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arXiv:1406.1408 · Physical Review B

Power-law approach to steady state in open lattices of noninteracting electrons

Mariya V. Medvedyeva, Stefan K. Kehrein

Published Nov 11, 2014• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.90.205410

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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 \ensuremathτ\ensuremath−3/2{\ensuremath{\tau}}^{\ensuremath{-}3/2}\ensuremathτ\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.
The Born-Markov approximation is widely used to study the dynamics of open quantum systems coupled to external baths. Using Keldysh formalism, we show that the dynamics of a system of bosons (fermions) linearly coupled to a noninteracting bosonic (fermionic) bath falls outside this paradigm if the bath spectral function has nonanalyticities as a function of frequency. In this case, we show that the dissipative and noise kernels governing the dynamics have distinct power-law tails. The Green's functions show a short-time ``quasi''-Markovian exponential decay before crossing over to a power-law tail governed by the nonanalyticity of the spectral function. We study a system of bosons (fermions) hopping on a one-dimensional lattice, where each site is coupled linearly to an independent bath of noninteracting bosons (fermions). We obtain exact expressions for the Green's functions of this system, which show power-law decay $\ensuremath{\sim}|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}3/2}$. We use these to calculate the density and current profile, as well as unequal-time current-current correlators. While the density and current profiles show interesting quantitative deviations from Markovian results, the current-current correlators show qualitatively distinct long-time power-law tails $|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}3}$ characteristic of non-Markovian dynamics. We show that the power-law decays survive in the presence of interparticle interaction in the system, but the crossover time scale is shifted to larger values with increasing interaction strength.
Abstract Relaxation processes in quantum systems coupled to external environments represent one of the most fundamental nonequilibrium phenomena in condensed matter physics. The Lindblad master equation provides a powerful framework for characterizing such open quantum dynamics. In this work, we systematically investigate how different types of quantum jump operators and system geometries influence the Liouvillian gap and the properties of the nonequilibrium steady state (NESS) in finite-size systems. We demonstrate that, due to the intricate structure of the Liouvillian superoperator, multiple NESSs with unphysical characteristics can emerge. The physically meaningful steady state must instead be understood as a superposition of these NESSs that collectively satisfy the required physical constraints. Furthermore, we find that the Liouvillian gap does not necessarily increase monotonically with the system–environment coupling strength. Instead, it can exhibit a nontrivial peak structure, corresponding to a minimum in the relaxation time. The magnitude of this peak is closely related to the symmetry properties of the system. Our results provide a deeper understanding of nonequilibrium behavior in finite quantum systems and offer new insights into the design and control of open quantum dynamics.
We develop a method of calculating the nonequilibrium steady state (NESS) of an open quantum system that is weakly coupled to reservoirs in different equilibrium states. We describe the system using a Redfield-type quantum master equation (QME). We decompose the Redfield QME into a Lindblad-type QME and the remaining part R\mathcal{R}R. Regarding the steady state of the Lindblad QME as the unperturbed solution, we perform a perturbative calculation with respect to R\mathcal{R}R to obtain the NESS of the Redfield QME. The NESS thus determined is exact up to the first order in the system-reservoir coupling strength (pump/loss rate), which is the same as the order of validity of the QME. An advantage of the proposed method in numerical computation is its applicability to systems larger than those in methods of directly solving the original Redfield QME. We apply the method to a noninteracting fermion system to obtain an analytical expression of the NESS density matrix. We also numerically demonstrate the method in a nonequilibrium quantum spin chain.
We develop a method of calculating the nonequilibrium steady state (NESS) of an open quantum system that is weakly coupled to reservoirs in different equilibrium states. We describe the system using a Redfield-type quantum master equation (QME). We decompose the Redfield QME into a Lindblad-type QME and the remaining part R\mathcal{R}R. Regarding the steady state of the Lindblad QME as the unperturbed solution, we perform a perturbative calculation with respect to R\mathcal{R}R to obtain the NESS of the Redfield QME. The NESS thus determined is exact up to the first order in the system-reservoir coupling strength (pump/loss rate), which is the same as the order of validity of the QME. An advantage of the proposed method in numerical computation is its applicability to systems larger than those in methods of directly solving the original Redfield QME. We apply the method to a noninteracting fermion system to obtain an analytical expression of the NESS density matrix. We also numerically demonstrate the method in a nonequilibrium quantum spin chain.
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