Algorithms for real-frequency evaluation of diagrammatic expansions

Taheridehkordi, Amir (2020) Algorithms for real-frequency evaluation of diagrammatic expansions. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

We compute perturbative expansions of the self-energy and spin susceptibility functions in real-frequency space for the two-dimensional Hubbard model using a new solution method. Each term of the expansion, represented by a Feynman diagram, is translated into its mathematical representation, which includes two types of summations: momentum-space (spatial) and frequency-space (temporal). We introduce algorithmic Matsubara integration (AMI), a method which utilizes the residue theorem to perform the Matsubara frequency summations and store the result in a symbolic form. This method provides the exact result (up to machine precision) at minimal computational expense. We then combine AMI with the Monte Carlo methods to sample diagram topologies in the expansion and to perform momenta summations. To optimize the Monte Carlo integration procedure we group the diagrams according to the symmetry of their integrands determined by the graph invariant transformations (GITs). Since the result of AMI (up to momentum sums) is analytic in terms of the external Matsubara frequency, temperature (T), chemical potential (μ), and Hubbard on-site potential (U), the analytic continuation to the real-frequency axis can be performed symbolically at any point of T − μ − U phase space, even at T = 0 which has been inaccessible in standard methods due to non-ergodicity of the Monte Carlo sampling in this temperature regime. We compare our results to other numerical methods in the parameter regimes where the perturbative expansion is convergent, and finally, benchmark our results on the real-frequency axis.

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