Algorithmic Matsubara integration for Hubbard-like models

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.

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“…Following similar steps as those in Ref. 35 , we first define the Hartree-shifted bare Green's function of the model…”

Section: A Symbolic Algorithmmentioning

confidence: 99%

Real-frequency diagrammatic Monte Carlo at finite temperature

Vučičević¹,

Ferrero

2020

Phys. Rev. B

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“…To clarify this, we provide in Supplemental Material the array representation of a particular third order selfenergy diagram as an example. Starting with the array representation (6) and following the AMI procedure 19 we construct and store the AMI result. A typical AMI result contains many terms, which are represented as nested arrays.…”

Section: Ami Proceduresmentioning

confidence: 99%

“…The only roadblock to doing so is the complexity of the analytic equations. This roadblock has recently been overcome by the method of algorithmic Matsubara integration (AMI) 19 that in principle allows for the symbolic evaluation of the Matsubara sums for arbitrarily complex Feynman diagrams with minimal computational expense. The analytic result of AMI can be evaluated at any temperature and the analytic continuation is trivialized since it can be imposed symbolically: iν n → ν +i0 + .…”

Section: Introductionmentioning

confidence: 99%

Optimal grouping of arbitrary diagrammatic expansions via analytic pole structure

2020

Self Cite

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We present a general method to optimize the evaluation of Feynman diagrammatic expansions, which requires the automated symbolic assignment of momentum/energy conserving variables to each diagram. With this symbolic representation, we utilize the pole structure of each diagram to automatically sort the Feynman diagrams into groups that are likely to contain nearly equal or nearly cancelling diagrams, and we show that for some systems this cancellation is exact. This allows for a potentially massive cancellation during the numerical integration of internal momenta variables, leading to an optimal suppression of the 'sign problem' and hence reducing the computational cost. Although we define these groups using a frequency space representation, the equality or cancellation of diagrams within the group remains valid in other representations such as imaginary time used in standard diagrammatic Monte Carlo. As an application of the approach we apply this method, combined with algorithmic Matsubara integration (AMI) [Phys. Rev. B 99, 035120 (2019)] and Monte Carlo methods, to the Hubbard model self-energy expansion on a 2D square lattice up to sixth order which we evaluate and compare with existing benchmarks. arXiv:1911.11129v1 [cond-mat.str-el]

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“…In this case, our previous fast algorithm [11] could be used to generate all EDDs. The conventional DiagMC relies on the Matsubara formalism, but recently, a new algorithm was proposed to simplify the evaluation of diagrams [12][13][14]. In particular, this development revolves around reducing the integration of diagrams in space-time to only space.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, this development revolves around reducing the integration of diagrams in space-time to only space. Our method [12] uses some basic definitions in graph theory, whereas the work of [13,14] is a direct application of residue theorem to each diagram. These methods reduce the calculation time and allow us to evaluate the grand potential or self-energy in the finite temperature of each order in MBPT.…”

Section: Introductionmentioning

confidence: 99%

Finite and High-temperature series expansion via many-body perturbation theory

Tag¹,

Boudiar²,

Mansour³

et al. 2021

Preprint

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We present a new algorithm to evaluate the grand potential at finite and high-temperature series expansion via many-body perturbation theory. This algorithm allows us to formulate each order as a divided difference. Further, we apply this algorithm to the Heisenberg spin-1/2 XXZ chain. We obtain all coefficients of the high-temperature expansion of the free energy and susceptibility per site of this model up to sixth order.

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Algorithmic Matsubara integration for Hubbard-like models

Cited by 50 publications

References 23 publications

Real-frequency diagrammatic Monte Carlo at finite temperature

Real-frequency diagrammatic Monte Carlo at finite temperature

Optimal grouping of arbitrary diagrammatic expansions via analytic pole structure

Finite and High-temperature series expansion via many-body perturbation theory

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