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arXiv:1910.03138 · Physical Review A

Thermalization and its breakdown for a large nonlinear spin

Shane P. Kelly, Eddy Timmermans, Shan-Wen Tsai

Published Nov 6, 2020• 15 citations
arXivHTMLPDF

Abstract

By developing a semiclassical analysis based on the eigenstate thermalization hypothesis, we determine the long time behavior of a large spin evolving with a nonlinear Hamiltonian. Despite integrable classical dynamics, we find the eigenstate thermalization hypothesis for the diagonal matrix elements of observables is satisfied in the majority of eigenstates, and thermalization of long time averaged observables is generic. The exception is an unusual mechanism for the breakdown of thermalization based on an unstable fixed point in the classical dynamics. Using the semiclassical analysis, we derive how the equilibrium values of observables encode properties of the initial state. This analysis shows an unusual memory effect in which the remembered initial state property is not conserved in the integrable classical dynamics. We conclude with a discussion of relevant experiments and the potential generality of this mechanism for long time memory and the breakdown of thermalization.

DOI
10.1103/physreva.102.052210

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By developing a semiclassical analysis based on the eigenstate thermalization hypothesis, we determine the long time behavior of a large spin evolving with a nonlinear Hamiltonian. Despite integrable classical dynamics, we find the eigenstate thermalization hypothesis for the diagonal matrix elements of observables is satisfied in the majority of eigenstates, and thermalization of long time averaged observables is generic. The exception is an unusual mechanism for the breakdown of thermalization based on an unstable fixed point in the classical dynamics. Using the semiclassical analysis, we derive how the equilibrium values of observables encode properties of the initial state. This analysis shows an unusual memory effect in which the remembered initial state property is not conserved in the integrable classical dynamics. We conclude with a discussion of relevant experiments and the potential generality of this mechanism for long time memory and the breakdown of thermalization.
We investigate the relation between the classical ergodicity and the quantum eigenstate thermalization in the fully connected Ising ferromagnets. In the case of spin-1/2, an expectation value of an observable in a single-energy eigenstate coincides with the long-time average in the underlying classical dynamics, which is a consequence of the Wentzel-Kramers-Brillouin approximation. In the case of spin-1, the underlying classical dynamics is not necessarily ergodic. In that case, it turns out that, in the thermodynamic limit, the statistics of the expectation values of an observable in the energy eigenstates coincides with the statistics of the long-time averages in the underlying classical dynamics starting from random initial states sampled uniformly from the classical phase space. This feature seems to be a general property in semiclassical systems, and the result presented here is crucial in discussing equilibration, thermalization, and dynamical transitions of such systems.
Quantum thermalization of closed systems -- i.e. the relaxation of local properties to their thermodynamic value along the Hamiltonian evolution -- is generically expected in non-integrable systems at finite energy density. In this work we show that quantum spin systems with at least one spin component coupled through long-range ferromagnetic interactions, and a second spin component coupled to a field, cannot in general thermalize to a paramagnetic state whenever they have a finite, negative energy per spin. The long-range interactions we consider decay as a power law of the distance, ∼r−α\sim r^{-α}∼r−α with α<dα< dα<d, where ddd is the number of spatial dimensions of the system. The lack of thermalization is generic to an infinite family of Hamiltonians, and it is not associated to any form of integrability. It can instead be understood by the anomalous features of the thermodynamics of the system in the paramagnetic phase, which cannot be reflected in the properties of the eigenstates. This leads to a breakdown of eigenstate typicality and of the eigenstate thermalization hypothesis. Interestingly, the non-thermal states emerging at long times in the dynamics display long-range multipartite entanglement, elevating the breakdown of thermalization to a resource.
Deriving conditions under which a macroscopic system thermalizes directly from the underlying quantum many-body dynamics of its microscopic constituents is a long-standing challenge in theoretical physics. The well-known eigenstate thermalization hypothesis (ETH) is presumed to be a key mechanism, but has defied rigorous verification for generic systems thus far. A weaker variant (weak ETH), by contrast, is provably true for a large variety of systems, including even many integrable models, but its implications with respect to the problem of thermalization are still largely unexplored. Here we analytically demonstrate that systems satisfying the weak ETH exhibit thermalization for two very natural classes of far-from-equilibrium initial conditions: the overwhelming majority of all pure states with a preset nonequilibrium expectation value of some given local observable, and the Gibbs states of a Hamiltonian which subsequently is subject to a quantum quench in the form of a sudden change of some local system properties.
Abstract In this work we study a scenario where unitary quantum dynamics in a many-body interacting system is restricted to a single excitation subspace. We ask how dynamics within to such a subspace may in general differ from predictions of the eigenstate thermalization hypothesis (ETH). We show that for certain initial states and observables, if thermalization occurs, it will not fulfil other key predictions of the ETH; instead following differing generic behaviours. We show this by analysing long-time fluctuations, two-point correlation functions, and the out-of-time-ordered correlator; analytically detailing deviation from ETH predictions. We derive instead an ETH-like relation, with non-random off-diagonals for matrix elements of observables, with correlations which alter long-time behaviour and constrain dynamics. Further, we analytically compute the time-dependence of the decay to equilibrium, showing it is proportional to the survival probability of the initial state. We finally note the conditions studied are common in many physical scenarios, such as under the rotating-wave approximation. We show numerically our predictions are robust to perturbations which break this approximation.
We study a fully connected quantum spin model resonantly coupled to a small environment of noninteracting spins, and investigate how initial state properties are remembered at long times. We find memory of initial state properties, in addition to the total energy, that are not conserved by the dynamics. This memory occurs in the middle of the spectrum where an eigenstate quantum phase transition (ESQPT) occurs as a function of energy. The memory effect at that energy in the spectrum is robust to system-environment coupling until the coupling changes the energy of the ESQPT. This work demonstrates the effect of ESQPT memory as independent of integrability and suggests a wider generality of this mechanism for preventing thermalization at ESQPTs.
The eigenstate thermalization hypothesis (ETH) is a successful theory that establishes the criteria for ergodicity and thermalization in isolated quantum many-body systems. In this work, we investigate the thermalization properties of a spin-1/21/21/2 XXZ chain with linearly inhomogeneous interactions. We demonstrate that introduction of the inhomogeneous interactions leads to an onset of quantum chaos and thermalization, which, however, becomes inhibited for sufficiently strong inhomogeneity. To exhibit ETH, and to display its breakdown upon varying the strength of interactions, we probe statistics of energy levels and properties of matrix elements of local observables in eigenstates of the inhomogeneous XXZ spin chain. Moreover, we investigate the dynamics of the entanglement entropy and the survival probability which further evidence the thermalization and its breakdown in the considered model. We outline a way to experimentally realize the XXZ chain with linearly inhomogeneous interactions in systems of ultracold atoms. Our results highlight a mechanism of emergence of ETH due to insertion of inhomogeneities in an otherwise integrable system and illustrate the arrest of quantum dynamics in the presence of strong interactions.