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arXiv:1805.10161 · The Journal of Chemical Physics

Extending the hierarchical quantum master equation approach to low temperatures and realistic band structures

André Erpenbeck, Christopher Hertlein, C. Schinabeck, Michael Thoss

Published Aug 9, 2018• 46 citations
arXivHTMLPDF

Abstract

The hierarchical quantum master equation (HQME) approach is an accurate method to describe quantum transport in interacting nanosystems. It generalizes perturbative master equation approaches by including higher-order contributions as well as non-Markovian memory and allows for the systematic convergence to the numerically exact result. As the HQME method relies on a decomposition of the bath correlation function in terms of exponentials, however, its application to systems at low temperatures coupled to baths with complexer band structures has been a challenge. In this publication, we outline an extension of the HQME approach, which uses re-summation over poles and can be applied to calculate transient currents at a numerical cost that is independent of temperature and band structure of the baths. We demonstrate the performance of the extended HQME approach for noninteracting tight-binding model systems of increasing complexity as well as for the spinless Anderson-Holstein model.

DOI
10.1063/1.5041716

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The hierarchical quantum master equation (HQME) approach is an accurate method to describe quantum transport in interacting nanosystems. It generalizes perturbative master equation approaches by including higher-order contributions as well as non-Markovian memory and allows for the systematic convergence to the numerically exact result. As the HQME method relies on a decomposition of the bath correlation function in terms of exponentials, however, its application to systems at low temperatures coupled to baths with complexer band structures has been a challenge. In this publication, we outline an extension of the HQME approach, which uses re-summation over poles and can be applied to calculate transient currents at a numerical cost that is independent of temperature and band structure of the baths. We demonstrate the performance of the extended HQME approach for noninteracting tight-binding model systems of increasing complexity as well as for the spinless Anderson-Holstein model.
We present a hierarchical quantum master equation (HQME) approach, which allows a numerically exact simulation of nonequilibrium transport in general open quantum systems involving multiple bosonic and fermionic environments. The performance of the method is demonstrated for a model of a nanosystem, which involves interacting electronic and vibrational degrees of freedom and is coupled to fermionic and bosonic baths. The results show the intricate interplay of electronic and vibrational degrees of freedom in this nonequilibrium transport scenario for both voltage and thermally driven transport processes. Furthermore, the use of importance criteria to improve the efficiency of the method is discussed.
The hierarchical equations of motion (HEOM) approach is an accurate method to simulate open system quantum dynamics, which allows for systematic convergence to numerically exact results. To represent effects of the bath, the reservoir correlation functions are usually decomposed into summation of multiple exponential terms in the HEOM method. Since the reservoir correlation functions become highly non-Markovian at low temperatures or when the bath has complex band structures, a present challenge is to obtain accurate exponential decompositions that allow efficient simulation with the HEOM. In this work, we employ the barycentric representation to approximate the Fermi function and hybridization functions in the frequency domain. This method, by approximating these functions with optimized rational decomposition, greatly reduces the number of basis functions in decomposing the reservoir correlation functions, which further allows the HEOM method to be applied to ultralow temperature and general band structures. We demonstrate the efficiency, accuracy, and long-time stability of this decomposition scheme by applying it to the Anderson impurity model in the low-temperature regime with the Lorentzian and tight-binding hybridization functions.
Abstract The hierarchical equations of motion (HEOM) provide a numerically exact approach for simulating the dynamics of open quantum systems coupled to a harmonic bath. However, its applicability has traditionally been limited to specific spectral forms and relatively high temperatures. Recently, an extended version called Free-Pole HEOM (FP-HEOM) has been developed to overcome these limitations. In this study, we demonstrate that the FP-HEOM method can be systematically employed to investigate higher order master equations by truncating the FP-HEOM hierarchy at a desired tier. We focus on the challenging scenario of the spin-boson problem with a sub-Ohmic spectral distribution at zero temperature and analyze the performance of the corresponding master equations. Furthermore, we compare the memory kernel for population dynamics obtained from the exact FP-HEOM dynamics with that of the approximate Non-Interacting-Blip Approximation (NIBA).
Within the exact framework established recently, which is a successful marriage\n\nbetween the time dependent density functional theory for open electronic system\n\nand quantum dissipation theory formulated in the hierarchical equations of motion,\n\nan entirely new scheme is proposed in this thesis to simulate the time-dependent\n\nquantum transport in nano-devices at both zero and finite temperature equally\n\nwithout relying on the pole structure of the Fermi distribution function. Neither\n\ndoes it depend on any non-unique parametrization of the line-width matrix, hence,\n\nthis new practical approach can be integrated with the first principles simulations\n\nseamlessly.\n\nBeyond the exact framework, a reliable method which works under the Wide-\n\nBand-Limit approximation at zero temperature is also developed. At the price of\n\nloss of some non-Markovian memory effects on the dynamics, a set of equations\n\nof motion which terminates at the first tier instead of the second tier is obtained.\n\nBenefiting from the latest advancement of numerical analysis, a hybrid fourth-order\n\nRunge-Kutta algorithm is proposed to solve this set of equations of motion which\n\ncomprises stiff ones. Based on this result, an alternative scheme is considered to\n\ndeal with the same approximation at finite temperature.\n\nAs an illustration of these new approaches, the transient current of the one\n\ndimensional tight-binding periodical chain with and without a single impurity, driven\n\nby some time alternating and/or static bias voltages, are investigated. The influence\n\nof temperature and switch-on rate of bias voltage is exemplified. Particularly, in the\n\none dimensional tight-binding chain with a single impurity which breaks its perfect\n\nperiodicity, an asymmetry between the left and right transient current is found.\n\nComparison between the results under the Wide-Band-Limit approximation and\n\nthose with the exact description is carried out.
A generalized quantum master equation theory that governs the exact, nonperturbative quantum dissipation and quantum transport is formulated in terms of hierarchically coupled equations of motion for an arbitrary electronic system in contact with electrodes under either a stationary or a nonstationary electrochemical potential bias. The theoretical construction starts with the influence functional in path integral, in which the electron creation and annihilation operators are Grassmann variables. Time derivatives on the influence functionals are then performed in a hierarchical manner. Both the multiple-frequency dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting hierarchical equations of motion formalism is in principle exact and applicable to arbitrary electronic systems, including Coulomb interactions, under the influence of arbitrary time-dependent applied bias voltage and external fields. Both the conventional quantum master equation and the real-time diagrammatic formalism of Schon and co-workers can be readily obtained at well defined limits of the present theory. We also show that for a noninteracting electron system, the present hierarchical equations of motion formalism terminates at the second tier exactly, and the Landuer-Buttiker transport current expression is recovered. The present theory renders an exact and numerically tractable tool to evaluate various transient and stationary quantum transport properties of many-electron systems, together with the involving nonperturbative dissipative dynamics.
In this Communication, we propose the time-domain Prony fitting decomposition (t-PFD) as an accurate and efficient exponential series method, applicable to arbitrary bath correlation functions. The resulting numerical efficiency of hierarchical equations of motion (HEOM) formalism is greatly optimized, especially in low temperature regimes that would be inaccessible with other methods. For demonstration, we calibrate the present t-PFD against the celebrated Padé spectrum decomposition method, followed by converged HEOM evaluations on the single-impurity Anderson model system.