A coordinate Bethe ansatz approach to the calculation of equilibrium and nonequilibrium correlations of the one-dimensional Bose gas

Abstract

We use the coordinate Bethe ansatz to exactly calculate matrix elements between eigenstates of the Lieb-Liniger model of one-dimensional bosons interacting via a two-body delta-potential. We investigate the static correlation functions of the zero-temperature ground state and their dependence on interaction strength, and analyze the effects of system size in the crossover from few-body to mesoscopic regimes for up to seven particles. We also obtain time-dependent nonequilibrium correlation functions for five particles following quenches of the interaction strength from two distinct initial states. One quench is from the noninteracting ground state and the other from a correlated ground state near the strongly interacting Tonks-Girardeau regime. The final interaction strength and conserved energy are chosen to be the same for both quenches. The integrability of the model highly constrains its dynamics, and we demonstrate that the time-averaged correlation functions following quenches from these two distinct initial conditions are both nonthermal and moreover distinct from one another.

Nonequilibrium dynamics of a one-dimensional Bose gas via the coordinate Bethe ansatz

“…[7,73] should disappear in the thermodynamic limit. However, results for local correlation functions at zero temperature [165,262] and our results for g GTBA (0) ∝ γ −2 obtained in the calculations of Ref. [7,8] lies in between the thermal scaling g…”

“…[165] and introduces a computational method for calculating matrix elements in the Lieb-Liniger model via the coordinate Bethe ansatz. We compute several ground-state correlation functions for system sizes of up to seven particles and characterize the finite-size scaling of the system, before turning to a non-equilibrium scenario.…”

“…[255]. For this purpose, we developed a symbolic integration algorithm, which will be presented elsewhere [165].…”

“…where the sum is over all eigenstates |{λ j } ofĤ, and the C {λ j } ≡ {λ j }|ψ 0 are the overlaps between the initial state |ψ 0 and these eigenstates, which we calculate from their coordinatespace representations ζ {λ j } ({x i }) [165]. 2 We note, however, that only those states |{λ j } that have zero total momentum, j λ j = 0 [cf.…”

“…[165]. The computational expense incurred in evaluating these matrix elements increases exponentially with the particle number N , placing a strong practical constraint on the system sizes we can describe with our coordinate…”