t)ψ(~r, t)|ψ0〉, (7.1)where i = x, y, z. The moment of inertia is important as it tells one aboutthe size of the expanding gas in a given direction, as a function of time.In Ref. [3], the expansion dynamics of a resonant Fermi gas, from an ini-tially anisotropic harmonic trap, were studied experimentally. The resultingflow is denoted as elliptic flow, as it is ultimately anisotropic. This is to becontrasted to compressional flow, where the dynamics are isotropic. Thisexperiment was analysed using a variational ansatz and the hydrodynamicequations of motion, a semiclassical technique. Although this approach wasable to reproduce the experimental observations, this approach does notidentify the effect of scale invariance. In this chapter we will examine theexact consequences of scale and conformal invariance on the dynamics.In order to investigate the expansion dynamics, it is ideal to consider themore general quantity:〈rirj〉(t) = 13〈r2δi,j +Qi,j〉(t), (7.2)607.1. Definitions for the One-Body Density Matrixwhere r2 is the moment of inertia, or monopole moment, and Qi,j is atraceless matrix, known as the quadrupole moment:〈Qi,j〉(t) =∫d3r(3rirj − r2δi,j)〈ψ0|ψ†(~r, t)ψ(~r, t)|ψ0〉. (7.3)This decomposition is ideal as it allows one to calculate the moment of inertiain any direction, and it splits the dynamics into two angular momentumchannels: l = 0 and l = 2.Although in previous chapters we studied the dynamics using a wavefunction approach in an expanding reference frame, it can be difficult toobtain quantitative predictions for many body systems. This is because onestill needs to solve for the eigenstates of a resonantly interacting Fermi gasin an harmonic potential in order to use the comoving frame.Instead, it is better to introduce a new quantity, the one-body densitymatrix:ρ(~r, ~r′, t) = 〈ψ0|eiHtψ†(~r′)ψ(~r)e−iHt|ψ0〉. (7.4)As we will see in the following section, it is possible to explicitly evaluate therole of scale invariance on the dynamics of the density matrix. In particular,we show that the density matrix satisfies an emergent conformal symmetryin the long time limit. This emergent symmetry is equivalent to the longtime dynamics in the comoving frame, when the dynamical phase freezes,and is the reason for the time dependent rescaling of local observables. In theremainder of this chapter, we will apply the density matrix to the problemof compressional and elliptic flow and make concrete predictions about thepossible dynamics that can occur. We will show that this conformal sym-metry fixes the leading long time behaviour of the moment of inertia. Thecorrections to this leading behaviour will depend on the initial conditions,and has not been examined previously. We will conclude this chapter bycomparing our methods to the semiclassical approach employed in Ref. [3].7.1 Definitions for the One-Body Density MatrixWe begin by explicitly determining the role of scale and conformal invari-ance on the dynamics of the one-body density matrix. We can once againsplit the motion into a term governed by the scale invariant Hamiltonian,Hs, and a term governed by the deviation, δH. To do this, we use the inter-action representation discussed in Chapter 5, and insert two complete setsof conformal tower states:617.1. Definitions for the One-Body Density Matrixρ(~r, ~r′, t) =∑m,nρ˜s n,m(~r, ~r′, t)Γ˜m,n(t)ρ˜s n,m(~r, ~r′, t) = e−i(En−Em)τ(t)〈n|eiHstψ†(~r′)ψ(~r)e−iHst|m〉Γ˜m,n(t) = ei(En−Em)τ(t)〈m|U(t)|ψ0〉〈ψ0|U †(t)|n〉, (7.5)where:U(t) = Te−i∫ t0dt′eiHst′δHe−iHst′U(t ω−1) ≈ exp[−i 1(√ωa)α(ωt)α−1α− 1 V˜]α > 1≈ exp[−i 1√ωalog(ωt)V˜]α = 1. (7.6)In Eqs. 7.5 and 7.6, τ(t) = ω−1 tan(ωt) and T is the time ordering operator.Finally we note that:〈n|V˜ |m〉 = 1ω1−α/2ei(En−Em)pi/2ω〈n|δh|m〉. (7.7)A full derivation of the effects of scale and conformal symmetry on theone-body density matrix is shown in Appendix G. Here we note that onecan use the fact that the conformal tower states are eigenstates of Hs +ω2C, to obtain a differential equation for the one-body density matrix nearresonance:[(1 + ω2t2)∂t + ω2t(~r · ∇r + ~r · ∇r + d) + iω r′2 − r22]ρ(~r, ~r′, t)=1 + ω2t2t log(ωt)1a∂∂a−1ρ(~r, ~r′, t) +∑m,ni (En − Em) ρ˜s n,m(~r, ~r′, t)Γ˜m,n(t).(7.8)The first line of Eq. 7.8 is nothing more than the generator of conformaltransformations [78]. It states that the for scale invariant systems, the den-sity matrix is an eigenfunction of the generator of conformal transformations,with zero eigenvalue. As a result, the long time dynamics are constrainedby conformal symmetry, not scale symmetry.627.2. Isotropic Trap and Compressional FlowUsing Eq. 7.8, it is possible to write a differential equation describingthe dynamics of 〈rirj〉(t):[(1 + ω2t2)∂t − 2 ω2t− 1 + ω2t2t log(ωt)1a∂∂a−1]〈rirj〉(t)= ω2t2∑m,ni (En − Em)∫d3r rirj ρ˜s n,m(~r, ~r, 0)Γ˜m,n(t).(7.9)In order to obtain Eq. 7.8, we have used the conformal property of thedensity matrix:ρ˜s n,m(~r, ~r, t) =(1ωt)3ρ˜s n,m(~rωt,~rωt, ω−1)≈(1ωt)3ρ˜s n,m(~rωt,~rωt, 0). (7.10)In the remainder of this chapter, we will investigate the solutions ofEq. 7.9 for compressional and elliptic flows.7.2 Isotropic Trap and Compressional FlowFirst consider the case of a resonantly interacting Fermi gas initially placedin an isotropic harmonic trap. In this case, angular momentum is a goodquantum number, and will be conserved throughout the expansion. In par-ticular, we will be focused on isotropic initial conditions.7.2.1 Scale Invariance and Compressional FlowSince the initial gas is isotropic, the quadrupole moment vanishes, and willremain so for all times. All that remains is the isotropic moment of inertia.The differential equation for the moment of inertia is given by:[(1 + ω2t2)∂t − 2ω2t]〈r2〉(t)= ω2t2∑m,ni (En − Em)∫d3r r2 ρ˜s n,m(~r, ~r, 0)Γ˜m,n(t).(7.11)637.2. Isotropic Trap and Compressional FlowThe leading long time solution can be obtained by focusing on the emergentconformal symmetry, or equivalently by neglecting the term on the righthand side. The solution reads:〈r2〉(t) =(tt′)2〈r2〉(t′), (7.12)where t, t′  ω−1. This is consistent with the analysis in the comoving ref-erence frame, Eq. 4.19, since the moment of inertia has a scaling dimensionof ∆r2 = −2. Moreover, we can now see that the approximate rescaling isdue to the emergence of conformal symmetry in the long time limit, and isindependent of the microscopic details of the system, and the initial condi-tions.The corrections to these long time asymptotics will depend on the ma-trix, Γ˜(t). For scale invariant systems, the time dependence of Γ˜(t) is trivialas the dynamical phase will saturate. As a result, the occupation of a givenconformal tower state is conserved during the expansion. For the purpose ofour discussions, we will be focused on initial conditions that can be describedby a real symmetric matrix:Γ˜m,n(0) = Γ˜n,m(0), (7.13)which is true for initial states that are invariant under time reversal sym-metry.With this in mind, we can now consider different initial conditions andhow the long time dynamics differ. First we examine a diagonal ensembleof conformal tower states: Γ˜m,n(0) = δm,nP (n). This can be easily achievedfor Fermi gases initially in thermal equilibrium at resonance. In this case,the right hand side of Eq. 7.11 vanishes for all times. The exact solution isthen:〈r2〉(t) = (1 + ω2t2)〈r2〉(0). (7.14)This is just the trivial rescaling dynamics, and is equivalent to the hydro-dynamic solution, see Appendix H.In general, the initial density matrix will involve states from multipleconformal towers. Here we will assume the case that the system is in anarbitrary superposition of s-wave states. This assumption is used to ensurethat the Fermi gas is initially isotropic. Such a system can be obtained byquenching a non-interacting Fermi gas in the ground state to resonance. Forthis situation, Eq. 7.11 simplifies to:647.2. Isotropic Trap and Compressional Flow[(1 + ω2t2)∂t − 2ω2t]〈r2〉(t)= ω3t∑m,n(−1)n−m4(n−m)2∫d3r r2 ρ˜s n,m(~r, ~r, 0)Γ˜m,n(0),(7.15)where we have used the fact that τ(t) ≈ pi/(2ω) and En −Em = 2(n−m)ωfor states within a conformal tower. A generic solution to Eq. 7.15 is of theform:〈r2〉(t) ≈[v2t2 +(v2ω2− F02ω)]〈r2〉(0). (7.16)where we have neglected higher powers of (ωt)−1, and defined:F0 =1〈r2〉(0)∑n,m(−1)n−m4(n−m)2∫d3r r2ρ˜s n,m(~r, ~r, 0)Γ˜m,n(0). (7.17)The primary difference between Eqs. 7.14 and 7.16 is the relative velocity:vrel = limt→∞√〈r2〉(t)〈r2〉(0)t2 . (7.18)For generic initial conditions, the relative velocity is no longer pinned to thetrap frequency, ω. In fact, in Appendix H we show using the Heisenbergequation of motion that the relative velocity is given by:vrel =√2〈Hs〉〈r2〉(0) . (7.19)7.2.2 Broken Scale Invariance and Compressional FlowWhen scale invariance is broken, it is necessary to use Eq. 7.8. Again, weonly focus on low energy, s-wave interactions, which will only mix conformaltower states with the same angular momentum. For diagonal ensembles, thesolution is given by:〈r2〉(t) ≈[1 + ω2t2] [1 +G(1alog(ωt))]〈r2〉(0). (7.20)657.3. Elliptic Flowfor some function G(x). For generic initial conditions, the solution Eq. 7.8is given by:〈r2〉(t) ≈[v2t2 +(v2ω2− F02ω)] [1 +G(1alog(ωt))]〈r2〉(0). (7.21)Both these solutions are modified by some function of log(ωt)/a, which isconsistent with the analysis presented in Chapter 5.7.3 Elliptic FlowIf the system is initially prepared in an anisotropic trap, the dynamics in eachdirection will differ. However, due to the emergent conformal symmetry, thelong time dynamics for each direction must still be given by Eq. 7.12. Thequestion then is how do the long time dynamics differ in comparison tocompressional flow?7.3.1 Scale Invariance and Elliptic FlowTo begin, we note that for an anisotropic trap, there will be at most threediffering trap frequencies, labeled as ωi with i = x, y, z. For this situation,angular momentum is no longer a good quantum number initially. As aresult, the initial state must be a superposition of different conformal towers.The density matrix can no longer be diagonal in this case. Although thiscould be true for compressional flow, we note that the integral:∫d3rr2i ρs n,m(~r, ~r, t), (7.22)can couple states together with the same angular momentum, or with angu-lar momentum that differ by two. The zero angular momentum channel willbe identical to the isotropic trap as the conformal towers are decoupled fromone another. However, the quadrupole moment will now be non-zero dueto the anisotropy of the initial trap. For the remainder of this section, wewill focus on the dynamics of the quadrupole moment. For scale invariantsystems, the differential equation for the quadrupole moment is:[(1 + ω2t2)∂t − 2ω2t]〈Qi,j〉(t)= −ω2t2∑m,n(En − Em) sin((En − Em) pi2ω)∫d3r Qi,j ρ˜s n,m(~r, ~r, 0)Γ˜m,n(0),667.3. Elliptic Flow+ ωt∑m,n(En − Em)2 cos((En − Em) pi2ω)∫d3r Qi,j ρ˜s n,m(~r, ~r, 0)Γ˜m,n(0).(7.23)A generic solution to Eq. 7.23 is given as:〈Qi,j〉(t) ≈[v2i,jt2 + F i,j2 t+v2i,jω2− Fi,j02ω], (7.24)where we note that:F i,j2 =∑m,n(En − Em) sin((En − Em) pi2ω)·∫d3r Qi,j ρ˜s n,m(~r, ~r, 0)Γ˜m,n(0)F i,j0 =∑m,n(En − Em)2 cos((En − Em) pi2ω)·∫d3r Qi,j ρ˜s n,m(~r, ~r, 0)Γ˜m,n(0), (7.25)and v2i,j are traceless symmetric tensors.The leading correction to the asymptotic dynamics is no longer of O(1),but now linear in t. The linear term arises from the interference betweendifferent conformal towers, something that is not present in an isotropicsystem. This is the main difference between the two cases, and allows oneto delineate the different initial conditions.In the case of azimuthal symmetry, we can simplify Eq. 7.24 appreciably.One can show that the quadrupole tensor will simplify to the form:〈Qi,j〉(t) = 3Q(t)(eziezj − 13δi,j)Q(t) ≈ v2Qt2 +AQt+BQ, (7.26)where vQ, AQ, and BQ are constants, and ez is the unit vector in the az-imuthally symmetric direction.7.3.2 Broken Scale Invariance and Elliptic FlowIf the Fermi gas is slightly away from resonance, the results of the previoussection will be modified by a log-periodic beat:677.4. Comparison to the Scaling Solution Ansatz〈Qi,j〉(t) =[v2i,jt2 + F i,j2 t+(v2i,jω2− Fi,j02ω)] [1 +G(1alog(ωt))]. (7.27)for some function G (t).7.4 Comparison to the Scaling Solution AnsatzThe results presented previously provide the scaling of the long time dynam-ics of (nearly) scale invariant systems initially confined in harmonic traps. Inthis section we will contrast the density matrix calculation with the scalingsolution of the hydrodynamic equations of motion. In order to do this, wewill consider an azimuthally symmetric system. We define the aspect ratioas the ratio of the moments of inertia in the two independent directions:σz,x =√〈r2z〉(t)〈r2x〉(t). (7.28)For a non-interacting Fermi gas released from an anisotropic trap, theaspect ratio saturates to unity. This is because the kinetic energy in agiven direction is proportional to the trap frequency in that direction, ωiwhile the initial size scales as ω−2i in the Thomas-Fermi limit. However, atresonance, the situation is different, the interaction distributes the energyanisotropically, and the emergent conformal symmetry states that the aspectratio saturates to a finite number:σz,x =〈r2〉(t)−Q(t)〈r2〉(t) + 2Q(t)≈ v2〈r2〉(0)− v2Qv2〈r2〉(0) + 2v2Q−(AQv2〈r2〉(0)− v2Q+2AQv2〈r2〉(0) + 2v2Q)1t. (7.29)In Ref. [3], a unitary Fermi gas was prepared in an anisotropic trap withfrequencies: ωy = 2.7ωx, and ωz = 33ωx. In their experiment they consid-ered the aspect ratio between the x and y directions. In their experiment,the aspect ratio for a non-interacting gas was observed to saturate at unity.For the resonant case, the aspect ratio was not observed to saturate at a687.5. Summaryfinite value, but its growth was indeed slowing down at large times. Theanalysis provided here is consistent with the experimental findings. More-over, we show that the aspect ratio will indeed saturate, due to the emergentconformal symmetry.In order to give more credence to the density matrix approach, we nowcompare Eq. 7.29 to a time dependent scaling ansatz that satisfies the hy-drodynamic equations of motion. This approach was used in Ref. [3] toanalyse their experimental findings. Here we reproduce their calculation.The time-dependent scaling ansatz for the moment of inertia in a givendirection is given by:〈r2i 〉(t) = b2i (t)〈r2i 〉(0), (7.30)where bi(t) is an unspecified scaling factor. In Appendix H we show thatthe scaling ansatz is a solution to the hydrodynamic equations of motion ifthe bi(t) factors satisfy:b¨i(t) =ω2i(bx(t)by(t)bz(t))2/3 bi(t)−2 〈αs〉〈r2i 〉(0)(b˙i(t)bi(t)− 13(b˙x(t)bx(t)+b˙y(t)by(t)+b˙z(t)bz(t))),(7.31)where 〈αs〉 is the trap averaged shear viscosity coefficient. This equationwas studied in Refs. [3] for finite shear viscosity and in Ref. [45], in theabsence of shear viscosity. The solution for the moment of inertia in a givendirection is qualitatively similar in both cases.The solution to the aspect ratio for an anisotropy in the x-y plane isshown in Fig. 7.1. The saturation at a non-unity value is clearly visible,and the leading correction is consistent with Eq. 7.29. In the inset, we showthe solution for 〈y2〉(t). A numerical fit for the moment of inertia along they-direction shows that it is also consistent with Eq. SummaryIn this chapter we examined the expansion dynamics of a Fermi gas either ator near unitarity from an initial harmonic trap. For a scale invariant Hamil-tonian, the long time dynamics are ultimately controlled by the emergenceof conformal symmetry. This emergent conformal symmetry is the expla-nation for the long time dynamics being equivalent to a time dependentrescaling. This conclusion is independent of the microscopic details of thesystem, and the initial conditions. However, the correction to this limit will697.5. Summary0.5 1.0 1.5 2.0t (ms)50010001500<y2>(t)50 100 150 200t (ms)σy,x〈αs 〉=0〈αs 〉=0.1〈αs 〉=0.5〈αs 〉=1〈αs 〉=2〈αs 〉=3〈αs 〉=4Figure 7.1: The solution to the aspect ratio for various shear viscositiesfor a unitary Fermi gas. These results were obtained by numerically solvingEq. 7.31. We have used the experimental parameters in Ref. [3], ωx = 2pi∗230Hz, ωy = 2.7ωx, ωz = 33ωx. For any value of the shear viscosity, the aspectratio saturates. In the inset, the dynamics of the moment of inertia in they direction is shown. This solution fits well with Eq. 7.24.sensitively depend on the initial conditions, and whether the harmonic trapis isotropic or not. For the case of isotropic traps, the interference betweenstates within a single conformal tower give a O(t−2) correction, while foranisotropic traps, the interference between differing conformal towers givesa O(t−1) correction. The leading correction to the asymptotic dynamics aresummarized in Table 7.1.We compared these results to the experimental study presented in Ref. [3].In this study, the experimental observations for the aspect ratio were com-pared to a scaling ansatz to the hydrodynamic equations of motion. Weshowed that the semiclassical approach is indeed consistent with the densitymatrix approach. However, the hydrodynamical approach does not clearlystate the role of conformal invariance; namely the long time behaviour ofthe moment of inertia, and the connection between the leading long timecorrection and the interference of conformal towers.Although Ref. [3] also studied the effect of broken scale invariance onelliptic flow was studied experimentally, the theoretical description was leftat the hydrodynamic level. It would be interesting to see if the deviationfrom scale invariance follows the scaling shown in Eqs. 7.27.707.5. Summary〈r2i 〉(t)/〈r2i 〉(0) Isotropic Trap Anisotropic TrapDiagonal Γ(0) 1 + ω2t2 NAGeneric Γ(0) v2t2 +B v2t2 +At+BTable 7.1: The leading and next leading order time dependence for themoment of inertia: 〈r2i 〉(t), for a unitary Fermi gas. The time dependencedepends on whether the initial harmonic confinement is isotropic, with fre-quency ω, or anisotropic, and whether the initial conditions are a diagonal,or generic non-diagonal ensemble of conformal tower states. The exact co-efficients, v, A, and B, can be found in Eqs. 7.14, 7.16, and 7.24.71Chapter 8ConclusionsThe focus of this thesis was to understand the role of scale invariance onnon-relativistic quantum dynamics, and the effects of explicitly breakingscale invariance. We addressed this using a quantum variational approachfor two dimensional bosons, and by using exact symmetry arguments appliedto Fermi gases in three spatial dimensions.For two dimensional Bose gases, we developed a quantum variationalapproach that assigned a wave function to the size of the condensate. Thisapproach allows one to treat the quantum fluctuations that are present forattractive interactions. For this system, we showed that the continuousscale invariance was broken, but replaced by a discrete scale invariance.This discrete scale invariance is manifest in the dynamics by a logarithmicrise in the density at short distances 3.24, and log periodic beats 3.25.We improved the preceding analysis by identifying the exact implicationsof scale invariance on the non-relativistic dynamics of quantum systems inChapter 4. We used the SO(2,1) symmetry, which connects the dynamics toscale and conformal transformations, to show that there is an ideal basis toexpand the dynamics in, the conformal tower basis. The time evolution ofthese states is related to a trivial time dependent rescaling. Using this basis,we showed that the long time dynamics of local observables is equivalent toa time dependent rescaling, 4.19.All of these results were for scale invariant systems. An important ques-tion to ask is what happens to the scale invariant dynamics when the sym-metry is explicitly broken. Will the asymptotic time dependent rescalingpersist or will it be broken by the broken scale invariance? In Chapter 5 weshowed that there are two types of deviations from a scale invariant point:irrelevant deviations which do not break the scaling dynamics in the longtime limit, and relevant dynamics which do. The relevancy of a deviationcan be related to the beta-function; a thermodynamic property. In particu-lar we showed that for deviations with scaling dimensions, α ≥ 1, where α isrelated to the derivative of the beta-function, see Eq. 5.17, the effect of theinteractions can not be ignored in the long time limit. We performed a non-perturbative calculation that gives the leading long time behaviour; the scale72Chapter 8. Conclusionsinvariant dynamics are modified by a function of tα−1/aα. The quantitativeaspects of the dynamics are contained in the so-called V-matrix, Eq. 5.23,which is a dimensionless, universal matrix that depends only on the numberof particles. This is the main result of this thesis, as it categorizes the typesof deviations, and allows for a scaling analysis of non-equilibrium dynamics.The formalism developed here was applied to two systems, the expansionof an ensemble of two-body systems, Chapter 6, and to the compressionaland elliptic flow of a Fermi gas, Chapter 7, both in three spatial dimensions.Both these systems are experimentally viable, and we examined the signa-tures of scale invariance, and its breaking on the mean size of the systems.In particular, we showed that for the unitary Fermi gas, this approach isequivalent to the hydrodynamic approach studied in Ref. [3]. However, ourapproach highlights the exact consequences of scale and conformal symme-try on the asymptotic dynamics and the leading long time correction, whichis hidden in the scaling solution.Our approach for studying dynamics is quite general and can be appliedto a variety of atomic systems, such as quantum gases in other dimensions,and with p-wave interactions. For these systems, the non-equilibrium dy-namics can be categorized by the relevancy of the deviation from scale in-variance. Again, all the quantitative information on the dynamics will becontained in the V-matrix.For future studies, there are a number of open questions pertaining tothis V-matrix. First of all, it would be interesting to investigate what kind ofstructure is present in the V-matrix, and whether some form of approxima-tions, like random matrix theory, can be applied to the dynamics. Secondly,it would be interesting to study different dynamical phenomena. In thisthesis we focused primarily on the effect scale invariance has on the ex-pansion of quantum gases. However, the field of dynamics is rich with acountless number of phenomena. The implications of conformal symmetryon transport dynamics, Floquet physics and others has still not been fullyaddressed. It will be interesting to press further and to have a clear picturehow conformal symmetry restricts other dynamical phenomena.73Bibliography[1] J. Maki, S. J. Jiang, and F. Zhou. Discrete scale invariant quantumdynamics and universal quantum beats in bose gases. Euro. Phys.Lett., 118:56002, 2017.[2] J. Maki, L.M. Zhao, and F. Zhou. Nonperturbative dynamical ef-fects in nearly-scale-invariant systems: The action of breaking scaleinvariance. Phys. Rev. A, 98:013602, 2018.[3] C.Cao, E. Elliott, J. Joseph, H. Wu, J. Petricka, T. Schafer, and J. E.Thomas. Universal quantum viscosity in a unitary fermi gas. Science,331:58–61, 2011.[4] S. Deng, Z.Y. Shi, P. Diao, Q. Yu, H. Zhai, R. Qi, and H. Wu. Observa-tion of the efimovian expansion in scale-invariant fermi gases. Science,353:371–374, 2016.[5] I. Bloch, J. Dalibard, and W. Zwerger. Many-body physics with ul-tracold gases. Rev. Mod. Phys., 80:885–964, 2008.[6] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga. Feshbach resonancein ultracold gases. Rev. Mod. Phys., 82:1225–1286, 2010.[7] R. Folman, P. Kruger, D. Cassettari, B. Hessmo, T. Maier, andJ. Schmiedmayer. Controlling cold atoms using nanofabricated sur-faces: Atom chips. Phys. Rev. Lett., 84:4749, 2000.[8] G. Bensky, R. Amsuss, J. Majer, D. Petrosyan, J. Schmiedmayer, andG. Kurizki. Controlling quantum information processining in hybridsystems on a chip. Quantum Inf. Process, 10:10:1037, 2011.[9] J. Rushton, R. Roy, J. Bateman, and M. Himsworth. A dynamicmagento-optical trap for atom chips. New J. Phys., 18:113020, 2016.[10] J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner.Quantum simulation of antiferromagnetic spin chains in an opticallattice. Nature, 472:307–312, 2011.74Bibliography[11] I. Bloch, J. Dalibard, and S. Nascimbene. Quantum simulations withultracold quantum gases. Nat. Phys., 08:267–276, 2012.[12] N. T. Zinner and A. S. Jensen. Comparing and constrasting nucleiand cold atomic gases. J. Phys. G, 40:053101, 2013.[13] I. M. Georgescu, S. Ashhab, and F. Nori. Quantum simulations. Rev.Mod, Phys., 86:153, 2014.[14] U. Ernst, J. Schuster, F. Schreck, A. Marte, A. Kuhn, and G. Rempe.Free expansion of a bose-einstein condensate from an ioffe-pritchardmagnetic trap. Apl. Phys. B, 67:719–722, 1998.[15] E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A.Cornell, and C. E. Wieman. Dynamics of collapsing and explodingbose-einstein condensates. Nature, 412:295–299, 2001.[16] K. M. O’Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E.Thomas. Observation of a strongly-interacting degenerate fermi gas.Science, 298:2179–2182, 2002.[17] E. Elliott, J. A. Joseph, and J. E. Thomas. Observation of confor-mal symmetry breaking and scale invariance in expanding fermi gases.Phys. Rev. Lett, 112:040405, 2014.[18] D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.Cornell. Collective excitations of a bose-einstein condensate in a dilutegas. Phys. Rev. letts., 77:420, 1996.[19] D. M. Stamper-Kurn, H. J. Meisner, S. Inouye, M. R. Andrew, andW. Ketterle. Collisionless and hydrodynamic excitations of a bose-einstein condensate. Phys. Rev. letts., 81:500, 1998.[20] J. Kinast, A. Turlapov, and J. E. Thomas. Breakdown of hydrodyam-ics in the radial breathing mode of a strongly interacting fermi gas.Phys. Rev. A, 70:051401, 2004.[21] A. Altmeyer, S. Riedl, C. Kohstall, M. J. Wright, R. Geursen,M. Bartenstein, C. Chin, J. Hecker Denschlag, and R. Grimm. Pre-cision measurements of collective oscillations in the bec-bcs crossover.Phys. Rev. letts., 98:040401, 2007.[22] K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet.Formation and propagation of matter-wave soliton trains. Nature,417:150–153, 2002.75Bibliography[23] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore. Nonequi-librium dynamics of closed interacting quantum systems. Rev. Mod.Phys., 83:863–883, 2011.[24] P. Makotyn, C. E. Klauss, D. L. Goldberger, E. A. Cornell, and D. S.Jin. Universal dynamics of a degenerate bose gas. Nat. phys., 10:116–119, 2014.[25] S. Deng, P. Diao, F. Li, Q. Yu, S. Yu, and H. Wu. Observation ofdynamical super-efimovian expansion in a unitary fermi gas. Phys.Rev. Lett, 120:125301, 2018.[26] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl,I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer. Relaxationand prethermalization in an isolated quantum system. Science, 337:1318–1322, 2012.[27] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Luschen, M. H. Fischer,R. Vosk, E. Altman, U. Schneider, and I. Bloch. Observation of many-body localization of interacting fermions in a quasi-random opticallattice. Science, 30, 2015.[28] A. M. Kaufmann, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M.Preiss, and M. Greiner. Quantum thermalization through entangle-ment in an isolated many-body system. Science, 353:794–800, 2016.[29] P. Bordia, H. Luschen, U. Schneider, M. Knap, and I. Bloch. Periodi-cally driving a many-body localized quantum system. Nat. Phys., 13:460–464, 2017.[30] C. J. Kennedy, W. C. Burton, W. C. Chung, and W. Ketterle. Ob-servation of bose-einstein condensation in a strong synthetic magneticfield. Nat. Phys., 11:859–864, 2015.[31] F. Meinert, M. J. Mark, K. Lauber, A. J. Daley, and H. C. Nagerl.Floquet engineering of correlated tunneling in the bose-hubbard modelwith ultracold atoms. Phys. Rev. Lett., 116:205301, 2016.[32] L. W. Clark, A. Gaj, L. Feng, and C. Chin. Collective emission ofmatter-wave jets from driven bose-einstein condensates. Nature, 551:356–359, 2017.[33] J. H. V. Nguyen, D. Luo, and R. G. Hulet. Formation of matter-wavesoliton trains by modulational instability. Science, 356:422–426, 2017.76Bibliography[34] A. Eckardt. Colloquium: Atomic gases in periodically driven opticallattices. Rev. Mod. Phys., 89:011004, 2017.[35] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol.One dimensional bosons: From condensed matter systems to ultracoldgases. Rev. Mod. Phys., 853:1405, 2011.[36] M. Collura, S. Sotiriadis, and P. Calabrese. Quench dynamics of atonks-girardeau gas released from a harmonic trap. J. Stat. Mech,2013:P09025, 2013.[37] M. Collura, M. Kormos, and P. Calabrese. Quantum quench in aharmonically trapped one-dimensional bose gas. Phys. Rev. A, 97:033609, 2018.[38] J. C. Zill, T. M. Wright, K. V. Kheruntsyan, T. Gasenzer, and M. J.Davis. Quantum quench dynamics of the attractive one-dimensionalbose gas via the coordinate bethe ansatz. SciPost Phys., 4:011, 2018.[39] A. Bulgac, Y.L. Luo, P. Magierski, K. J. Roche, and Y. Yu. Real-timedynamics of quantized vortices in a unitary fermi superfluid. Science,10:1288–1291, 2011.[40] A. Bulgac, Y.L. Luo, and K. J. Roche. Quantum shock waves anddomain walls in the real-time dynamics of a superfluid unitary fermigas. Phys. Rev. Lett., 108:150401, 2012.[41] F. Ancilotto, M. Rossi, L. Salasnich, and F. Toigo. Quenched dynamicsof the momentum distribution of the unitary bose gas. Few-Body Syst.,56:801–807, 2015.[42] Y. Kagan, E. L. Surkov, and G. V. Shlyapnikov. Evolution of a bose-condensed gas under variations of the confining potential. Phys. Rev.A, 54:R1753, 1996.[43] Y. Castin and R. Dum. Bose-einstein condensates in time dependenttraps. Phys. Rev. Lett., 77:5315, 1996.[44] F. Dalfovo, C. Minniti, S. Stringari, and L. Pitaevskii. Nonlineardynamics of a bose condensed gas. Phys. Lett. A, 227:259–264, 1997.[45] C. Menotti, P. Pedri, and S. Stringari. Expansion of an interactingfermi gas. Phys. Rev. Lett., 89:250402, 2002.77Bibliography[46] K. G. Wilson. The renormalization group and critical phenomena.Rev. Mod. Phys., 55:583, 1983.[47] H. Nishimori and G. Ortiz. Elements of Phase Transitions and CriticalPhenomena. Oxford University Press, Oxford, 2011.[48] S. Weinberg. The Quantum Theory of Fields. Cambridge UniversityPress, Cambridge, 2005.[49] M. Tinkham. Group Theory and Quantum Mechanics. Dover, NewYork, 1992.[50] L. P. Kadanoff. Scaling laws for ising models near tc. Physics, 2:263,1966.[51] P.C. Hohenberg and B. I. Halperin. Theory of dynamic critical phe-nomena. Rev. Mod. Phys., 49:435, 1977.[52] J. H. Brown and G. B. West. Scaling in Biology. Oxford UniversityPress, Oxford, 2000.[53] T. Yefsah, R. Desbuquois, L. Chomaz, K. J. Gunter, and J. Dalibard.Exploring the thermodynamics of a two-dimensional bose gas. Phys.Rev. Lett., 107:130401, 2011.[54] C.L. Hung, X. Zhang, N. Gemelke, and C. Chin. Observation of scaleinvariance and universality in two-dimensional bose gases. Nature,470:236–239, 2011.[55] E. Vogt, M. Feld, B. Frohlich, D. Pertot, M. Koschorreck, and M. Kohl.Scale invariance and viscosity of a two-dimensional fermi gas. Phys.Rev. Lett., 108:070404, 2012.[56] W. Li and T.L. Ho. Bose gases near unitarity. Phys. Rev. Lett., 108:195301, 2012.[57] S. J. Jiang, W.M. Liu, G. Semenoff, and F. Zhou. Universal bose gasesnera resonance: A rigorous solution. Phys. Rev. A, 89:033614, 2014.[58] M. J.H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein. Re-vealing the superfluid lambda transition in the universal thermody-namics of a unitary fermi gas. Science, 335:563–567, 2012.[59] T. Kinoshita, T. Wenger, and D. S. Weiss. Observation of a one-dimensional tonks-girardeau gas. Science, 305:1125–1128, 2004.78Bibliography[60] Z.D. Zhang, G. E. Astrakharchik, D. C. Aveline, S. Choi, H. Perrin,T. H. Bergeman, and M Olshanii. Breakdown of scale invariance in thevicinity of the tonks-girardeau limit. Phys. Rev. A, 89:063616, 2014.[61] W. Rohringer, D. Fischer, F. Steiner, I. E. Mazets, J. Schmiedmayer,and M Trupke. Non-equilibrium scale invariance and shortcuts toadiabaticity in a one-dimensional bose gas. Scientific Reports, 5:9820,2015.[62] S.J. Jiang and F. Zhou. Two-dimensional fermi gases near a p-waveresonance: Effect of quantum fluctuations. Phys. Rev. A, 97:063606,2018.[63] J. E Thomas, J. Kinast, and A. Turlapov. Virial theorem and univer-sality in a unitary fermi gas. Phys. Rev. Lett., 95:120402, 2005.[64] Y. Nishida and D. T. Son.  expansion for a fermi gas at infinitescattering length. Phys. Rev. Lett., 97:050403, 2006.[65] E. Taylor and M. Randeria. Apparent low-energy scale invariance intwo-dimensional fermi gases. Phys. Rev. Lett., 109:135301, 2012.[66] L. P. Pitaevskii amd A. Rosch. Breathing modes and hidden symmetryof trapped atoms in two dimensions. Phys. Rev. A, 89:063616, 1997.[67] J. Maki, M. Mohammadi, and F. Zhou. Quantum n-boson states andquantized motion of solitonic droplets: Universal scaling properties inlow dimensions. Phys. Rev. A, 90:063609, 2014.[68] S. Ebrahim and D. Blume. Broken scale invariance in time-dependenttrapping potentials. Phys. Rev. A, 94:063639, 2016.[69] Y. Castin. Exact scaling transform for a unitary quantum gas in atime dependent harmonic potential. Comp. Rend. Phys., 5:407–410,2004.[70] F. Werner and Y. Castin. Unitary gas in an isotropic harmonic trap:symmetry properties and applications. Phys. Rev. A, 74:053604, 2006.[71] V. Gritsev, P. Barmettler, and E. Demler. Scaling approach to quan-tum non-equilibrium dynamics of many-body systems. New J. Phys.,12:113005, 2010.79Bibliography[72] C. Qu, L. P. Pitaevskii, and S. Stringari. Expansion of harmoni-cally trapped interacting particles and time dependence of the contact.Phys. Rev. A, 94:063635, 2016.[73] A. Minguzzi and D. M. Gangardt. Exact coherent states of a har-monically confined tonks-girardeau gas. Phys. Rev. Lett., 94:240404,2005.[74] S. Moroz. Scale-invariant fermi gas in a time-dependent harmonicpotential. Phys. Rev. A, 86:011601(R), 2012.[75] S. Zhang and A. J. Leggett. Universal properties of the ultracold fermigas. Phys. Rev. A, 79:023601, 2009.[76] S. Tan. Energetics of a strongly correlated fermi gas. Ann. Phys., 323:2952–2970, 2008.[77] S. Tan. Large momentum part of a strongly correlated fermi gas. Ann.Phys., 323:2971–2986, 2008.[78] C. R. Hagen. Scale and conformal transformations in galilean-covariant field theory. Phys. Rev. D, 5:377, 1972.[79] Y. Nishida and D. T. Son. Nonrelativistic conformal field theories.Phys. Rev. D, 76:086004, 2007.[80] M. Shick. Two-dimensional system of hard-core bosons. Phys. Rev.A, 3:1067, 1971.[81] V. N. Popov. On the theory of the superfluidity of two- and one-dimensional bose systems. Theor. Math. Phys., 11:565–573, 1972.[82] D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov. Bose-einsteincondensation in quasi-2d trapped gases. Phys. Rev. Lett., 84:2551,2000.[83] M. S. Mashayekhi, J.S. Bernier, D. Borzov, J.L. Song, and F. Zhou.Two-dimensional bose gases near resonance: Competing two-body andthree-body effects. Phys. Rev. Lett., 110:145301, 2013.[84] O. Bergman. Nonrelativistic field-theoretic scale anomaly. Phys. Rev.D, 46:5474, 1992.[85] M. Olshanii, H. Perrin, and V. Lorent. Example of a quantum anomalyin the physics of ultracold gases. Phys. Rev. Lett., 105:095302, 2010.80Bibliography[86] J. Hofmann. Quantum anomaly, universal relations, and breathingmode of a two-dimensional fermi gas. Phys. Rev. Lett., 108:185303,2012.[87] C. Gao and Z. Yu. Breathing mode of two-dimensional atomic fermigases in harmonic traps. Phys. Rev. A, 86:043609, 2012.[88] R. P. Feynman. Statistical Mechanics: A set of Lectures. W. A.Benjamin Inc., 1972.[89] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari. Theory ofbose-einstein condensation in trapped gases. Rev. Mod. Phys., 71:463,1999.[90] V. Efimov. Energy levels arising from resonant two-body forces in athree-body system. Phys. Lett. B, 33:563–564, 1970.[91] J. G. Esteve. Anomalies in conservation laws in the hamiltonian for-malism. Phys. Rev. D, 34:674, 1986.[92] R. S. Ward. Anomalies in conservation laws in quantum mechanics.Phys. Rev. D, 36:640, 1987.[93] G. N. J. Ananos, H. E. Camblong, C. Gorrichategui, E. Hernandez,and C. R. Ordonez. Anomalous commutator algebra for conformalquantum mechanics. Phys. Rev. D, 67:045018, 2003.[94] J. B. McGuire. Study of exactly soluble one-dimensional n-body prob-lems. J. Math. Phys., 6:622–636, 1964.[95] H. W. Hammer and D. T. Son. Universal properties of two-dimensionalboson droplets. Phys. Rev. Lett., 93:250408, 1986.[96] D. Blume. Threshold behavior of bosonic two-dimensional few-bodysystems. Phys. Rev. B, 72:094510, 2005.[97] U. Niederer. The maximal kinematical invariance group of the freeschrodinger equation. Helv. Phys. Acta, 45:802, 1972.[98] E. Braaten and L. Platter. Exact relations for a strongly interactingfermi gas from the operator product expansion. Phys. Rev. Lett., 100:205301, 2008.81[99] A. Widera, F. Gerbier, S. Folling, T. Gericke, O. Mandel, and I. Bloch.Coherent collisional spin dynamics in optical lattices. Phys. Rev. Let.,95:190405, 2005.[100] F. Serwane, G. Zurn, T. Lompe, T. B. Ottenstein, A. N. Wenz, andS. Jochim. Deterministic preparation of a tunable few-fermion system.Science, 332:336–338, 2011.[101] E. Taylor and M. Randeria. Viscosity of strongly interacting quantumfluids: Spectral functions and sum rules. Phys. Rev. A, 81:053610,2010.[102] T. Enss, R. Haussmann, and W. Zwerger. Viscosity and scale invari-ance in the unitary fermi gas. Ann. Phys., 326:770–796, 2010.82Appendix AThe Action of Scale andConformal TransformationsIn this appendix we present explicit calculations showing the action of scaleand conformal transformations on the field operator, ψ(~r, t), and its equationof motion:i∂tψ(~r, t) = [Hs, ψ(~r, t)] . (A.1)Here we consider a scale invariant Hamiltonian, as well as the generatorof scale transformations:D = −i∫ddrψ†(~r)(d2+ ~r · ∇r)ψ(~r), (A.2)and the generator of conformal transformations:C =∫ddr r2ψ†(~r)ψ(~r). (A.3)These three operators have the following commutation relations:[Hs, C] = −iD [D,Hs] = 2iHs [D,C] = −2iC. (A.4)A.1 Scale TransformationFirst, consider a scale transformation:ψb(~r, t) = eiDbψ(~r, t)e−iDb. (A.5)Taking the derivative with respect to b will give:ddbψb(~r, t) = ieiDbeiHst[e−iHstDeiHst, ψ(~r)]e−iHste−iDb. (A.6)Using the commutation relations in Eq. A.4 and the identity:83A.1. Scale TransformationeibABe−ibA = B + ib [A,B] +(ib)22![A, [A,B]] + ..., (A.7)it is possible to write:e−iHstDeiHst = D − 2tHs. (A.8)Eq. A.8 simplifies the calculation of the equation of motion to commutatorsbetween ψ(~r) and time independent operators, D and Hs. These commuta-tors can be easily worked out to:[D,ψ(†)(~r)]= −i(d2 + ~r · ∇r)ψ(†)(~r),[Hs, ψ(†)(~r)]= −ie−iHst∂tψ(†) (~r, t) eiHst. (A.9)Substituting Eq. A.9 into Eq. A.6 yields the following differential equa-tion:(∂∂b+∂~r(b)∂b· ∇r + ∂t(b)∂b∂t)ψb(~r, t) = −(d2+ ~r · ∇r + 2t∂t)ψb(~r, t).(A.10)It is possible to equate like terms on each side of Eq. A.10 and define newcoordinates via:∂~r(b)∂b= −~r ∂t(b)∂b= −2t. (A.11)The above equations are solved by:~r(b) = ~re−b t(b) = te−2b. (A.12)Finally, combining this all together, one obtains the desired result:ψb(~r, t) = e−db/2ψ(~re−b, te−2b). (A.13)We now consider applying the scale transformation to the equation ofmotion. It is straightforward to show that the transformed equation ofmotion is:i∂tψb(~r, t) =[eiDbHse−iDb, ψb(~r, t)]. (A.14)Substituting Eq. A.13 into the transformed equation of motion gives:84A.2. Conformal Transformatione−2b∂t′ψ(~r′, t′) =[eiDbHse−iDb, ψ(~r′, t′)], (A.15)where ~r′ = e−b~r, and t′ = te−2b.At this point, it is necessary to see how the Hamiltonian changes undera scale transformation. This can be readily worked out through Eq. A.7:eiDbHse−iDb = e−2bHs. (A.16)Eq. A.16 states that the Hamiltonian has the same scaling as the time deriva-tive. The result is that the equation of motion will remain unchanged duringa scale transformation. This is the definition of scale invariance.A.2 Conformal TransformationWe apply the same technique to the conformal transformation. Defining thetransformed field as:ψb(~r, t) = eiCbψ(~r, t)e−iCb, (A.17)one can obtain the relation:ddbψb(~r, t) = ieiCbeiHst[e−iHstCeiHst, ψ(~r)]e−iHste−iCb. (A.18)Following the identities in Eq. A.4 and Eq. A.7, one obtains:ddbψb(~r, t) = ieiCbeiHst[C −Dt+ t2Hs, ψ(~r)]e−iHste−iCb, (A.19)The commutators between the field operator and Hs and D were evaluatedin Eq. A.9. In addition one can show:[C,ψ(~r)] = −r22ψ(~r)[C,ψ†(~r)]=r22ψ†(~r). (A.20)Substituting in these commutators gives a differential equation for the trans-formed field operator:85A.2. Conformal Transformation(∂∂b+∂t(b)∂b∂t +∂~r(b)∂b)ψb(~r, t)=(t2∂t + t(d2+ ~r · ∇r)− ir22)ψb(~r, t). (A.21)As in the case for scale transformations, we can define new coordinates thatsatisfy:∂~r(b)∂b= rt∂t(b)∂b= t2. (A.22)Solving the coordinates gives:~r(b) =~r1− bt t(b) =t1− bt . (A.23)The last piece is to determine how the field rescales. This is also straightforward to do [78], and in summary the conformal transformation acting onthe field gives:ψb(~r, t) = (1− bt)−d/2e−ir22b1−btψ(~r1− bt ,t1− bt). (A.24)Now let us consider the equation of motion:i∂tψb(~r, t) =[eiCbHse−iCb, ψb(~r, t)]. (A.25)The time derivative acting on ψb(~r, t) will generate spurious terms:i∂tψb(~r, t) = (1− bt)−d/2e−ir22b1−bt(b1− bt(d2+ ~r′ · ∇r′)−ir′2b22+1(1− bt)2∂t′)ψ(~r′, t′), (A.26)where ~r′ = r/(1 − bt) and t′ = t/(1 − bt). These spurious terms will becancelled by the transformed Hamiltonian.In order to study the transformed Hamiltonian, it turns out to be advan-tageous to write the Hamiltonian in terms of the transformed field operatorsψb(~r, t). The result is:86A.2. Conformal TransformationeiCbHse−iCb = eiCbeiHstHse−iHste−iCb=1(1− bt)2Hs +b1− btD(t) + b2C(t), (A.27)where we note H, C(t), and D(t), are now written in terms of the operators:ψ(†)(~r′, t′). Performing the commutators gives:[eiCbHse−iCb, ψb(~r, t)]= (1− bt)−d/2e−i r22b1−bt[1(1− bt)2 i[Hs, ψ(~r′, t′)]+(b1− bt(d2+ ~r′ · ∇r′)− ir′2b22)ψ(~r′, t′)](A.28)Comparison between Eqs. A.26 and A.28 show that all the spurious termsvanish and the equation of motion is left invariant:i∂t′ψ(r′, t′) =[Hs, ψ(r′, t′)]. (A.29)87Appendix BDerivation of the QuantumVariational ApproachB.1 The Effective ActionIn the following sections of this appendix, we provide a detailed derivationof the effective theory:n(~r, t) =∫dλ Nλ2f(rλ) |ψ(λ, t)|2∫dλ |ψ(λ, t)|2 . (B.1)starting from:n(~r, t) =∫Dψ(~x)Dψ′(~x) 〈ψ0|eiHt|{ψ(~x)}〉〈{ψ′(~x)}|e−iHt|ψ0〉[〈ψ∗(~r)ψ′(~r) 〈{ψ(~x)}|{ψ′(~x)}〉] . (B.2)The states |{ψ(~x}〉 are the eigenstates of the annihilation operator ψˆ(~r):ψˆ(~r)|{ψ(~x}〉 = ψ(~r)|{ψ(~x}〉, defined on a discretized lattice with sites: ~x.For the current system, we are only interested in the coherent states nor-malized to the number of particles N . That is, in the continuum limit:∫d2r|ψ(~r)|2 = N .The matrix element〈{ψ(~x)}|e−iHt|ψ0〉can be written in terms of afunctional integral [88]:〈{ψ(~x)}|e−iHt|ψ0〉=∫ ′DφeiS , (B.3)where S is the action for a non-relativistic Bose gas:S =∫d2r∫ t0dt′ψ∗(~r, t)(i∂t +∇2r2)ψ(~r, t)− g2|ψ(~r, t)|4 (B.4)88B.1. The Effective Actionand∫ ′Dψ denotes the sum over all field configurations ψ(~r, t) which satisfythe following boundary conditions:ψ(~r, T ) = ψ(~r),ψ(~r, 0) = ψ0(~r). (B.5)When the number of particles at each point in our discretized space islarge |ψ(~r)|2  1, it is possible to simplify Eq. B.2 by noting that the overlapbetween two coherent states approaches a functional delta function:〈{ψ(~x)}|{ψ′(~x)}〉 = exp[∫d2x(ψ∗(~x)ψ′(~x)− 12|ψ(~x)|2 − 12|ψ′(~x)|2)]≈ Π~x δ(ψ(~x)− ψ′(~x))≡ δ ({ψ′(~x)} − {ψ(~x)}) . (B.6)Eq. B.6 states that the value of the two fields ψ(~r) and ψ′(~r) are equivalentat each point in space. In the continuum limit, this is equivalent to a deltafunction enforcing the two field configurations to be identical. This resultsimplifies Eq. B.2 to:n(~r, t) =∫Dψ(~x) |〈{ψ(~x)}|e−iHt|ψ0〉|2|ψ(~r)|2. (B.7)In this work it will be advantageous to rewrite Eqs. B.3, B.4, and B.7 interms of two new fields; the density field, ρ(~r), and phase field θ(~r). Thesetwo fields are related to ψ(~r) by:ψ(~r) =√ρ(~r)eiθ(~r) (B.8)In terms of the density and phase field, Eq. B.7 can be written as:n(~r, t) =∫Dρ(~x)∫Dθ(~x) |〈{ρ(~x)}, {θ(~x)}|e−iHt|ψ0〉|2ρ(~r)∫Dρ(~x)∫Dθ(~x) | 〈{{ρ(~x)}, {θ(~x)}|e−iHt|ψ0〉 |2 .(B.9)where |{φ(~x)}〉 = |{ρ(~x)}, {θ(~x)}〉. The matrix element〈{ρ(~x)}, {θ(~x)}|e−iHt|ψ0〉can also be expressed in terms of the density and phase fields:89B.2. Scale Invariance at the Semiclassical Level〈{ρ(~x)}, {θ(~x)}|e−iHt|ψ0〉=∫ ′Dρ(~x, t)Dθ(~x, t)eiS , (B.10)where the action is given by:S = −∫ t0dt′∫d2r [ρ(~r, t)∂tθ(~r, t)+12∇r√ρ(~r, t) · ∇r√ρ(~r, t) +ρ(~r, t)2∇rθ(~r, t) · ∇rθ(~r, t)+g2ρ2(~r, t)](B.11)These manipulations are exact and do not require any approximations.B.2 Scale Invariance at the Semiclassical LevelIn this section we examine the semiclassical solution to the dynamics andthe role of scale invariance. The semiclassical solution is obtained by mini-mizing this action and only considering the semiclassical contribution to thedynamics. This approach gives the standard hydrodynamic description of aBose gas:0 = ∂tθ(~r, t)− 12∇2r√ρ(~r, t)√ρ(~r, t)+12(∇rθ(~r, t))2 + gρ(~r, t),0 = ∂tρ(~r, t) +∇r · (ρ(~r, t)∇rθ(~r, t)).(B.12)Both of the hydrodynamic equations are invariant under the transfor-mation:~r′ = e−b~r t′ = e−2bt, (B.13)and are scale invariant. This implies that a generic solution to this equationof motion also satisfies:ρ(~r′, t′, {λ′}) = e−2bρ(~r, t, {λ}).θ(~r′, t′, {λ′}) = θ(~r, t, {λ}). (B.14)90B.3. Coarse Grained DynamicsThe set of parameters {λ} represent any additional scales introduced by theinitial conditions. These length scales explicitly break the scale invariance,and need to be rescaled alongside the spatial and temporal coordinates in theproblem: {λ′} = e−b{λ}. That is, the scale invariance relates the dynamicsfrom different initial conditions to one another.For the remainder of this discussion we consider the case when the initialconditions introduces a single length scale into the problem, λ0. This casewas previously studied for repulsive interactions in Ref. [42]. At t = 0,Eq. B.14 implies that the density field can be written as:ρ(~r, 0, λ0) =Nλ20f(rλ0), (B.15)The scale invariance does not predict the function f(x) or λ0, but they canbe determined by the initial conditions of the system.We then choose a time dependent scaling solution for the density andphase fields:ρ(~r, t) =Nλ(t)f(rλ(t))θ(~r, t) =r22λ˙(t)λ(t). (B.16)This solution can be shown to be consistent with Eq. B.12, if λ(t) satisfies:mλ¨(t) =Vλ3(t). (B.17)This is the scale invariant equation of motion for a classical particle in a r−2potential. The general solution for attractive interactions is:λ(t) = |V |2E2E√ 1m|V | t+√1 +2Eλ20|V |2 − 11/2. (B.18)B.3 Coarse Grained DynamicsTo understand the quantum dynamics contained in Eq. B.7, we performa coarse graining procedure. This is accomplished by splitting the fields inEqs. B.4 and B.7 into long wavelength isotropic degrees of freedom, n(r, t) ≈91B.3. Coarse Grained Dynamicsρλ(~r, t) and θλ(~r, t), and short wavelength fluctuations, δρ(~r, t) and δθ(~r, t).This expansion is controllable in the limit of dense condensates, ρλ(~r, t) 1,and leads to an effective theory describing the isotropic long wavelengthdynamics of the system.Motivated by the discussion in the previous appendix, we choose to workwith a single parameter ansatz for the isotropic long wavelength degrees offreedom:ρλ(~r, t) =Nλ2(t)f(rλ(t)),θλ(~r, t) =r22λ˙(t)λ(t)+ η(t), (B.19)where f(x) is a normalizable isotropic function that is regular at the origin,and η(t) is a time dependent phase that is irrelevant to the following dis-cussion. The parameter λ(t) is the time dependent size of the condensate.In this approach, all the dynamical information is encoded in λ(t), and wewish to derive an effective theory for this single parameter.This specific form of the phase field, θλ(~r, t), is chosen in order to satisfythe conservation law, the second line in Eq. B.12. The dynamics of the phasefield are treated semiclassically, and is of little importance for the remainderof this discussion. However, no restrictions are placed on the density field.Applying Eq. B.19 to Eq. B.11 results in the following zeroth order ac-tion:Sλ =∫ t0dt′12mλ˙2 − V2λ2(B.20)where m = C1N and V = C3gN2 + C2N . The coefficients C1, C2, and C3can be calculated once the function f(x) has been specified. Some examplesof these coefficients can be found in Ref. [67].The main effect of the short wavelength fluctuations to the dynamics isto modify the matrix element, Eq. B.10, or equivalently the action, Eq. B.20.To generate the correction to Eq. B.20 we expand Eq, B.11 to second orderin δρ(~r, t) and δθ(~r, t). Since the phase field is chosen to satisfy the semi-classical equation of motion, Eq. B.12, the fluctuations δθ(~r, t) will appearat O(δθ2(~r, t)), while the fluctuations in the density, δρ(~r, t) will appear atO (δρ(~r, t)).In principle there are fluctuations of linear order in δρ(~x, t) since wedo not minimize the action with respect to density. However, these short92B.3. Coarse Grained Dynamicswavelength fluctuations represent anisotropic modes that are orthogonal tothe motion described by λ. Therefore any term linear in the fluctuationsmust vanish.The remaining quadratic fluctuations can then be integrated out in orderto derive an action in terms of the slow degrees of freedom, λ(t). In principlethere is no limitation to integrating out these fluctuations, however in prac-tice this can be quite a challenge. In order to obtain an estimate of thesefluctuations, we assume that the isotropic modes ρλ(~x, t) and θλ(~x, t) areapproximately constant over the length and time scales associated with thefluctuations. By neglecting the spatial and temporal dependence of the slowdegrees of freedom, the action for the fluctations becomes diagonal whenexpanded in the definite angular momentum basis:δρ(~x, t) =∑k,`Nk,`J`(kx)ei`φ√2piδρk,`(t)δθ(~x, t) =∑k,`Nk,`J`(kx)ei`φ√2piδθk,`(t) (B.21)where k and ` specify the radial mode and angular momentum, respectively,Nk,` is the normalization factor associated with each radial mode, and J`(kx)is the Bessel function of order `. An explicit calculation of the fluctuationsis given in Ref. [67].The effect of the fluctuations is to act as a background field upon whichthe long wave dynamics occur. These fluctuations introduce a correction tothe action, δS. δS contains both real and imaginary terms. The real partof δS renormalizes the coefficients C1, C2, C3, and the coupling constant g,while the imaginary part implies that the system under consideration has afinite lifetime. These corrections to the action are suppressed in the limit|g|  1 which is the focus of this work. These corrections are thoroughlydiscussed in Ref. [67].After integrating out the fluctuations, the expression for the time evolveddensity is given by:n(~r, t) =∫dλNλ2f(rλ)|〈ψλ|e−iHt|{ψ0}〉|2〈ψλ|e−iHt|ψ0〉 =∫ λ(t)=λλ(0)=λ0Dλ(t)ei∫ t0dt′ 12mλ˙2(t)− V2λ(t)2+iδS,(B.22)93B.4. Need for Quantizationwhere λ0 represents the size of the condensate which is in the state |ψ0〉 .Eq. B.22 is equivalent to the quasi-unitary time evolution of a wavefunction ψ(λ, t) ≡ 〈λ|e−iHλt|λ0〉 under the the Hamiltonian Hλ:Hλ =P 2λ2m+V2λ2+ i Im δHλ (B.23)where λ is now an operator with eigenstates |λ〉 and Pλ the conjugate mo-mentum: [λ, Pλ] = i. m and V are now defined via the renormalized con-stants C1, C2, and C3. The correction, i Im δHλ, is the imaginary contri-bution due to the anisotropic fluctuations:Im δHλ =C4g2N22λ2. (B.24)Finally, this equivalence between the full dynamics and the effectivequantum mechanical model, Eq. B.23 allows one to recast Eq. B.9 into thedesired result:ρ(~r, t) =∫dλ ρ(~r, λ)|ψ(λ, t)|2∫dλ |ψ(λ, t)|2 . (B.25)B.4 Need for QuantizationIn this section, we highlight the need for quantization. This is most readilydone by examining the fluctuations around a semiclassical path. Here weconsider the semiclassical path of a particle with no initial kinetic energy.The solution to Eq. B.18 will be periodic. For half a period, the solution isgiven by:λsc(t) = λ0√1− (2t/T )2 (B.26)where T = 2λ20√|V | is the semiclassical period.To understand the effect of fluctuations around this semiclassical path,we calculate the autocorrelation of the fluctuations and compare that tothe semiclassical path: 〈δλ(t)δλ(t)〉/λ20. To calculate this it is necessary toexpand Eq. B.20 around this solution to quadratic order:δS =∫ t0dt′m2˙δλ2(t)− 32Vλ4sc(t)δλ2(t). (B.27)94B.4. Need for QuantizationIn order to evaluate the correlator, it is ideal to expand the fluctuations insine modes;δλ(t) =1√T∑nsin(npitT)δλn. (B.28)From expanding the action in terms of sine modes one finds that the actionis proportional to T−2. This implies that:〈δλ(t)δλ(t)〉λ20∝ 1TT 2λ20=1√m|V | (B.29)where√m|V | = √C1N (C2N + C3gN2).We note that the strength of the fluctuations is not controlled by thesize of the classical path, but by a scale independent parameter. Therefore,depending on the value of 1/√m|V |, full quantization might be necessary.We choose to work in this regime.95Appendix CQuantum Anomaly and theHeisenberg Equation ofMotionIn this section we derive the quantum anomaly using the equation of motion.To begin we define an operator in the Heisenberg representation:O(t) = eiH†tOe−iHt. (C.1)Here we note that although the Hamiltonian, H, is Hermitian, it may not beself-adjoint, i.e. H and H† may not act on the same subspaces. Therefore,a more correct form of the Heisenberg equation of motion is [91–93]:i∂tO(t) = i [H,O(t)] + i(H† −H)O(t). (C.2)We define the final term as the anomalous piece:A =(H† −H)O(t). (C.3)Here we consider the Hamiltonian for a two-dimensional Bose (or Fermi)gas:H =∫d2rψ†(~r)(−∇2r2)ψ(~r) +g2∫d2rψ†(~r)ψ†(~r)ψ(~r)ψ(~r). (C.4)The anomaly associated with scale transformations will then have the form:A = i∫d2r∫d2r′∇r2i(ψ†(~r) ∇rψ(~r)−∇rψ†(~r) ψ(~r))ψ†(~r′)(d2+ ~r′ · ∇r′)ψ(~r′).(C.5)As one can see the anomaly depends on the current operator:96Appendix C. Quantum Anomaly and the Heisenberg Equation of MotionJ =∫d2r12i(ψ†(~r) ∇rψ(~r)−∇rψ†(~r) ψ(~r)). (C.6)For potentials with ill-defined ultraviolet physics, the current operator can besingular, leading to non-zero anomalies. For the remainder of this section, wewill illustrate this physics by applying this formalism to two-body problems-wave scattering in two spatial dimensions.Consider two bosons interacting via a short range square well potential:V (r) = −V0θ(r0 − r). (C.7)The s-wave wave function for the system can be written in terms of thecenter-of-mass, ~R, and relative, ~r, coordinates:ψ(~R,~r) =1√Ωei~P ·~Rχ(r), (C.8)where Ω is the volume of the system, P is the center of mass momentum,and χ(r) is the relative wave function. The s-wave relative wave functionsatisfies the Schrodinger equation:(−∂2r + V (r))χ(r) = Eχ(r). (C.9)Although the physical potential has a finite range, this physical potentialis often replaced with a contact interaction. In order to do this, one mustapply an boundary condition at r = r0 that reproduces the low energyscattering of the true potential. For two-dimensional systems this conditionis given by:χ(r0) ∝ log(r0a2D), (C.10)where a2D is the effective scattering length for the two bosons. The effectivescattering length is related to the bare parameters by:a2D =r02eγEe2V0r20 , (C.11)where γE is Euler’s gamma constant. Note that this length scale is relatedto the bound state energy by Eb = −a−22D.Since we replaced the physical non-scale invariant potential with a scaleinvariant contact interaction, the consequence will be a non-zero anomaly.It is only necessary to focus on the relative coordinate’s contribution to the97Appendix C. Quantum Anomaly and the Heisenberg Equation of Motionanomaly, as the center of mass motion is non-interacting. For this two-bodysystem the anomaly is given by:A =∫d2rχ(r)(∇2r)(~r · ∇r)χ(r)−∇2rχ(r)(~r · ∇r)χ(r)=∫d~S [χ(r)∇r (~r · ∇r)χ(r)−∇rχ(r) · ~r · ∇rχ(r)] , (C.12)where the integrals are over the surface of the space.As mentioned previously, the anomaly is caused by neglecting the physicsinside the potential. Therefore we focus on the contribution from shortdistances. Applying Eq. C.10 to Eq. C.12, one can show that [65, 84–87]:A = 2pi. (C.13)The fact that A is non-zero implies that the scale invariance is broken.However, if one were to consider the physics inside the potential, the scaleinvariance would be explicitly broken, but the anomaly would be zero. Theanomaly comes from treating the system with a scale invariant effectivemodel.98Appendix DExistence of ConformalTowersIn this appendix, we show that conformal symmetry means that the eigen-states of a quantum system are organized in terms of towers, where thestates within a tower are evenly spaced by two harmonic oscillator units.To begin we note that for non-relativistic quantum systems. The con-formal symmetry is a subgroup of the Galilean group, which is the largestsymmetry class for the non-relativistic Schrodinger equation [97]. The alge-bra is formed by the following operators:N =∫d3r n(~r) Pi =∫d3r ~rji(~r) Mi,j =∫d3r (rijj(~r)− rjji(~r)) ,(D.1)Ki =∫d3r rin(~r) C =∫d3rr22n(~r) D =∫d3r ~r~j(~r), (D.2)where:n(~r) = ψ†(~r)ψ(~r)~j(~r) = − i2(ψ†(~r)∇ψ(~r)−∇ψ†(~r)ψ(~r)), (D.3)are the particle density and probability current, and the field operators,ψ(†)(~r) satisfy either bosonic or fermionic statistics:[ψ(~r), ψ†(~r′)]± = δ3(~r − ~r′). (D.4)The remaining operators are associated with particle number, N , momen-tum, Pi, angular momentum Mi,j , Galilean boosts, Ki, conformal transfor-mations, C, and scale transformations, D.A subgroup of these operators is called the conformal group. The con-formal group consists of three operators, a scale invariant Hamiltonian, Hs,the dilation operator, D, and the conformal operator, C. This conformal99Appendix D. Existence of Conformal Towersgroup forms a subalgebra, the so(2,1) algebra. For the conformal group, onecan show the following commutation relations:[Hs, C] = −iD [D,C] = −2iC [D,Hs] = 2iHs. (D.5)Next consider an operator:O(~r) = ei~P ·~rO(0)e−i ~P ·~r, (D.6)such that: [D,O(0)] = i∆OO(0), and [C,O(0)] = [Ki, O(0)] = 0. Such anoperator is called a primary operator. Here we consider the state:|ψ〉 = e−Hs/ωO(0)|vac〉, (D.7)where |vac〉 is the vacuum state. This state is an eigenstate of the Hamilto-nian, Hs +ω2C. For the following discussion we will set the trap frequency,ω, to unity. To see that |ψ〉 is an eigenstate of Hs + C, note:(Hs + C)|ψ〉 = e−Hs(Hs + eHsCe−Hs)O(0)|vac〉 (D.8)Using the identity:eHsCe−Hs = C + [Hs, C] +12[Hs, [Hs, C]] + .... (D.9)and using the commutation relations contained in Eq. D.5 and the definitionof O(0), one obtains:(Hs + C)|ψ〉 = e−Hs(C − iD)O(0)|vac〉 = ∆OO(0)|ψ〉. (D.10)Therefore, |ψ〉 is an eigenstate of Hs+C, with eigenvalue equal to ∆O. Thisis true for any primary operator O(0).Given the states |ψ〉, it is possible to generate the remaining spectrumof a conformal tower. Consider the operators:L± = Hs − C ± iD. (D.11)These operators satisfy:[L−, L+] = 4(Hs + C) [L±, Hs + C] = 2L±. (D.12)Eq. D.12 is identical to the commutation relations for the ladder operatorsof a non-interacting harmonic oscillator. The only difference is that theyraise and lower the energy by two harmonic units.100D.1. Application of Conformal Tower Spectrum to a 1D Harmonic OscillatorTo show that |ψ〉 is the base of the tower, let us consider the action ofL− on this state:L−|ψ〉 = e−HseHs(Hs − C − iD)e−HsO(0)|vac〉= −e−HsCO(0)|vac〉= 0. (D.13)As can be seen by Eq. D.13, all states |ψ〉 constructed from a primaryoperator, O, form the bases of the conformal towers. The remaining states inthe spectrum can be constructed by the application of the raising operator.This proves that the spectrum of a conformally symmetric quantum systemcan be decomposed into a series of conformal towers. The states in eachtower are evenly spaced by two harmonic units.In order to to label the towers, we note that:[L±, N ] = 0, (D.14)where N is the total number of particles. As well, for rotationally invariantsystems:[Hs,Mi,j ] = [C,Mi,j ] = [D,Mi,j ] = 0, (D.15)or:[L±,Mi,j ] = 0. (D.16)Therefore a given conformal tower can be labelled by the total numberof particles, and their angular momentum, or equivalently for one spatialdimension, by the parity. As a result there are infinitely many towers. Inour analysis we will always consider a fixed number of particles. In this case,the conformal towers can be labelled by their angular momentum, or parity,alone.D.1 Application of Conformal Tower Spectrumto a 1D Harmonic OscillatorThe spectrum for a 1D harmonic oscillator is given by:En =(n+12)ω. (D.17)101D.1. Application of Conformal Tower Spectrum to a 1D Harmonic OscillatorAs can be seen, for a fixed number of particles, the spectrum can be decom-posed into two conformal towers: one for even and odd parity. For a singleparticle, the energies for the even and odd single particle conformal towers,respectively, are:En,e =(2n+12)ωEn,o =(2n+32)ω.(D.18)The creation operators for a particle in the lowest even and odd confor-mal tower states:ψ†0,e =∫dx(ωpi)1/4e−ωx22 ψ†(x)ψ†0,o =∫dx√2(ωpi)1/4x√ωe−ωx22 ψ†(x). (D.19)We note that the Fourier Transform of Eq. D.19 is:ψ†0,e =∫dk2pi√2(piω)1/4e−k22ωψ†(k)ψ†0,o =∫dk2pi− 2i(piω)1/4 k√ωe−k22ωψ†(k). (D.20)Upon comparison to Eq. D.7, one can read off:Oe(0) =√2(piω)1/4ψ†(0)Oo(0) = 2(piω)1/4 1√ωlimx→0∂∂xψ†(x). (D.21)A straightforward calculation shows that:[D,Oe(0)] = i12Oe(0)[D,Oo(0)] = i32Oo(0), (D.22)which are the correct ground state energies for the even and odd conformaltowers.102Appendix EComoving Reference Frameand Time Dependent TrapsIn this appendix we discuss the use of the comoving reference frame to scaleinvariant gases placed in time dependent traps. The Hamiltonian for thissystem is given by:H =N∑i=1−12∇2i +12N∑i,j=1Vs(~ri − ~rj) +N∑i=1ω2(t)2r2i= Hs + ω2(t)C. (E.1)We employ the trial many body wave function:ψ({~ri, σi}, t) = 1λ3N/2(t)ei2∑Ni=1r2iλ˙(t)λ(t)φ({ ~riλ(t), σi}, τ(t)). (E.2)Upon substituting into the Schrodinger equation, one obtains:i∂τ∂t∂∂τφ ({~xi, σi}, τ(t)) =(∑i[−121λ2(t)∇˜2i +x2i2(λ¨(t)λ(t) + ω2(t)λ2(t))]+12∑i,jVs(λ(t) (~xi − ~xj))φ ({~xi, σi}, τ(t)) .(E.3)To make further simplifications, we define:∂τ(t)∂t=1λ2(t)λ¨(t) =ω20λ3(t)− ω2(t)λ(t), (E.4)noindent where the boundary conditions for λ(t) are chosen such that:103E.1. Quench of The Trapping potentialλ(0) = 1 λ˙(0) = 0 (E.5)Solving Eq. E.4 and substituting into Eq. E.3 gives a Schrodinger equa-tion for the effective wave function:i∂∂τφ({~xi, σi}, τ) = H˜φ({~xi, σi}, τ),H˜ =N∑i=1[−12∇˜2i + ω20x2i2]+12N∑i,j=1Vs(~xi − ~xj). (E.6)The Hamiltonian in the comoving frame is simply a scale invariant gas in atime independent harmonic trap.Eq. E.6 is conformally invariant, and the spectrum of the system canbe written in terms of the conformal tower spectra discussed in AppendixD. Therefore, in the laboratory frame, any system prepared in one of theseconformal tower states will exhibit the trivial dynamics contained in Eq. E.4.E.1 Quench of The Trapping potentialAs a first example, consider a quench of the trapping potential:ω(t) = ωi t < 0= ωf t ≥ 0, (E.7)where ωf < ωi. The solution was derived in Ref. [74]. Here we show theresult for λ(t) and τ(t):λ(t) = λ0√√√√1 + ω2i − ω2fω2fsin2(ωf t)τ(t) =1ωiλ20arctan(ωiωftan(ωf t)). (E.8)The solution for λ(t) is shown in Fig. E.1 has two main features. First thesolution oscillates at frequency ωf , and secondly, for ωf t  1, the solutionis equivalent to free space expansion.104E.2. Efimovian Expansion2 4 6 8 10ωf t12345λ2(t)/λ02Figure E.1: Solution for λ(t) for the dynamics of a quench trapping potential.The ratio of the trapping potentials is given by ωi = 5ωf . For conveniencewe plot λ2(t) as a function of time.E.2 Efimovian ExpansionIn this section, we consider a trap that has a trap frequency:ω(t) = ωi t < t0=t0tωi t ≥ t0. (E.9)The solution for λ(t) and τ(t) are:λ(t) = λ0(1 + s20)tt0s201− 1√1 + s20cos(s0 log(tt0)− arctan(s0))1/2τ(t) =t0λ202√1 + s20arctan(2 + s20) tan(s02 log(tt0)− s0)s0√1 + s20 , (E.10)where s0 =√4(ωit0)2 − 1.105E.2. Efimovian Expansion5 10 15 20Log(t/t0 )11000106109Log(λ2(t))Figure E.2: The dynamics of the moment of inertia for a scale invariantquantum gas inside an expanding trap. The time dependence of the trapis given by Eq. E.9.The dynamics of the moment of inertia are given byEq. E.11, and are in strong agreement with the observations in Ref. [4].E.2.1 Experimental Detection of Efimovian ExpansionThe dynamics of a scale invariant Fermi gas in a time dependent har-monic trap have been studied experimentally in Ref. [4], and theoreticallyin Ref. [68]. In this experiment the trap evolves according to Eq. E.9, andthe dynamics of the moment of inertia were examined. In this section wediscuss the dynamics using the comoving reference frame.Consider a scale invariant Fermi gas of N particles placed in an isotropicharmonic potential of frequency ωi. At zero temperatures, the gas can beconsidered to be in the many body ground state, which necessarily is aconformal tower state. The moment of inertia is then defined to be:〈rˆ2〉(t) =∫ddr1λd(t)∫ddr2λd(t)...∫ddrNλd(t)r21∣∣∣∣φ0 ({ riλ(t)})∣∣∣∣2= λ2(t)〈rˆ2〉(0), (E.11)where φ0({~xi}), is the many body wave function for the ground state.The dynamics of the moment of inertia are shown in Fig. E.2. Eq. E.11is an accurate description of the dynamics, and faithfully reproduces theexperiments in Ref. [4].106Appendix FTwo-Body SolutionIn this appendix we derive the deviation, δH˜, from the transformed scaleinvariant Hamiltonian, H˜s, in the expanding comoving frame. The approachwe employ here is applicable for both the non-interacting quantum gas withan impurity, and the relative dynamics of the two-body problem. The onlydifference is that in the two-body problem, one uses the reduced mass forthe two particles. In both cases, the physical interaction will be some shortranged, spherically symmetric potential, V (r).We begin with the radial Schrodinger equation in the co-moving frame:i∂τχl(x, τ) = H˜χl(x, τ),H˜ = −12∂2x +12x2 +l(l + 1)2x2+ λ2(τ)V (xλ(τ)),λ(τ) = λ0 sec(τ), (F.1)where we have set the (reduced) mass to unity, and Yl,m(~x) is the sphericalharmonic with angular quantum number l and projection quantum number,m. The radial wave function, χl(x, τ), is related to the full wave functionvia: φl,m(~x, τ) = Yl,m(xˆ)χl(x, τ)/x, and is properly normalized:∫ ∞0dx|χl(x, τ)|2 = 1. (F.2)In what follows we will only focus on the zero angular momentum, or s-wave, scattering of this potential, as higher angular momentum scatteringis suppressed by a factor of (√Er0)2l, where r0 is the range of the potential,and E is the energy.For specificity, we will consider the potential to be a square well of depth:V0λ2(τ), and range: r0/λ(τ). This potential is consistent with the timedependence of the interaction in the coming frame: λ2(τ)V (xλ(τ)), andcaptures all the essential physics at low energies. It is important to notethat the range and depth of the potential are changing at a rate set by λ(τ),which is much slower than the energy scale set by the finite range of the107Appendix F. Two-Body Solutionpotential, r0. This implies that we can use the adiabatic approximation. Inthis approximation, the effect of the finite scattering length is to impose thetime-dependent boundary condition at the range of the potential [76, 77]:χ′(r0/λ(τ))χ(r0/λ(τ))= −λ(τ)a. (F.3)As discussed in the main text, we split the effective Hamiltonian in thecomoving frame, H˜, into the effective Hamiltonian at the scale invariantfixed point, H˜s, and a deviation, δH˜:H˜ = H˜s + δH˜H˜s = −12∂2x +12x2 + λ2(τ)Vs(λ(τ)x), (F.4)and Vs(x) is a scale invariant potential, and we have set the trap frequencyto unity. In this analysis the quantities of interest are the matrix elementsof the deviation:δH˜(τ) = λ2(τ)V (xλ(τ))− λ2(τ)Vs(λ(τ)x)δH˜(τ) = λ2(τ)V (xλ(τ))− Vs(x), (F.5)with respect to the eigenstates of H˜s. In Eq. F.5, we have used the factthat the system possesses scale invariance at a fixed point, i.e. Vs(λx) =λ−2Vs(x).We first evaluate the deviation from the resonant fixed point. The matrixelements of Eq. F.5 near resonance can be determined by examining the zeroangular momentum Schrodinger equation at, and near, resonance:Er,nχr,n(x) =(−12∂2x +12x2 + Vres(x))χr,n(x)Em(τ)χm(x, τ) =(−12∂2x +12x2 + λ2(τ)V(xλ(τ)))χm(x, τ). (F.6)The top and bottom lines correspond to the resonant and off resonantSchrodinger equations, respectively. The states χr,n(x) and χm(x, τ) arethe eigenstates of the system with energy Er,n and Em(τ), and quantumnumbers n and m, for the resonant, and off resonant Hamiltonians, respec-tively.108Appendix F. Two-Body SolutionAt this stage one can multiply the resonant (off-resonant) Schrodingerequation by the state χm(x, τ) (χr,n(x)), and integrate over the range of thepotential, r0/λ(τ). The difference between the two Schrodinger equationsis:∫ r0/λ(τ)0dxχr,n(x)(Em(τ)− Er,n)χm(x, τ) =∫ r0/λ(τ)0dxχr,n(x)[−12∂2x +12x2 + λ2(τ)V (xλ(τ))]χm(x, τ)−∫ r0/λ(τ)0dxχm(x, τ)[−12∂2x +12x2 + Vres(x)]χr,n(x) (F.7)To obtain the deviation operator, we expand the difference to first order in1/a. To this order the expansion of Eq. F.7 gives:∫ r0/λ(τ)0dx(λ2(τ)V (xλ(τ))− Vres(x))χr,m(x)χr,n(x) =∫ r0/λ(τ)0dx {(Em(τ)− Er,n)χm(x, τ)χr,n(x)}− λ(τ)2aχr,m(r0/λ(τ))χr,n(r0/λ(τ)).In Eq. F.8 we have used Eq. F.3 to evaluate the difference in kinetic energies.In the second line of Eq. F.8, the quantities Em(τ)−Er,n and χr,n(x)χm(x, τ)are to be expanded to first order in λ(τ)/a. The integrals themselves will beproportional to r0/λ(τ), for r0  λ0. Therefore the second line in Eq. F.8will be a correction of order r0/a to the matrix elements, which is negligiblein the large scattering length limit. After neglecting the terms proportionalto O(r0/a), the expression for the deviation becomes:〈m|δH˜|m〉 =∫ r0/λ(τ)0(λ2(τ)V (xλ(τ))− Vres(x))χr,m(x)χr,n(x)= −λ(τ)2aχr,m(r0/λ(τ))χr,n(r0/λ(τ)).To simplify the deviation further, we note that the resonant eigenstatesoutside the potential are given by:109Appendix F. Two-Body Solutionχr,n(x) = 〈x|n〉 =√2φh.o,2n(x) Er,n = 2n+ 1/2, (F.10)where φh.o,n(x) is the normalized one-dimensional harmonic oscillator wavefunction with quantum number n = 0, 1, 2, ..., and with the harmonic lengthscale set to unity. For x 1, the resonant eigenstates are constant near theorigin. The continuity of the wave function at the boundary allows one tosimplify the deviation to:〈m|δH˜|n〉 = −λ(τ)2aχr,m(0)χr,n(0)=λ(τ)afmfn (F.11)where:fn =√2pi1/4(2n− 1)!!√(2n)!. (F.12)Therefore the scaling of the deviation is α = 1.For weak interactions, we compare the non- and weakly- interactingSchrodinger equations in the co-moving frame:E0,nχ0,n(x) =(−12∂2x +12x2)χ0,n(x)Em(τ)χm(x, τ) =(−12∂2x +12x2 + λ2(τ)V (xλ(τ)))χm(x, τ).(F.13)Here χ0,n(x) is a non-interacting eigenstate with energy E0,n and quantumnumber n = 0, 1, 2, ...:χ0,n(x) = 〈x|n〉 =√2φh.o,2n+1(x) E0,n = 2n+ 3/2. (F.14)A calculation identical to the resonant case yields:〈m|δH˜|n〉 = a2λ(τ)χ′0,m(0)χ′0,n(0)=aλ(τ)gmgn, (F.15)110F.1. Beat Amplitudes for Moment of Inertia and Contactwhere:gn = (−1)n 1pi1/412n−1√(2n+ 1)!n!. (F.16)The approach used here is similar to Refs. [75–77]. The matrix elementsof Eq. F.11 and F.15 have an important connection to the thermodynamiccontact first examined in Ref. [76, 77]. If one were to consider just theground state expectation value of the deviation, Eq. F.5, one would simplyobtain the contact. Eq. F.5 is then a natural extension of the idea of contactto a matrix. The relationship between the breaking of scale invariance andthe contact in equilibrium physics has been discussed in Ref. [86].F.1 Beat Amplitudes for Moment of Inertia andContactIn this appendix we report the analytic expressions for the long time, ωt 1,or equivalently, τ ≈ pi/2, dynamics for the moment of inertia of an non-interacting quantum gas in the presence of a scale invariant external poten-tial, and for the contact in the two-body problem, both near resonance.The moment of inertia for N -particles is defined as:rˆ2 =1NN∑i=1r2i =λ2(t)NN∑i=1x2i= λ2(t)〈x2〉(τ(t)), (F.17)where 〈x2〉 is the moment of inertia calculated in the comoving frame. Thelong time dynamics of the moment of inertia, in the comoving frame, hasthe following form:limt→∞〈x2〉(t) ≈ A+B sin(v1√ωalog(ωt))1ωt+D sin2(v21√ωalog(ωt)).(F.18)The coefficients A, B, and D are found by explicitly evaluating the expec-tation value. Here we quote the result:A =nmax∑{ni},{mi}=0∑P,Q(±1)P+QN !ψ({mi})ψ({ni})111F.1. Beat Amplitudes for Moment of Inertia and Contact 12NN∑j=1(4nQj + 1)∏k=1,NδmPk ,nQk1NN∑i=j√(2nQj + 1)(2nQj + 1)N∏k 6=j=1δmPk ,nQk δmPj ,nQj+1 (F.19)B =N∑{ni},{mi}=0(±1)P+QN !12NN∑j=1√(2nQj + 1)(2nQj + 1) N∏k 6=j=1δmPk ,nQk δmPj ,nQj+1(ψ({ni})〈mP1 , ...mPN |Pv|ψ0〉 − ψ({mi})〈nQ1 , ..., nQN |Pv|ψ0〉)](F.20)D =N∑{ni},{mi}=0(±1)P+QN !2NN∑j=1(4nQj + 1) ∏k=1,NδmPk ,nQk(〈nQ1 , ..., nQN |Pv|ψ0〉〈mP1 , ...,mPN |Pv|ψ0〉−〈mP1 , ...,mPN |Pv|ψ0〉ψ({ni})− 〈nQ1 , ..., nQN |Pv|ψ0〉ψ({mi}))]+N∑{ni},{mi}=0(±1)P+QN !2NN∑j=1[√(2nQj + 1)(2nQj + 1)N∏k 6=j=1δmPk ,nQk δmPj ,nQj+1 [〈mP1 , ...,mPN |Pv|ψ0〉ψ({ni})+〈nQ1 , ..., nQN |Pv|ψ0〉ψ({mi})− 〈nQ1 , ..., nQN |Pv|ψ0〉〈mP1 , ...,mPN |Pv|ψ0〉]](F.21)In Eqs. F.19, F.20, and F.21, |ψ0〉, is the fully symmetrized initial state:|ψ0〉 =∑{ni}[ψ({ni}) 1√N !∑P(±1)P |nP1 , ..., nPN 〉](F.22)The many body states are expanded in the single particle basis: |n1, ..., nN 〉,where ni = 0, 1, ..., nmax, with nmax = 1/(ωr20). The quantities ψ({ni}) arethe expansion coefficients for the many body states; they do not dependon the specific permutation of the indices, but rather the combination ofindices. For this reason, the summations over the many body states is re-stricted to distinct combinations of single particle indices. The symmetry isfixed by summing over all the permutations of a given set of single particleindices, specified by the summation over P .112F.1. Beat Amplitudes for Moment of Inertia and ContactFor the non-interacting quantum gas, the deviation from scale invariance,V˜ will only have a single non-zero eigenvalue of value v:v =nmax∑n=0f2n fn =√2pi1/4(2n− 1)!!√(2n)!. (F.23)For N = 1, the state with eigenvalue v is non-degenerate, however for multi-ple particle there are a number of degenerate states with eigenvalue, v. Theoperator, Pv, in Eqs. F.19, F.20, and F.21 is simply the projection operatoronto all the eigenstates of the N -body deviation, V˜I , with eigenvalue v.Similarly, we quote the results for the contact of the relative motion inthe interacting two-body problem. Near resonance, the asymptotic form ofthe contact is:limt→∞C(t) ≈Eλ(t)+Fλ(t)sin2(v21√ωaln(tλ20)). (F.24)The coefficients, E and F , are given by:E =∣∣∣∣∣nmax∑n=0fn√pi2(−1)nCn(0)∣∣∣∣∣2, (F.25)F =∣∣∣∣∣nmax∑n=0fn√pi(−1)n〈n|v〉〈v|ψ0〉∣∣∣∣∣2−nmax∑n,n′=0pi2fnf′n(−1)n−n′· (C ′n(0)〈n|v〉〈v|ψ0〉+ Cn(0)〈n′|v〉〈v|ψ0〉) .where again, Cn(0) here are the expansion coefficients for the relative mo-tion, v and fn are defined in Eq. F.23.113Appendix GThe Density Matrix andConformal SymmetryG.1 The Density Matrix and Scale InvarianceHere we consider the dynamics of the one-body density matrix, or simplydensity matrix:ρ(~r, ~r′, t) = 〈ψ0|eiHtψ†(~r′)ψ(~r)e−iHt|ψ0〉, (G.1)where H = Hs+δH is the Hamiltonian of the nearly scale invariant system,Hs is the scale invariant Hamiltonian, and δH is the deviation. As we didpreviously, it is ideal to expand the full Hamiltonian around Hs:ρ(~r, ~r′, t) = 〈ψ0|eiHte−iHstψ†(~r′, t)ψ(~r, t)eiHste−iHt|ψ0〉, (G.2)where ψ(~r, t) = eiHstψ(~r)e−iHst is the field operator time evolved by thescale invariant Hamiltonian, Hs. The operator eiHste−iHt can be shown tosatisfy:U(t) = eiHste−iHt = Te−i∫ t0dt′eiHst′δHe−iHst′. (G.3)where T is the time ordering operator. Inserting two sets of conformal towerstates allows one to separate the scale invariant dynamics from the dynamicsgoverned by the deviation:ρ(~r, ~r′, t) =∑m,nρs n,m(~r, ~r′, t)Γm,n(t)ρs n,m(~r, ~r′, t) = 〈n|eiHstψ†(~r′)ψ(~r)e−iHst|m〉Γm,n(t) = 〈m|U(t)|ψ0〉〈ψ0|U(t)|n〉. (G.4)114G.1. The Density Matrix and Scale InvarianceG.1.1 Scale Invariant DynamicsFirst consider the scale invariant piece of the density matrix, ρs n,m(~r, ~r′, t).Here it is again possible to exploit the properties of the conformal towerstates. Since the conformal tower states are eigenstates of the Hamiltonian,Hs + ω2C, one can write the scale invariant piece as:ρs n,m(~r, ~r′, t) = e−i(En−Em)η· 〈n|ei(Hs+ω2C)ηeiHstψ†(~r′)ψ(~r)e−iHste−i(Hs+ω2C)η|m〉.(G.5)However, Eq. G.5 is valid for arbitrary value of η. This leads to the followingrelation:∂∂ηρs n,m(~r, ~r′, t) = 0= −i(En − Em)ρs n,m(~r, ~r′, t)+ i〈n|eiHst[e−iHst(Hs + ω2C)eiHst, ψ†(~r′)ψ(~r)]e−iHst|m〉.(G.6)One can use the relation:eiHst(Hs + ω2C)e−iHst =(1 + ω2t2)Hs − ω2tD + ω2C, (G.7)to obtain:∂∂ηρs n,m(~r, ~r′, t) = 0=[(1 + ω2t2)∂t + ω2t(~r · ∇r + ~r′ · ∇r′ + d)+ i ω2r′2 − r22− i(En − Em)]ρs n,m(~r, ~r′, t).(G.8)Note to derive Eq. G.8 we have employed the commutators:[C,ψ(†)(~r)]= ±r22ψ(†)(~r)[D,ψ(†)(~r)]=i2(2~r · ∇r + d)ψ(†)(~r), (G.9)115G.1. The Density Matrix and Scale Invariancewhere ± refers to whether the operator is a creation operator or annihilationoperator, respectively.Eq. G.8 is reminiscent of the operator for the generator of time dependentconformal transformations 3:GC(~r, ~r′, t) = t2∂t + t(~r · ∇r + ~r′ · ∇r′ + d)+ ir′2 − r22. (G.10)In fact, if we perform a gauge transformation:ρ˜s n,m(~r, ~r′, t) = e−i(En−Em)τ(t)ρs n,m(~r, ~r′, t), (G.11)where dτ(t)/dt = 1/(1 + ω2t2), the equation of motion for ρ˜s n,m(~r, ~r′, t) issimply the generator of conformal transformations. Stated differently, thedensity matrix is an eigenfunction of the generator for conformal transfor-mations, with zero eigenvalue. Therefore the density matrix is a conformallyinvariant function. In conclusion, although the system obeys a scale invari-ant Hamiltonian, the dynamics produce an emergent conformal symmetry.The price for this gauge transformation is that the matrix, Γm,n(t), ac-quires a trivial time dependence:Γ˜m,n(t) = ei(En−Em)τ(t)Γm,n(t). (G.12)G.1.2 Broken Scale InvarianceAlthough Eq. G.8 puts a restraint on the scale invariant part of the densitymatrix, it doesn’t restrict the full density matrix. In order to obtain adifferential equation for the full density matrix, it is necessary to evaluate thetime dependence of the matrix, Γ(t), or equivalently, the time dependenceof the operator, U(t). Thankfully, the evaluation of U(t) is identical to theevaluation of the expansion coefficients in Chapter 5. The result is:U(t ω−1) ≈ exp[−i 1(√ωa)α(ωt)α−1α− 1 V˜]α > 1≈ exp[−i 1√ωalog(ωt)V˜]α = 1. (G.13)where again:V˜ =1ω1−α/2eiHspi2ω δhe−iHspi2ω (G.14)3See Appendix A for the derivation of the generator of conformal transformations116G.1. The Density Matrix and Scale InvarianceAs was the case for the expansion coefficients, the matrix, Γ(t), is afunction of tα−1/aα, where α is the scaling of the deviation. This allows oneto make the identification(1 + ω2t2)∂tΓ(t) =α− 1t1aα(1 + ω2t2)∂∂a−αΓ(t) α > 1=1 + ω2t2t log(ωt)1a∂∂a−1Γ(t) α = 1. (G.15)The differential equation for the density matrix then satisfies:[(1 + ω2t2)∂t + ω2t(~r · ∇r + ~r · ∇r + d) + iω r′2 − r22]ρ(~r, ~r′, t)=α− 1t1 + ω2t2aα∂∂a−αρ(~r, ~r′, t) +∑m,ni (En − Em) ρ˜s n,m(~r, ~r′, t)Γ˜m,n(t).(G.16)for α > 1, or:[(1 + ω2t2)∂t + ω2t(~r · ∇r + ~r · ∇r + d) + iω r′2 − r22]ρ(~r, ~r′, t)=1 + ω2t2t log(ωt)1a∂∂a−1ρ(~r, ~r′, t) +∑m,ni (En − Em) ρ˜s n,m(~r, ~r′, t)Γ˜m,n(t).(G.17)for α = 1.G.1.3 Dynamics of Local ObservablesConsider an observable, O, with scaling dimensions, ∆O:O =∫d3r O(~r)ψ†(~r)ψ(~r). (G.18)The dynamics for this operator can be written in terms of the density matrix:〈O〉(t) =∫d3r O(~r) ρ(~r, ~r, t). (G.19)117G.1. The Density Matrix and Scale InvarianceUsing Eq. G.17, or equivalently Eq. G.16, it is possible to obtain a differentialequation for 〈O〉(t), which is valid in the limit of long times. For example,if α = 1, one would obtain:[(1 + ω2t2)∂t −∆Oω2t]〈O〉(t) = 1 + ω2t2t log(ωt)1a∂∂a−1〈O〉(t)+ i∑n,m(En − Em)∫ddr O(~r)ρ˜sn,m(~r, ~r, t)Γ˜m,n(t).(G.20)For scale invariant systems, the long time solution to Eq. G.20 is:〈O〉(t) ≈(t′t)∆O〈O〉(t′), (G.21)for some t, t′  ω−1. This result is equivalent to the results in the co-moving reference frame, Eq. 4.19. Similarly, for systems with broken scaleinvariance, the result is:〈O〉(t) = t−∆OF(log(ωt)a), (G.22)again, for α = 1. Similar results for α ≥ 1 are straightforward to obtain.118Appendix HHydrodynamic andHeisenberg Equation ofMotion for Compressionaland Elliptic FlowIn this appendix we examine alternate approaches to understanding com-pressional and elliptic flow. The results here are consistent with the densitymatrix formalism discussed in Chapter 7, but do not highlight the facets ofscale and conformal symmetry.In both compressional and elliptic flow, the quantities of interest are themoments of inertia along a given direction:〈r2i 〉(t) =1N∫d3rr2i n(r, t), (H.1)where i = x, y, z, and n(r, t) is the density of the fluid. For compressionalflow, the motion is isotropic, while it is anisotropic for elliptic flow.We will study the dynamics of this operator using the hydrodynamic andHeisenberg equations of motion.H.1 Scaling Solution to the Hydrodynamic FlowThe hydrodynamic equations of motion are given by:∂tn(~r, t) +∇ · (~v(~r, t)n(~r, t)) = 0 (H.2)n(~r, t) [∂t + ~v(~r, t) · ∇] vi(~r, t) = −∂iP +∑j∂j(ησi,j + ζBσ′δi,j)− n(~r, t)∂iU(~r, t) (H.3)where ~v(~r, t) is the velocity field, P is the pressure, η and ζB are the shear119H.1. Scaling Solution to the Hydrodynamic Flowand bulk viscosity coefficients, respectfully, U(~r, t) is an external potential,and:σi,j = ∂jvi(~r, t) + ∂ivj(~r, t)− 23δi,j∇ · ~v(~r, t)σ′ = ∇ · ~v(~r, t). (H.4)Using Eqs. H.2 and H.3, one can show that:d〈r2i 〉(t)dt= 〈rivi〉(t)12d2〈r2i 〉(t)dt2=1N∫d3r P (t) +1N∫d3r(η σi,i + ζBσ′) (t)− 〈ri∂iU〉(t).(H.5)We will use this equation to understand the dynamics of a scale invariantFermi gas. We will assume that the density is initially of the Thomas-Fermiform, and that the dynamics can be captured by a time dependent rescaling:n(~r, t) =1bx(t)by(t)bz(t)16pi2(2µ− ω2xx2b2x(t)− ω2yy2b2y(t)− ω2zz2b2z(t))3/2vi(~r, t) =b˙i(t)bi(t)riµ = (6Nωxωyωz)1/3 . (H.6)where bi(t) are time dependent scaling factors that satisfy:bi(0) = 1 b′i(0) = 0. (H.7)The dynamics of the moment of inertia is solely encapsulated in the timedependent rescaling factors:〈r2i 〉(t) = b2i (t)〈r2i 〉(0) 〈r2i 〉(0) =µ4ω2i(H.8)For scale invariant systems initially in harmonic traps, it is possible tomake a number of simplifications. The first is that the bulk viscosity, ζB,vanishes for scale invariant systems [101]. Secondly, the Pressure can berelated to the energy density, E via:120H.1. Scaling Solution to the Hydrodynamic FlowP (t) =32E ∝ n2/3(~r, t)=1(bx(t)by(t)bz(t))2/3P (0) (H.9)Initially, the gas is in equilibrium in the harmonic trap. Therefore, Eq. H.5can be used to relate the initial pressure to the trap energy:3N∫d3r P (0) = ω2i 〈r2i 〉(0). (H.10)Combining all this information gives a set of coupled differential equationsfor the scaling factors, bi(t):b¨i(t) =ω2i(bx(t)by(t)bz(t))2/3 bi(t)−2 〈αs〉〈r2i 〉(0)(b˙i(t)bi(t)− 13(b˙x(t)bx(t)+b˙y(t)by(t)+b˙z(t)bz(t))),(H.11)where 〈αs〉 is the trap averaged shear viscosity coefficient:〈αs〉 = 1N∫d3r η. (H.12)H.1.1 Isotropic ExpansionFor isotropic expansion, Eq. H.11 reduces to:b¨(t) =ω2b3(t). (H.13)We have encountered this equation of motion before, and the solution forthe moment of inertia is given by:〈r2〉(t) = (1 + ω2t2)〈r2〉(0). (H.14)This is just the dynamics for a system either prepared in an conformal towerstate, or prepared in a thermal ensemble.H.1.2 Anisotropic ExpansionFor anisotropic expansion, it is necessary to evaluate Eq. H.11 in its entirety.This has been done in Fig. 7.1 for a variety of shear viscosity coefficients.For all cases, the long dynamics of the moment of inertia can be fit to:121H.2. Heisenberg Equation of motion〈r2i 〉(t) ≈ (v2t2 +At+B)〈r2i 〉(0). (H.15)This is consistent with the emergent conformal symmetry, and the densitymatrix analysis presented in Chapter 7.H.2 Heisenberg Equation of motionWe now analyse the expansion dynamics of the motion of the moment ofinertia under a scale invariant Hamiltonian, using the Heisenberg equationsof motion. First consider the equation of motion for the moment of inertia,〈r2〉(t). After the trap is released, the equation of motion for the momentof inertia can be found to be:d2dt2〈r2〉(t) = 4〈Hs〉. (H.16)The solution for this equation is simply:〈r2〉(t) =(1 +2〈Hs〉〈r2〉(0) t2)〈r2〉(0). (H.17)If the system is in equilibrium, it possible to relate the energy of the gasto its initial size. Consider the initial Hamiltonian:Hs +∫12 ∑i=x,y,zω2i r2iψ†(~r)ψ(~r). (H.18)The resulting equation of motion for the moment of inertia is then:d2dt2〈r2〉(t) = 4Hs − 12∑i=x,y,zω2i 〈r2i 〉(t) . (H.19)However, in its initial state, the gas is stationary, which gives the followingcondition:〈Hs〉 = 12∑i=x,y,zω2i 〈r2i 〉(0), (H.20)For isotropic traps this simply reduces to 〈Hs〉 = ω2〈r2〉(0)/2, or equiv-alently:〈r2〉(t) = (1 + ω2t2). (H.21)122H.2. Heisenberg Equation of motionThis result is a consequence of the Feynman-Hellmann theorem, and is againakin to the dynamics of a quantum system in a diagonal ensemble of con-formal tower states. For generic conditions, we can not relate the initialenergy to the trapping potential. In this case, the relative velocity is nolonger pinned to the trap frequency.123


Citation Scheme:


Citations by CSL (citeproc-js)

Usage Statistics



Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            async >
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:


Related Items