Bilinear Majorana representations for spin operators with spin magnitudes $S>1/2$

TL;DR

This paper classifies bilinear Majorana representations for higher spin operators, identifying efficient types that avoid unphysical sectors and demonstrating their usefulness in exactly solvable models.

Contribution

It introduces a new classification of Majorana representations for spins greater than 1/2, highlighting two types and their properties, including the avoidance of unphysical sectors.

Findings

Two types of Majorana representations identified

Type 2 reproduces known spin-1/2 and 3/2 representations

Unphysical sectors are minimal for spins s=1,2, and manageable for s>2

Abstract

We present a classification of bilinear Majorana representations for spin-$S$ operators, based on the real irreducible matrix representations of SU(2). We identify two types of such representations: While the first type can be straightforwardly mapped onto standard complex fermionic representations of spin-$S$ operators, the second type realizes spin amplitudes $S=s(s+1)/4$ with $s\in\mathbb{N}$ and can be considered particularly efficient in representing spins via fermions. We show that for $s=1$ and $s=2$ this second type reproduces known spin-$1/2$ and spin-$3/2$ Majorana representations and we prove that these are the only bilinear Majorana representations that do not introduce any unphysical spin sectors. While for $s>2$, additional unphysical spin spaces are unavoidable they are less numerous than for more standard complex fermionic representations and carry comparatively small spin amplitudes. We apply our Majorana representations to exactly solvable small spin clusters and confirm that their low energy properties remain unaffected by unphysical spin sectors, making our representations useful for auxiliary-particle based methods.