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Physical Review A

Monte Carlo Bethe-ansatz approach for the study of the Lieb-Liniger model

Zhehao Zhang, Yi-Cong Yu, Yang-Yang Chen, Song Cheng, Xi‐Wen Guan

Published Mar 27, 2024• 2 citations
Publisher

Abstract

We have developed a Monte Carlo algorithm to explore the equilibrium and out-of-equilibrium properties of the Lieb-Liniger model. This Monte Carlo Bethe-ansatz (MCBA) algorithm has enabled us to successfully reconstruct statistical ensembles for equilibrium or postquench dynamics, thereby facilitating the calculation of macroscopic quantities of integrable models. Our results substantiate the validity of the (quench) thermodynamic Bethe-ansatz equation from the perspective of first-principles statistical physics. Additionally, we have employed this method to study the generalized Gibbs ensemble in relation to the postquench dynamics of the Lieb-Liniger model. Furthermore, we have demonstrated the MCBA algorithm's capacity to calculate correlations using Bethe-ansatz wave functions. Our approach offers an efficient methodology for the investigation of the equilibrium and out-of-equilibrium properties of integrable systems.

DOI
10.1103/physreva.109.033320

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We have developed a Monte Carlo algorithm to explore the equilibrium and out-of-equilibrium properties of the Lieb-Liniger model. This Monte Carlo Bethe-ansatz (MCBA) algorithm has enabled us to successfully reconstruct statistical ensembles for equilibrium or postquench dynamics, thereby facilitating the calculation of macroscopic quantities of integrable models. Our results substantiate the validity of the (quench) thermodynamic Bethe-ansatz equation from the perspective of first-principles statistical physics. Additionally, we have employed this method to study the generalized Gibbs ensemble in relation to the postquench dynamics of the Lieb-Liniger model. Furthermore, we have demonstrated the MCBA algorithm's capacity to calculate correlations using Bethe-ansatz wave functions. Our approach offers an efficient methodology for the investigation of the equilibrium and out-of-equilibrium properties of integrable systems.
We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.
We present a comprehensive review on the state-of-the-art of the approximate analytic approaches describing the finite-temperature thermodynamic quantities of the Lieb-Liniger model of the one-dimensional (1D) Bose gas with contact repulsive interactions. This paradigmatic model of quantum many-body-theory plays an important role in many areas of physics-thanks to its integrability and possible experimental realization using, e.g., ensembles of ultracold bosonic atoms confined to quasi-1D geometries. The thermodynamics of the uniform Lieb-Liniger gas can be obtained numerically using the exact thermal Bethe ansatz (TBA) method, first derived in 1969 by Yang and Yang. However, the TBA numerical calculations do not allow for the in-depth understanding of the underlying physical mechanisms that govern the thermodynamic behavior of the Lieb-Liniger gas at finite temperature. Our work is then motivated by the insights that emerge naturally from the transparency of closed-form analytic results, which are derived here in six different regimes of the gas and which exhibit an excellent agreement with the TBA numerics. Our findings can be further adopted for characterising the equilibrium properties of inhomogeneous (e.g., harmonically trapped) 1D Bose gases within the local density approximation and for the development of improved hydrodynamic theories, allowing for the calculation of breathing mode frequencies which depend on the underlying thermodynamic equation of state. Our analytic approaches can be applied to other systems including impurities in a quantum bath, liquid helium-4, and ultracold Bose gas mixtures.
We apply Quantum Monte Carlo technique to analyze the non equlibrium state of a trapped 1d Bose gas just after the quenching of the confining potential. As a matter of fact we solve the time dependent Schroedinger equation for the system of one-dimensional bosons interacting via delta potential in an infinite square well (namely Lieb-Liniger model) using Feynman-Kac path integral Monte Carlo technique. These 1d systems are extremely interesting and worth investigating in the context of non-equilibrium dynamics of interacting many body systems. Even though the systems can be realized experimentally and are exactly solvable by Bethe Ansatz, the diffusion Monte Carlo is proven to be more efficient in most circumstances than other mean value techniques as the numerical method can incorporate the finite interaction very easily. Using N particle ground state wavefunction for one-dimensional hard core bosons in a harmonic trap, we develop an algorithm to calculate density. We also observe the change in the density distribution by changing the length of the hard wall box. After an increase in the box length, we still get a self-similar density distribution.
By combining classical Monte Carlo and Bethe ansatz techniques we devise a numerical method to construct the Truncated Generalized Gibbs Ensemble (TGGE) for the spin-1/2 isotropic Heisenberg (XXXXXXXXX) chain. The key idea is to sample the Hilbert space of the model with the appropriate GGE probability measure. The method can be extended to other integrable systems, such as the Lieb-Liniger model. We benchmark the approach focusing on GGE expectation values of several local observables. As finite-size effects decay exponentially with system size, moderately large chains are sufficient to extract thermodynamic quantities. The Monte Carlo results are in agreement with both the Thermodynamic Bethe Ansatz (TBA) and the Quantum Transfer Matrix approach (QTM). Remarkably, it is possible to extract in a simple way the steady-state Bethe-Gaudin-Takahashi (BGT) roots distributions, which encode complete information about the GGE expectation values in the thermodynamic limit. Finally, it is straightforward to simulate extensions of the GGE, in which, besides the local integral of motion (local charges), one includes arbitrary functions of the BGT roots. As an example, we include in the GGE the first non-trivial quasi-local integral of motion.
Determining the dynamics of interacting integrable many-particle quantum systems at finite times after homogeneous quantum quenches is a long-standing challenge. We present a Monte Carlo sampling scheme that numerically evaluates the Lehmann representation for time-dependent expectation values of local operators, allowing us to access system sizes and times significantly beyond the reach of existing methods. The approach accommodates both the full Lehmann sum and the Quench Action formalism. We benchmark against exact results for non-interacting lattice and continuum models and short-time results at weak interactions, finding excellent agreement. We apply the method to quantum quenches from a Bose-Einstein condensate in the repulsive Lieb-Liniger model and determine the time evolution of the order parameter for a wide range of interaction strengths. We discuss the emergence of a "sign problem" for more general dynamical correlators and setups.