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\nof Redfield and Lindblad master equations for open quadratic systems of n\nfermions in terms of diagonalization of 4n x 4n matrix. Non-equilibrium thermal\ndriving in terms of the Redfield equation is analyzed in detail. We explain how\nto compute all physically relevant quantities, such as non-equilibrium\nexpectation values of local observables, various entropies or information\nmeasures, or time evolution and properties of relaxation. We also discuss how\nto exactly treat explicitly time dependent problems. The general formalism is\nthen applied to study a thermally driven open XY spin 1/2 chain. We find that\nrecently proposed non-equilibrium quantum phase transition in the open XY chain\nsurvives the thermal driving within the Redfield model. In particular, the\nphase of long-range magnetic correlations can be characterized by\nhypersensitivity of the non-equilibrium-steady state to external (bath or bulk)\nparameters. Studying the heat transport we find negative thermal conductance\nfor sufficiently strong thermal driving, as well as non-monotonic dependence of\nthe heat current on the strength of the bath coupling.\n