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arXiv:1602.03065 · Journal of Statistical Mechanics Theory and Experiment

Real-time dynamics in a strongly interacting bosonic hopping model: global quenches and mapping to the XX chain

Balázs Pozsgay, Viktor Eisler

Published May 1, 2016• 30 citations
arXivHTMLPDF

Abstract

Abstract. We study the time evolution of an integrable many-particle system, described by the q -boson Hamiltonian in the limit of strong bosonic interactions q → ∞ . It is shown that, for a particular class of pure initial states, the analytical calculation of certain observables simplifies considerably. Namely, we provide exact formulas for the calculation of the Loschmidt-echo and the emptiness formation probability, where the computational time scales polynomially with the particle number. Moreover, we construct a non-local mapping of the q -boson model to the XX spin chain, and show how this can be utilized to obtain the time evolution of various local bosonic observables for translationally invariant initial states. The results obtained via the bosonic and fermionic picture show perfect agreement. In the infinite volume and large time limits, we rigorously verify the prediction of the generalized Gibbs ensemble for homogeneous initial Fock states.

DOI
10.1088/1742-5468/2016/05/053107

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Abstract. We study the time evolution of an integrable many-particle system, described by the q -boson Hamiltonian in the limit of strong bosonic interactions q → ∞ . It is shown that, for a particular class of pure initial states, the analytical calculation of certain observables simplifies considerably. Namely, we provide exact formulas for the calculation of the Loschmidt-echo and the emptiness formation probability, where the computational time scales polynomially with the particle number. Moreover, we construct a non-local mapping of the q -boson model to the XX spin chain, and show how this can be utilized to obtain the time evolution of various local bosonic observables for translationally invariant initial states. The results obtained via the bosonic and fermionic picture show perfect agreement. In the infinite volume and large time limits, we rigorously verify the prediction of the generalized Gibbs ensemble for homogeneous initial Fock states.
We study the time evolution of an integrable many-particle system, described by the q-boson Hamiltonian in the limit of strong bosonic interactions q ->infinity. It is shown that, for a particular class of pure initial states, the analytical calculation of certain observables simplifies considerably. Namely, we provide exact formulas for the calculation of the Loschmidt-echo and the emptiness formation probability, where the computational time scales polynomially with the particle number. Moreover, we construct a non-local mapping of the q-boson model to the XX spin chain, and show how this can be utilized to obtain the time evolution of various local bosonic observables for translationally invariant initial states. The results obtained via the bosonic and fermionic picture show perfect agreement. In the infinite volume and large time limits, we rigorously verify the prediction of the generalized Gibbs ensemble for homogeneous initial Fock states.
We consider quantum quenches in the so-called q -boson lattice model. We argue that the Generalized Eigenstate Thermalization Hypothesis holds in this model, therefore the Generalized Gibbs Ensemble (GGE) gives a valid description of the stationary states in the long time limit. For a special class of initial states (which are the pure Fock states in the local basis) we are able to provide the GGE predictions for the resulting root densities. We also give predictions for the long-time limit of certain local operators. In the q → ∞ limit the calculations simplify considerably, the wave functions are given by Schur polynomials and the overlaps with the initial states can be written as simple determinants. In two cases we prove rigorously that the GGE prediction for the root density is correct. Moreover, we calculate the exact time dependence of a physical observable (the one-site Emptiness Formation Probability) for the quench starting from the state with exactly one particle per site. In the long-time limit the GGE prediction is recovered.
We study the nonequilibrium time evolution of the spin-1/2 anisotropic Heisenberg (XXZ) spin chain, with a choice of dimer product and Néel states as initial states. We investigate numerically various short-ranged spin correlators in the long-time limit and find that they deviate significantly from predictions based on the generalized Gibbs ensemble (GGE) hypotheses. By computing the asymptotic spin correlators within the recently proposed quench-action formalism [Phys. Rev. Lett. 110, 257203 (2013)], however, we find excellent agreement with the numerical data. We, therefore, conclude that the GGE cannot give a complete description even of local observables, while the quench-action formalism correctly captures the steady state in this case.
One of the fundamental principles of statistical physics is that only partial information about a system's state is required for its macroscopic description. This is not only true for thermal ensembles, but also for the unconventional ensemble, known as Generalized Gibbs Ensemble (GGE), that is expected to describe the relaxation of integrable systems after a quantum quench. By analytically studying the quench dynamics in a prototypical one-dimensional critical model, the massless free bosonic field theory, we find evidence of a novel type of equilibration characterized by the preservation of an enormous amount of memory of the initial state that is accessible by local measurements. In particular, we show that the equilibration retains memory of non-Gaussian initial correlations, in contrast to the case of massive free evolution which erases all such memory. The GGE in its standard form, being a Gaussian ensemble, fails to predict correctly the equilibrium values of local observables, unless the initial state is Gaussian itself. Our findings show that the equilibration of a broad class of quenches whose evolution is described by Luttinger liquid theory with an initial state that is non-Gaussian in terms of the bosonic field, is not correctly captured by the corresponding bosonic GGE, raising doubts about the validity of the latter in general one-dimensional gapless integrable systems such as the Lieb-Liniger model. We also propose that the same experiment by which the GGE was recently observed [Langen et al., Science 348 (2015) 207-211] can also be used to observe its failure, simply by starting from a non-Gaussian initial state.
We address the computation of the Loschmidt echo in interacting integrable spin chains after a quantum quench. We focus on the massless regime of the XXZ spin-1/2 chain and present exact results for the dynamical free energy (Loschmidt echo per site) for a special class of integrable initial states. For the first time we are able to observe and describe points of non-analyticities using exact methods, by following the Loschmidt echo up to large real times. The dynamical free energy is computed as the leading eigenvalue of an appropriate Quantum Transfer Matrix, and the non-analyticities arise from the level crossings of this matrix. Our exact results are expressed in terms of “excited-state” thermodynamic Bethe ansatz equations, whose solutions involve non-trivial Riemann surfaces. By evaluating our formulas, we provide explicit numerical results for the quench from the Néel state, and we determine the first few non-analytic points.
Abstract In an m particle quantum system, the rank of interactions and the nature of particles (fermions or bosons) can strongly affect the dynamics of the system. To explore this, we study the nonequilibrium dynamics with particles in a one-body mean-field and quenched by an interaction of body-rank k = 2, 3, …, m . Using fermionic embedded Gaussian orthogonal ensembles (EGOEs) and bosonic embedded Gaussian orthogonal ensembles (BEGOEs) of one plus k -body interactions (also the unitary variants EGUE and BEGUE), it is seen that the short time decay of the survival probability of many-particle systems is given by the Fourier transform of the generating function of the q -Hermite polynomials. Deriving the formulas for q for both fermion and boson systems as a function of m , k and the number of single particle states N , we have verified that the Fourier transform of agrees very well with the numerical ensemble calculations for both fermion and boson systems. These results bridge the gap between the known results for k = 2 and k = m .