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arXiv:2112.14311

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

Mohamed Amine Tag, Abid Boudiar, Mohamed El-Hadi Mansour, Abdelkader Hafdallah, Chafia Bendjeroudib

Published Dec 28, 2021• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2112.14311

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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.
Abstract We present a new algorithm to evaluate the grand potential at high and finite temperatures using many-body perturbation theory. This algorithm enables us to calculate the contribution of any Hugenholtz or Feynman vacuum diagrams and formulate the results as a sum of divided differences. Additionally, the proposed method is applicable to any interaction in any dimension, allowing us to calculate thermodynamic quantities efficiently at any given temperature, particularly at high temperatures.Furthermore, 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 the sixth order.
The Heisenberg XXZ model is a chain model with nearest-neighbor interactions. Its thermodynamics is exactly obtained via Bethe ansatz. Recently, we developed a method to derive the high-temperature expansion of the grand potential per site of translationally invariant chain models, with periodic boundary conditions. Here we apply this approach to the XXZ model with periodic boundary conditions for the Ising limit case (t = 0) and the free fermion case (delta = 0 and h = 0), obtaining results in agreement with the literature. In this new way of obtaining the coefficients of the high-temperature expansion of the grand potential, the coefficients are derived from an auxiliary function written only in terms of open connected sub-chains.
We consider an arbitrary translationally invariant chain model with nearest neighbors interaction and satisfying periodic boundary condition. The approach developed here allows a thermodynamic description of the chain model directly in terms of grand potential per site. This thermodynamic function is derived from an auxiliary function constructed only from open connected subchains. In order to exemplify its application and how this approach works we consider the Heisenberg XXZ model. We obtain the coefficients of the high temperature expansion of the free energy per site of the model up to third order.
We present symbolic algorithms designed to investigate the perturbative expansions of the d-Hubbard model. These methods are part of recent developments in many-body perturbation theory that aim to reformulate Feynman diagrams in terms of divided differences. The direct application of this technique to the one-dimensional Hubbard model yields the coefficients of the grand potential up to the sixth order, expressed in terms of both the interacting potential (U expansions) and high-temperature expansions (β expansions). A key feature of this approach is the ability to merge the β expansions to any desired order. To verify our analytical results, we compare the derived magnetic susceptibility with the exact solution of the quantum transfer matrix method in the half-filled case.
Abstract We investigated the two-dimensional Hubbard model using many-body perturbation theory, focusing on the systematic computation of thermodynamic properties. The contributions of generalized Hugenholtz (Feynman) diagrams were evaluated symbolically and reformulated using divided differences, enabling efficient and precise calculations. We derived the coefficients of the grand potential expansion up to the eighth order in both the interaction potential (U-expansion) and the inverse temperature (β-expansion). Additionally, we presented results for magnetic susceptibility and heat capacity at half-filling, providing critical insights into the thermodynamic behavior of the model in this regime.
New algorithm of the finite lattice method is presented to generate the high-temperature expansion series of the Ising model. It enables us to obtain much longer series in three dimensions when compared not only to the previous algorithm of the finite lattice method but also to the standard graphical method. It is applied to extend the high-temperature series of the simple cubic Ising model from beta^{26} to beta^{46} for the free energy and from beta^{25} to beta^{32} for the magnetic susceptibility.