Density matrix of chaotic quantum systems
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THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
Density matrix of chaotic quantum systems Xinxin Yang and Pei Wang a Department of Physics, Zhejiang Normal University, Jinhua 321004, P.R. China
Received 8 February 2020 / Received in final form 15 August 2020 / Accepted 9 September 2020 Published online 14 October 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. The nonequilibrium dynamics in chaotic quantum systems denies a fully understanding up to now, even if thermalization in the long-time asymptotic state has been explained by the eigenstate thermalization hypothesis which assumes a universal form of the observable matrix elements in the eigenbasis of Hamiltonian. It was recently proposed that the density matrix elements have also a universal form, which can be used to understand the nonequilibrium dynamics at the whole time scale, from the transient regime to the long-time steady limit. In this paper, we numerically test these assumptions for density matrix in the models of spins.
1 Introduction The nonequilibrium dynamics of quantum many-body systems keeps on attracting attention of both experimentalists [1] and theorists [2,3]. For integrable systems, the case-by-case study of exact solutions revealed exotic properties of quantum states driven out of equilibrium. The long-time asymptotic state is far from thermal equilibrium, but should be described by the generalized Gibbs ensemble [4]. On the other hand, for systems whose classical counterparts are chaotic, it is widely believed that they thermalize in the long time limit [5]. But, the dynamics in the transient and intermediate time scale is still hard to explore, due to the lack of a reliable analytical or numerical method. The study of dynamics in quantum chaotic systems dated back to the early days of quantum mechanics, when the question has been raised as to how the statistical properties of equilibrium ensembles arise from the unitary evolution of a pure quantum state [6,7]. A breakthrough was made in 1950s by Wigner [8–10], who stated that the statistics of the eigenenergies of a chaotic system should be as same as that of a random matrix whose level spacing follows the Wigner–Dyson distribution. This statement was verified by both experiments and numerical simulations [11–15]. On the other hand, the level spacing satisfies a Poisson distribution in an integrable system, according to Berry and Tabor [16]. In the random matrix theory (RMT), the eigenstates of Hamiltonian are considered to be random vectors in the Hilbert space. This oversimplified picture ignores the dependence of eigenstate on the eigenenergy, and then fails to explain why the observable is a function of energy a
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or temperature of the system. The eigenstate thermalization hypothesis (ETH) [17–19] overcomes this problem by assuming a generic form of the matrix elements of observable operators in the eigenbasis of Hamiltonian: O ¯ δαβ + D− 21 E ¯ fO ω, E ¯ Rαβ Oαβ = O E ,
(1
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