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On the steady state correlation functions of open interacting systems

Horia D. Cornean, Valeriu Moldoveanu, Claude‐Alain Pillet

Published Jan 1, 2013• 15 citations
OpenAlex

Abstract

We address the existence of steady state Green-Keldysh correlation functions of interacting fermions in mesoscopic systems for both the partitioning and partition-free scenarios. Under some spectral assumptions on the non-interacting model and for sufficiently small interaction strength, we show that the system evolves to a NESS which does not depend on the profile of the time-dependent coupling strength/bias. For the partitioned setting we also show that the steady state is independent of the initial state of the inner sample. Closed formulae for the NESS two-point correlation functions (Green-Keldysh functions), in the form of a convergent expansion, are derived. In the partitioning approach, we show that the 0th order term in the interaction strength of the charge current leads to the Landauer-Buettiker formula, while the 1st order correction contains the mean-field (Hartree-Fock) results.

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We address the existence of steady state Green-Keldysh correlation functions of interacting fermions in mesoscopic systems for both the partitioning and partition-free scenarios. Under some spectral assumptions on the non-interacting model and for sufficiently small interaction strength, we show that the system evolves to a NESS which does not depend on the profile of the time-dependent coupling strength/bias. For the partitioned setting we also show that the steady state is independent of the initial state of the inner sample. Closed formulae for the NESS two-point correlation functions (Green-Keldysh functions), in the form of a convergent expansion, are derived. In the partitioning approach, we show that the 0th order term in the interaction strength of the charge current leads to the Landauer-Buettiker formula, while the 1st order correction contains the mean-field (Hartree-Fock) results.
Under certain conditions, we prove the existence of a steady-state transport regime for interacting mesoscopic systems coupled to reservoirs (leads). The partitioning and partition-free scenarios are treated on an equal footing. Our time-dependent scattering approach is exact and proves, among other things, the independence of the steady-state quantities from the initial state of the sample. Closed formulas for the steady-state current amenable for perturbative calculations with regard to the interaction strength are also derived. In the partitioning case, we calculate the first-order correction and recover the mean-field (Hartree-Fock) results.
We construct a non-equilibrium steady state and calculate the corresponding current for a mesoscopic Fermi system in the partition-free setting. To this end we study a small sample coupled to a finite number of semi-infinite leads. Initially, the whole system of quasi-free fermions is in a grand-canonical equilibrium state. At t = 0 we turn on a potential bias on the leads and let the system evolve. We study how the charge current behaves in time and how it stabilizes itself around a steady-state value, which is given by a Landauer-type formula.
Full understanding of the relaxation mechanisms and far-from-equilibrium transport in modern mesoscopic structures requires that such systems be treated as open. We therefore generalize some of the core elements of the Kadanoff-Baym-Keldysh nonequilibrium Green's function formalism, inherently formulated for closed systems, to treatment of an open system, coupled with its environment. We define the two-time correlation functions and analyze the influence of the memory effects on the open-system transport. In the transient regime, the two-time correlation functions clearly show four distinct terms: a closed-system-like term, an entanglement term, and two memory terms that depend explicitly on the initial state of the environment. We show that it is not possible to completely eliminate the influence of the environment by a fortunate choice of the initial state, and approximating the system as closed is valid only in the limit of negligible system-environment coupling, which is never the case in the transient regime. We derive the transport equations for transients that properly account for the system-environment coupling. On the other hand, we address the important issue of transport in a far-from-equilibrium steady state. We show that, once a steady state is reached, the balance between the driving and relaxation forces implies that the two-time correlation functions regain a closed-system-like form, but with an effective, modified system Hamiltonian, and with the system statistical operator unrelated to that of the initial state. We emphasize that the difference between the transient and the far-from-equilibrium steady-state regimes, crucial for theoretical investigation of nonequilibrium quasiparticle transport, effectively lies within the different relative magnitude of the combined entanglement and memory terms with respect to the closed-system-like term in two-time correlation functions.
Schwinger Keldysh field theory is a widely used paradigm to study nonequilibrium dynamics of quantum many-body systems starting from a thermal state. We extend this formalism to describe nonequilibrium dynamics of quantum systems starting from arbitrary initial many-body density matrices. We show how this can be done for both bosons and fermions, and for both closed and open quantum systems, using additional sources coupled to bilinears of the fields at the initial time, calculating Green's functions in a theory with these sources, and then taking appropriate set of derivatives of these Green's functions with respect to initial sources to obtain physical observables. The set of derivatives depends on the initial density matrix. The physical correlators in a dynamics with arbitrary initial conditions do not satisfy Wick's theorem, even for noninteracting systems. However, our formalism constructs intermediate ``$n\ensuremath{-}\mathrm{particle}$ Green's functions'' that obey Wick's theorem and provide a prescription to obtain physical correlation functions from them. This allows us to obtain analytic answers for all physical many-body correlation functions of a noninteracting system even when it is initialized to an arbitrary density matrix. We use these exact expressions to obtain an estimate of the violation of Wick's theorem and relate it to presence of connected multiparticle initial correlations in the system. We illustrate this new formalism by calculating density and current profiles in many-body fermionic and bosonic open quantum systems initialized to nontrivial density matrices. We have also shown how this formalism can be extended to interacting many-body systems.
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.
A numerical approach is presented that allows to compute nonequilibrium steady-state properties of strongly correlated quantum many-body systems. The method is imbedded in the Keldysh Green's function formalism and is based upon the idea of the variational cluster approach as far as the treatment of strong correlations is concerned. It appears that the variational aspect is crucial as it allows for a suitable optimization of a ``reference'' system to the nonequilibrium target state. The approach is neither perturbative in the many-body interaction nor in the field, that drives the system out of equilibrium, and it allows to study strong perturbations and nonlinear responses of systems in which also the correlated region is spatially extended. We apply the presented approach to nonlinear transport across a strongly correlated quantum wire described by the fermionic Hubbard model. We illustrate how the method bridges to cluster dynamical mean-field theory upon coupling two baths containing an increasing number of uncorrelated sites.