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Lefschetz thimble Monte Carlo for many body theories: application to the repulsive Hubbard model away from half filling

Abhishek Mukherjee, Marco Cristoforetti

TL;DR

The paper tackles the sign problem in fermionic many-body Monte Carlo by reformulating the path integral on Lefschetz thimbles attached to saddle points, ensuring a constant imaginary part of the action $\\Im \\mathcal{S}$ and a localized weight $e^{-\\Re \\mathcal{S}}$. It provides a concrete algorithm to identify dominant phases, sample on the associated thimbles, and incorporate a residual Jacobian factor $e^{\\mathcal{R}}$ for reweighting. The method is applied to the repulsive Hubbard model away from half filling at intermediate temperatures, yielding results in excellent agreement with state-of-the-art cluster DMFT benchmarks such as DCA and DQMC. This approach non-perturbatively includes quantum corrections about mean-field solutions and offers a practical route to study other strongly correlated systems where the sign problem hinders conventional Monte Carlo techniques, with potential extension to low-temperature regimes where the dominant phase is not known a priori.

Abstract

Recently, a new method, based on stochastic integration on the surfaces of steepest descent of the action, was introduced to tackle the sign problem in quantum field theories. We show how this method can be used in many body theories to perform fully non-perturbative calculations of quantum corrections about mean field solutions. We discuss an explicit algorithm for implementing our method, and present results for the repulsive Hubbard model away from half-filling at intermediate temperatures. Our results are consistent with those from the recent state of the art cluster dynamical mean field theory calculations.

Lefschetz thimble Monte Carlo for many body theories: application to the repulsive Hubbard model away from half filling

TL;DR

The paper tackles the sign problem in fermionic many-body Monte Carlo by reformulating the path integral on Lefschetz thimbles attached to saddle points, ensuring a constant imaginary part of the action and a localized weight . It provides a concrete algorithm to identify dominant phases, sample on the associated thimbles, and incorporate a residual Jacobian factor for reweighting. The method is applied to the repulsive Hubbard model away from half filling at intermediate temperatures, yielding results in excellent agreement with state-of-the-art cluster DMFT benchmarks such as DCA and DQMC. This approach non-perturbatively includes quantum corrections about mean-field solutions and offers a practical route to study other strongly correlated systems where the sign problem hinders conventional Monte Carlo techniques, with potential extension to low-temperature regimes where the dominant phase is not known a priori.

Abstract

Recently, a new method, based on stochastic integration on the surfaces of steepest descent of the action, was introduced to tackle the sign problem in quantum field theories. We show how this method can be used in many body theories to perform fully non-perturbative calculations of quantum corrections about mean field solutions. We discuss an explicit algorithm for implementing our method, and present results for the repulsive Hubbard model away from half-filling at intermediate temperatures. Our results are consistent with those from the recent state of the art cluster dynamical mean field theory calculations.

Paper Structure

This paper contains 6 sections, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Functional integral formulation for many body theories
  3. Quantum corrections via integration on the paths of steepest descent
  4. The Monte Carlo algorithm
  5. Results for the Hubbard model away from half filling
  6. Conclusion

Figures (2)

  • Figure 1: (Color online) Double occupation vs number density for the repulsive Hubbard model. Our results are shown with filled red squares, while the results from DCA DMFT Leblanc2013 are shown with filled blue circles. The error bars, where not visible, are smaller than the size of the symbols. The lines are meant as a guide to the eye.
  • Figure 2: Double occupation and effective hopping parameter vs temperature at $\mu/t=-1.0$ for the repulsive Hubbard model. The errorbars, where not visible, are smaller than the size of the symbols. The lines are meant as a guide to the eye.