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Chinese Journal of Chemical Physics

Adiabatic terminator for fermionic hierarchical equations of motion

Daochi Zhang, Xu Ding, Hou-Dao Zhang, Xiao Hong Zheng, YiJing Yan

Published Dec 1, 2021• 12 citations
Publisher

Abstract

The hierarchical equation of motion method has become one of the most popular numerical methods for describing the dissipative dynamics of open quantum systems linearly coupled to environment. However, its applications to systems with strong electron correlation are largely restrained by the computational cost, which is mainly caused by the high truncation tier L required to accurately characterize the strong correlation effect. In this work, we develop an adiabatic terminator by decoupling the principal dissipation mode with the fastest dissipation rate from the slower ones. The adiabatic terminator leads to substantially enhanced convergence with respect to L as demonstrated by the numerical tests carried out on a single impurity Anderson model. Moreover, the adiabatic terminator alleviates the numerical instability problems in the long-time dissipative dynamics.

DOI
10.1063/1674-0068/cjcp2110212

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The hierarchical equation of motion method has become one of the most popular numerical methods for describing the dissipative dynamics of open quantum systems linearly coupled to environment. However, its applications to systems with strong electron correlation are largely restrained by the computational cost, which is mainly caused by the high truncation tier L required to accurately characterize the strong correlation effect. In this work, we develop an adiabatic terminator by decoupling the principal dissipation mode with the fastest dissipation rate from the slower ones. The adiabatic terminator leads to substantially enhanced convergence with respect to L as demonstrated by the numerical tests carried out on a single impurity Anderson model. Moreover, the adiabatic terminator alleviates the numerical instability problems in the long-time dissipative dynamics.
Investigations of strongly correlated quantum impurity systems (QIS), which exhibit diversified novel and intriguing quantum phenomena, have become a highly concerning subject in recent years. The hierarchical equations of motion (HEOM) method is one of the most popular numerical methods to characterize QIS linearly coupled to the environment. This review provides a comprehensive account of a formally rigorous and numerical convergent HEOM method, including a modeling description of the QIS and an overview of the fermionic HEOM formalism. Moreover, a variety of spectrum decomposition schemes and hierarchal terminators have been proposed and developed, which significantly improve the accuracy and efficiency of the HEOM method, especially in cryogenic temperature regimes. The practicality and usefulness of the HEOM method to tackle strongly correlated issues are exemplified by numerical simulations for the characterization of nonequilibrium quantum transport and strongly correlated Kondo states as well as the investigation of nonequilibrium quantum thermodynamics.
The fermionic hierarchical equations of motion (HEOM) approach has found wide application in the exploration of open quantum systems, and extensive efforts have been committed to improving its efficiency and accuracy in practical calculations. In this work, by scrutinizing the stationary-state and dynamic properties of Kondo-correlated quantum impurity systems, we show that the strength of Kondo correlation induced by the system-environment entanglement primarily determines the converged hierarchical truncation tier of the HEOM method. This complements the rule of thumb regarding the positive correlation between the height of hierarchy and system-environment coupling strength. These insights will provide useful guidelines for developing a more sophisticated fermionic HEOM method for the investigation of many-body open quantum systems.
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.
The hierarchical equations of motion (HEOM) method has become one of the most popular methods for the studies of the open quantum system. However, its applicability to systems at ultra-low temperatures is largely restrained by the enormous computational cost, which is caused by the numerous exponential functions required to accurately characterize the non-Markovian memory of the reservoir environment. To overcome this problem, a Fano spectrum decomposition (FSD) scheme has been proposed recently [Cui et al., J. Chem. Phys. 151, 024110 (2019)], which expands the reservoir correlation functions using polynomial-exponential functions and hence greatly reduces the size of the memory basis set. In this work, we explicitly establish the FSD-based HEOM formalisms for both bosonic and fermionic environments. The accuracy and efficiency of the FSD-based HEOM are exemplified by the calculated low-temperature dissipative dynamics of a spin-boson model and the dynamic and static properties of a single-orbital Anderson impurity model in the Kondo regime. The encouraging numerical results highlight the practicality and usefulness of the FSD-based HEOM method for general open systems at ultra-low temperatures.
The hierarchical equations of motion (HEOM) theory is in principle exact for describing the dissipative dynamics of quantum systems linearly coupled to Gaussian environments. In practice, the hierarchy needs to be truncated at a finite tier. We demonstrate that, for general systems described by the fermionic HEOM, the (n+L̃)th-tier truncation with L̃=2NσNν yields the exact density operators up to the nth tier. Here, Nσ = 2 for fermionic systems and Nν is the system degrees of freedom. For noninteracting systems, L̃ is further reduced by half. Such an exact termination pattern originates from the Pauli exclusion principle for fermions, and it holds true regardless of the system-environment coupling strength, the number of coupling reservoirs, or the specific scheme employed to unravel the environment memory contents. The relatively small L̃ emphasizes the nonperturbative nature of the HEOM theory. We also propose a simplified HEOM approach to further reduce the memory cost for practical calculations.
Recent technological advancement in scanning tunneling microscopes has enabled the measurement of spin-field and spin-spin interactions in single atomic or molecular junctions with an unprecedentedly high resolution. Theoretically, although the fermionic hierarchical equations of motion (HEOM) method has been widely applied to investigate the strongly correlated Kondo states in these junctions, the existence of low-energy spin excitations presents new challenges to numerical simulations. These include the quest for a more accurate and efficient decomposition for the non-Markovian memory of low-temperature environments and a more careful handling of errors caused by the truncation of the hierarchy. In this work, we propose several new algorithms, which significantly enhance the performance of the HEOM method, as exemplified by the calculations on systems involving various types of low-energy spin excitations. Being able to characterize both the Kondo effect and spin excitation accurately, the HEOM method offers a sophisticated and versatile theoretical tool, which is valuable for the understanding and even prediction of the fascinating quantum phenomena explored in cutting-edge experiments.