Computational Many-Body Physics via $\mathcal M_{2^q}$ Algebra

TL;DR

This paper introduces a method to represent many-fermion operators as matrices using an algebraic isomorphism, enabling explicit calculations for complex fermionic systems.

Contribution

It develops an analytic matrix representation of many-fermion operators via an isomorphism between CAR and $ ext{M}_{2^q}$ algebras, with practical code implementations.

Findings

Explicit matrix representations for fermionic operators

Implementation of Hubbard-type Hamiltonians

Enhanced computational tools for many-body physics

Abstract

The many-body Hamiltonians and other fermionic physical observables are expressed in terms of fermionic creation and annihilation operators, which form the algebra of canonical anti-commutation relations (CAR). In this work we use a canonical isomorphism between CAR and $\mathcal M_{2^\infty}$ algebras to derive analytic matrix representations of many-fermion operators. Code-lines implementing these matrix representations are supplied and Hubbard-type Hamiltonians are worked out explicitly.