Real-frequency diagrammatic Monte Carlo at finite temperature

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 interactionexpansion 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 Σ(ω) 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.

The Weak, the Strong and the Long Correlation Regimes of the Two-Dimensional Hubbard Model at Finite Temperature

“…In this work, we introduce and use a novel version of the diagrammatic Monte Carlo (DiagMC) method [81][82][83][84][85][86][87][88][89]. DiagMC has been used for lattice and continuum models with short and long-range interactions [90][91][92][93], for interacting topological models [94], for real-time propagation [95][96][97][98][99][100][101][102], and in combination with extensions of DMFT [103][104][105]. The main idea of DiagMC is to write a diagrammatic expansion for intensive physical quantities and to sample the diagrams of this expansion with a Monte Carlo procedure.…”

Charge transport in the Hubbard model at high temperatures: triangular versus square lattice

“…With this work we also made a benchmark of several state-of-the-art numerical methods for solving the Hubbard model and calculating the conductivity at high temperatures. This may be a useful reference for calculations of conductivity using a recent approach that calculates perturbatively the correlation functions directly on the real frequency axis, [56][57][58][59] thus eliminating a need for analytical continuation, while going beyond the calculation on the 4 × 4 cluster. The CTINT algorithm has been implemented using the TRIQS toolbox.…”

Fermionic Sign Structure of High-order Feynman diagrams in a Many-fermion System

“…In this approach, one calculates the physical observable in terms of diagrammatic expansions with Monte Carlo samplings. The methods have been applied to solve a series of important problems in the Hubbard model [7][8][9][10][11][12][13][14][15][16] , many-electron problem [17][18][19][20] , Fermi polaron problem 6,[21][22][23][24][25][26][27][28] and frustrated spin systems [29][30][31] . The conventional approaches stochastically sample individual diagrams and therefore suffer from the severe sign problem caused by the massive sign cancellation between the diagrams.…”

“…The conventional approaches stochastically sample individual diagrams and therefore suffer from the severe sign problem caused by the massive sign cancellation between the diagrams. Recently, a new generation of the algorithms, which sample the summed weight of groups of diagrams, has been developed 15,19,[32][33][34] . We will refer them to cluster DiagMC algorithms.…”