Finite-temperature many-body perturbation theory for electrons: Algebraic recursive definitions, second-quantized derivation, linked-diagram theorem, general-order algorithms, grand canonical and canonical ensembles

Abstract

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 downwards, 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 towards the finite-temperature full-configuration-interaction limit unless the series diverges. Normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree--Fock and first-order many-body perturbation theories.

arXiv comment29 pages, 21 figures, 8 tables

References in corpus (24)

• Heat-bath Configuration Interaction: An efficient selected CI algorithm inspired by heat-bath sampling
• The Density Matrix Renormalization Group in Chemistry and Molecular Physics: Recent Developments and New Challenges
• Towards the solution of the many-electron problem in real materials: equation of state of the hydrogen chain with state-of-the-art many-body methods
• Efficient Implementation of Ab Initio Quantum Embedding in Periodic Systems: Density Matrix Embedding Theory
• Dynamical mean-field theory from a quantum chemical perspective
• Accurate exchange-correlation energies for the warm dense electron gas
• Second-order Green's function perturbation theory for periodic systems
• Direct comparison of many-body methods for realistic electronic Hamiltonians
• Finite-temperature Auxiliary-Field Quantum Monte Carlo: Self-Consistent Constraint and Systematic Approach to Low Temperatures
• A Phaseless Auxiliary-Field Quantum Monte Carlo Perspective on the Uniform Electron Gas at Finite Temperatures: Issues, Observations, and Benchmark Study
• Exploring connections between statistical mechanics and Green's functions for realistic systems. Temperature dependent electronic entropy and internal energy from a self-consistent second-order Green's function
• Mottness in High-Temperature Copper-Oxide Superconductors
• Thermofield Theory for Finite-Temperature Quantum Chemistry
• Finite-temperature second-order many-body perturbation theory revisited
• Theory of Non-Interacting Fermions and Bosons in the Canonical Ensemble
• Finite-Temperature Many-Body Perturbation Theory in the Canonical Ensemble
• Finite-temperature many-body perturbation theory in the grand canonical ensemble
• Wave function methods for canonical ensemble thermal averages in correlated many-fermion systems
• Converging finite-temperature many-body perturbation theory that conserves the average number of electrons
• High-Level Coupled-Cluster Energetics by Monte Carlo Sampling and Moment Expansions: Further Details and Comparisons
• Unveiling the Finite Temperature Physics of Hydrogen Chains via Auxiliary Field Quantum Monte Carlo
• Numerical Evidence Invalidating Finite-Temperature Many-Body Perturbation Theory
• Koopmans' theorem in statistical Hartree-Fock theory
• Low-temperature breakdown of many-body perturbation theory for thermodynamics

Cited by in corpus (9)

• Nonconvergence of the Feynman-Dyson diagrammatic perturbation expansion of propagators
• General solution to the Kohn-Luttinger nonconvergence problem
• Finite-temperature many-body perturbation theory for vibrations: Recursions, algebraic reduction, second-quantized reduction, diagrammatic rules, linked-diagram theorem, finite-temperature self-consistent field, and general-order algorithm
• Thermal mean-field theories
• Piecewise Interaction Picture Density Matrix Quantum Monte Carlo
• Electronic specific heat capacities and entropies from density matrix quantum Monte Carlo using Gaussian process regression to find gradients of noisy data
• Thermal quasi-particle theory
• In-Medium Similarity Renormalization Group at Finite Temperature
• Complex analysis of divergent perturbation theory at finite temperature