[Paper Review] Low-rank quantics tensor train representations of Feynman diagrams for multiorbital electron-phonon models

Hirone Ishida, Natsuki Okada+2 more|arXiv (Cornell University)|May 10, 2024

Advanced NMR Techniques and ApplicationsChemistry3 citations

TL;DR

This paper introduces a low-rank quantics tensor train (QTT) representation combined with Tensor Cross Interpolation (TCI) to efficiently compute Feynman diagrams in multiorbital electron-phonon models. It demonstrates exponential convergence of integration error with respect to computational cost and robustness to discontinuities and symmetry breaking, enabling high-resolution, sign-problem-free numerical integration of self-energy diagrams.

ABSTRACT

Feynman diagrams are an essential tool for simulating strongly correlated electron systems. However, stochastic quantum Monte Carlo sampling suffers from the sign problem, particularly when solving a multiorbital quantum impurity model. Recently, two approaches have been proposed for efficient numerical treatment of Feynman diagrams: Tensor Cross Interpolation (TCI) to replace stochastic sampling and the Quantics Tensor Train (QTT) representation for compressing space-time dependence. One of the remaining challenges is the nontrivial task of identifying low-rank structures in weak-coupling Feynman diagrams for multiorbital electron-phonon systems. In particular, the traditional TCI algorithm faces an ergodicity problem, which prevents it from fully exploring the multiorbital space. To address this, we incorporate a new algorithm called global search, which resolves this issue. By combining this approach with QTT, we uncover low-rank structures and achieve efficient numerical integration with exponential resolution in time and faster-than-power-law convergence of error relative to computational cost. Additionally, our approach does not require the division of discontinuous regions necessary in non-quantics TCI.

Motivation & Objective

• To investigate whether low-rank tensor train (TT) structures exist in self-energy Feynman diagrams for multiorbital electron-phonon systems.
• To assess the robustness of Tensor Cross Interpolation (TCI) for integrating complex integrands with both continuous and discrete degrees of freedom.
• To evaluate whether TCI can effectively handle phonon contributions, which often exacerbate the sign problem in quantum Monte Carlo (QMC) methods.
• To determine if the quantics tensor train (QTT) representation enables exponential convergence in error with respect to computational cost.

Proposed method

• Discretization of imaginary-time variables using the quantics tensor train (QTT) format to achieve exponential resolution in time.
• Embedding discrete degrees of freedom—such as spin-orbital indices and phonon vibrational modes—into the same tensor train as continuous time variables.
• Application of Tensor Cross Interpolation (TCI) to learn and integrate high-dimensional Feynman diagram integrands via low-rank TT approximations.
• Use of a hierarchical low-rank structure to compress the integrand and enable efficient numerical integration without domain partitioning.
• Evaluation of interpolation error and convergence rate as a function of bond dimension χ, with error normalized to the maximum function value.
• Validation of results through projection onto an IR basis, confirming expected decay of expansion coefficients.

Figure 1 : (a) Second-order skeleton diagrams for the electron self-energy $\Sigma^{(2)}$ and phonon self-energy $\Pi^{(2)}$ , as well as tensor train (TT) representations of their integrands. (b) Bond dimensions $\chi_{\ell}$ for TCI tolerance $\epsilon=10^{-3}$ for typical parameters set above/bel

Experimental results

Research questions

• RQ1 Do low-rank tensor train (TT) structures exist in the integrands of self-energy Feynman diagrams for multiorbital electron-phonon models?
• RQ2 Is TCI robust for interpolating integrands with mixed continuous and discrete degrees of freedom, including those with discontinuities?
• RQ3 Can TCI effectively handle phonon contributions that worsen the sign problem in stochastic QMC simulations?
• RQ4 Does the QTT representation enable exponential convergence of integration error with respect to computational cost?

Key findings

• The integrands for electron and phonon self-energies in second-order diagrams exhibit low-rank structures, with bond dimensions χ ≈ 35–55 for TCI tolerance ε = 10⁻³, significantly below the incompressible limit.
• Interpolation error decreases exponentially with increasing bond dimension χ, with convergence rates exceeding O(1/n_s^a) for a > 1/2, indicating superior performance over standard QMC sampling.
• The presence of a superconducting transition or external field (α) increases the required χ to reach ε = 10⁻² by roughly a factor of two, due to nonzero components in the self-energy.
• Even with discontinuities in the self-energy integrand, the QTT representation maintains compactness and enables exponential error decay, eliminating the need for domain partitioning.
• The reconstructed self-energy shows no discretization artifacts, and its τ dependence exhibits smooth, high-resolution behavior consistent with exponential resolution.
• The method successfully integrates both Σ⁽²⁾ and Π⁽²⁾ diagrams with amplitudes much smaller than Σ⁽¹⁾ and Π⁽¹⁾, validating the one-shot approximation at second order.

Figure 2 : (a) $T$ dependence of the absolute maximum of the gap function with no external field ( $\alpha=0$ ). (b) $\alpha$ dependence of the absolute maximum of the gap function at $T=0.016$ . (c) $\Sigma^{(1)}$ and $\Pi^{(1)}$ computed at $T=0.05,~{}0.016$ with $\alpha=0$ . (d) $\Sigma^{(2)}$ an