Exact solution of Markovian master equations for quadratic Fermi systems: thermal baths, open XY spin chains and non-equilibrium phase transition

Abstract

We generalize the method of third quantization to a unified exact treatment of Redfield and Lindblad master equations for open quadratic systems of n fermions in terms of diagonalization of 4n × 4n matrix. Non-equilibrium thermal driving in terms of the Redfield equation is analyzed in detail. We explain how to compute all physically relevant quantities, such as non-equilibrium expectation values of local observables, various entropies or information measures, or time evolution and properties of relaxation. We also discuss how to exactly treat explicitly time dependent problems. The general formalism is then applied to study a thermally driven open XY spin 1/2 chain. We find that recently proposed non-equilibrium quantum phase transition in the open XY chain survives the thermal driving within the Redfield model. In particular, the phase of long-range magnetic correlations can be characterized by hypersensitivity of the non-equilibrium-steady state to external (bath or bulk) parameters. Studying the heat transport we find negative thermal conductance for sufficiently strong thermal driving, as well as non-monotonic dependence of the heat current on the strength of the bath coupling.

Relaxation times of dissipative many-body quantum systems

“…For γ < γ c we see that the gap (red pluses) is proportional to γ (horizontal line in Fig. 7) and therefore perturbation series (13) holds. The convergence radius γ c shrinks algebraically with L in the gapless phase ∆ < 1, at ∆ = 0.5 it decays as γ c 4.1/L 1.2 , although we can not exclude asymptotic ∼ 1/L scaling, while it is exponentially small in the gapped phase ∆ > 1.…”

“…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. Understanding how the gap scales with the system size is therefore of fundamental importance.…”

From Thermal Rectifiers to Thermoelectric Devices

“…Particularly, in analogy with the studies at the classical level the relation between the validity of Fourier law and the onset of quantum chaos has been investigated in recent years [123,105,102,137,115,116].…”

Nonequilibrium Phase Transition in a Periodically DrivenXYSpin Chain

“…[8]), so many effects can be analyzed exactly or in great detail. For example, quantum phase transitions in nonequilibrium steady states have been observed either in quasi-free [5,9], or strongly interacting [6], or even dissipative [7,10] quantum systems in one dimension. …”

“…[9] for the formulation in compatible notation). We focus on quasifree dynamics where the Hamiltonian is given in terms of a quadratic form H = j,k w j H j,k w k ≡ w · Hw with antisymmetric imaginary matrix H and linear Lindblad operators…”