A Mapping between the Spin and Fermion Algebra

TL;DR

This paper develops a formalism to map spin algebra to fermionic operators and vice versa, extending previous work and enabling the expression of complex Hamiltonians in different algebraic frameworks.

Contribution

It introduces a new formalism for expressing spin algebra in terms of fermionic operators and vice versa, extending previous mappings to higher spin values and multiple fermion flavors.

Findings

Derived a formalism for spin-fermion algebra mapping.

Extended the mapping to systems with multiple fermion flavors.

Explored simplification of Hamiltonians using the mapping.

Abstract

We derive a formalism to express the spin algebra $\mathfrak{su}(2)$ in a spin $s$ representation in terms of the algebra of $L$ fermionic operators that obey the Canonical Anti-commutation Relations. We also give the reverse direction of expressing the fermionic operators as polynomials in the spin operators of a single spin. We extend here to further spin values the previous investigations by Dobrov [J.Phys.A: Math. Gen 36 L503, (2003)] who in turn clarified on an inconsistency within a similar formalism in the works of Batista and Ortiz [Phys.\ Rev.\ Lett. 86, 1082 (2001)]. We then consider a system of $L$ fermion flavors and apply our mapping in order to express it in terms of the spin algebra. Furthermore we investigate a possibility to simplify certain Hamiltonian operators by means of the mapping.