Abstract
The past years have seen a revived interest in the diagrammatic Monte Carlo (DiagMC) methods for interacting fermions on a lattice. A promising recent development allows one to now circumvent the analytical continuation of dynamic observables in DiagMC calculations within the Matsubara formalism. This is made possible by symbolic algebra algorithms, which can be used to analytically solve the internal Matsubara frequency summations of Feynman diagrams. In this paper, we take a different approach and show that it yields improved results. We present a closed-form analytical solution of imaginary-time integrals that appear in the time-domain formulation of Feynman diagrams. We implement and test a DiagMC algorithm based on this analytical solution and show that it has numerous significant advantages. Most importantly, the algorithm is general enough for any kind of single-time correlation function series, involving any single-particle vertex insertions. Therefore, it readily allows for the use of action-shifted schemes, aimed at improving the convergence properties of the series. By performing a frequency-resolved action-shift tuning, we are able to further improve the method and converge the self-energy in a non-trivial regime, with only 3-4 perturbation orders. Finally, we identify time integrals of the same general form in many commonly used Monte Carlo algorithms and therefore expect a broader usage of our analytical solution.
Highlights
Finding controlled solutions of the Hubbard model is one of the central challenges in condensed matter physics [1,2,3,4]
We present a closed-form analytical solution of imaginary-time integrals that appear in the time-domain formulation of Feynman diagrams
Monte Carlo (MC) algorithms are often plagued by two notorious problems: the fermionic sign problem and the analytical continuation of frequency-dependent quantities in calculations based on the Matsubara formalism [5,6,7,8]
Summary
Finding controlled solutions of the Hubbard model is one of the central challenges in condensed matter physics [1,2,3,4]. The analytical solution is general and applies to all diagram topologies that would appear in the transformed series proposed in Refs. We show that even if a full convergence is not possible with a single choice of the action-tuning parameter, one can choose the optimal tuning parameter for each frequency independently [46] Such a frequency-resolved resummation can be used to improve the solution and in some cases systematically eliminate the nonphysical features that appear in the result due to the truncation of the series at a finite order. We restrict to thermal equilibrium and paramagnetic phases with full lattice symmetry
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