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arXiv:hep-th/9301130

Bosonisation in any dimension

H. S. Sharatchandra

Published Jan 31, 1993• 1 citation
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Abstract

The mechanism underlying any bosonisation or fermionisation is exposed.It is shown that any local theory of fermions on a lattice in any spatial dimension greater than one is equivalent to a local theory of Ising spins coupled to a $Z_{2}$ gauge field.There is a close relation to the description of anyons using a Chern-Simons term.

DOI
10.48550/arxiv.hep-th/9301130

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The mechanism underlying any bosonisation or fermionisation is exposed.It is shown that any local theory of fermions on a lattice in any spatial dimension greater than one is equivalent to a local theory of Ising spins coupled to a Z2Z_{2}Z2​ gauge field.There is a close relation to the description of anyons using a Chern-Simons term.
We describe an n-dimensional (n≥2) analog of the Jordan-Wigner transformation, which maps an arbitrary fermionic system to Pauli matrices while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization gives a duality between any fermionic system in arbitrary n spatial dimensions and a new class of (n-1)-form Z₂ gauge theories in n dimensions with a modified Gauss’s law. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z₂ lattice gauge theory with Dirac cones in the spectrum. This bosonization formalism has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold, a key feature for defining fermions. A new formula for Stiefel-Whitney homology classes on lattices is derived. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2+1)D or Steenrod Square terms for general dimensions. Finally, we apply the bosonization to construct various bosonic or fermionic symmetry-protectedtopological (SPT) phases. It has been shown that supercohomology fermionic SPT phases are dual to bosonic higher-group SPT phases.
I show that Ising models and Z(2)Z(2)Z(2) gauge theories, in any number of dimensions, are equivalent to systems of fermions with local interactions only. This is done in the Hamiltonian formalism by rewriting the Hamiltonians in terms of locally coupled fermion operators. One advantage of the fermionic formalism is that duality transformations become trivial. All unsolved models have products of four or more fermion operators as terms in their Hamiltonians.
The well-known Jordan-Wigner transformation maps an arbitrary system of fermions on a one-dimensional lattice to a system of spins. An essential property of this transformation is that it preserves the locality of observables. This transformation has been recently extended to two-dimensional lattices, where it maps an arbitrary system of fermions to a gauge theory, with locality preserved. Here, the authors construct a bosonization map for arbitrary systems of fermions in three dimensions. As an application, the authors design a model of interacting spins on a cubic lattice, which is exactly soluble and is equivalent to a tight-binding model of free fermions with Dirac points.
We devise a unitary transformation that replaces the fermionic degrees of freedom of lattice gauge theories by (hard-core) bosonic ones. The resulting theory is local and gauge invariant, with the same symmetry group. The method works in any spatial dimension and can be directly applied, among others, to the gauge groups G=U(N)G=U(N)G=U(N) and SU(2N)SU(2N)SU(2N), where $N\ensuremath{\in}\mathbb{N}$. For SU(2N+1)SU(2N+1)SU(2N+1), one can also carry out the transformation after introducing an extra idle Z2{\mathbb{Z}}_{2}Z2​ gauge field, so that the resulting symmetry group trivially contains Z2{\mathbb{Z}}_{2}Z2​ as a normal subgroup. Those results have implications in the field of quantum simulations of high-energy physics models.
We prove, working in the Hamiltonian formalism, that a U(1) Chern-Simons theory coupled to fermions on a lattice can be mapped exactly onto a theory of interacting lattice anyons. This map does not involve any singular gauge transformations, and is everywhere well defined. We also prove that, when the statistics parameter is an odd integer so that the anyons are bosons, the ground state, which consists of a condensate of bound pairs of flux tubes and fermions, breaks phase invariance. The ensuing long range order implies that the system is an unconventional superfluid.