Unitary Dynamics of Strongly Interacting Bose Gases withthe Time-Dependent Variational Monte Carlo Method in Continuous Space

Giuseppe CarleoInstitute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland

Lorenzo CevolaniLaboratoire Charles Fabry, Institut d’Optique, CNRS, Univ Paris Sud 11,

2 avenue Augustin Fresnel, F-91127 Palaiseau cedex, France

Laurent Sanchez-PalenciaLaboratoire Charles Fabry, Institut d’Optique, CNRS, Univ Paris Sud 11, 2 avenue Augustin Fresnel,

F-91127 Palaiseau cedex, Franceand Centre de Physique Théorique, Ecole Polytechnique, CNRS, Univ Paris-Saclay,

F-91128 Palaiseau, France

Markus HolzmannLPMMC, UMR 5493 of CNRS, Université Grenoble Alpes, F-38042 Grenoble, France

and Institut Laue-Langevin, BP 156, F-38042 Grenoble Cedex 9, France(Received 19 December 2016; revised manuscript received 6 April 2017; published 8 August 2017)

We introduce the time-dependent variational Monte Carlo method for continuous-space Bose gases. Ourapproach is based on the systematic expansion of the many-body wave function in terms of multibodycorrelations and is essentially exact up to adaptive truncation. The method is benchmarked by comparisonto an exact Bethe ansatz or existing numerical results for the integrable Lieb-Liniger model. We first showthat the many-body wave function achieves high precision for ground-state properties, including energyand first-order as well as second-order correlation functions. Then, we study the out-of-equilibrium, unitarydynamics induced by a quantum quench in the interaction strength. Our time-dependent variationalMonte Carlo results are benchmarked by comparison to exact Bethe ansatz results available for a smallnumber of particles, and are also compared to quench action results available for noninteracting initialstates. Moreover, our approach allows us to study large particle numbers and general quench protocols,previously inaccessible beyond the mean-field level. Our results suggest that it is possible to find correlatedinitial states for which the long-term dynamics of local density fluctuations is close to the predictions of asimple Boltzmann ensemble.

DOI: 10.1103/PhysRevX.7.031026 Subject Areas: Atomic and Molecular Physics,Computational Physics,Quantum Physics

I. INTRODUCTION

The study of equilibration and thermalization propertiesof complex many-body systems is of fundamental interestfor many areas of physics and natural sciences [1]. Forsystems governed by classical physics, an exact solution ofNewton’s equations of motion is often numerically feasible,using, for instance, molecular-dynamics simulations. Forquantum systems, the mathematical structure of the time-dependent Schrödinger equation is instead fundamentally

more involved. Quantum Monte Carlo algorithms, the defacto tool for simulating quantum many-body systems atthermal equilibrium [2–4], cannot be directly used to studytime-dependent unitary dynamics. Out-of-equilibriumproperties are then often treated on the basis of approx-imations drastically simplifying the microscopic physics.Irreversibility is either enforced with an explicit breaking ofunitarity, e.g., within the quantum Boltzmann approach, orthe dynamics is reduced to mean-field description usingtime-dependent Hartree-Fock and Gross-Pitaevskiiapproaches. Although these approaches may qualitativelydescribe thermalization [5,6], their range of validity cannotbe assessed because genuine quantum correlations andentanglement are ignored.For specific systems, exact dynamical results can be

derived. This is the case for integrable 1D models, forwhich Bethe ansatz (BA) solutions exist [7]. However, also

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

PHYSICAL REVIEW X 7, 031026 (2017)

2160-3308=17=7(3)=031026(12) 031026-1 Published by the American Physical Society

in this case many open questions still persist. For example,the exact evaluation of correlation functions for out-of-equilibrium dynamics is at present an unsolved problem.As a result, despite important theoretical and experimentalprogress [8–15], a complete picture of thermalization (or itsabsence), e.g., based on general quench protocols [16], isstill missing.Numerical methods for strongly interacting systems face

important challenges as well. Numerical renormalizationgroup and density-matrix renormalization group (DMRG)approaches provide an essentially exact description ofarbitrary 1D lattice systems in and out of equilibrium[17–20], but they have less predictive power when appliedto continuous-space systems. On one hand, multiscaleextensions of the DMRG optimization scheme to the limitof continuous-space lattices [21] are, to date, limited torelatively small system sizes [22]. On the other hand,efficient ground-state optimization schemes for continuousquantum field matrix product states (cMPS) [23] have beenintroduced only very recently [24], and applications toquantum dynamics are still to be realized. A furtherformidable challenge is the efficient extension of theseapproaches to higher dimensions, which is a fundamentallyhard problem.Another class of methods for strongly interacting systems

is based on variational Monte Carlo (VMC) methods,combining highly entangled variational states with robuststochastic optimization schemes [25]. Such approaches havebeen successfully applied to the description of continuousquantum systems, in any dimension and not only in 1D[26,27]. More recently, out-of-equilibrium dynamics hasbecome accessible with the extension of these methods toreal-time unitary dynamics, within the time-dependent varia-tional Monte Carlo (tVMC) method [28,29]. So far, thetVMC approach has been developed for lattice systems withbosonic [28,29], spin [30–32], and fermionic [33] statistics,yielding a description of dynamical properties with anaccuracy often comparable with MPS-based approaches.In this paper, we extend the tVMC approach to access

dynamical properties of interacting quantum systems incontinuous space. Our approach is based on a systematicexpansion of the wave function in terms of few-body Jastrow correlation functions. Using the 1D Lieb-Liniger model as a test case, we first show that the inclusionof high-order correlations allows us to systematicallyapproach the exact BA ground-state energy. Our resultsimprove by orders of magnitude on previously publishedVMC and cMPS results, and are in line with the latest state-of-the-art developments in the field. We further computesingle-body and pair correlation functions, hardly acces-sible by current BA methods. We then calculate the timeevolution of the contact pair correlation function followinga quench in the interaction strength. For the noninteractinginitial state, we benchmark our results to exact BAcalculations available for a small number of bosons and

further compare to the quench action approach for largesystems approaching the thermodynamic limit. Finally, weapply our method to the study of general quenches fromarbitrary initial states, for which no exact results in thethermodynamic limit are currently available.

II. METHOD

A. Expansion of the many-body wave function

Consider a nonrelativistic quantum system of N identicalbosons in d dimensions, and governed by the first-quantization Hamiltonian

H ¼ −1

2

XNi¼1

∇2i þ

XNi¼1

v1ðxiÞ þ1

2

Xi≠j

v2ðxi; xjÞ; ð1Þ

where v1ðxÞ and v2ðx; yÞ are, respectively, a one-bodyexternal potential and a pairwise interparticle interaction[34]. Without loss of generality, a time-dependent N-bodystate can be written as ΦðX; tÞ ¼ exp ½UðX; tÞ�, whereX ¼ x1; x2;…; xN is the ensemble of particle positionsand U is a complex-valued function of the N-particlecoordinates, RN×d → C. Since the Hamiltonian Eq. (1)contains only two-body interactions, it is expected that anexpansion of U in terms of few-body Jastrow functionscontaining at most m-body terms rapidly convergestowards the exact solution. Truncating this expansion upto a certain order,M ≤ N, leads to the Bijl-Dingle-Jastrow-Feenberg expansion [35–38],

UðMÞðX; tÞ ¼Xi¼1

u1ðxi; tÞ þ1

2!

Xi≠j

u2ðxi; xj; tÞ

þ � � � 1

M!

Xi1≠i2≠…iM

uMðxi1 ; xi2 ;…; xiM ; tÞ; ð2Þ

where umðr; tÞ are functions of m particle coordinates,r ¼ xi1 ; xi2 ;…; xim , and of the time t. A global constraint onthe function UðX; tÞ is given by particle statistics. Inthe bosonic case, we demand that Uðx1; x2;…; xNÞ ¼Uðxσð1Þ; xσð2Þ;…; xσðNÞÞ, for all particle permutations σ.In general, the functions umðr; tÞ can have an arbitrarilycomplex dependence on the m particle coordinates, whichcan prove problematic for practical applications.Nonetheless, a simplified functional dependence can oftenbe imposed, resulting from the two-body character of theinteractions in the original Hamiltonian. For m ≥ 3,umðr; tÞ can be conveniently factorized in terms of generaltwo-particle vector and tensor functions followingRef. [26]. Details of this approach and the present imple-mentation are presented in Appendixes A and F.An appealing property of the many-body expansion

Eq. (2) is that it is able to describe intrinsically nonlocalcorrelations in space. For instance, the two-bodyu2ðxi; xj; tÞ, as well as any m-body function umðr; tÞ,

GIUSEPPE CARLEO et al. PHYS. REV. X 7, 031026 (2017)

031026-2

can be long range in the particle separation jxi − xjj. Thisnonlocal spatial structure allows for a correct description ofgapless phases, where a two-body expansion may alreadycapture all the universal features, in the sense of therenormalization group approach [39]. This is in contrastwith the MPS decomposition of the wave function, which isintrinsically local in space.

B. Time-dependent variational Monte Carlo method

The time evolution of the variational state Eq. (2) isentirely determined by the time dependence of the Jastrowfunctions umðr; tÞ. In order to establish optimal equationsof motion for the variational parameters, we note that thefunctional derivative of UðX; tÞ with respect to the varia-tional, complex-valued, Jastrow functions umðr; tÞ,

δUðX; tÞδumðr; tÞ

≡ ρmðrÞ; ð3Þ

yields the m-body density operators

ρmðrÞ ¼1

m!

Xi1≠i2≠���im

Yj

δðxij − rjÞ: ð4Þ

The expectation values of the operators ρm over thestate jΦðtÞi give the instantaneous m-body correlations.For instance, hρ2ðr1; r2Þit ¼

Pi<jhδðxi − r1Þδðxj − r2Þit,

where h� � �it ¼ hΦðtÞj…jΦðtÞi=hΦðtÞjΦðtÞi, is propor-tional to the two-point density-density correlation function.We can then express the time derivative of the truncated

variational state UðMÞðX; tÞ using the functional deriva-tives, Eq. (3), as a sum of the few-body density operators upto the truncation M; i.e.,

∂tUðMÞðX; tÞ ¼XMm¼1

ZdrρmðrÞ∂tumðr; tÞ: ð5Þ

The exact wave function satisfies the Schrödingerequation i∂tUðX; tÞ ¼ ElocðX; tÞ, where ElocðX; tÞ ¼ðhXjHjΦðtÞi=hXjΦðtÞiÞ is the so-called local energy.The optimal time evolution of the truncated Bijl-Dingle-Jastrow-Feenberg expansion Eq. (2) can be derivedimposing the Dirac-Frenkel time-dependent variationalprinciple [40,41]. In geometrical terms, this amountsto minimizing the Hilbert-space norm of the residuals

RðMÞðX; tÞ≡ ∥i∂tUðMÞðX; tÞ − EðMÞloc ðX; tÞ∥, thus yielding

a variational many-body state as close as possible to theexact one [42]. The minimization can be performedexplicitly and yields a closed set of integro-differentialequation for the Jastrow functions umðr; tÞ:

XMp¼1

Zdr0

δhρmðrÞitδupðr0; tÞ

∂tupðr0; tÞ ¼ −iδhHit

δumðr; tÞ: ð6Þ

In practice, these equations are numerically solved for thetime derivatives ∂tupðr0; tÞ at each time step t. The expect-ation values taken over the time-dependent state h·it, whichenter Eq. (6), are found via a stochastic sampling of theprobability distribution ΠðX; tÞ ¼ jΦðX; tÞj2. This is effi-ciently achieved by means of the Metropolis-Hastingsalgorithm, as per conventional Monte Carlo schemes(see Appendix B for details). It then yields the full timeevolution of the truncated Bijl-Dingle-Jastrow-Feenbergstate Eq. (2) after time integration.The tVMC approach as formulated here provides, in

principle, an exact description of the real-time dynamics ofthe N-body system. The essential approximation lies,however, in the truncation of the Bijl-Dingle-Jastrow-Feenberg expansion to the M most relevant terms. Inpractical applications, the M ¼ 2 or M ¼ 3 truncation isoften sufficient. Systematic improvement beyondM ¼ 3 ispossible [26], but may require substantial computationaleffort.

III. LIEB-LINIGER MODEL

As a first application of the continuous-space tVMCapproach, we consider the Lieb-Liniger model [43]. On onehand, some exact results and numerical data are available,allowing us to benchmark the Jastrow expansion and thetVMC approach. On the other hand, several aspects of theout-of-equilibrium dynamics of this model are unknown,which we compute here for the first time using the tVMCapproach.The Lieb-Liniger model describes N interacting bosons

in one dimension with contact interactions. It correspondsto the Hamiltonian Eq. (1) with v1ðxÞ ¼ 0 and v2ðx; yÞ ¼gδðx − yÞ, where g is the coupling constant. Here, weconsider periodic boundary conditions over a ring of lengthL and the particle density n ¼ N=L. The density depend-ence is as usual expressed in terms of the dimensionlessparameter γ ¼ mg=ℏ2n. The Lieb-Liniger model is theprototypal model of continuous one-dimensional stronglycorrelated gas exactly solvable by the Bethe ansatz [43].This model is experimentally realized in ultracold atomicgases strongly confined in one-dimensional optical traps,and several studies on out-of-equilibrium physics havealready been realized [14,44–48].As a result of translation invariance, we have

u1ðx; tÞ ¼ const, and the first nontrivial term in themany-body expansion Eq. (2) is the two-body, translationinvariant, function u2ðjx − yj; tÞ. To compute the functionalderivatives of the many-body wave function, we proceedwith a projection of the continuous Jastrow fields umðr; tÞonto a finite-basis set. Here, we find it convenient torepresent both the field Hamiltonian Eq. (1) and the Jastrowfunctions on a uniform mesh of spacing a, leading to nvar ¼½L=ð2aÞ� variational parameters for the two-body Jastrowterm. In the following, our results are extrapolated to thecontinuous limit, corresponding to a → 0. The finite-basis

UNITARY DYNAMICS OF STRONGLY-INTERACTING BOSE … PHYS. REV. X 7, 031026 (2017)

031026-3

projection aswell as the numerical time integration of Eq. (6)are detailed in Appendix C and benchmarked against exactdiagonalization results for N ¼ 3 in Appendix E.

A. Ground-state properties

To assess the quality of truncated Bijl-Dingle-Jastrow-Feenberg expansions for the Lieb-Liniger model, we startour analysis considering ground-state properties. An exactsolution can be found from the Bethe ansatz and givesaccess to exact ground-state energies and local properties[43]. Other nonlocal properties are substantially moredifficult to extract from the BA solution, and unbiasedresults for ground-state correlation functions have not beenreported so far. In order to determine the best possiblevariational description of the ground state within our many-body expansion, different strategies are possible. A firstpossibility is to consider the imaginary-time evolutionjΨðτÞi ¼ e−τHjΨ0i, which systematically converges tothe exact ground state in the limit τ ≫ Δ1, where Δ1 ¼E1 − E0 is the gap with the first excited state on a finitesystem and provided that the trial state Ψ0 is nonorthogonalto the exact ground state. Imaginary-time evolution in thevariational subspace can be implemented considering theformal substitution t → −iτ in the tVMC equations[Eq. (6)]. The resulting equations are equivalent to thestochastic reconfiguration approach [49]. However, directminimization of the variational energy can be significantlymore efficient, in particular, for systems becoming gaplessin the thermodynamic limit, where Δ1 ∼ polyð1=NÞ. Giventhe gapless nature of the Lieb-Liniger model, we find itcomputationally more efficient to adopt a Newton methodto minimize the energy variance [50].For the ground state, the many-body expansion truncated

atM ¼ 2 is exact not only in the noninteracting limit γ ¼ 0but also in the Tonks-Girardeau limit γ → ∞. In thisfermionized limit, the wave function can be written as

the modulus of a Vandermonde determinant of planewaves, corresponding to the two-body Jastrow functionu2ðrÞ ¼ log ðsin rπ=LÞ [51]. To assess the overall quality ofpair wave functions for ground-state properties, we startcomparing the variational ground-state energies E obtainedfor M ¼ 2 with the exact BA result. In Fig. 1(a), we showthe relative error ΔE=E as a function of the interactionstrength γ. We find that the relative error is lower than 10−4

for all values of γ and the accuracy of our two-body Jastrowfunction is superior to previously published variationalresults based on either cMPS [23] or VMC [52] methods[see Fig. 1(a) for a quantitative comparison]. Notice that theimprovement with respect to previous VMC results is dueto the larger variational freedom of our u2ðrÞ function,which is not restricted to any specific functional form asdone in Ref. [52].Even though the accuracy reached by the two-body

Jastrow function may already be sufficient for mostpractical purposes for all values of γ, we also considerhigher-order terms with M ¼ 3; see Fig. (2)(a). Theintroduction of the third-order term yields a sizableimprovement such that the maximum error is about 3orders of magnitude smaller than original cMPS results[23], and features a similar accuracy of recently reportedcMPS results [24]. Overall, our approach reaches a pre-cision on a continuous-space system, which is comparableto state-of-the-art MPS or DMRG results for gaplesssystems on a lattice [53].Finally, to further assess the quality of our ground-state

ansatz beyond the total energy, we also study nonlocalproperties of the ground-state wave function, which arenot accessible by existing exact BA methods. In Figs. 1(b)and 1(c), we show, respectively, our results for theoff-diagonal part of the one-body density matrix,g1ðrÞ ∝ hΨ†ðrÞΨð0Þi, and for the pair correlation function,g2ðrÞ ∝ hΨ†ðrÞΨðrÞΨ†ð0ÞΨð0Þi, whereΨðrÞ is the bosonic

(a) (b) (c)

FIG. 1. Ground-state properties of the Lieb-Liniger model as obtained from different variational approaches: (a) relative accuracy ofthe ground-state energy, (b) one-body density matrix, (c) density-density pair correlation functions. Uð2Þ and Uð3Þ denote results for thetwo- and three-body expansion (present work), Uð2Þ

AG is the parametrized two-body Jastrow state of Ref. [52], cMPS1 results are fromRef. [23], and cMPS2 are those very recently reported in Ref. [24]. Distances r are in units of the inverse density 1=n. Our variationalresults are obtained for N ¼ 100 particles. Finite-size corrections on local quantities are negligible, and very mildly affect the reportedlarge-distance correlations. Overall statistical errors are of the order of symbol sizes for (a) and line widths for (b) and (c).

GIUSEPPE CARLEO et al. PHYS. REV. X 7, 031026 (2017)

031026-4

field operator. We find an overall excellent agreement withthe results that have been obtained with cMPS in Ref. [24],except for some small deviations at large values of r, whichwe attribute to residual finite-size effects in our approach.Wefind that the addition of the three-body terms does notsignificantly change correlation functions. Already at thetwo-body level, the present results are statistically indistin-guishable from exact results obtained using our implemen-tation [54] of the worm algorithm [55] and for the samesystem (not shown).

B. Quench dynamics

Having assessed the quality of the ansatz for local andnonlocal ground-state properties, we now turn to the studyof the out-of-equilibrium properties of the Lieb-Linigermodel. We focus on the description of the unitary dynamicsinduced by a global quantum quench of the interactionstrength, from an initial value γi to a final value γf. ExactBA results are available only in the case of a noninteractinginitial state (γi ¼ 0). Even in this case, the dynamical BAequations can be exactly solved only for a modest numberof particles with further truncation in the number of energyeigenmodes [56,57], N ≲ 10, since the complexity of theBA solution increases exponentially with the number ofparticles. Simplifications in the thermodynamic limit areexploited by the quench action [58], and have recently beenapplied to quantum quenches starting from a noninteractinginitial state [57]. In the following, we first compare ourtVMC results to these existing results, and then present newresults for quenches following a nonvanishing initialinteraction strength.To assess the quality of the time-dependent wave

function, we compare our results for the evolution of localdensity-density correlations g2ð0; tÞ with the truncated BAresults obtained in Refs. [56,59] for a small number ofparticles, N ≃ 6. Appendix E also provides further

validation of our method accessing g2ðr; tÞ at nonvanishingdistances. The comparison shown in Fig. 2(a) shows anoverall good agreement. The tVMC and BA results areindistinguishable for weak interactions (γf ¼ 1) at the scaleof the figure; similarly, for this interaction strength, tVMCcalculations based on Uð2Þ and Uð3Þ agree with each otherup to possible dephasing effects not visible within our noiselevel. For larger interactions, we notice systematic but smalldifferences between BA and tVMC with M ¼ 2 or M ¼ 3results. These differences amount to a small increase in theamplitude of the oscillations. This effect tends to increasewith the interaction strength, being hardly visible for γ ¼ 1and more pronounced for γ ¼ 10. However, these oscil-lations result at large times from the discrete mode structuredue to the very small number of particles. They vanish inthe physical thermodynamic limit. In turn, the comparisonat small particle numbers indicates an accuracy betterthan a few percent for time-averaged quantities in theasymptotic large time limit up to γ ¼ 10, with results atM ¼ 3 systematically improving on the M ¼ 2 case.Concerning small and intermediate time scales, we donot observe systematic deviations between the tVMCresults and the BA solution. In particular, the relaxationtimes are remarkably well captured by the tVMC approach.On the basis of this comparison and of the comparison forthree particles with exact diagonalization results presentedin Appendix E, we conclude that tVMC approach allowsaccurate quantitative studies of both the relaxation andequilibration dynamics. This careful benchmarking nowallows us to confidently apply the tVMC approach regimesthat are inaccessible to exact BA, namely large but finite Nand long times, as well as the case of nonvanishing initialinteractions.Let us consider relaxation of density correlations for a

large number of particles, close to the thermodynamic limit(here, we useN ¼ 100). As we show in Fig. 2(b), we notice

(a) (b)

FIG. 2. Time-dependent expectation value of local two-body correlations after a quantum quench from a noninteracting state, γi ¼ 0,to γf . (a) tVMC results are compared with BA results obtained for a small number of particles [56,59]. The correlation function isrescaled to have g2ð0; 0Þ ¼ 1. (b) tVMC results for N ¼ 100 particles and γf ¼ 1, 2, 4, 8 (from top to bottom) compared to the quenchaction (QA) predictions from Ref. [57] (dashed lines), to the Boltzmann thermal averages at the effective temperature T⋆ (dot-dashedlines), and to the GGE thermal averages prediction (rightmost dashed lines). Statistical error bars on tVMC data are of the order of theline widths.

UNITARY DYNAMICS OF STRONGLY-INTERACTING BOSE … PHYS. REV. X 7, 031026 (2017)

031026-5

that the amplitudes of the large-time oscillations, attributedto the discrete mode spectrum, are now drastically sup-pressed compared to the quenches with N ¼ 6. After aninitial relaxation phase, the quantity g2ð0; tÞ approaches astationary value. Comparing our curves with those obtainedwith the quench action method, we find a qualitatively goodagreement, albeit a general tendency to underestimate thequench action predictions is observed.We now turn to quenches from interacting initial states

(γi ≠ 0) to different interacting final states for which noresults have been obtained by means of exact BA nor thesimplified quench action method so far. In Fig. 3, we showthe asymptotic equilibrium values obtained with our tVMCapproach for quantum quenches from γi ¼ 1 (left-handpanel) and γi ¼ 4 (right-hand panel) to several values ofγf. Since, by the variational theorem, the ground state ofHi

gives an upper bound for the ground-state energy ofHf, thesystem is pushed into a linear combination of excited states ofthe final Hamiltonian. For systems able to thermalize to theBoltzmann ensemble (BE), relaxation to a stationary statedescribed by the density matrix ρT⋆ ¼ e−Hf=T⋆

, at aneffective temperature T⋆, would occur. Comparing the sta-tionary value g2ð0Þ of our tVMCcalculations at long times tothe thermal values of the pair correlation functions gT

⋆2 ð0Þ, a

necessary condition for simple Boltzmann thermalization isgiven by g2 ¼ gT

⋆2 . The effective temperature T⋆ is deter-

mined by imposing the energy expectation value of the finalHamiltonian Hf in the ground state Φ0ðγiÞ of the initialHamiltonian, hHfiT⋆ ¼ hΦ0ðγiÞjHfjΦ0ðγiÞi. Here, the ther-mal expectationvalue, hHfiT⋆ at the equilibrium temperatureT⋆ is computed from the Yang-Yang BA equations [60]. Thequantity hHfiT⋆ then depends on a single parameter T⋆,which is fitted to match the value of hΦ0ðγiÞjHfjΦ0ðγiÞi.

As we show in Fig. 2(b), Boltzmann thermalizationcertainly does not occur in the case for the Lieb-Linigermodel when quenching from a noninteracting state, γi ¼ 0,where we find g2 ≠ gT

⋆2 . This can be understood in terms of

the existence of dynamically conserved charges (beyondenergy and density conservation), which can yield anequilibrium value substantially different from the BEprediction. In particular, it is widely believed that thegeneralized Gibbs ensemble (GGE) is the correct thermaldistribution approached after the quench [61,62]. Severalconstructive approaches for the GGE have been putforward in past years [8,9,57], and the quench actionpredictions reported in Fig. 2(b) converge to the GGEpredictions for the thermal values. In Fig. 2(b), we alsoshow the thermal GGE values gGGE2 (rightmost dashedlines), and note that our results are much closer to the GGEpredictions than the simple BE. Deviations from theasymptotic GGE results are observed at large γf, a regimein which the accuracy of our approach is still sufficient toresolve the difference between the BE and the long-termequilibration value.For correlated initial ground states γi ≠ 0, GGE predic-

tions are fundamentally harder to obtain than for thenoninteracting initial states, and the BE is the onlyreference thermal distribution we can compare with at thisstage. From our results we observe that the differencebetween g2 and the simple BE prediction gT

⋆2 is quantita-

tively reduced; see Fig. 3. In particular, for γi ¼ 4, thestationary values g2 are quantitatively close to the onespredicted by the Boltzmann thermal distribution at theeffective temperature T⋆. Even though this quantitativeagreement is likely to be coincidental, the regimes ofparameter quenches we study here provide guidance forfuture experimental studies. In particular, it will be of greatinterest to understand whether a crossover from a stronglynon-Boltzmann to a close-to-Boltzmann thermal behaviormight occur as a function of the initial interaction strengthalso for other local observables.

IV. CONCLUSIONS

In this paper, we introduce a novel approach to thedynamics of strongly correlated quantum systems incontinuous space. Our method is based on a correlatedmany-body wave function systematically expanded interms of reduced m-body Jastrow functions. The unitarydynamics in the subspace of these correlated states isrealized using the time-dependent variational Monte Carlomethod. We demonstrate the possibility of performingcalculations up to the three-body level, m ≤ 3, for theLieb-Liniger model, for both static and dynamical proper-ties. The improvement from m ¼ 2 to m ¼ 3 provides aninternal criterion to judge the validity of our resultswhenever exact results are unavailable. Benchmarkingthe tVMC approach with exact or numerical approaches

(a) (b)

FIG. 3. Time-dependent expectation value of local two-bodycorrelations after a quantum quench from the interacting groundstate at γi ¼ 1 (a) and γi ¼ 4 (b). Long-term dynamical averages(red continuous lines) are compared to thermal averages at theeffective temperature set by energy conservation (black dashedlines). Uncertainties on the thermal averages are of the order ofthe line width, and are larger for small values of γf .

GIUSEPPE CARLEO et al. PHYS. REV. X 7, 031026 (2017)

031026-6

whenever available, we find a very good agreement withexisting results. For static properties, our approach is at thelevel of state-of-the-art MPS techniques in lattice systemsand of the latest cMPS results for interacting gases. Fordynamical properties, we investigate, for the first time,general interaction quenches, which are at the momentunaccessible to Bethe ansatz approaches. Since the generalstructure of our t-VMC method does not depend on thedimensionality of the system, it can be directly applied tobosonic systems in higher dimensions with a polynomialincrease in computational cost. The methods we presenthere therefore pave the way to accurate out-of-equilibriumdynamics of two- and three-dimensional quantum gasesand fluids beyond mean-field approximations.

ACKNOWLEDGMENTS

We acknowledge discussions with M. Dolfi, J. DeNardis, M. Fagotti, T. Osborne, and M. Troyer. We thankM. Ganahl for providing the cMPS results in Fig. 1, J. Zillfor the Bethe ansatz results in Fig. 2(a), and J. De Nardis forthe quench actions results in Fig. 2(b). This research wassupported by the Marie Curie IEF Program (FP7/2007–2013 Grant Agreement No. 327143), the EuropeanResearch Council Starting Grant “ALoGlaDis” (FP7/2007-2013 Grant Agreement No. 256294) and AdvancedGrant “SIMCOFE” (FP7/2007-2013 Grant AgreementNo. 290464), the European Commission FET-ProactiveQUIC (H2020 Grant No. 641122), the French ANR-16-CE30-0023-03 (THERMOLOC), and the Swiss NationalScience Foundation through NCCR QSIT. It was per-formed using HPC resources from GENCI-CCRT/CINES (Grant No. c2015056853).

APPENDIX A: FUNCTIONAL STRUCTUREOF MANY-BODY TERMS

The local residuals RðMÞðX; tÞ ¼ i∂tUðMÞðX; tÞ −EðMÞloc ðX; tÞ are vanishing if the Schrödinger equation is

exactly satisfied by the many-body wave function truncated

at some order M. The local energy EðMÞloc ðX; tÞ, however,

may contain effective interaction terms involving a numberof bodies larger thanM, which leads to a systematic error inthe truncation. However, the structure of these additionalterms stemming from the local energy can be systemati-cally used to deduce the functional structure of the higher-order terms. For example, the one-body truncated localenergy reads, for one-dimensional particles,

Eð1ÞlocðX; tÞ ¼ −

1

2

Xi

f½∂xiu1ðxi; tÞ�2 þ ∂2xiu1ðxi; tÞg

þXi

v1ðxiÞ þ1

2

XNi≠j

v2ðxi; xjÞ; ðA1Þ

and contains a two-body term that cannot be accounted forexactly by u1. Introduction of a symmetric two-bodyJastrow factor u2ðxi; xj; tÞ then leads to

Eð2ÞlocðX; tÞ ¼ Eð1Þ

locðX; tÞþ

−1

2

Xi≠j

½∂xiu1ðxi; tÞ∂xiu2ðxi; xj; tÞ�þ

−1

2

Xi≠j

∂2xiu2ðxi; xj; tÞþ

−1

2

Xi≠j

Xk≠i

∂xiu2ðxi; xj; tÞ∂xiu2ðxi; xk; tÞ:

ðA2Þ

In the latter expression, one recognizes an effective two-body term which can be accounted for by u2 and anadditional three-body term in the form of a product of two-body functions. The functional form of the three-bodyJastrow term can therefore be deduced from this additionalterm and formed accordingly:

u3ðxi; xj; xk; tÞ ¼ u3ðxi; xj; tÞu3ðxj; xk; tÞ; ðA3Þ

with two-body functions u3ðxi; xj; tÞ containing new varia-tional parameters to be determined. Upon pursuing thisapproach, the expansion can be systematically pushed tohigher orders and the functional structure of the higher-order functions inferred. The same constructive approachwe discuss here is also valid for the Schrödinger equation inimaginary time, ∂τUðX; τÞ ¼ −Elocðx; τÞ, and has beensuccessfully used to infer the functional structure forground-state properties [63].

APPENDIX B: MONTE CARLO SAMPLING

In order to solve the tVMC equations of motion, Eq. (6),expectation values of some given operator O need to becomputed over the many-body wave functionΦðX; tÞ. Thisis achieved by means of Monte Carlo sampling of theprobability distribution ΠðXÞ ¼ jΦðXÞj2 (in the following,we omit explicit reference to the time t, assuming that allexpectation values are taken over the wave function at agiven fixed time). An efficient way of sampling the givenprobability distribution is to devise a Markov chain ofconfigurations Xð1Þ;Xð2Þ;…;XðNc − 1Þ;XðNcÞ, whichare distribute according to ΠðXÞ. Quantum expectationvalues of a given operator O can then be obtained asstatistical expectation values over the Markov chain as

hΦjOjΦihΦjΦi ≃ 1

Nc

XNc

i¼1

Oloc(XðiÞ); ðB1Þ

UNITARY DYNAMICS OF STRONGLY-INTERACTING BOSE … PHYS. REV. X 7, 031026 (2017)

031026-7

where OlocðXÞ ¼ ðhXjOjΦi=hXjΦiÞ, and the equivalenceis achieved in the limit Nc → ∞.The Markov chain is realized by the Metropolis-Hastings

algorithm. Given the current state of the Markov chainXðiÞ, a configuration X0 is generated according to a giventransition probability T(XðiÞ → X0). The proposed con-figuration is then accepted [i.e., Xðiþ 1Þ ¼ X0] withprobability

A(XðiÞ → X0) ¼ min

�1;

ΠðX0ÞΠ(XðiÞ)

T(X0 → XðiÞ)T(XðiÞ → X0)

�;

ðB2Þ

otherwise, it is rejected and Xðiþ 1Þ ¼ XðiÞ.In the present tVMC calculations, we use simple

transition probabilities in which a single particle is dis-placed, while leaving all the other particle positionsunchanged. In particular, a particle index p is chosen withuniform probability 1=N and the position of particle p isthen displaced according to x0p ¼ xp þ ηΔ, where ηΔ is arandom number uniformly distributed in ½−ðΔ=2Þ; ðΔ=2Þ�.The amplitude Δ is an adjustable parameter, and it cantypically be chosen to be of the order of the averageinterparticle distance. With this choice, the transitionprobability is simply

Tðxp → x0pÞ ¼1

NΔ; ðB3Þ

and the acceptance probability is therefore given by themereratio of the probability distributions, ΠðX0Þ=Π(XðiÞ).

APPENDIX C: FINITE-BASIS PROJECTION

The numerical solution of the equations of motion[Eq. (6)] requires the projection of the Jastrow fieldsumðr; tÞ onto a finite basis. The continuous variable r isreduced to a finite set of P values for each order m,ðm; rÞ → ðr1;m; r2;m;…; rP;mÞ. We introduce a super-indexK spanning all possible values of the discrete variables ri;m.The complete set of variational parameters resulting fromthe projection on the finite basis can then be written asuKðtÞ and the associated functional derivatives read ρKðtÞ.The integro-differential equations [Eq. (6)] are then

brought to the algebraic form,

XK0

SK;K0 _uK0 ðtÞ ¼ −ihElocðtÞρKðtÞi; ðC1Þ

where we introduce the Hermitian correlation matrix:

SK;K0 ðtÞ ¼ ∂hρKðtÞit∂uK0

¼ hρKðtÞρK0 ðtÞit − hρKðtÞithρK0 ðtÞit: ðC2Þ

At a given time, all the expectations values in Eq. (C1) canbe explicitly computed with the stochastic approachdescribed in Appendix B. We are therefore left with alinear system in the nvar unknowns _uKðtÞ, which needs tobe solved at each time t.In the presence of a large number of variational parameters

nvar, the solution of the linear system can be achieved usingiterative solvers, e.g., conjugate gradient methods, which donot need to explicitly form the matrix S. Calling niter thenumber of iterations needed to obtain a solution for the linearsystem, the computational cost to solve Eq. (C1) is OðM ×nvar × niterÞ as opposed to the OðM × n2varÞ operationsneeded by a standard solver in which the matrix S is formedexplicitly. In the present work, we resort to the minimalresidual method, which is a variant of the Lanczos method,working in the Krylov subspace spanned by the repeatedaction of the matrix S onto an initial vector. In typicalapplications we obtain that niter ≪ nvar, and several thou-sands of variational parameters can be efficiently treated.This is of fundamental importance when the continuous(infinite-basis) limit must be taken, for which nvar → ∞.Once the unknowns _uKðtÞ are determined, we can solve

numerically the first-order differential equations given inEq. (6) for given initial conditions uKð0Þ. In the presentwork, we adopt an adaptive fourth-order Runge-Kuttascheme for the integration of the differential equations.

APPENDIX D: LATTICE REGULARIZATIONFOR THE LIEB-LINIGER MODEL

We consider a general wave function Ψðx1; x2;…; xNÞfor N one-dimensional particles, governed by the Lieb-Liniger Hamiltonian. By means of the variationalMonte Carlo method, we want to sample jΦðXÞj2. Thisis achieved via a lattice regularization, i.e.,

ZdXjΦðXÞj2 ≃ X

l1;l2…lN

jΦðl1; l2;…; lNÞj2;

where li ¼ f0; a;…; L − ag are discrete particle positions,athe lattice spacing, L the box size and Ns ¼ 1þ L=a thenumber of lattice sites. As a discretizedHamiltonian, we take

HaΦðl1…lNÞ ¼ −ℏ2

2ma2Xi

�4

3½Φðl1…li − a;…; lNÞ þΦðl1…li þ a;…; lNÞ�

−1

12½Φðl1…li − 2a;…; lNÞ þΦðl1…li þ 2a;…; lNÞ� þ−

5

2Φðl1…li;…; lNÞ

�þΦðl1…lNÞ

ga

Xi<j

δðli; ljÞ:

GIUSEPPE CARLEO et al. PHYS. REV. X 7, 031026 (2017)

031026-8

The first terms constitute just the fourth-order approximationof theLaplacianvia central finite differences,whereas the lastterm corresponds to the two-body delta interaction part.With this discretization, a two-body Jastrow factor reads

u2ðxij; tÞ ¼ u2ðli; lj; tÞ;

where u2ða; b; tÞ is a time-dependent matrix of sizeNs × Ns, which, in 1D and in the presence of translationalsymmetry, depends only on distða − bÞ; i.e., it has Ns=2variational parameters.

APPENDIX E: BENCHMARK STUDYFOR N = 3 ON A LATTICE

Here, we use exact diagonalization of a Hamiltonianwithin a given finite basis for a quantitative test of ourmethod. Exact diagonalization is limited to very smallsystems on a finite basis, and we choose a system containingN ¼ 3 particles on L=a ¼ 40 lattice sites as a simple, buthighly nontrivial reference. In contrast to our comparisonwith BA methods, all observables can be accessed by exactdiagonalization, and we use the off-diagonal one-bodydensity matrix g1ðr; tÞ and the pair correlation functiong2ðr; tÞ at different distances r ¼ jx1 − x2j of two particlesafter time t, where the system is quenched from the non-interacting initial state, γi ¼ 0, to a final interaction, γf > 0,to provide a benchmark on a more general observable.We first benchmark the influence of the time-step lattice

size discretization Δt error on g2. From Fig. 4(a), we seethat the tVMC dynamics is stable over a long time and thetime step error can be brought to convergence. Further, wesee that for final interaction γf ¼ 4, the truncation at thetwo-body level Uð2Þ introduces only a small systematicerror, mainly a dephasing effect, which is almost negligibleat the scale of the figure. Because of the stochastic noise ofthe Monte Carlo integration, tVMC introduces additionalhigh-frequency oscillations which are, however, well

separable from the deterministic propagation. The ampli-tude of these high-frequency oscillations also quantifies thepurely statistical error of our data.Whereas exact diagonalization is limited to rather small

basis sets, we can access a much larger basis within tVMCmethod. In Fig. 4(b), we show results within the Uð2Þapproximation with L=a ¼ 80 and L=a ¼ 160 with timediscretization Δt ∼ ðL=aÞ2. We see that the basis settruncation in general introduces a dephasing at largeenough time.The systematic error of Uð2Þ increases towards quenches

to stronger interaction strength and becomes more visiblefor γf ¼ 8; see Fig. 5(a). However, even in this case, themost important effect remains to be a simple dephasing; asmall shift of averaged quantities is probable, but difficultto quantify precisely. Introducing general three-bodyJastrow fields Uð3Þ, described in detail in Appendix F,the systematic error for N ¼ 3 can be fully eliminated.In Fig. 5(b), we also benchmark the possibility of

calculating the off-diagonal one-body density matrix aftera quench. Here, the systematic error of Uð2Þ is morepronounced at smaller γf in the long range and long timeregime.From our study of the three-particle problem, we con-

clude that truncation of the many-body wave function at thelevel of U2 may provide an excellent approximation forg2ðr; tÞ for quenches involving not too strong interactionstrengths, γ ≲ 5. The systematic error due to the Uð2Þtruncation is mainly a dephasing at large times involvingsmall relative errors of time-averaged quantities. Similarsystematic dephasing errors will occur for too large timediscretization or basis set truncation.Since our method provides a parametrization of the full

wave function for a given time, many different observablescan be evaluated via usual Monte Carlo methods. However,the quality of different observables may vary and dependmore sensitively on the inclusion of higher-order

(a) (b)

FIG. 4. Time-dependent expectation value of the two-body correlations after a quantum quench from a noninteracting state, γi ¼ 0, toγf ¼ 4 at three different distances, jx1 − x2j ¼ 0, L=10, L=4. Here, the system is on a lattice with L=a ¼ 40 lattice sites. The full line isobtained by exact diagonalization (ED) of the Hamiltonian; the other curves are from tVMC results truncated at the level of Uð2Þ. In (a),we show the convergence with different time-step discretization. In (b), we show the approach to the continuum for tVMC simulationsusing Uð2Þ for discretizations L=a ¼ 40, 80, and 160.

UNITARY DYNAMICS OF STRONGLY-INTERACTING BOSE … PHYS. REV. X 7, 031026 (2017)

031026-9

correlations UðnÞ with n > 2, as in the case of the single-body density matrix. Although these higher-order terms arecomputationally expensive, the scaling is not exponential,and we explicitly show that calculations with n ¼ 3 arefeasible. We note that the computational complexity maybe further reduced by functional forms adapted to theproblem [26].

APPENDIX F: GENERAL STRUCTURE OF Uð3Þ

For a general time-dependent wave function, we have togo beyond the usual ground-state structure of the three-body Jastrow term given in Eq. (A3). Here, we providedetails of our three-body term in a general form beyond thepresent application in one dimension.Introducing M basis functions, baðrÞ, a ¼ 1;…;M, we

can introduce many-body vectors [26], Baiα ¼

Pjr

αijb

aðrijÞ,where α ¼ 1;…; D indicates the summation over direc-tions and i ¼ 1;…; N. The variational parameters of ageneral three-body structure can then be written in terms ofa matrix wab, such that

Xi≠j≠k

u3ðr1; r2; r3Þ ¼Xab

wabWab; Wab ¼Xiα

BaiαB

biα:

ðF1Þ

In order to reduce the variational parameters (∼M2), wemay perform a singular value decomposition of the matrixwab to reduce the effective degrees of freedom.

[1] Nature Physics Insight on Non-equilibrium physics, Nat.Phys. 11, 103 (2015).

[2] D. Ceperley, Path Integrals in the Theory of CondensedHelium, Rev. Mod. Phys. 67, 279 (1995).

[3] L. Pollet, Recent Developments in Quantum Monte CarloSimulations with Applications for Cold Gases, Rep. Prog.Phys. 75, 094501 (2012).

[4] T. Plisson, B. Allard, M. Holzmann, G. Salomon, A. Aspect,P. Bouyer, and T. Bourdel, Coherence Properties of a Two-Dimensional Trapped Bose Gas around the SuperfluidTransition, Phys. Rev. A 84, 061606 (2011).

[5] N. G. Berloff and B. V. Svistunov, Scenario of StronglyNonequilibrated Bose-Einstein Condensation, Phys. Rev. A66, 013603 (2002).

[6] N. Navon, A. L. Gaunt, R. P. Smith, and Z. Hadzibabic,Emergence of a Turbulent Cascade in a Quantum Gas,Nature (London) 539, 72 (2016).

[7] M. Gaudin, La Fonction d’Onde de Bethe (Masson, Paris,1983).

[8] J.-S. Caux and R. M. Konik, Constructing the GeneralizedGibbs Ensemble after a Quantum Quench, Phys. Rev. Lett.109, 175301 (2012).

[9] J.-S. Caux and F. H. L. Essler, Time Evolution of LocalObservables After Quenching to an Integrable Model, Phys.Rev. Lett. 110, 257203 (2013).

[10] M. Collura, S. Sotiriadis, and P. Calabrese, Equilibration ofa Tonks-Girardeau Gas Following a Trap Release, Phys.Rev. Lett. 110, 245301 (2013).

[11] S. Trotzky, Y-A. Chen, A. Flesch, I. P. McCulloch, U.Schollwöck, J. Eisert, and I. Bloch, Probing the RelaxationTowards Equilibrium in an Isolated Strongly CorrelatedOne-Dimensional Bose Gas, Nat. Phys. 8, 325 (2012).

[12] T. Langen, R. Geiger, M. Kuhnert, B. Rauer, and J.Schmiedmayer, Local Emergence of Thermal Correlationsin an Isolated Quantum Many-Body System, Nat. Phys. 9,640 (2013).

[13] D. Greif, G. Jotzu, M. Messer, R. Desbuquois, and T.Esslinger, Formation and Dynamics of AntiferromagneticCorrelations in Tunable Optical Lattices, Phys. Rev. Lett.115, 260401 (2015).

(a) (b)

FIG. 5. (a) Time-dependent expectation value of the two-body correlations g2ðr; tÞ after a quantum quench from a noninteracting state,γi ¼ 0, to γf ¼ 8 at three different distances, jx1 − x2j ¼ 0, L=10, L=4. Here, the system is on a lattice with L=a ¼ 40 lattice sites. Thefull line is obtained by exact diagonalization of the Hamiltonian; the other curves are from tVMC results truncated at the level ofUð2Þ orUð3Þ. In contrast to γf ¼ 4 shown in Fig. 4, systematic differences of Uð2Þ compared to the exact results are more visible here; the exactdynamics is recovered by inclusion of three-body terms Uð3Þ into the tVMC wave function. (b) Off-diagonal single-particle densitymatrix g1ðr; tÞ at three different distances, r ¼ L=10, L=4, and L=2. The exact results (ED–black dashed line) are for all purposesindistinguishable from the Uð3Þ curves.

GIUSEPPE CARLEO et al. PHYS. REV. X 7, 031026 (2017)

031026-10

[14] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B.Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler,and J. Schmiedmayer, Relaxation and Prethermalization inan Isolated Quantum System, Science 337, 1318 (2012).

[15] M. Cominotti, D. Rossini, M. Rizzi, F. Hekking, and A.Minguzzi, Optimal Persistent Currents for InteractingBosons on a Ring with a Gauge Field, Phys. Rev. Lett.113, 025301 (2014).

[16] P. Calabrese and J. Cardy, Time Dependence of CorrelationFunctions Following a Quantum Quench, Phys. Rev. Lett.96, 136801 (2006).

[17] S. R. White, Density Matrix Formulation for QuantumRenormalization Groups, Phys. Rev. Lett. 69, 2863 (1992).

[18] S. R. White and A. E. Feiguin, Real-Time Evolution Usingthe Density Matrix Renormalization Group, Phys. Rev. Lett.93, 076401 (2004).

[19] A. J. Daley, C. Kollath, U. Schollwock, and G. Vidal, Time-Dependent Density-Matrix Renormalization-Group UsingAdaptive Effective Hilbert Spaces, J. Stat. Mech. (2004)P04005.

[20] F. B. Anders and A. Schiller, Real-Time Dynamics inQuantum-Impurity Systems: A Time-Dependent NumericalRenormalization-Group Approach, Phys. Rev. Lett. 95,196801 (2005).

[21] M. Dolfi, B. Bauer, M. Troyer, and Z. Ristivojevic,Multigrid Algorithms for Tensor Network States, Phys.Rev. Lett. 109, 020604 (2012).

[22] M. Dolfi, A. Kantian, B. Bauer, and M. Troyer, MinimizingNonadiabaticities in Optical-Lattice Loading, Phys. Rev. A91, 033407 (2015).

[23] F.Verstraete and J. I.Cirac,ContinuousMatrixProduct Statesfor Quantum Fields, Phys. Rev. Lett. 104, 190405 (2010).

[24] M. Ganahl, J. Rincon, and G. Vidal, Continuous MatrixProduct States for Quantum Fields: An Energy Minimiza-tion Algorithm., Phys. Rev. Lett. 118, 220402 (2017).

[25] C. J. Umrigar, J. Toulouse, C. Filippi, S. Sorella, and R. G.Hennig, Alleviation of the Fermion-Sign Problem by Opti-mization of Many-Body Wave Functions, Phys. Rev. Lett.98, 110201 (2007).

[26] M. Holzmann, B. Bernu, and D.M. Ceperley, Many-BodyWavefunctions for Normal Liquid 3He, Phys. Rev. B 74,104510 (2006).

[27] M. Taddei, M. Ruggeri, S. Moroni, and M. Holzmann,Iterative Backflow Renormalization Procedure for Many-Body Ground-State Wave Functions of Strongly InteractingNormal Fermi Liquids, Phys. Rev. B 91, 115106(2015).

[28] G. Carleo, F. Becca, M. Schiro, and M. Fabrizio, Locali-zation and Glassy Dynamics Of Many-Body QuantumSystems., Sci. Rep. 2, 243 (2012).

[29] G. Carleo, F. Becca, L. Sanchez-Palencia, S. Sorella, and M.Fabrizio, Light-Cone Effect and Supersonic Correlations inOne- and Two-Dimensional Bosonic Superfluids, Phys.Rev. A 89, 031602 (2014).

[30] L. Cevolani, G. Carleo, and L. Sanchez-Palencia, ProtectedQuasilocality in Quantum Systems with Long-Range Inter-actions, Phys. Rev. A 92, 041603 (2015).

[31] B. Blaß and H. Rieger, Test of Quantum Thermalization inthe Two-Dimensional Transverse-Field Ising Model, Sci.Rep. 6, 38185 (2016).

[32] G. Carleo and M. Troyer, Solving the Quantum Many-BodyProblem with Artificial Neural Networks, Science 355, 602(2017).

[33] K. Ido, T. Ohgoe, and M. Imada, Time-Dependent Many-Variable Variational Monte Carlo Method for Nonequili-brium Strongly Correlated Electron Systems, Phys. Rev. B92, 245106 (2015).

[34] We have conveniently set the particle mass and the reducedPlanck constant to unity.

[35] A. Bijl, The Lowest Wave Function of the Symmetrical ManyParticles System, Physica (Amsterdam) 7, 869 (1940).

[36] R. B. Dingle, The Zero-Point Energy of a System ofParticles, Philos. Mag. 40, 573 (1949).

[37] R. Jastrow, Many-Body Problem with Strong Forces, Phys.Rev. 98, 1479 (1955).

[38] E. Feenberg, Theory of Quantum Fluids, Pure and AppliedPhysics (Academic Press, New York, 1969).

[39] C. L. Kane, S. Kivelson, D. H. Lee, and S. C. Zhang,General Validity of Jastrow-Laughlin Wave Functions,Phys. Rev. B 43, 3255 (1991).

[40] P. A.M. Dirac,Note on Exchange Phenomena in the ThomasAtom, Math. Proc. Cambridge Philos. Soc. 26, 376 (1930).

[41] I. Frenkel, Wave Mechanics: Advanced General Theory, inThe International Series of Monographs on NuclearEnergy: Reactor Design Physics Vol. 2 (Clarendon Press,Oxford, 1934).

[42] The natural norm induced by a quantum Hilbert space is theFubini-Study norm, which is gauge invariant and thereforeinsensitive to the unknown normalizations of the quantumstates we are dealing with here.

[43] E. H. Lieb and W. Liniger, Exact Analysis of an InteractingBose Gas. I. The General Solution and the Ground State,Phys. Rev. 130, 1605 (1963).

[44] H. Moritz, T. Stoferle, M. Kohl, and T. Esslinger, ExcitingCollective Oscillations in a Trapped 1D Gas, Phys. Rev.Lett. 91, 250402 (2003).

[45] T. Kinoshita, T. Wenger, and D. S. Weiss, A QuantumNewton’s Cradle, Nature (London) 440, 900 (2006).

[46] J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodgman,S. Langer, I. P. McCulloch, F. Heidrich-Meisner, I. Bloch,and U. Schneider, Expansion Dynamics of InteractingBosons in Homogeneous Lattices in One and Two Dimen-sions, Phys. Rev. Lett. 110, 205301 (2013).

[47] B. Fang, G. Carleo, A. Johnson, and I. Bouchoule, Quench-Induced Breathing Mode of One-Dimensional Bose Gases,Phys. Rev. Lett. 113, 035301 (2014).

[48] G. Boéris et al., Mott Transition for Strongly InteractingOne-Dimensional Bosons in a Shallow Periodic Potential,Phys. Rev. A 93, 011601 (2016).

[49] S. Sorella, Generalized Lanczos Algorithm for VariationalQuantum Monte Carlo, Phys. Rev. B 64, 024512 (2001).

[50] C. J. Umrigar and C. Filippi, Energy and Variance Opti-mization of Many-Body Wave Functions, Phys. Rev. Lett.94, 150201 (2005).

[51] M. Girardeau, Relationship between Systems of Impen-etrable Bosons and Fermions in One Dimension, J. Math.Phys. (N.Y.) 1, 516 (1960).

[52] G. E. Astrakharchik and S. Giorgini, Correlation Functionsand Momentum Distribution of One-Dimensional BoseSystems, Phys. Rev. A 68, 031602 (2003).

UNITARY DYNAMICS OF STRONGLY-INTERACTING BOSE … PHYS. REV. X 7, 031026 (2017)

031026-11

[53] P. Pippan, S. R. White, and H. G. Evertz, Efficient Matrix-Product State Method for Periodic Boundary Conditions,Phys. Rev. B 81, 081103 (2010).

[54] G. Carleo, G. Boéris, M. Holzmann, and L. Sanchez-Palencia, Universal Superfluid Transition and TransportProperties of Two-Dimensional Dirty Bosons, Phys. Rev.Lett. 111, 050406 (2013).

[55] M. Boninsegni, N. Prokof’ev, and B. Svistunov, WormAlgorithm for Continuous-Space Path Integral Monte CarloSimulations, Phys. Rev. Lett. 96, 070601 (2006).

[56] J. C. Zill, T. M. Wright, K. V. Kheruntsyan, T. Gasenzer,and M. J. Davis, Relaxation Dynamics of the Lieb-LinigerGas Following an Interaction Quench: A CoordinateBethe-Ansatz Analysis, Phys. Rev. A 91, 023611 (2015).

[57] J. De Nardis, L. Piroli, and J.-S. Caux, Relaxation Dynamicsof Local Observables in Integrable Systems., J. Phys. A 48,43FT01 (2015).

[58] J.-S. Caux and F. H. L. Essler, Time Evolution of LocalObservables After Quenching to an Integrable Model, Phys.Rev. Lett. 110, 257203 (2013).

[59] J. C. Zill, T. M. Wright, K. V. Kheruntsyan, T. Gasenzer, andM. J. Davis, A Coordinate Bethe Ansatz Approach to theCalculation of Equilibrium and Nonequilibrium Correla-tions of the One-Dimensional Bose Gas, New J. Phys. 18,045010 (2016).

[60] C. N. Yang and C. P. Yang, Thermodynamics of aOne-Dimensional System of Bosons with Repulsive Delta-Function Interaction, J. Math. Phys. (N.Y.) 10, 1115(1969).

[61] M. Rigol, Fundamental Asymmetry in Quenches betweenIntegrable and Nonintegrable Systems, Phys. Rev. Lett.116, 100601 (2016).

[62] M. Kollar and M. Eckstein, Relaxation of a One-Dimensional Mott Insulator after an Interaction Quench,Phys. Rev. A 78, 013626 (2008).

[63] M. Holzmann, D. M. Ceperley, C. Pierleoni, and K. Esler,Backflow Correlations for the Electron Gas and MetallicHydrogen, Phys. Rev. E 68, 046707 (2003).

GIUSEPPE CARLEO et al. PHYS. REV. X 7, 031026 (2017)

031026-12