Rapid mixing implies exponential decay of correlations

Abstract

We provide an analysis of the correlation properties of spin and fermionic systems on a lattice evolving according to open system dynamics generated by a local primitive Liouvillian. We show that if the Liouvillian has a spectral gap which is independent of the system size, then the correlations between local observables decay exponentially as a function of the distance between their supports. We prove, furthermore, that if the Log-Sobolev constant is independent of the system size, then the system satisfies clustering of correlations in the mutual information - a much more stringent form of correlation decay. As a consequence, in the latter case we get an area law (with logarithmic corrections) for the mutual information. As a further corollary, we obtain a stability theorem for local distant perturbations. We also demonstrate that gapped free-fermionic systems exhibit clustering of correlations in the covariance and in the mutual information. We conclude with a discussion of the implications of these results for the classical simulation of open quantum systems with matrix-product operators and the robust dissipative preparation of topologically ordered states of lattice spin systems.Comment: 25 pages, 2 figures, replaced by final versio

Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

“…It has also been shown that in 1D exponential clustering of correlations already implies an area law [7]. For local Liouvillians, general area laws (in terms of entropic measures suitable for mixed states) can be derived for stationary states [33], again using Lieb-Robinson bounds.…”

“…[55] and has been made rigorous and largely generalized in Ref. [33]: If a local Liouvillian is primitive (that is, if its stationary state has full rank) and has a spectral gap which is independent of the system size, then correlation functions between local observables again decay exponentially as a function of the distance between their supports.…”

“…For the case of a unique stationary state the spectral gap of the Liouvillian is a measure of the speed of convergence [33] towards this stationary state. Evolving some state ρ from s to r ≥ s and then from r to t ≥ r also yields ρ s (t) and hence the propagator has the composition property…”

“…Stability of stationary states: Inspired by the stability results on Hamiltonian ground states, Lieb-Robinson bounds have also been used to prove the stability of stationary states of certain local Liouvillians [13,33].…”

Relaxation times of dissipative many-body quantum systems

“…Namely, if the gap is finite (so-called rapidly mixing systems) one can show that this implies a clustering of correlations in the steady state [8,9], meaning that local observables are uncorrelated on a scale larger than ∼ 1/g. Rapid mixing also implies the stability of steady state to local perturbations [10][11][12]. If the gap on the other hand closes in the thermodynamic limit this can lead to a nonequilibrium phase transition [13][14][15][16][17][18] and can result in a non-exponential relaxation [19,20] towards a steady state.…”