Quenching the Anisotropic Heisenberg Chain: Exact Solution and Generalized Gibbs Ensemble Predictions

Abstract

General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. We study quenches in integrable spin-1=2 chains in which we evolve the ground state of the antiferromagnetic Ising model with the anisotropic Heisenberg Hamiltonian. For this nontrivially interacting situation, an application of the first-principles-based quench-action method allows us to give an exact description of the postquench steady state in the thermodynamic limit. We show that a generalized Gibbs ensemble, implemented using all known local conserved charges, fails to reproduce the exact quench-action steady state and to correctly predict postquench equilibrium expectation values of physical observables. This is supported by numerical linked-cluster calculations within the diagonal ensemble in the thermodynamic limit. [6] all the way to atomic-scale isolated quantum systems [7]. Much recent experimental and theoretical activity has been focused on the latter, raising fundamental questions as to whether, how, and to what state such systems relax under unitary time evolution following a sudden quantum quench . From this work, two scenarios for equilibration have emerged, one applicable to models having only a few local conserved quantities, the other relevant to integrable models characterized by an infinite number of local conserved charges. In the former, thermalization to a Gibbs ensemble is the rule [11], while in the latter, equilibration to a so-called generalized Gibbs ensemble (GGE) [9,10] is generally thought to occur, in particular for lattice spin systems [12][13][14][15][16][17][18][19][20].In this Letter, we study a quench in which the second scenario breaks down. Our initial state, defined as a purely antiferromagnetic (spin-1=2 Néel) state, is let to evolve unitarily in time according to the XXZ spin chain Hamiltonian. This is a physically meaningful quench protocol, which can, in principle, be implemented using cold atoms [43][44][45][46][47]. We provide a thermodynamically exact solution for the steady state reached long after the quench, derived directly from microscopics using the recently proposed quench-action method [48]. The solution takes the form of a set of distributions of quasimomenta that completely characterizes the macrostate representing the steady state, from which observables ...

Recursive formulas for the overlaps between Bethe states and product states in XXZ Heisenberg chains

“…Compact solutions would be very useful for analytical studies of the quench dynamics of the XXZ model based on the representative state approach [9], on the same lines of those presented in Refs. [30,31] for the dimer and Néel states respectively. This would allow to confront with the GGE solution [34] for a wide class of states.…”

“…(19) for the scalar product, from Eqs. (29), (30), and (34), we finally arrive at the recursive formula…”

“…The predictions based on GGE resisted all tests in many models both having a free-particle representation [10,11,[14][15][16][17][18][19][20][21][22][23][24] and being genuinely interacting [25][26][27][28][29], until very recently when it has been found that for the XXZ spin chain, after a quench from the (symmetrized) Néel and dimer states, the obtained stationary values [30][31][32] disagree with the predictions of the GGE built with all the known local charges [33,34] (see also Refs. [35][36][37]).…”

Non-equilibrium dynamics of isolated quantum systems

“…From these overlaps, the infinite time limit of the evolution has been explicitly calculated for the Neel state in Ref. [65,66] and for the dimer state in [67], again exploiting the representative state approach of Ref. [52].…”

Stationary ensemble approximations of dynamic quantum states: Optimizing the generalized Gibbs ensemble

“…Unfortunately, it is not always clear which conserved quantities have to be taken into account [18][19][20][21][22]. Actually, the linearity of quantum mechanics implies that the set of conserved quantities contains all projectorsP n on eigenstates of the Hamiltonian H = n ǫ nPn .…”