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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum Monte Carlo study of tunneling diffusion in a dissipative multistate system

Chi H. Mak, Reinhold Egger

Published Mar 1, 1994• 26 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.49.1997

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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 T2K\ensuremath−1{\mathit{T}}^{2\mathit{K}\mathrm{\ensuremath{-}}1}T2K\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.
In simple ferromagnetic quantum Ising models characterized by an effective double-well energy landscape the characteristic tunneling time of path-integral Monte Carlo (PIMC) simulations has been shown to scale as the incoherent quantum-tunneling time, i.e., as $1/{\mathrm{\ensuremath{\Delta}}}^{2}$, where $\mathrm{\ensuremath{\Delta}}$ is the tunneling gap. Since incoherent quantum tunneling is employed by quantum annealers (QAs) to solve optimization problems, this result suggests that there is no quantum advantage in using QAs with respect to quantum Monte Carlo (QMC) simulations. A counterexample is the recently introduced shamrock model (Andriyash and Amin, arXiv:1703.09277), where topological obstructions cause an exponential slowdown of the PIMC tunneling dynamics with respect to incoherent quantum tunneling, leaving open the possibility for potential quantum speedup, even for stoquastic models. In this work we investigate the tunneling time of projective QMC simulations based on the diffusion Monte Carlo (DMC) algorithm without guiding functions, showing that it scales as $1/\mathrm{\ensuremath{\Delta}}$, i.e., even more favorably than the incoherent quantum-tunneling time, both in a simple ferromagnetic system and in the more challenging shamrock model. However, a careful comparison between the DMC ground-state energies and the exact solution available for the transverse-field Ising chain indicates an exponential scaling of the computational cost required to keep a fixed relative error as the system size increases.
This work is motivated by the fact that the investigation of non-equilibrium phenomena in strongly correlated electron systems has developed into one of the most active and exciting branches of condensed matter physics as it provides rich new insights that could not be obtained from the study of equilibrium situations. However, a theoretical description of those phenomena is missing. Therefore, in this thesis, we develop a numerical method that can be used to study two minimal models -- the Hubbard model and the Anderson impurity model with general parameter range and time dependence. We begin by introducing the theoretical framework and the general features of the Hubbard model. We then describe the dynamical mean field theory (DMFT), which was first invented by Georges in 1992. It provides a feasible way to approach strongly correlated electron systems and reduces the complexity of the calculations via a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. We employ the non-equilibrium extension of DMFT and map the Hubbard model to the single impurity Anderson model (SIAM). Since the fundamental component of the DMFT method is a solver of the single impurity Anderson model, we continue with a description of the formalism to study the real-time dynamics of the impurity model staring at its thermal equilibrium state. We utilize the non-equilibrium strong-coupling perturbation theory and derive semi-analytical approximation methods such as the non-crossing approximation (NCA) and the one-crossing approximation (OCA). We then use the Quantum Monte-Carlo method (QMC) as a numerically exact method and present proper measurements of local observables, current and Green's functions. We perform simulations of the current after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. The current exhibits short equilibrium times and saturates upon the decrease of temperature at all times, indicating Kondo behavior both in the transient regime and in the steady state. However, this bare QMC solver suffers from a dynamical sign problem for long time propagations. To overcome the limitations of this bare treatment, we introduce the ``Inchworm algorithm'', based on iteratively reusing the information obtained in previous steps to extend the propagation to longer times and stabilize the calculations. We show that this algorithm greatly reduces the required order for each simulation and re-scales the exponential challenge to quadratic in time. We introduce a method to compute Green's functions, spectral functions, and currents for inchworm Monte Carlo and show how systematic error assessments in real time can be obtained. We illustrate the capabilities of the algorithm with a study of the behavior of quantum impurities after an instantaneous voltage quench from a thermal equilibrium state. We conclude with the applications of the unbiased inchworm impurity solver to DMFT calculations. We employ the methods for a study of the one-band paramagnetic Hubbard model on the Bethe lattice in equilibrium, where the DMFT approximation becomes exact. We begin with a brief introduction of the Mott metal insulator phase diagram. We present the results of both real time Green's functions and spectral functions from our nonequilibrium calculations. We observe the metal-insulator crossover as the on-site interaction is increased and the formation of a quasi-particle peak as the temperature is lowered. We also illustrate the convergence of our algorithms in different aspects.
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 $i{\ensuremath{\varepsilon}}_{n}\ensuremath{\rightarrow}{i0}^{+}$. We apply the analysis to the dc conductance through a region with a few resonance states.
The tunneling dynamics of the spin-boson problem has been computed using discretized path-integral simulations for temperatures T and couplings, i.e., the Kondo parameter \ensuremath{\alpha}, spanning the entire T-\ensuremath{\alpha} plane. The inherent problem of alternating weights has been solved using a combination of the stationary-phase Monte Carlo method and contour-distortion techniques. A transition from coherent to purely incoherent relaxation was observed for the spin correlation function. The time correlation functions and the location of the coherent-incoherent boundary on the T-\ensuremath{\alpha} plane are well described by the noninteracting-blip approximation. In the deep-tunneling limit of large \ensuremath{\alpha}, low T, and high bath frequency, the system relaxes exponentially, with its relaxation time constant following a power-law temperature dependence, in accord with perturbation theory. At higher T and low bath frequency, the relaxation time crosses over to a classical Arrhenius temperature dependence, reflecting the onset of activated processes. For a narrow region within 1/21, numerical results suggest that the system undergoes incoherent relaxation, with a short-time exponential decay, followed by a long-time tail of the power-law type. The short-time exponential relaxation time follows a peculiar power-law temperature dependence, with the relaxation rate increasing as a function of decreasing temperature.
A discretized path integral formulation of the real-time dynamics applicable to a general external field is derived for the driven spin-boson model to extend the recently developed real-time Quantum Monte Carlo algorithm to driven systems. Numerical results are obtained for a monochromatic periodic driving force and Ohmic dissipation with a high-frequency cutoff. The impact of an external driving field on the dissipative dynamics is examined for systems with a time-dependent bias as well as a time-dependent interstate coupling matrix element. This method is easily generalized to multistate tight-binding models.
Ground-state properties of the one-dimensional Kondo lattice model are investigated using the quantum Monte Carlo (QMC) simulation method. To reduce the difficulty of the negative sign problem, we employ the variationally optimized singlet wave function. Since the improvement of Monte Carlo sampling efficiency is achieved in the half-filling case, we can perform the simulations on large systems (up to 26-site system). The QMC data on the local spin correlations are closely compared with those of the variational Monte Carlo (VMC) calculations as well as with the exact diagonalization data of the Ruderman-Kittel-Kasuya-Yoshida (RKKY) lattice model. We find that the local spin correlation decays exponentially with the correlation length ${\ensuremath{\xi}}_{\mathit{f}\mathit{f}}$\ensuremath{\sim}2.54 in the QMC data, while ${\ensuremath{\xi}}_{\mathit{f}\mathit{f}}$\ensuremath{\sim}1.08 in the VMC results (J/t=1). This implies the absence of the long-range order (the existence of the spin gap) and the importance of intersite correlation effects which are not taken into account in the VMC results. On the other hand, the system size dependence is almost absent in the Kondo lattice model results, which contrasts the behavior in the RKKY data.