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arXiv:2604.22568 · SciPost Physics Core

On truncations of hierarchical equations of motion for finite-dimensional systems

Vasilii Vadimov

quant-phmath-phSubmitted Apr 24, 2026• Announced Apr 27, 2026• 0 citations• Revised Aug 17, 2026 (v2)
v1 · Apr 24, 2026v2 · Aug 17, 2026
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Abstract

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.

Journal reference
SciPost Phys. Core 9, 062 (2026)
DOI
10.21468/SciPostPhysCore.9.3.062
Comments
23 pages, 1 figure Submission to SciPost

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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) 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 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.
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).
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.
The hierarchical equations of motion (HEOM) provide a numerically exact approach for computing the reduced dynamics of a quantum system linearly coupled to a bath. We have found that HEOM contains temperature-dependent instabilities that grow exponentially in time. In the case of continuous-bath models, these instabilities may be delayed to later times by increasing the hierarchy dimension; however, for systems coupled to discrete, nondispersive modes, increasing the hierarchy dimension does little to alleviate the problem. We show that these instabilities can also be removed completely at a potentially much lower cost via projection onto the space of stable eigenmodes; furthermore, we find that for discrete-bath models at zero temperature, the remaining projected dynamics computed with few hierarchy levels are essentially identical to the exact dynamics that otherwise might require an intractably large number of hierarchy levels for convergence. Recognizing that computation of the eigenmodes might be prohibitive, e.g., for large or strongly coupled models, we present a Prony filtration algorithm that may be useful as an alternative for accomplishing this projection when diagonalization is too costly. We present results demonstrating the efficacy of HEOM projected via diagonalization and Prony filtration. We also discuss issues associated with the non-normality of HEOM.