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arXiv:1607.05231 · Journal of Statistical Mechanics Theory and Experiment

A theory of nonequilibrium steady states in quantum chaotic systems

Pei Wang

Published Sep 1, 2017• 4 citations
arXivHTMLPDF

Abstract

Abstract Nonequilibrium steady state (NESS) is a quasistationary state, in which exist currents that continuously produce entropy, but the local observables are stationary everywhere. We propose a theory of NESS under the framework of quantum chaos. In an isolated quantum system whose density matrix follows a unitary evolution, there exist initial states for which the thermodynamic limit and the long-time limit are noncommutative. The density matrix ρ ^ of these states displays a universal structure. Suppose that | α 〉 and | β 〉 are different eigenstates of the Hamiltonian with energies E α and E β , respectively. 〈 α | ρ ^ | β 〉 behaves as a random number which has zero mean. In thermodynamic limit, the variance of 〈 α | ρ ^ | β 〉 is a smooth function of | E α − E β | , scaling as 1 / | E α − E β | 2 in the limit | E α − E β | → 0 . If and only if

DOI
10.1088/1742-5468/aa85c0

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Abstract Nonequilibrium steady state (NESS) is a quasistationary state, in which exist currents that continuously produce entropy, but the local observables are stationary everywhere. We propose a theory of NESS under the framework of quantum chaos. In an isolated quantum system whose density matrix follows a unitary evolution, there exist initial states for which the thermodynamic limit and the long-time limit are noncommutative. The density matrix ρ ^ of these states displays a universal structure. Suppose that | α 〉 and | β 〉 are different eigenstates of the Hamiltonian with energies E α and E β , respectively. 〈 α | ρ ^ | β 〉 behaves as a random number which has zero mean. In thermodynamic limit, the variance of 〈 α | ρ ^ | β 〉 is a smooth function of | E α − E β | , scaling as 1 / | E α − E β | 2 in the limit | E α − E β | → 0 . If and only if
The nonequilibrium dynamics of generic quantum systems denies a fully understanding up to now, even if the thermalization in the long-time asymptotic state has been explained by the eigenstate thermalization hypothesis which proposes a universal form of the observable matrix elements in the eigenbasis of the Hamiltonian. In this paper, we study the form of the quantum state, i.e., of the density matrix elements. We propose that the density matrix has also a universal form in chaotic systems, which is used to understand the nonequilibrium dynamics in the whole time scale, from the transient regime to the long-time steady limit, and then extends the applicability of eigenstate thermalization hypothesis to true nonequilibrium phenomena such as nonequilibrium steady states. Our assumption is numerically tested in various models, and its intimate relation to the eigenstate thermalization hypothesis is discussed.
The fluctuations of macroscopic observables in quantum systems which are in a nonequilibrium steady state are studied rigorously in the thermodynamic limit. In particular, the nonequilibrium steady state (NESS) of a quantum spin system that is composed of two infinite reservoirs at different temperatures coupled in a bounded region is considered. Under suitable assumptions on the interaction and the asymptotics of the fluctuations, a nonequilibrium central limit theorem for the fluctuations of the NESS expectation value of the empirical average of observables is proven by showing the analyticity of the corresponding moment generating function in a neighbourhood of the origin using the cluster expansion method. Furthermore, the large deviation principle is proven for a class of observables by establishing the existence and differentiability of the corresponding moment generating function.
We address the question of how a nonequilibrium steady state (NESS) is reached in the Lindbladian dynamics of an open quantum system. We develop an expansion of the density matrix in terms of the NESS excitations, each of which has its own (exponential) decay rate. However, when the decay rates tend to zero for many NESS excitations (the spectral gap of the Liouvillian is closed in the thermodynamic limit), the long-time dynamics of the system can exhibit a power-law behavior. This relaxation to NESS expectation values is determined by the density of states close to zero spectral gap and the value of the operator in these states. We illustrate this main idea on the example of the lattice of noninteracting fermions coupled to Markovian leads at infinite bias voltage. The current comes towards its NESS value starting from a typical initial state as ${\ensuremath{\tau}}^{\ensuremath{-}3/2}$. This behavior is universal and independent of the space dimension.
We consider Lindblad dynamics of quantum spin systems on infinite lattices and define a nonequilibrium steady state (NESS) and a time-averaged nonequilibrium steady state (TANESS) on the basis of C∗C^*C∗-algebraic formalism. Generically, the NESS on an infinite system does not equal the thermodynamic limit of NESSs on finite systems. We give a sufficient condition that they coincide with each other, in terms of both a condition number, which quantifies the normality of a Liouvillian, and some spectral gaps on finite subsystems. To appreciate the importance of the condition number, we provide an example in which the spectral gaps have nonzero lower bounds uniformly for any finite subsystems but a thermodynamic limit and a long-time limit (or a long-time average) do not commute with each other.
We study the nonequilibrium steady state (NESS) in a quantum system in contact with two heat baths at different temperatures. We use a time-independent perturbative expansion with respect to the coupling with the two heat baths to obtain the density matrix for the NESS. In particular, we show an explicit representation of the density matrix for the reflection symmetric and weakly nonequilibrium case. We also calculate the expectation value of the energy current and show that the Kubo formula holds in this case.
Understanding the emergence of chaos in many-body quantum systems away from semi-classical limits, particularly in spatially local interacting spin Hamiltonians, has been a long-standing problem. In these intrinsically quantum regimes, quantum chaos has been primarily understood through the correspondence between the eigensystem statistics of midspectrum eigenstates and the universal statistics described by random matrix theory (RMT). However, this correspondence no longer holds for finite-temperature eigenstates. Here we show that the statistical properties of finite-temperature eigenstates of quantum chaotic Hamiltonians can be accurately described by pure random states constrained by a local charge, with the average charge density of the constrained random state ensemble playing the same role as the average energy density of the eigenstates. By properly normalizing the energy density using a single Hamiltonian-dependent parameter that quantifies the typical energy per degree of freedom, we find excellent agreement between the entanglement entropy statistics of eigenstates and that of constrained random states. Interestingly, in small pockets of Hamiltonian parameter phase space which we previously identified as `maximally chaotic' [PRX 14, 031014 (2024)], we find excellent agreement not only at the level of the first moment, including O(1) corrections, but also at the level of statistical fluctuations. These results show that notions of maximal chaos -- in terms of how much randomness eigenstates contain -- can still be defined at finite temperature in physical Hamiltonian models away from semi-classical and large-NNN limits.