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arXiv:2408.05520 · The European Physical Journal Special Topics

On stability issues of the HEOM method

Malte Krug, Jürgen T. Stockburger

Published Sep 4, 2023• 13 citations
arXivHTMLPDF

Abstract

Abstract The Hierarchical Equations of Motion (HEOM) method has become one of the cornerstones in the simulation of open quantum systems and their dynamics. It is commonly referred to as a non-perturbative method. Yet, there are certain instances, where the necessary truncation of the hierarchy of auxiliary density operators seems to introduce errors which are not fully controllable. We investigate the nature and causes of this type of critical error both in the case of pure decoherence, where exact results are available for comparison, and in the spin-boson system, a full system-reservoir model. We find that truncating the hierarchy to any finite size can be problematic for strong coupling to a dissipative reservoir, in particular when combined with an appreciable reservoir memory time.

DOI
10.1140/epjs/s11734-023-00972-9

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Abstract The Hierarchical Equations of Motion (HEOM) method has become one of the cornerstones in the simulation of open quantum systems and their dynamics. It is commonly referred to as a non-perturbative method. Yet, there are certain instances, where the necessary truncation of the hierarchy of auxiliary density operators seems to introduce errors which are not fully controllable. We investigate the nature and causes of this type of critical error both in the case of pure decoherence, where exact results are available for comparison, and in the spin-boson system, a full system-reservoir model. We find that truncating the hierarchy to any finite size can be problematic for strong coupling to a dissipative reservoir, in particular when combined with an appreciable reservoir memory time.
The hierarchical equations of motion (HEOM) approach is one of the most powerful methods avail- able to simulate the dynamics of open quantum systems. Although the HEOM approach has been shown to be stable and efficient in many parameter regimes and physical scenarios, it is also well known that the unavoidable finite truncation of the hierarchy can lead to fundamental numeri- cal instabilities for bosonic environments. In this work, we apply recently developed techniques to analyze these issues. We use an auxiliary Fock-space picture of the hierarchy to first identify the unbalanced hierarchy-raising terms responsible for the strong transient amplification associated with the non-normal algebraic structure of the HEOM generator. We then apply a non-unitary similarity transformation that converts these terms into balanced combinations of hierarchy-raising and hierarchy-lowering terms, thereby suppressing their numerical impact. We first demonstrate the efficacy of this approach via a spin-boson model with a simple Brownian oscillator spectral den- sity and subsequently extend the analysis to a structured spectral density, showing for both cases that the transformed HEOM approach allows simulations of much larger system-bath coupling than would otherwise be possible.
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.
We study truncations of hierarchical equations of motion (HEOM) for finite-dimensional open quantum systems. We prove that for finite-dimensional approximations constructed with a Schur-complement type of terminator, the spectrum converges to that of the full HEOM as the truncation depth increases. We also prove that this approximation is free of spectral pollution: sufficiently deep truncations do not produce spurious unstable modes, provided the exact HEOM is stable. We illustrate the results for the spin-boson model.
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).
Several recent advancements for the hierarchical equations of motion (HEOM) approach are reported. First, we propose an a priori estimate for the optimal number of basis functions for the reservoir memory decomposition. Second, we make use of the sparsity of auxiliary density operators (ADOs) and propose two ansatzs to screen out all the intrinsic zero ADO elements. Third, we propose a new truncation scheme by utilizing the time derivatives of higher-tier ADOs. These novel techniques greatly reduce the memory cost of the HEOM approach, and thus enhance its efficiency and applicability. The improved HEOM approach is applied to simulate the coherent dynamics of Aharonov-Bohm double quantum dot interferometers. Quantitatively accurate dynamics is obtained for both noninteracting and interacting quantum dots. The crucial role of the quantum phase for the magnitude of quantum coherence and quantum entanglement is revealed.
The Hierarchical equations of motion (HEOM) method is an important non-perturbative technique, allowing numerically exact treatment of open quantum systems with strong coupling and non-Markovian memory. However, its encoding of bath memory into auxiliary density operators often limits direct access to detailed bath information. In contrast, the reaction-coordinate (RC) mapping allows direct and transparent access to the dominant collective bath mode, but its perturbative and often Markovian treatment of the residual bath restricts its reliability. In this work, we introduce RC-HEOM, a hybrid method that unifies the strengths of both approaches by combining RC mapping with a fully non-perturbative HEOM description of the residual bath. RC-HEOM simultaneously retains exact non-Markovian memory and access to the RC mode, which enables analysis of system-RC information. Applying this method to the Anderson impurity models, we directly track the emergence of the Kondo singlet from the growth of the Kondo resonance and uncover a nontrivial RC-mediated coherence revival. These results demonstrate that RC-HEOM is a promising method for characterizing open quantum systems in regimes that are difficult to access with conventional master-equation methods.