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arXiv:0910.0195 · New Journal of Physics

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

Tomaž Prosen, Bojan Žunkovič

Published Feb 26, 2010• 150 citations
arXivHTMLPDF

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 a 4 n ×4 n matrix. Non-equilibrium thermal driving in terms of the Redfield equation is analyzed in detail. We explain how one can compute all the 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 the 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 differential thermal conductance for sufficiently strong thermal driving as well as non-monotonic dependence of the heat current on the strength of the bath coupling.

DOI
10.1088/1367-2630/12/2/025016

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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 a 4 n ×4 n matrix. Non-equilibrium thermal driving in terms of the Redfield equation is analyzed in detail. We explain how one can compute all the 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 the 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 differential thermal conductance for sufficiently strong thermal driving as well as non-monotonic dependence of the heat current on the strength of the bath coupling.
Abstract. The Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4n × 4n matrix, provided that all Lindblad bath operators are linear in the fermionic variables. The method is applied to the explicit construction of non-equilibrium steady states and the calculation of asymptotic relaxation rates in the far from equilibrium problem of heat and spin transport in a nearest neighbor Heisenberg XY spin 1/2 chain in a transverse magnetic field. PACS numbers: 02.30.Ik, 03.65.Yz, 05.30.Fk, 75.10.PqA general method to solve master equations for quadratic open Fermi systems 2 1.
The Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4 n × 4 n matrix, provided that all Lindblad bath operators are linear in the fermionic variables. The method is applied to the explicit construction of non-equilibrium steady states (NESS) and the calculation of asymptotic relaxation rates in the far from equilibrium problem of heat and spin transport in a nearest neighbour Heisenberg XY spin-1/2 chain in a transverse magnetic field.
We present the Born-Markov approximated Redfield quantum master equation (RQME) description for an open system of noninteracting particles (bosons or fermions) on an arbitrary lattice of NNN sites in any dimension and weakly connected to multiple reservoirs at different temperatures and chemical potentials. The RQME can be reduced to the Lindblad equation, of various forms, by making further approximations. By studying the N=2N=2N=2 case, we show that RQME gives results which agree with exact analytical results for steady-state properties and with exact numerics for time-dependent properties over a wide range of parameters. In comparison, the Lindblad equations have a limited domain of validity in nonequilibrium. We conclude that it is indeed justified to use microscopically derived full RQME to go beyond the limitations of Lindblad equations in out-of-equilibrium systems. We also derive closed-form analytical results for out-of-equilibrium time dynamics of two-point correlation functions. These results explicitly show the approach to steady state and thermalization. These results are experimentally relevant for cold atoms, cavity QED, and far-from-equilibrium quantum dot experiments.
We build an exact framework to evaluate heat, energy, and particle transport between Gaussian reservoirs mediated by a quadratic quantum system. By combining full counting statistics with newly developed non-Markovian master equation approaches, we introduce an effective master equation whose solution can be used to generate arbitrary moments of the heat statistics for any number of reservoirs. This theory applies equally to fermionic and bosonic systems, holds at arbitrarily strong coupling, and resolves out-of-equilibrium transient dynamics determined by the system's initial state. In the steady-state, weak-coupling limit, we recover results analogous to those of the well-known Landauer-Büttiker formalism. We conclude our discussion by demonstrating an application of the method to a prototypical fermionic system. Our results uncover a regime of transient negative heat conductance contingent upon the initial system preparation, providing a clear signature of non-trivial out-of-equilibrium dynamics.
Explicit solution for the 2-point correlation function in a non-equilibrium steady state of a nearly isotropic boundary-driven open XY spin 1/2 chain in the Lindblad formulation is provided. A non-equilibrium quantum phase transition from exponentially decaying correlations to long-range order is discussed analytically. In the regime of long-range order a new phenomenon of correlation resonances is reported, where the correlation response of the system is unusually high for certain discrete values of the external bulk parameter, e.g. the magnetic field.
Abstract We formulate and discuss the explicit computation of dynamic correlation functions in open quadratic fermionic systems which are driven and dissipated by Lindblad jump processes that are linear in canonical fermionic operators. Dynamic correlators are interpreted in terms of a local quantum quench where the pre-quench state is a non-equilibrium steady state, i.e. a fixed point of the Liouvillian. As an example, we study the XY spin 1/2 chain and quadratic Majorana chains with boundary Lindblad driving, whose dynamics exhibit asymmetric (skewed) light cone behaviour. We also numerically treat the two-dimensional XY model and the XY spin chain with additional Dzyaloshinskii–Moriya interactions. The latter exhibits a new non-thermal phase transition which can be understood in terms of bifurcations of the quasi-particle dispersion relation. Finally, considering in some detail the periodic quadratic fermionic ring with dissipation at a single (arbitrary) site, we present analytical expressions for first order corrections (in the strength of dissipation) to the spectrum and non-equilibrium steady state (NESS) correlation functions.