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Model of bosonization by flux attachment on hamiltonian lattices of arbitrary dimension

Arkadiusz Bochniak, Błażej Ruba

Published Mar 15, 2020• 1 citation
OpenAlex

Abstract

We present and prove the correctness of a bosonization prescription for fermionic lattice models in arbitrary dimensions. Our bosonized model is subject to constraints, which are interpreted in the language of lattice $\mathbb{Z}_2$ gauge theory. Complete solutions of the constraints is found in the case of even-even spatial lattices. Further possible relations with other topologically non-trivial lattice models are discussed.

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We present and prove the correctness of a bosonization prescription for fermionic lattice models in arbitrary dimensions. Our bosonized model is subject to constraints, which are interpreted in the language of lattice Z2\mathbb{Z}_2Z2​ gauge theory. Complete solutions of the constraints is found in the case of even-even spatial lattices. Further possible relations with other topologically non-trivial lattice models are discussed.
Flux-attached theories are a novel class of lattice gauge theories whose gauge constraints involve both electric and magnetic operators. Like ordinary gauge theories, they possess confining phases. Unlike ordinary gauge theories, their confinement does not imply a trivial gapped vacuum. This paper will offer three lessons about the confining phases of flux-attached ZK\mathbb Z_KZK​ theories in two spatial dimensions. First, on an arbitrary orientable lattice, flux attachment that satisfies a simple, explicitly derived criterion leads to a confining theory whose low-energy behavior is captured by an action of a general Chern-Simons form. Second, on a square lattice, this criterion can be solved, and all theories that satisfy it can be enumerated. The simplest such theory has an action given by a difference of two Chern-Simons terms, and it features a kind of subsystem symmetry that causes its topological entanglement entropy to behave pathologically. Third, the simplest flux-attached theory on a square lattice that does not satisfy the above criterion is exactly solvable when the gauge group is Z2\mathbb Z_2Z2​. On a torus, its confined phase possesses a twofold topological degeneracy that stems from a sum over spin structures in a dual fermionic theory. This makes this flux-attached Z2\mathbb Z_2Z2​ theory an appealing candidate for a microscopic description of a U(1)2\mathrm U(1)_2U(1)2​ Chern-Simons theory.
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.
Despite being less known, local bosonizations of fermionic systems exist in spatial dimensions higher than 1. Interestingly, the dual bosonic systems are subject to local constraints, as in theories with gauge freedom. These constraints effectively implement long distance exchange interactions. In this work, we study in detail one such system, proposed a long time ago. Properties of the constraints are elaborated for two-dimensional, rectangular lattices of arbitrary sizes. For several small systems, the constraints are solved analytically. It is checked that spectra of reduced spin Hamiltonians agree with the original fermionic ones. The equivalence is extended to fermions in the presence of background Wegner Z2{\mathbb{Z}}_{2}Z2​ fields coupling to fermionic parity. This is illustrated by an explicit calculation for a particular configuration of Wegner's variables. Finally, a possible connection with the recently proposed web of dualities is discussed.
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.
Lattice gauge theories are a powerful language to theoretically describe a variety of strongly correlated systems, including frustrated magnets, high-Tc{T}_{c}Tc​ superconductors, and topological phases. However, in many cases gauge fields couple to gapless matter degrees of freedom, and such theories become notoriously difficult to analyze quantitatively. In this paper we study several examples of Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories with gapless fermions at finite density, in one and two spatial dimensions, that are either exactly soluble or whose solution reduces to that of a known problem. We consider complex fermions (spinless and spinful) as well as Majorana fermions and study both theories where Gauss' law is strictly imposed and those where all background charge sectors are kept in the physical Hilbert space. We use a combination of duality mappings and the Z2{\mathbb{Z}}_{2}Z2​ slave-spin representation to map our gauge theories to models of gauge-invariant fermions that are either free, or with on-site interactions of the Hubbard or Falicov-Kimball type that are amenable to further analysis. In 1D, the phase diagrams of these theories include free-fermion metals, insulators, and superconductors, Luttinger liquids, and correlated insulators. In 2D, we find a variety of gapped and gapless phases, the latter including uniform and spatially modulated flux phases featuring emergent Dirac fermions, some violating Luttinger's theorem.
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 class of (n -1)-form 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 spatial manifold.A 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 space-time action.