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arXiv:1308.3087 · Journal of Statistical Mechanics Theory and Experiment

The dynamical free energy and the Loschmidt echo for a class of quantum quenches in the Heisenberg spin chain

Balázs Pozsgay

Published Oct 31, 2013• 99 citations
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Abstract

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.

DOI
10.1088/1742-5468/2013/10/p10028

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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.
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 We consider the computation of the Loschmidt echo after quantum quenches in the interacting XXZ Heisenberg spin chain both for real and imaginary times. We study two-site product initial states, focusing in particular on the Néel and tilted Néel states. We apply the quantum transfer matrix (QTM) approach to derive generalized TBA equations, which follow from the fusion hierarchy of the appropriate QTM’s. Our formulas are valid for arbitrary imaginary time and for real times at least up to a time t 0 , after which the integral equations have to be modified. In some regimes, t 0 is seen to be either very large or infinite, allowing to explore in detail the post-quench dynamics of the system. As an important part of our work, we show that for the Néel state our imaginary time results can be recovered by means of the quench action approach, unveiling a direct connection with the quantum transfer matrix formalism. In particular, we show that in the zero-time limit, the study of our TBA equations allows for a simple alternative derivation of the recently obtained Bethe ansatz distribution functions for the Néel, tilted Néel and tilted ferromagnet states.
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.
We investigate quench dynamics in the quantum S=1/2S=1/2S=1/2 XXZ antiferromagnetic chain with staggered and anisotropic interactions in the flat-band limit. We consider global quenches that interchange odd- and even-bond strengths in a fully dimerized chain, yielding a postquench Hamiltonian with couplings only on even links. Due to the vanishing group velocity of excitations, the system fails to relax, providing an ideal setting to study dynamical topological quantum phase transitions. We show that the zzz-component of the interactions, controlled by the anisotropy ΔΔΔ, acts as a relevant term that generates additional dynamical quantum phase transitions and induces rich structures in the Loschmidt echo. These transitions are symmetry-protected and robust for quenches between distinct dimerization patterns. Using a Bell-basis representation, we derive closed-form results for entanglement entropies, Loschmidt echoes for finite and infinite systems, and identify both ΔΔΔ-independent and ΔΔΔ-dependent critical times. Quantum simulations on IBM Quantum processors demonstrate that key observables can be reliably extracted on noisy intermediate-scale devices without error mitigation.