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arXiv:1304.5374 · Journal of Statistical Mechanics Theory and Experiment

The generalized Gibbs ensemble for Heisenberg spin chains

Balázs Pozsgay

Published Jul 4, 2013• 104 citations
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Abstract

We consider the generalized Gibbs ensemble (GGE) in the context of global quantum quenches in XXZ Heisenberg spin chains. Embedding the GGE into the quantum transfer matrix formalism, we develop an iterative procedure to fix the Lagrange multipliers and to calculate predictions for the long-time limit of short-range correlators. The main idea is to consider truncated GGEs with only a finite number of charges and to investigate the convergence of the numerical results as the truncation level is increased. As an example we consider a quantum quench situation where the system is initially prepared in the Néel state and then evolves with an XXZ Hamiltonian with anisotropy Δ > 1. We provide predictions for short-range correlators and gather numerical evidence that the iterative procedure indeed converges. The results show that the system retains memory of the initial condition, and there are clear differences between the numerical values of the correlators as calculated from the purely thermal and generalized Gibbs ensembles.

DOI
10.1088/1742-5468/2013/07/p07003

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We consider the generalized Gibbs ensemble (GGE) in the context of global quantum quenches in XXZ Heisenberg spin chains. Embedding the GGE into the quantum transfer matrix formalism, we develop an iterative procedure to fix the Lagrange multipliers and to calculate predictions for the long-time limit of short-range correlators. The main idea is to consider truncated GGEs with only a finite number of charges and to investigate the convergence of the numerical results as the truncation level is increased. As an example we consider a quantum quench situation where the system is initially prepared in the Néel state and then evolves with an XXZ Hamiltonian with anisotropy Δ > 1. We provide predictions for short-range correlators and gather numerical evidence that the iterative procedure indeed converges. The results show that the system retains memory of the initial condition, and there are clear differences between the numerical values of the correlators as calculated from the purely thermal and generalized Gibbs ensembles.
We consider the Generalized Gibbs Ensemble (GGE) in the context of global quantum quenches in XXZ Heisenberg spin chains. Embedding the GGE into the Quantum Transfer Matrix formalism we develop an iterative procedure to fix the Lagrange-multipliers and to calculate predictions for the long-time limit of short-range correlators. The main idea is to consider truncated GGE’s with only a finite number of charges and to investigate the convergence of the numerical results as the truncation level is increased. As an example we consider a quantum quench situation where the system is initially prepared in the Néel state and then evolves with an XXZ Hamiltonian with anisotropy ∆ > 1. We provide predictions for short range correlators and gather numerical evidence that the iterative procedure indeed converges. The results show that the system retains memory of the initial condition, and there are clear differences between the numerical values of the correlators as calculated from the purely thermal and the Generalized Gibbs ensembles.
Abstract We revisit the question of whether and how the steady states arising after non-equilibrium time evolution in integrable models (and in particular in the XXZ spin chain) can be described by the so-called generalized Gibbs ensemble (GGE). Whereas it is known that the micro-canonical ensemble built on a complete set of charges correctly describes the long-time limit of local observables, it has been shown recently by Ilievski et al that the corresponding canonical ensemble is not well defined, and instead a different canonical ensemble was proposed in terms of particle occupation number operators. Here we provide an alternative construction by considering truncated GGEs (tGGEs) that include only a finite number of local and quasi-local conserved operators. It is shown that the tGGEs can approximate the steady states with arbitrary precision, i.e. all physical observables are exactly reproduced in the infinite truncation limit. We trace back the problems encountered in defining an untruncated GGE to the dependence of the associated Lagrange multipliers on the truncation index. Conversely, we show that this problem may be circumvented by considering a new set of (quasi)local charges which are linear combinations of the standard ones, and whose associated Lagrange multipliers are well-defined state functions. Our general arguments are applied to concrete quench situations in the XXZ chain, where the initial states are simple two-site or four-site product states. Depending on the quench we find that numerical results for the local correlators can be obtained with remarkable precision using truncated GGEs with only 10–100 charges.
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.
We study quantum quenches in the XXZ spin-1/21/21/2 Heisenberg chain from families of ferromagnetic and antiferromagnetic initial states. Using Bethe ansatz techniques, we compute short-range correlators in the complete generalized Gibbs ensemble (GGE), which takes into account all local and quasilocal conservation laws. We compare our results to exact diagonalization and numerical linked cluster expansion calculations for the diagonal ensemble, finding excellent agreement and thus providing a very accurate test for the validity of the complete GGE. Furthermore, we use exact diagonalization to compute the diagonal entropy in the postquench steady state. We show that the Yang-Yang entropy for the complete GGE is consistent with twice the value of the diagonal entropy in the largest chains or the extrapolated result in the thermodynamic limit. Finally, the complete GGE is quantitatively contrasted with the GGE built using only the local conserved charges (local GGE). The predictions of the two ensembles are found to differ significantly in the case of ferromagnetic initial states. Such initial states are better suited than others considered in the literature to experimentally test the validity of the complete GGE and contrast it to the failure of the local GGE.
In integrable many-particle systems, it is widely believed that the stationary state reached at late times after a quantum quench can be described by a generalized Gibbs ensemble (GGE) constructed from their extensive number of conserved charges. A crucial issue is then to identify a complete set of these charges, enabling the GGE to provide exact steady-state predictions. Here we solve this long-standing problem for the case of the spin-1/2 Heisenberg chain by explicitly constructing a GGE which uniquely fixes the macrostate describing the stationary behavior after a general quantum quench. A crucial ingredient in our method, which readily generalizes to other integrable models, are recently discovered quasilocal charges. As a test, we reproduce the exact postquench steady state of the Néel quench problem obtained previously by means of the Quench Action method.
Integrable quantum systems evolving from non-equilibrium initial states do not thermalize to conventional Gibbs ensembles (GE). Instead, at long times they relax to generalized Gibbs ensembles (GGEs), which incorporate the full set of local and quasi-local conserved quantities. While GGEs have been extensively studied in the literature, a quantitative analytic characterization of how different they are from ordinary GEs is still lacking. In this work, we address this question by employing the concept of athermality, which we define within quantum resource theory as the relative entropy between a given state and the closest thermal state. By means of integrability techniques we compute the athermality for several quantum quenches in paradigmatic integrable models, including the free XY spin chain, the interacting Lieb-Liniger model, the XXZ spin chain, and the harmonic chain. We find that often the athermality becomes anomalously small when the post-quench Hamiltonian is critical in its ground state, despite probing physics at a finite energy density. We also prove that it systematically develops a singularity at criticality, which is inherited from the entropy of the GGE.