Real-frequency Diagrammatic Monte Carlo at Finite Temperature

TL;DR

This paper introduces a real-frequency diagrammatic Monte Carlo method that directly computes dynamic responses at finite temperature, avoiding the ill-posed analytical continuation from imaginary to real frequencies.

Contribution

The authors develop a novel Monte Carlo technique that performs internal Matsubara summations analytically, yielding results directly on the real-frequency axis, improving accuracy over traditional methods.

Findings

Applied to the doped Hubbard model, revealing pseudogap signatures.

Showed limitations of maximum entropy method on truncated series.

Demonstrated the method's potential for complex correlated systems.

Abstract

Diagrammatic expansions are a central tool for treating correlated electron systems. At thermal equilibrium, they are most naturally defined within the Matsubara formalism. However, extracting any dynamic response function from a Matsubara calculation ultimately requires the ill-defined analytical continuation from the imaginary- to the real-frequency domain. It was recently proposed [Phys. Rev. B 99, 035120 (2019)] that the internal Matsubara summations of any interaction-expansion diagram can be performed analytically by using symbolic algebra algorithms. The result of the summations is then an analytical function of the complex frequency rather than Matsubara frequency. Here we apply this principle and develop a diagrammatic Monte Carlo technique which yields results directly on the real-frequency axis. We present results for the self-energy $\Sigma(\omega)$ of the doped 32x32 cyclic square-lattice Hubbard model in a non-trivial parameter regime, where signatures of the pseudogap appear close to the antinode. We discuss the behavior of the perturbation series on the real-frequency axis and in particular show that one must be very careful when using the maximum entropy method on truncated perturbation series. Our approach holds great promise for future application in cases when analytical continuation is difficult and moderate-order perturbation theory may be sufficient to converge the result.