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Overlap distributions for quantum quenches in the anisotropic Heisenberg chain

Mazza, P., Jean-Marie Stéphan, Canovi, E., Vincenzo Alba, Brockmann, M., Masudul Haque

Published Jan 1, 2016• 44 citations
OpenAlex

Abstract

The dynamics after a quantum quench is determined by the weights of the initial state in the eigenspectrum of the final Hamiltonian, i.e. by the distribution of overlaps in the energy spectrum. We present an analysis of such overlap distributions for quenches of the anisotropy parameter in the one-dimensional anisotropic spin-1/2 Heisenberg model (XXZ chain). We provide an overview of the form of the overlap distribution for quenches from various initial anisotropies to various final ones, using numerical exact diagonalization. We show that if the system is prepared in the antiferromagnetic Neel state (infinite anisotropy) and released into a non-interacting setup (zero anisotropy, XX point) only a small fraction of the final eigenstates gives contributions to the post-quench dynamics, and that these eigenstates have identical overlap magnitudes. We derive expressions for the overlaps, and present the selection rules that determine the final eigenstates having nonzero overlap. We use these results to derive concise expressions for time-dependent quantities (Loschmidt echo, longitudinal and transverse correlators) after the quench. We use perturbative analyses to understand the overlap distribution for quenches from infinite to small nonzero anisotropies, and for quenches from large to zero anisotropy.

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The dynamics after a quantum quench is determined by the weights of the initial state in the eigenspectrum of the final Hamiltonian, i.e. by the distribution of overlaps in the energy spectrum. We present an analysis of such overlap distributions for quenches of the anisotropy parameter in the one-dimensional anisotropic spin-1/2 Heisenberg model (XXZ chain). We provide an overview of the form of the overlap distribution for quenches from various initial anisotropies to various final ones, using numerical exact diagonalization. We show that if the system is prepared in the antiferromagnetic Neel state (infinite anisotropy) and released into a non-interacting setup (zero anisotropy, XX point) only a small fraction of the final eigenstates gives contributions to the post-quench dynamics, and that these eigenstates have identical overlap magnitudes. We derive expressions for the overlaps, and present the selection rules that determine the final eigenstates having nonzero overlap. We use these results to derive concise expressions for time-dependent quantities (Loschmidt echo, longitudinal and transverse correlators) after the quench. We use perturbative analyses to understand the overlap distribution for quenches from infinite to small nonzero anisotropies, and for quenches from large to zero anisotropy.
Abstract The dynamics after a quantum quench is determined by the weights of the initial state in the eigenspectrum of the final Hamiltonian, i.e. by the distribution of overlaps in the energy spectrum. We present an analysis of such overlap distributions for quenches of the anisotropy parameter in the one-dimensional anisotropic spin-1/2 Heisenberg model (XXZ chain). We provide an overview of the form of the overlap distribution for quenches from various initial anisotropies to various final ones, using numerical exact diagonalization. We show that if the system is prepared in the antiferromagnetic Néel state (infinite anisotropy) and released into a non-interacting setup (zero anisotropy, XX point) only a small fraction of the final eigenstates gives contributions to the post-quench dynamics, and that these eigenstates have identical overlap magnitudes. We derive expressions for the overlaps, and present the selection rules that determine the final eigenstates having nonzero overlap. We use these results to derive concise expressions for time-dependent quantities (Loschmidt echo, longitudinal and transverse correlators) after the quench. We use perturbative analyses to understand the overlap distribution for quenches from infinite to small nonzero anisotropies, and for quenches from large to zero anisotropy.
The steady state after a quantum quench from the Néel state to the anisotropic Heisenberg model for spin chains is investigated. Two methods that aim to describe the postquench non-thermal equilibrium, the generalized Gibbs ensemble and the quench action approach, are discussed and contrasted. Using the recent implementation of the quench action approach for this Néel-to-XXZ quench, we obtain an exact description of the steady state in terms of Bethe root densities, for which we give explicit analytical expressions. Furthermore, by developing a systematic small-quench expansion around the antiferromagnetic Ising limit, we analytically investigate the differences between the predictions of the two methods in terms of densities and postquench equilibrium expectation values of local physical observables. Finally, we discuss the details of the quench action solution for the quench to the isotropic Heisenberg spin chain. For this case we validate the underlying assumptions of the quench action approach by studying the large-system-size behavior of the overlaps between Bethe states and the Néel state.
We consider the time evolution after quantum quenches in the spin-12\frac{1}{2}21​ Heisenberg XXZ quantum spin chain with Ising-type anisotropy. The time evolution of short-distance spin-spin correlation functions is studied by numerical tensor network techniques for a variety of initial states, including N\'eel and Majumdar-Ghosh states and the ground state of the XXZ chain at large values of the anisotropy. The various correlators appear to approach stationary values, which are found to be in good agreement with the results of exact calculations of stationary expectation values in appropriate generalized Gibbs ensembles. In particular, our analysis shows how symmetries of the post-quench Hamiltonian that are broken by particular initial states are restored at late times.
We consider a class of global quantum quenches in the Heisenberg XXZ spin chain, where the initial states are given by products of local two-site states. The two main examples are the Néel state and the dimer state. We derive an exact analytic result for the ‘Loschmidt echo per site’ at imaginary times and also consider the analytic continuation back to real times. As a by-product we obtain an exact result for the ‘overlap per site’ between the Néel state and the ground state of the XXZ Hamiltonian in the massive regime.
Recent experimental achievements in controlling ultracold gases in optical lattices open a new perspective on quantum many-body physics. In these experimental setups, it is possible to study coherent time evolution of isolated quantum systems. These dynamics reveal new physics beyond the low-energy properties that are usually relevant in solid-state many-body systems. In this paper, we study the time evolution of antiferromagnetic order in the Heisenberg chain after a sudden change of the anisotropy parameter, using various numerical and analytical methods. As a generic result, we find that the order parameter, which can show oscillatory or non-oscillatory dynamics, decays exponentially except for the effectively non-interacting case of the XX limit. For weakly ordered initial states, we also find evidence for an algebraic correction to the exponential law. The study is based on numerical simulations using a numerical matrix product method for infinite system sizes (iMPS), for which we provide a detailed description and an error analysis. Additionally, we investigate in detail the exactly solvable XX limit. These results are compared to approximative analytical approaches including an effective description by the XZ model as well as by mean-field, Luttinger-liquid and sine-Gordon theories. The comparison reveals which aspects of non-equilibrium dynamics can, as in equilibrium, be described by low-energy theories and which are the novel phenomena specific to quantum quench dynamics. The relevance of the energetically high part of the spectrum is illustrated by means of a full numerical diagonalization of the Hamiltonian.
We develop an analytical approach for the study of the quench dynamics of the anisotropic Heisenberg model (XXZ model) on the infinite line. We present the exact time-dependent wave functions after a quench in an integral form for any initial state and for any anisotropy Δ by means of a generalized Yudson contour representation. We calculate the evolution of several observables from two particular initial states: starting from a local Néel state we calculate the time evolution of the antiferromagnetic order parameter-staggered magnetization; starting from a state with consecutive flipped spins (1) we calculate the evolution of the local magnetization and express it in terms of the propagation of magnons and bound state excitations, and (2) we predict the evolution of the induced spin currents. These predictions can be confronted with experiments in ultracold gases in optical lattices. We also show how the "string" solutions of Bethe ansatz equations emerge naturally from the contour approach.