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International Journal of Modern Physics C

Automatic computing of the grand potential in finite temperature many-body perturbation theory

Mohamed Amine Tag, Mohammed Mansour

Published Sep 9, 2019• 6 citations
Publisher

Abstract

A new program created in C/C[Formula: see text] language generates automatically the analytic expression of grand potential and prints it in Latex2e format and in textual data. Another code created in Mathematica language can translate the textual data into a mathematical expression and help any interested to evaluate the thermodynamic quantities in analytic or numeric forms.

DOI
10.1142/s0129183119501006

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A new program created in C/C[Formula: see text] language generates automatically the analytic expression of grand potential and prints it in Latex2e format and in textual data. Another code created in Mathematica language can translate the textual data into a mathematical expression and help any interested to evaluate the thermodynamic quantities in analytic or numeric forms.
Abstract In this work, we present a Python code that automatically calculates the integration over imaginary time of the contributions from many-body Feynman diagrams at finite temperature. The method is based on the recently developed finite-temperature many-body perturbation theory (FT-MBPT) and uses algorithms from graph theory. The code systematically generates the analytical expressions for the grand potential associated with each vacuum diagram, together with the corresponding symmetry factor, at a given perturbative order n . This framework can be used to compute the thermodynamic properties of fermionic and bosonic systems at a given temperature, including the zero-temperature limit.
A finite-temperature many-body perturbation theory is presented, which expands in power series the electronic grand potential, chemical potential, internal energy, and entropy on an equal footing. Sum-over-states and sum-over-orbitals analytical formulas for the second-order perturbation corrections to these thermodynamic properties are obtained in a time-independent, nondiagrammatic, algebraic derivation, relying on the sum rules of the Hirschfelder–Certain degenerate perturbation energies in a degenerate subspace as well as nine algebraic identities for the zeroth-order thermal averages of one- through four-indexed quantities and products thereof. They reproduce numerically exactly the benchmark data obtained as the numerical derivatives of the thermal-full-configuration-interaction results for a wide range of temperatures.
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.
We describe the first version (v1.0.0) of the code ADG that automatically (1) generates all valid Bogoliubov many-body perturbation theory (BMBPT) diagrams and (2) evaluates their algebraic expression to be implemented for numerical applications . This is achieved at any perturbative order p for a Hamiltonian containing both two-body (four-legs) and three-body (six-legs) interactions (vertices). The automated generation of BMBPT diagrams of order p relies on elements of graph theory, i.e., it is achieved by producing all oriented adjacency matrices of size ( p + 1 ) × ( p + 1 ) satisfying topological Feynman’s rules. The automated evaluation of BMBPT diagrams of order p relies both on the application of algebraic Feynman’s rules and on the identification of a powerful diagrammatic rule providing the result of the remaining p -tuple time integral. The diagrammatic rule in question constitutes a novel finding allowing for the straight summation of large classes of time-ordered diagrams at play in the time-independent formulation of BMBPT. Correspondingly, the traditional resolvent rule employed to compute time-ordered diagrams happens to be a particular case of the general rule presently identified. The code ADG is written in Python2.7 and uses the graph manipulation package NetworkX . The code is also able to generate and evaluate Hartree–Fock-MBPT (HF-MBPT) diagrams and is made flexible enough to be expanded throughout the years to tackle the diagrammatics used in various many-body formalisms that already exist or are yet to be formulated. Program summary Program Title: ADG Program Files doi: http://dx.doi.org/10.17632/6h4xrydwfb.1 Licensing provisions: GPLv3 Programming language: Python2.7 Nature of problem: As formal and numerical developments in many-body-perturbation-theory-based ab initio methods make higher orders reachable, producing and evaluating all the diagrams become rapidly undoable on a handmade basis as both their number and complexity grow quickly, making it prone to mistakes and oversights. Solution method : BMBPT diagrams are encoded as square matrices known as oriented adjacency matrices in graph theory, and then turned into graph objects using the NetworkX package. Checks on the diagrams and evaluation of their time-integrated expression are then done on a purely diagrammatic basis. HF-MBPT diagrams are produced and evaluated as well using the same principle.
A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh-Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams-renormalization diagrams-whose resolvent lines are displaced upward, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series toward the finite-temperature full-configuration-interaction limit unless the series diverges. The normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree-Fock and first-order many-body perturbation theories.