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arXiv:2303.11148 · Physical Review Research

Nonequilibration, synchronization, and time crystals in isotropic Heisenberg models

Peter Reimann, Patrick Vorndamme, Jürgen Schnack

Published Oct 12, 2023• 5 citations
arXivHTMLPDF

Abstract

Isotropic but otherwise largely arbitrary Heisenberg models in the presence of a homogeneous magnetic field are considered, including various integrable, nonintegrable, as well as disordered examples, and not necessarily restricted to one dimension or short-range interactions. Taking for granted that the nonequilibrium initial condition and the spectrum of the field-free model satisfy some very weak requirements, expectation values of generic observables are analytically shown to exhibit permanent long-time oscillations, thus ruling out equilibration. If the model (but not necessarily the initial condition) is translationally invariant, the long-time oscillations are moreover shown to exhibit synchronization in the long run, meaning that they are invariant under arbitrary translations of the observable. Analogous long-time oscillations are also recovered for temporal correlation functions when the system is already at thermal equilibrium from the outset, thus realizing a so-called time crystal.

DOI
10.1103/physrevresearch.5.043040

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Isotropic but otherwise largely arbitrary Heisenberg models in the presence of a homogeneous magnetic field are considered, including various integrable, nonintegrable, as well as disordered examples, and not necessarily restricted to one dimension or short-range interactions. Taking for granted that the nonequilibrium initial condition and the spectrum of the field-free model satisfy some very weak requirements, expectation values of generic observables are analytically shown to exhibit permanent long-time oscillations, thus ruling out equilibration. If the model (but not necessarily the initial condition) is translationally invariant, the long-time oscillations are moreover shown to exhibit synchronization in the long run, meaning that they are invariant under arbitrary translations of the observable. Analogous long-time oscillations are also recovered for temporal correlation functions when the system is already at thermal equilibrium from the outset, thus realizing a so-called time crystal.
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Local observables of closed many-body quantum systems are generally believed to quickly equilibrate after a quench, according to the eigenstate thermalization hypothesis. Here, the authors show that the paradigmatic Heisenberg quantum magnet violates this fundamental assumption. They analytically find measurable local correlation functions that show perpetual oscillations, while at the same time all single-particle observables equilibrate. They link their existence to the presence of dynamical symmetries and obtain a fractal dependence of their oscillation frequency on the interaction strength.
A discrete time crystal is an out-of-equilibrium phase of matter characterized by the spontaneous breaking of discrete time-translation symmetry. Using extensive numerical simulations based on matrix-product-state methods, we provide evidence for discrete time-crystalline behavior in strongly disordered spin chains with Heisenberg interactions, including the isotropic point, subject to periodic driving. Starting from a many-body localized regime, we observe that rotations induced by delta kicks produce a pronounced subharmonic response at half the drive frequency in spin observables. We investigate the stability of this response against rotation-angle errors through entanglement entropy, quantum Fisher information, short-range spin correlations, and restricted-control ergotropy. Increasing the rotation error reveals an intermediate dynamical regime separating the time-crystalline and Floquet-localized responses. In this regime, most observables exhibit signatures of weakly correlated dynamics reminiscent of Anderson localization.
Time crystals in periodically driven systems have initially been studied assuming either the ability to quench the Hamiltonian between different many-body regimes, the presence of disorder or long-range interactions. Here we propose the simplest scheme to observe discrete time crystal dynamics in a one-dimensional driven quantum system of the Ising type with short-range interactions and no disorder. The system is subject only to a periodic kick by a global magnetic field, and no extra Hamiltonian quenching is performed. We analyze the emerging time crystal stabilizing mechanisms via extensive numerics as well as using an analytic approach based on an off-resonant transition model. Due to the simplicity of the driven Ising model, our proposal can be implemented with current experimental platforms including trapped ions, Rydberg atoms, and superconducting circuits.
A theoretical approach to the one-dimensional isotropic Heisenberg model at low temperatures is presented which uses rotationally invariant combinations of spin operators only and is thus adapted to treat correctly effects resulting from the absence of long-range order in these systems. Analytic expressions for space- and time-dependent spin correlation functions are given for small values of TJS2\frac{T}{J{S}^{2}}JS2T​ and 1S\frac{1}{S}S1​. The results include an expression for the spin correlation function in the antiferromagnetic ground state and a discussion of changes in line shape, due to quantum effects, for short-wavelength magnons as observed in neutron scattering experiments.
The primary goals of this dissertaion are to describe and elucidate a new formalism to study the out of equilibrium dynamics of integrable models, and to apply this formalism to some specific problems. In particular, I describe the “quench dynamics” of a gas of interacting bosons in one dimension, and provide details of some preliminary work on the isotropic Heisenberg chain. The formalism, based on earlier work by Yudson, provides a way of getting around some of the difficulties involved in calculating the time evolution of arbitrary initial states evolved with integrable Hamiltonians. In addition to this, I aim to discuss some of the general aspects of quench dynamics in quantum systems. The description of the nonequilibrium dynamics of a given system depends significantly on initial states, and the time scales at which the system is probed compared with the various inherent time scales in the system. I present the experimental context and motivation for these studies, and survey the existing techniques and efforts at understanding the relaxation of systems that are far from their equilibrium states or ground states.