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Provably Efficient Long-Time Exponential Decompositions of Non-Markovian Gaussian Baths

Zhen Huang, Zhiyan Ding, Ke Wang, Jason Kaye, Xiantao Li, Lin Lin

Abstract

Gaussian baths are widely used to model non-Markovian environments, yet the cost of accurate simulation at long times remains poorly understood, especially when spectral densities exhibit nonanalytic behavior as in a range of realistic models. We rigorously bound the complexity of representing bath correlation functions on a time interval $[0,T]$ by sums of complex exponentials, as employed in recent variants of pseudomode and hierarchical equations of motion methods. These bounds make explicit the dependence on the maximal simulation time $T$, inverse temperature $β$, and the type and strength of singularities in an effective spectral density. For a broad class of spectral densities, the required number of exponentials is bounded independently of $T$, achieving time-uniform complexity. The $T$-dependence emerges only as polylogarithmic factors for spectral densities with strong singularities, such as step discontinuities and inverse power-law divergences. The temperature dependence is mild for bosonic environments and disappears entirely for fermionic environments. Thus, the true bottleneck for long-time simulation is not the simulation duration itself, but rather the presence of sharp nonanalytic features in the bath spectrum. Our results are instructive both for long-time simulation of non-Markovian open quantum systems, as well as for Markovian embeddings of classical generalized Langevin equations with memory kernels.

Provably Efficient Long-Time Exponential Decompositions of Non-Markovian Gaussian Baths

Abstract

Gaussian baths are widely used to model non-Markovian environments, yet the cost of accurate simulation at long times remains poorly understood, especially when spectral densities exhibit nonanalytic behavior as in a range of realistic models. We rigorously bound the complexity of representing bath correlation functions on a time interval by sums of complex exponentials, as employed in recent variants of pseudomode and hierarchical equations of motion methods. These bounds make explicit the dependence on the maximal simulation time , inverse temperature , and the type and strength of singularities in an effective spectral density. For a broad class of spectral densities, the required number of exponentials is bounded independently of , achieving time-uniform complexity. The -dependence emerges only as polylogarithmic factors for spectral densities with strong singularities, such as step discontinuities and inverse power-law divergences. The temperature dependence is mild for bosonic environments and disappears entirely for fermionic environments. Thus, the true bottleneck for long-time simulation is not the simulation duration itself, but rather the presence of sharp nonanalytic features in the bath spectrum. Our results are instructive both for long-time simulation of non-Markovian open quantum systems, as well as for Markovian embeddings of classical generalized Langevin equations with memory kernels.

Paper Structure

This paper contains 5 sections, 6 theorems, 71 equations, 3 figures, 1 table.

Table of Contents

  1. Setup
  2. Main result
  3. Temperature (in)dependence and finite frequency truncation
  4. Classical generalized Langevin equations with spectral singularities
  5. Numerical experiments in Fig. \ref{['fig:n_vs_t']}

Key Result

Theorem 1

Suppose that the effective spectral function $J_{\text{eff}}(\omega)$ has singularity of order $\alpha>-1$ at $\omega=\pm 1$ (see defn:singularity_type). Then there exist constants $A_\alpha,c>0$, independent of $t$, and a choice of $y_2\in (\theta_1,\frac{\pi}{4}-\theta_0)$ such that for any $0<h<1 Here $j_{M,\alpha}(t) = \min\{ \mathrm e^{-(\alpha+1)M}, (1+t)^{-(1+\alpha)}\}$ for $\alpha\neq 0$,

Figures (3)

  • Figure 1: Number of bath modes $N$ required to represent the bath of an Ohmic spin-boson model at zero temperature with fixed accuracy $\varepsilon = 0.01$ in $L^1$ norm, as a function of the maximum simulation time $T$.
  • Figure 2: (a,b) Contour transformation from the $\omega$-plane to the $z$-plane. The gray area is the holomorphic region $\Omega$ of $J_{\text{eff}}(\omega)$, and the hatched area is the analyticity strip used in our analysis. Corresponding contours are illustrated in the same color. (c) Comparison of the holomorphic region in our analysis and that in Ref. ThoennissVilkoviskiyAbanin2024.
  • Figure 3: (a) A system coupled to a fermionic periodic lattice. (b) Spectral density in the case of 1D, 2D and 3D lattices exhibiting the van Hove singularities, and corresponding complexity scalings in $T$. (c) Gapless and gapped spectral densities with power-law singularities at the band edge, and corresponding complexity scalings in $T$.

Theorems & Definitions (13)

  • Definition 1: Singularity order $\alpha$
  • Theorem 1: Pointwise error estimate
  • Corollary 2: Main result: $L^1$ error estimate
  • proof : Proof of Corollary \ref{['cor:main']} from Theorem \ref{['thm:pointwise_error']}
  • Corollary 3: $L^\infty$ error estimate
  • proof : Proof of Corollary \ref{['cor:linfty']} from Theorem \ref{['thm:pointwise_error']}
  • Lemma 1: Upper bound of $g(x-\mathrm iy;t)$ (\ref{['eq:g_definition']})
  • proof : Proof of \ref{['lem:g']}
  • proof : Proof of \ref{['thm:pointwise_error']}
  • Proposition 1
  • ...and 3 more