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Physical review. B, Condensed matter

Quantum Monte Carlo study of the transport through a small interacting system

Akira Oguri

Published Nov 15, 1997• 23 citations
Publisher

Abstract

A numerical analysis for evaluating the transport coefficient through a small interacting system is examined using the quantum Monte Carlo (QMC) method. To formulate the analysis, we consider a finite interacting system on a one-dimensional lattice that is connected to noninteracting leads, and investigate the expression of the dc conductance making use of the Kubo formula with the perturbation theory in the Coulomb interaction. At $T=0$ the dc conductance is written in terms of the transmission coefficient, which can be obtained numerically by extrapolating the QMC data for the Matsubara Green's function as $i{\ensuremath{\varepsilon}}_{n}\ensuremath{\rightarrow}{i0}^{+}$. We apply the analysis to the dc conductance through a region with a few resonance states.

DOI
10.1103/physrevb.56.13422

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A numerical analysis for evaluating the transport coefficient through a small interacting system is examined using the quantum Monte Carlo (QMC) method. To formulate the analysis, we consider a finite interacting system on a one-dimensional lattice that is connected to noninteracting leads, and investigate the expression of the dc conductance making use of the Kubo formula with the perturbation theory in the Coulomb interaction. At T=0T=0T=0 the dc conductance is written in terms of the transmission coefficient, which can be obtained numerically by extrapolating the QMC data for the Matsubara Green's function as i\ensuremathεn\ensuremath→i0+i{\ensuremath{\varepsilon}}_{n}\ensuremath{\rightarrow}{i0}^{+}i\ensuremathεn​\ensuremath→i0+. We apply the analysis to the dc conductance through a region with a few resonance states.
We describe microscopic theory for the quantum transport through finite interacting systems connected to noninteracting leads. It can be applied to small systems such as quantum dots, quantum wires, atomic chain, molecule, and so forth. The Keldysh formalism is introduced to study the nonlinear current-voltage characteristics, and general properties of the nonequilibrium Green's functions are provided. We apply the formulated to an out-of-equilibrium Anderson model that has been used widely for the Kondo effect in quantum dots. In the linear-response regime the Kubo formalism still has an advantage, because it relates the transport coefficients directly to the correlation functions defined with respect to the thermal equilibrium, and it has no ambiguities about a profile of the inner electric field. We introduce a many-body transmission coefficient, by which the dc conductance can be expressed in a Landauer-type form, quite generally. We also discuss transport properties of the Tomonaga-Luttinger liquid, which has also been an active field of research in this decade.
We study the effects of electron correlation on transport through an interacting region connected to multi-mode leads based on the perturbation expansion with respect to the inter-electron interaction. At zero temperature the conductance defined in the Kubo formalism can be written in terms of a single-particle Green's function at the Fermi energy, and it can be mapped onto a transmission coefficient of the free quasiparticles described by an effective Hamiltonian. We apply this formulation to a two-dimensional Hubbard model of finite size connected to two noninteracting leads. We calculate the conductance in the electron-hole symmetric case using the order U2U^2U2 self-energy. The conductance shows several maximums in the UUU dependence in some parameter regions of ty/txt_y/t_xty​/tx​, where txt_xtx​ (tyt_yty​) is the hopping matrix element in the xxx- (yyy-) directions. This is caused by the resonance occurring in some of the subbands, and is related with the UUU dependence of the eigenvalues of the effective Hamiltonian.
The quantum Monte Carlo (QMC) technique is an extremely powerful method to treat many-body systems. Usually the quantum Monte Carlo method has been applied in cases where the interaction potential has a simple analytic form, like the 1/r Coulomb potential. However, in a complicated environment as in a semiconductor heterostructure, the evaluation of the interaction itself becomes a nontrivial problem. Obtaining the potential from any grid-based finite-difference method for every walker and every step is infeasible. We demonstrate an alternative approach of solving the Poisson equation by a classical Monte Carlo calculation within the overall quantum Monte Carlo scheme. We have developed a modified "walk on spheres" algorithm using Green's function techniques, which can efficiently account for the interaction energy of walker configurations, typical of quantum Monte Carlo algorithms. This stochastically obtained potential can be easily incorporated with variational, diffusion, and other Monte Carlo techniques. We demonstrate the validity of this method by studying a simple problem, the polarization of a helium atom in the electric field of an infinite capacitor.
Real-time quantum Monte Carlo (QMC) methods have been used to study diffusion on a one-dimensional tight-binding lattice in a dissipative environment with Ohmic friction. For this system, the inherent sign problem of real-time QMC methods can be substantially reduced by employing a partial summation scheme, allowing direct calculations of long-time transport properties. At very low temperatures, the system undergoes a transition from a coherent transport mechanism to an incoherent mechanism as the Kondo parameter K goes through 1. For 0K1, perturbation theory which predicts a ${\mathit{T}}^{2\mathit{K}\mathrm{\ensuremath{-}}1}$ power-law temperature dependence for the diffusion coefficient D is grossly incorrect for low temperatures. Instead, D(T\ensuremath{\rightarrow}0)=0 for 0K1/2, and with increasing temperature D goes through a pronounced maximum. This maximum disappears for K>1/2 and D increases monotonically with increasing temperature. For all K1, perturbation theory becomes exact at sufficiently high temperatures, indicating that the transport mechanism eventually becomes incoherent. For K>1, perturbation theory is exact for all temperatures, and transport is always incoherent.
Quantum Monte Carlo (QMC) is an extremely powerful method to treat many-body systems. Usually quantum Monte Carlo has been applied in cases where the interaction potential has a simple analytic form, like the 1/r Coulomb potential. However, in a complicated environment as in a semiconductor heterostructure, the evaluation of the interaction itself becomes a non-trivial problem. Obtaining the potential from any grid-based finite-difference method, for every walker and every step is unfeasible. We demonstrate an alternative approach of solving the Poisson equation by a classical Monte Carlo within the overall quantum Monte Carlo scheme. We have developed a modified ''Walk On Spheres'' algorithm using Green's function techniques, which can efficiently account for the interaction energy of walker configurations, typical of quantum Monte Carlo algorithms. This stochastically obtained potential can be easily incorporated within popular quantum Monte Carlo techniques like variational Monte Carlo (VMC) or diffusion Monte Carlo (DMC). We demonstrate the validity of this method by studying a simple problem, the polarization of a helium atom in the electric field of an infinite capacitor.