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arXiv:1406.4312

Numerical determination of a non-equilibrium many-body statistical operator for quasi-bound electrons in a gated nanowire system

José María Castelo, K. M. Indlekofer

Published Jun 17, 2014• 0 citations
arXivHTMLPDF

Abstract

We present a numerical approach to construct a non-equilibrium many-body statistical operator $\hatρ_\mathrm{rel}$ for an adaptive subspace of relevant quasi-bound electronic states in a semiconductor nanowire-based field-effect transistor (NWFET). As a constraint for $\hatρ_\mathrm{rel}$, we assume that the single-particle density matrix $ρ_1$ is a given quantity, resulting from a non-equilibrium Green's function (NEGF) calculation for the NWFET for a given set of applied voltages. Two different orthonormal (ON) eigenbases for $\hatρ_\mathrm{rel}$ are considered: (A) a Slater determinant basis of natural orbitals (eigenstates of $ρ_1$) and (B) the eigenbasis of the projected many-body Hamiltonian $\hat{H}_\mathrm{rel}$ within a relevant Fock subspace of the system. As for the eigenvalues $w_n$ of $\hatρ_\mathrm{rel}$, we furthermore assume that $w_n$ have a generalized Boltzmann form, parameterized by effective electrochemical potentials of natural orbitals and a given temperature. From the determined $\hatρ_\mathrm{rel}$, in turn, one can calculate expectation values for any many-body observable within the relevant subspace. As an example, we analyze the electron density and the covariance of the density-density correlation function for representative electronic preparations of the NWFET.

DOI
10.48550/arxiv.1406.4312

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We develop a theoretical framework to determine distribution functions in nonequilibrium systems coupled to equilibrium reservoirs, by using the nonequilibrium Green's function technique. As a paradigmatic example, we consider the nonequilibrium distribution function in a nanowire under a bias voltage. We model the system as a tight-binding chain connected to reservoirs with different electrochemical potentials at both ends. For electron scattering processes in the wire, we consider both elastic scattering from impurities and inelastic scattering from phonons within the self-consistent Born approximation. We demonstrate that the nonequilibrium distribution functions, as well as the electrostatic potential profiles, in various scattering regimes are well described within our framework. This scheme will contribute to advancing our understanding of quantum many-body phenomena driven by nonequilibrium distribution functions that have different functional forms from the equilibrium ones.
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Theoretical descriptions of the non-equilibrium dynamics of quantum many-body systems essentially employ either (i) explicit treatments, relying on the truncation of the expansion of the many-body wave function, (ii) compressed representations of the many-body wave function, or (iii) evolution of an effective (downfolded) representation through Green's functions. In this work, we select representative cases of each of the methods and address how these complementary approaches capture the dynamics driven by intense field perturbations to non-equilibrium states. Under strong driving, the systems are characterized by strong entanglement of the single-particle density matrix and natural populations approaching those of a strongly interacting equilibrium system. We generate a representative set of results that are numerically exact and form a basis for a critical comparison of the distinct families of methods. We demonstrate that the compressed formulation based on similarity-transformed Hamiltonians (coupled-cluster approach) is practically exact in weak fields and, hence, weakly or moderately correlated systems. Coupled cluster, however, struggles for strong driving fields, under which the system exhibits strongly correlated behavior, as measured by the von Neumann entropy of the single-particle density matrix. The dynamics predicted by Green's functions in the (widely popular) G W approximation are less accurate, but improve significantly upon the mean-field results in the strongly driven regime.
In the quantum field theory with the vacuum being the ground state the Green functions are the vacuum expectation values of the chronological, retarded or advanced products of the field operators (Bjoken & Drell, 1964; Itzykson & Zuber, 1985; Peskin & Schroeder, 1995). They are the generalized functions of the real time variables ti (and also other spatial coordinates). For the application of the Green function technique to the study of the timeindependent phenomena in equilibrium many-body systems at a finite temperature, the Matsubara imaginary time Green functions were introduced and widely used (Abrikosov et al., 1975; Bruuns & Flensberg, 2004; Haken, 1976). They are the mean values over a statistical ensemble at a finite temperature of the chronological products of the imaginary timedependent operators. Both these types of Green functions are inadequate for the application to the study of the time-dependent phenomena in the many-body systems with a finite density and at a finite temperature, in particular the non-equilibrium systems. For the application to the study of the time-dependent dynamical processes in non-equilibrium many-body systems Keldysh (Keldysh, 1965) has introduced a more general class of timedependent Green functions at finite temperature and density. They are the mean values of the time-ordered products of quantum operators in the Heisenberg picture over statistical ensembles of many-body systems with finite densities and at finite temperatures (which may be non-vanishing). The simplest example is the two-point Green function
Abstract We review one of the most versatile theoretical approaches to the study of time-dependent correlated quantum transport in nano-systems: the non-equilibrium Green’s function (NEGF) formalism. Within this formalism, one can treat, on the same footing, inter-particle interactions, external drives and/or perturbations, and coupling to baths with a (piece-wise) continuum set of degrees of freedom. After a historical overview on the theory of transport in quantum systems, we present a modern introduction of the NEGF approach to quantum transport. We discuss the inclusion of inter-particle interactions using diagrammatic techniques, and the use of the so-called embedding and inbedding techniques which take the bath couplings into account non-perturbatively. In various limits, such as the non-interacting limit and the steady-state limit, we then show how the NEGF formalism elegantly reduces to well-known formulae in quantum transport as special cases. We then discuss non-equilibrium transport in general, for both particle and energy currents. Under the presence of a time-dependent drive—encompassing pump–probe scenarios as well as driven quantum systems—we discuss the transient as well as asymptotic behavior, and also how to use NEGF to infer information on the out-of-equilibrium system. As illustrative examples, we consider model systems general enough to pave the way to realistic systems. These examples encompass one- and two-dimensional electronic systems, systems with electron–phonon couplings, topological superconductors, and optically responsive molecular junctions where electron–photon couplings are relevant.