Exact bosonization in arbitrary dimensions

Yu-An Chen

doi:10.1103/physrevresearch.2.033527

Abstract

We extend the previous results of exact bosonization, mapping from fermionic operators to Pauli matrices, in 2d and 3d to arbitrary dimensions. This bosonization map gives a duality between any fermionic system in arbitrary $n$ spatial dimensions and a new class of $(n-1)$-form $\mathbb{Z}_2$ gauge theories in $n$ dimensions with a modified Gauss's law. This map preserves locality and has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold. A new formula for Stiefel-Whitney homology classes on lattices is derived. In the Euclidean path integral, this exact bosonization map is equivalent to introducing a topological "Steenrod square" term to the spacetime action.

Highlights

INTRODUCTIONIt is well known that every fermionic lattice system in 1D is dual to a lattice system of spins with a Z2 global symmetry (and vice versa)
AND SUMMARYIt is well known that every fermionic lattice system in 1D is dual to a lattice system of spins with a Z2 global symmetry
The nonexactness of the second Stiefel-Whitney class is the obstruction to determine this 1-chain E, which prevents us from defining a self-consistent bosonization map, which dualizes the even sector of fermionic Hilbert space to a Z2 gauge theory

Summary

INTRODUCTION

It is well known that every fermionic lattice system in 1D is dual to a lattice system of spins with a Z2 global symmetry (and vice versa). Generalized to higher dimensions: the ∪n−2 of two (n − 1)cochains acting on an n-simplex is the sum of the product of (n − 1)-cochains acting on its boundary (n − 1)-simplices with the same orientation. This geometry interpretation of higher cup products is crucial since it is further shown that this property coincides with the commutation relations of fermionic hopping operators. The nonexactness of the second Stiefel-Whitney class is the obstruction to determine this 1-chain E , which prevents us from defining a self-consistent bosonization map, which dualizes the even sector of fermionic Hilbert space to a Z2 gauge theory.

EXACT BOSONIZATION IN n DIMENSIONS

Commutation relations

Gauge constraints

Gauss’s law as boundary anomaly

Euclidean path integral of lattice gauge theories

CONCLUSIONS