Fermionization Transform for Certain Higher-Dimensional Quantum Spin Models

TL;DR

This paper introduces a generalized fermionization transform for higher-dimensional quantum spin models, enabling exact mapping of complex lattice Hamiltonians with multi-spin interactions to fermionic systems.

Contribution

It extends the Jordan-Wigner transform to higher dimensions by enlarging the Hilbert space with auxiliary spins, allowing exact fermionization of complex spin Hamiltonians.

Findings

Provides a new fermionization method for higher-dimensional models.

Suggests a spin-liquid state in a related ring-exchange model.

Simplifies analysis of complex multi-spin interactions.

Abstract

Proposed is a generalization of Jordan-Wigner transform that allows to exactly fermionize a large family of quantum spin Hamiltonians in dimensions higher than one. The key new steps are to enlarge the Hilbert space of the original model by adding to it a collection of stand-alone free spins and to use a combination of these auxiliary operators and the lattice spins to construct a proper fermion representation of the physical Hamiltonian. The transform is especially useful for lattice spin Hamiltonians, where two-spin interactions of XY-type are either absent or exist only within one-dimensional chains and where the chains are coupled via two-spin interactions of Ising type, ring-exchange terms, or more general multi-spin interactions that involve an even number of spin operators from each chain. Using the proposed fermionization method we provide a simple argument suggesting that a spin Hamiltonian closely-related to the ring-exchange model proposed by Paramekanti, Balents, and Fisher [Phys. Rev. B 66, 054526 (2002)] indeed realizes a spin-liquid state.