Spin-fermion mappings for even Hamiltonian operators

TL;DR

This paper generalizes the Jordan-Wigner transformation to arbitrary dimensions, providing local mappings between spin and fermionic Hamiltonians, and introduces new integrable models with known ground states.

Contribution

It offers a novel local identity-based framework for spin-fermion mappings applicable in any dimension, extending the classical Jordan-Wigner transformation.

Findings

Established local mappings between spin and fermionic models in arbitrary dimensions.

Derived new integrable spin Hamiltonians with known ground states.

Connected constrained-fermion models to spin 1 chains, revealing new solvable cases.

Abstract

We revisit the Jordan-Wigner transformation, showing that --rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators-- it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan-Wigner's result. It provides direct mapping of local quantum spin problems into local fermionic problems (and viceversa), under the (rather physical) requirement that the latter are described by Hamiltonian's which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermions models and spin 1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground state energies.