Accelerated calculation of impurity Green's functions exploiting the extreme Mpemba effect

TL;DR

This paper addresses the computational bottleneck of calculating equilibrium two-time impurity Green's functions in non-Markovian open quantum systems relevant to DMFT. It introduces a two-step strategy that combines the non-Markovian quantum Mpemba effect, to prepare a fast-thermalising initial state $\rho_f$, with a dynamical-map extrapolation based on a time-local generator $\hat{\mathcal{L}}_m$ to extend dynamics beyond the environment memory time. Across fermionic and bosonic impurity models (including RLM, two-coupled modes, SIAM, and spin-boson baths), the method delivers high-accuracy two-time correlations with substantial speedups, while clearly identifying regimes (notably persistent-memory or Kondo-like regimes) where extrapolation has limited advantage. The approach promises significant practical impact for DMFT impurity solvers by enabling efficient, long-time two-time dynamics with controlled accuracy.

Abstract

Simulating the dynamics of quantum impurity models remains a fundamental challenge due to the complex memory effects that arise from system-environment interactions. Of particular interest are two-time correlation functions of an impurity, which are central to the characterization of these many-body systems, and are a cornerstone of the description of correlated materials in dynamical mean field theory (DMFT). In this work, we extend our previous work on the extrapolation of single-time observables to demonstrate an efficient scheme for computing two-time impurity correlation functions, by combining the non-Markovian quantum Mpemba effect (NMQMpE) with a dynamical map-based framework for open quantum systems. Our method is benchmarked against exact and known accurate results in prototypical impurity models for both fermionic and bosonic environments, demonstrating significant computational savings compared to state-of-the-art methods.

Figures (9)

• Figure 1: Schematic of our methodology. The top timeline represents the evolution of the Choi state, which allows for the extraction of the dynamical map $\hat{\Lambda}(t)$ as detailed in the Appendix \ref{['appendix:Extracting map']}. This is needed for the calculation of $\hat{\rho}_{f}$ and the extrapolation. The bottom timeline represents the thermalisation from $t=t_0$ to $t=0$, at which point the system is perturbed by $\hat{A}$ and the combined system-environment setup undergoes unitary evolution until Eq.\ref{['eq:10']} is valid, at which point extrapolation is used for the remaining time evolution.
• Figure 2: Green's functions and thermalisation for the Resonant Level Model. (a) Schematic of the open system. (b) Thermalisation of the mode density $\langle \hat{f}^\dagger\hat{f}\rangle$ towards its steady-state value in $\hat{\rho}_{\infty}$ starting from $\hat{\rho}_{f}$, $\hat{\rho}_{\infty}$ and $\hat{\rho}_\mathbb1$. (c) $G(t)$ calculated exactly and using extrapolation from $t=25/D$, with the inset showing the maximum error for the extrapolation as a function of $\tau_{m}$. (d) Comparison of $A(\omega)$ calculated from the extrapolated $G(t)$ with the exact solution. Parameters: $\Gamma=0.05D,\beta=10/D,\mu=0,\epsilon=0.1D$, maximum bond dimension $\chi_{\textrm{max}}=300$, $N_{B} = 210$.
• Figure 3: Green's functions and thermalisation for two interacting fermionic modes. (a) Schematic of the two-mode open system. (b) Thermalisation in terms of trace distance starting from $\hat{\rho}_{f}$, $\hat{\rho}_{\infty}$ and the maximally mixed state. (c) $G_{1}(t)$ calculated using the direct MPS method versus extrapolation from $t=25/D$, with the inset showing the maximum error of the extrapolation as a function of $\tau_{m}$. (d) Comparison of $A_1(\omega)$ calculated from the extrapolated $G_1(t)$ with the near-exact direct MPS result. Parameters: $\Gamma = 0.1D, \beta = 10/D,\mu = 0.1D,t_c = 0.05D, U = 0.2D$, maximum bond dimension $\chi_{\textrm{max}}=300$, $N_{B} = 150$.
• Figure 4: Schematics for (a) the single impurity Anderson model and (b) the spin-boson model.
• Figure 5: Green's functions and spectral functions for the SIAM. (a) $G_\sigma(t)$ for $\beta=2.5/D$, calculated using direct MPS versus Eq. \ref{['eq:10']} for various choices of $\tau_{m}$. (b) The associated spectral function. (c) $G_\sigma(t)$ for $\beta=2.5/D$, calculated using direct MPS versus Eq. \ref{['eq:10']} for various choices of $\tau_{m}$. (d) The associated spectral function $A_\sigma(\omega)$. Parameters: $\Gamma=\pi D/20,\;U=5\Gamma,\;\mu=0$, $N_{B}=80$, maximum bond dimension $\chi_{\textrm{max}}=150$ (same as Ref. PhysRevB.104.014303).