Exact solution of three-dimensional (3D) spinless fermions

TL;DR

This paper presents an exact analytical solution for a 3D spinless fermionic model derived from the Ising model, providing formulas for eigenvalues, thermodynamics, and topological phases, with applications in magnetism and superconductivity.

Contribution

It introduces a novel exact solution for the 3D spinless fermion model using Clifford algebra, Fourier, and Bogoliubov transformations, extending methods from the 3D Ising model.

Findings

Derived formulas for eigenvalues and partition function.

Analyzed critical behaviors and topological phases.

Applicable to studies of magnetism, superfluidity, and topological materials.

Abstract

The three-dimensional (3D) Ising model is mapped into a 3D spinless fermionic model by the Jordan-Wigner transformation. The exact solution of the 3D model for spinless fermions is derived analytically by performing a diagonalization process consisting of the Clifford algebraic approach, the Fourier transformation and the Bogoliubov transformation. The Clifford algebraic approach is the same as that developed for the 3D Ising model, using a time average within the Jordan-von Neumann-Wigner framework, a linearization procedure and a local gauge transformation. The formulas for eigenvalues, partition function, subsequent thermodynamic properties and critical behaviors are presented. The dimensionality and the topological phases are investigated. The present results for many spinless fermions in a 3D lattice are applicable for studying the mechanisms of magnetism, superfluid, superconductors and topological materials.