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arXiv:quant-ph/0606216 · Europhysics Letters (EPL)

Thermalization through unitary evolution of pure states

Stein Olav Skrøvseth

Published Nov 22, 2006• 8 citations
arXivHTMLPDF

Abstract

The unitary time evolution of a critical quantum spin chain with an impurity is calculated, and the entanglement evolution is shown. Moreover, we show that the reduced density matrix of a part of the chain evolves such that the fidelity of its spectrum is very high with respect to a state in thermal equilibrium. Hence, a thermal state occurs through unitary time evolution in a simple spin chain with impurity.

DOI
10.1209/epl/i2006-10388-9

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The unitary time evolution of a critical quantum spin chain with an impurity is calculated, and the entanglement evolution is shown. Moreover, we show that the reduced density matrix of a part of the chain evolves such that the fidelity of its spectrum is very high with respect to a state in thermal equilibrium. Hence, a thermal state occurs through unitary time evolution in a simple spin chain with impurity.
We explore the dynamics of the entanglement entropy near equilibrium in highly entangled pure states of two quantum-chaotic spin chains undergoing unitary time evolution. We examine the relaxation to equilibrium from initial states with either less or more entanglement entropy than the equilibrium value, as well as the dynamics of the spontaneous fluctuations of the entanglement that occur in equilibrium. For the spin chain with a time-independent Hamiltonian and thus an extensive conserved energy, we find slow relaxation of the entanglement entropy near equilibration. Such slow relaxation is absent in a Floquet spin chain with a Hamiltonian that is periodic in time and thus has no local conservation law. Therefore, we argue that slow diffusive energy transport is responsible for the slow relaxation of the entanglement entropy in the Hamiltonian system.
Thermalization plays a central role in out-of-equilibrium physics of ultracold atoms or electronic transport phenomena. On the other hand, entanglement concepts have proven to be extremely useful to investigate quantum phases of matter. Here, it is argued that bipartite entanglement measures provide key information on out-of-equilibrium states and might therefore offer stringent thermalization criteria. This is illustrated by considering a global quench in an (extended) XXZ\mathit{XXZ}XXZ spin-1/2 chain across its (zero-temperature) quantum critical point. A nonlocal bipartition of the chain preserving translation symmetry is proposed. The time evolution after the quench of the reduced density matrix of the half-system is computed and its associated (time-dependent) entanglement spectrum is analyzed. Generically, the corresponding entanglement entropy quickly reaches a ``plateau'' after a short transient regime. However, in the case of the integrable XXZ\mathit{XXZ}XXZ chain, the low-energy entanglement spectrum still reveals strong time fluctuations. In addition, its infinite-time average shows strong deviations from the spectrum of a Boltzmann thermal density matrix. In contrast, when the integrability of the model is broken (by small next-nearest-neighbor couplings), the entanglement spectra of the time average and thermal density matrices become remarkably similar.
We study the dynamics of the entanglement spectrum, that is the time evolution of the eigenvalues of the reduced density matrices after a bipartition of a one-dimensional spin chain. Starting from the ground state of an initial Hamiltonian, the state of the system is evolved in time with a new Hamiltonian. We consider both instantaneous and quasi adiabatic quenches of the system Hamiltonian across a quantum phase transition. We analyse the Ising model that can be exactly solved and the XXZ for which we employ the time-dependent density matrix renormalisation group algorithm. Our results show once more a connection between the Schmidt gap, i.e. the difference of the two largest eigenvalues of the reduced density matrix and order parameters, in this case the spontaneous magnetisation.
Abstract Classical arguments for thermalization of isolated systems do not apply in a straightforward way to the quantum case. Recently, there has been interest in diagnostics of quantum chaos in many-body systems. In the classical case, chaos is a popular explanation for the legitimacy of the methods of statistical physics. In this work, we relate a previously proposed criteria of quantum chaos in the unitary time evolution operator to the entanglement entropy growth for a far-from-equilibrium initial pure state. By mapping the unitary time evolution operator to a doubled state, chaos can be characterized by suppression of mutual information between subsystems of the past and that of the future. We show that when this mutual information is small, a typical unentangled initial state will evolve to a highly entangled final state. Our result provides a more concrete connection between quantum chaos and thermalization in many-body systems.
We characterize the early stages of the approach to equilibrium in isolated quantum systems through the evolution of the entanglement spectrum. We find that the entanglement spectrum of a subsystem evolves with three distinct timescales. First, on an o(1) timescale, independent of system or subsystem size and the details of the dynamics, the entanglement spectrum develops nearest-neighbor level repulsion. The second timescale sets in when the light cone has traversed the subsystem. Between these two times, the density of states of the reduced density matrix takes a universal, scale-free 1/f form; thus, random-matrix theory captures the local statistics of the entanglement spectrum but not its global structure. The third time scale is that on which the entanglement saturates; this occurs well after the light cone traverses the subsystem. Between the second and third times, the entanglement spectrum compresses to its thermal Marchenko-Pastur form. These features hold for chaotic Hamiltonian and Floquet dynamics as well as a range of quantum circuit models.
We study the time evolution of the reduced density matrix of a system of spin-1/2 particles interacting with an environment of spin-1/2 particles.The initial state of the composite system is taken to be a product state of a pure state of the system and a pure state of the environment.The latter pure state is prepared such that it represents the environment at a given finite temperature in the canonical ensemble.The state of the composite system evolves according to the time-dependent Schrödinger equation, the interaction creating entanglement between the system and the environment.It is shown that independent of the strength of the interaction and the initial temperature of the environment, all the eigenvalues of the reduced density matrix converge to their stationary values, implying that also the entropy of the system relaxes to a stationary value.We demonstrate that the difference between the canonical density matrix and the reduced density matrix in the stationary state increases as the initial temperature of the environment decreases.As our numerical simulations are necessarily restricted to a modest number of spin-1/2 particles (< 36), but do not rely on time-averaging of observables nor on the assumption that the coupling between system and environment is weak, they suggest that the stationary state of the system directly follows from the time evolution of a pure state of the composite system, even if the size of the latter cannot be regarded as being close to the thermodynamic limit.