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Physical review. B, Condensed matter

Nonequilibrium linked cluster expansion for steady-state quantum transport

Petr Král

Published Sep 15, 1997• 22 citations
Publisher

Abstract

We generalize the linked cluster expansion of Matsubara Green functions into nonequilibrium situations. This allows us to compute nonequilibrium correlation functions and, consequently, physical observables for interacting quantum systems with intermediate interaction strengths. As a specific example, we study nonlinear dc transport in a resonant tunneling system with electron-phonon interaction. The first- and second-order nonequilibrium cluster approximations are explicitly evaluated. The results are compared with those obtained by the standard nonequilibrium Green functions approach within the Migdal approximation for the self-energy and in the low-density limit also with the known exact solution. For moderately strong electron-phonon interactions the first-order linked cluster approximation is more precise than the Migdal approach. The second-order approximation appears to be problematic in the presence of the Fermi sea.

DOI
10.1103/physrevb.56.7293

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We generalize the linked cluster expansion of Matsubara Green functions into nonequilibrium situations. This allows us to compute nonequilibrium correlation functions and, consequently, physical observables for interacting quantum systems with intermediate interaction strengths. As a specific example, we study nonlinear dc transport in a resonant tunneling system with electron-phonon interaction. The first- and second-order nonequilibrium cluster approximations are explicitly evaluated. The results are compared with those obtained by the standard nonequilibrium Green functions approach within the Migdal approximation for the self-energy and in the low-density limit also with the known exact solution. For moderately strong electron-phonon interactions the first-order linked cluster approximation is more precise than the Migdal approach. The second-order approximation appears to be problematic in the presence of the Fermi sea.
We implement a highly efficient strong-coupling expansion for the Green's function of the Hubbard model. In the limit of extreme correlations, where the onsite interaction is infinite, the evaluation of diagrams simplifies dramatically enabling us to carry out the expansion to the eighth order in powers of the hopping amplitude. We compute the finite-temperature Green's function analytically in the momentum and Matsubara frequency space as a function of the electron density. Employing Padé approximations, we study the equation of state, Kelvin thermopower, momentum distribution function, quasiparticle fraction, and quasiparticle lifetime of the system at temperatures lower than, or of the order of, the hopping amplitude. We also discuss several different approaches for obtaining the spectral functions through analytic continuation of the imaginary frequency Green's function, and show results for the system near half filling. We benchmark our results for the equation of state against those obtained from a numerical linked-cluster expansion carried out to the eleventh order.
The cluster perturbation theory (CPT) is one of the simplest but systematic quantum cluster approaches to lattice models of strongly correlated electrons with local interactions. By treating the interactions, in addition to the intercluster potential, as a perturbation, it is shown that the CPT can be reformulated as an all-order resummation of diagrams within standard weak-coupling perturbation theory where vertex corrections are neglected. This reformulation is shown to allow for a straightforward generalization of the CPT to the general nonequilibrium case using contour-ordered Green's functions. By solving the resulting generalized CPT equation on the discretized Keldysh-Matsubara time contour, the transient dynamics of an essentially arbitrary initial pure or mixed state can be traced. In this way, the time-dependent expectation values of one-particle observables can be obtained within an approximation that neglects spatial correlations beyond the extension of the reference cluster. The necessary computational effort is very moderate. A detailed discussion and simple test calculations are presented to demonstrate the strengths and the shortcomings of the proposed approach. The nonequilibrium CPT is systematic and is controlled in principle by the inverse cluster size. It interpolates between the noninteracting and the atomic or decoupled-cluster limit which are recovered exactly and is found to predict the correct dynamics at very short times in a general nontrivial case. The effects of initial-state correlations on the subsequent dynamics and the necessity to extend the Keldysh contour by the imaginary Matsubara branch are analyzed carefully and demonstrated numerically. It is further shown that the approach can describe the dissipation of spin and charge to an uncorrelated bath with an essentially arbitrary number of degrees of freedom.
We present a new approach to treat correlations in nonequilibrium quantum many-particle system. The method is based on ideas of configuration interaction theory of exact nonperturbative ground state electronic structure calculations. We use superoperator techniques in Liouville–Fock space and represent the nonequilibrium density matrix as a linear combination of all possible nonequilibrium quasiparticle excitations built on the appropriate reference state. As an example we consider the electron transport through the system with electron–phonon interaction. The concept of embedding (buffer zones between the reservoirs and the correlated quantum system) is used to derive an exact master equation for the reduced density matrix. Using approximate (truncated) expansion of the trial density matrix we obtain the linear system of equations for two-quasiparticle amplitudes. Then we compute the steady-state current and compare the result with other approaches. The current conserving property of the method is proven.
A single-time quantum transport equation, which includes effects beyond the quasiparticle approximation, is derived for Fermi systems in the framework of nonequilibrium real-time Green's functions theory. Ternary correlations are incorporated in the kinetic description via a cluster expansion for the self-energies (e.g., the transport vertex and the width) truncated at the level of three-body scattering amplitudes. A finite temperature/density formulation of the three-body problem is given. Corresponding three-body equations reduce to the well-known Faddeev equations in the vacuum limit. In equilibrium the equation of state contains virial corrections proportional to the third quantum virial coefficient.
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.
We develope a method of mapping quantum nonequilibrium steady-state to an effective equilibrium system and present an algorithm to calculate electron-transport using an equilibrium technique. A systematic implementation of boundary conditions in steady-state nonequilibrium is made in the statistical operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ constructed from scattering state operators. We explicitly demonstrate the equivalence of this method to nonequilibrium Green function techniques for a noninteracting quantum dot model. In electron-phonon coupled quantum dot systems, we formulate an algorithm to construct the statistical bias operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ and perform a full many-body calculation with the quantum Monte Carlo technique. The results coherently demonstrate various transport behaviors such as phonon dephasing, $I\ensuremath{-}V$ staircase, and phonon-assisted tunneling phenomena. This formulation makes the existing computational quantum many-body techniques applicable to quantum steady-state nonequilibrium problems, which will complement the theories based on the diagrammatic approach.