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Efficient and accurate tensor network algorithm for Anderson impurity problems

Zhijie Sun, Zhenyu Li, Chu Guo

TL;DR

The paper addresses the challenge of long-time dynamics in the Anderson impurity model by encoding bath effects into the Feynman-Vernon influence functional and representing it as a Grassmann MPS. It introduces the exact eTTI-IF method, which decomposes the hybridization into a sum of $n$ exponentials and builds the IF from $2n$ (imaginary time) or $8n$ (real time) small GMPSs, with a proven exact exponentiation (Theorem 1) and worst-case bond-dimension scaling of $\chi \sim 2^{2n}$ or $\chi \sim 2^{8n}$. The approach achieves substantial speedups while maintaining accuracy across two common bath spectral functions, with $n$ remaining modest (≈4) for times up to $t=100$, implying polynomial-time scaling in $t$ under mild assumptions. These results broaden practical access to AIM dynamics and hold promise for efficient multi-orbital DMFT simulations via GMPS-based IF representations.

Abstract

The Anderson impurity model (AIM) is of fundamental importance in condensed matter physics to study strongly correlated effects. However, accurately solving its long-time dynamics still remains a great numerical challenge. An emergent and rapidly developing numerical strategy to solve the AIM is to represent the Feynman-Vernon influence functional (IF), which encodes all the bath effects on the impurity dynamics, as a matrix product state (MPS) in the temporal domain. The computational cost of this strategy is basically determined by the bond dimension $χ$ of the temporal MPS. In this work, we propose an efficient and accurate method which, when the hybridization function in the IF can be approximated as the summation of $n$ exponential functions, can systematically build the IF as a MPS by multiplying $O(n)$ small MPSs, each with bond dimension $2$. Our method gives a worst case scaling of $χ$ as $2^{8n}$ and $2^{2n}$ for real- and imaginary-time evolution respectively. We demonstrate the performance of our method for two commonly used bath spectral functions, where we show that the actually required $χ$s are much smaller than the worst case.

Efficient and accurate tensor network algorithm for Anderson impurity problems

TL;DR

The paper addresses the challenge of long-time dynamics in the Anderson impurity model by encoding bath effects into the Feynman-Vernon influence functional and representing it as a Grassmann MPS. It introduces the exact eTTI-IF method, which decomposes the hybridization into a sum of exponentials and builds the IF from (imaginary time) or (real time) small GMPSs, with a proven exact exponentiation (Theorem 1) and worst-case bond-dimension scaling of or . The approach achieves substantial speedups while maintaining accuracy across two common bath spectral functions, with remaining modest (≈4) for times up to , implying polynomial-time scaling in under mild assumptions. These results broaden practical access to AIM dynamics and hold promise for efficient multi-orbital DMFT simulations via GMPS-based IF representations.

Abstract

The Anderson impurity model (AIM) is of fundamental importance in condensed matter physics to study strongly correlated effects. However, accurately solving its long-time dynamics still remains a great numerical challenge. An emergent and rapidly developing numerical strategy to solve the AIM is to represent the Feynman-Vernon influence functional (IF), which encodes all the bath effects on the impurity dynamics, as a matrix product state (MPS) in the temporal domain. The computational cost of this strategy is basically determined by the bond dimension of the temporal MPS. In this work, we propose an efficient and accurate method which, when the hybridization function in the IF can be approximated as the summation of exponential functions, can systematically build the IF as a MPS by multiplying small MPSs, each with bond dimension . Our method gives a worst case scaling of as and for real- and imaginary-time evolution respectively. We demonstrate the performance of our method for two commonly used bath spectral functions, where we show that the actually required s are much smaller than the worst case.

Paper Structure

This paper contains 8 sections, 28 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The path integral formalism
  4. Brief review of the GTEMPO method
  5. The exact TTI IF method
  6. Numerical results
  7. Conclusion
  8. Proof of Theorem 1

Figures (4)

  • Figure 1: Schematic illustration of (a) the partial IF method, (b) the TTI IF method and (c) the eTTI IF method. The empty circles mean that these sites (GVs) are not involved in the GMPS, while the light blue circles mean the opposite. The dark blue circles in (a) correspond to the first indices of the partial IF.
  • Figure 2: (a,b) Imaginary part of $G^>(t)$ for the semi-circular (a) and Lorentzian (b) BSFs, calculated using the partial IF method (red solid line), the eTTI IF method (blue solid line), and the TTI-IF method with $m=3,5,7$ (green dashed lines from lighter to darker). The analytical solutions are shown in black solid lines. The insets show the absolute errors of the results calculated using different methods against the analytical solutions. (c,d) The run time of different methods used to build the IF as GMPSs as a function of the total evolution time $t$ for the semi-circular (c) and Lorentzian (d) BSFs.
  • Figure 3: (a) The number of exponential functions $n$, required to achieve a tolerance $\varsigma$ for the semi-circular (blue solid line with circle) and Lorentzian (orange line with x) BSFs, where we have fixed $t=100$. (b) The number of exponential functions $n$ to achieve a fixed tolerance $\varsigma=10^{-5}$ as a function of $t$. (c,d) The maximum error of the imaginary part of $G^>(t)$ calculated using the eTTI IF method against the analytical solutions, as a function of the tolerance $\varsigma$ (c) and $t$ (d), respectively.
  • Figure 4: (a,c) Real (a) and imaginary (c) parts of $G^>(t)$ for the AIM with the semi-circular BSF. (b,d) Real (b) and imaginary (d) parts of $G^>(t)$ for the AIM with the Lorentzian BSF. The blue solid lines represent the eTTI IF results, while the green dashed lines from lighter to darker are the TTI IF results with $m=3,5,7$ respectively, similar to Fig. \ref{['fig:1']}. The insets show the absolute error of the TTI-IF and eTTI-IF results against the partial IF results.