Is bosonization possible in any number of dimensions?

Question

Is bosonization applicable to an arbitrary number of spacetime dimensions?

Answer 1

Exact bosonization exists in all dimensions, i.e., mapping from even fermionic operators to Pauli matrices (with certain constraints on Hilbert space) in arbitrary triangulation which preserves the algebra. The simplest example should be Bravyi-Kitaev superfast simulation of fermions mentioned in https://arxiv.org/abs/quant-ph/0003137. Their construction preserves the locality of operators and is valid for any graph.

We have a more formal way to derive the exact bosonization in

2d: https://arxiv.org/abs/1711.00515

3d: https://arxiv.org/abs/1807.07081

All dimensions: https://arxiv.org/abs/1911.00017

In (2+1)D, the exact bosonization is described by flux attachment. For example, given a toric code, we combine both charge excitation $e$ and flux excitation $m$ to form an emergent fermion $\epsilon = em$. This can help us to define an isomorphism between the fermionic operators and logical operators in the toric code subspace where only $\epsilon$ excitations are allowed. In other words, any fermionic theory in two dimensions is dual to a 1-form $\mathbb{Z}_2$ lattice gauge symmetry. If we go to spacetime picture, the above flux attachment approach can be described by the Chern-Simon term in spacetime action: $$\int A \cup \delta A$$ where $A$ is the 1-cochain representing the $\mathbb{Z}_2$ fields living on edges.

In (3+1)D, we used higher cup products to formulate the bosonization map in Hamiltonian level. The formula is analogous to flux attachment in (2+1)D. We found that the spacetime action (after Euclidean path integral) is: $$\int B\cup B + B \cup_1 \delta B$$ where B is the 2-cochain representing the $\mathbb{Z}_2$ fields living on faces. This action is called Steenrod square, which is used for the classification of fermionic SPT phases: https://arxiv.org/abs/1809.01112.

The statement above can be generalized to all dimensions easily.

Answer 2

The technique normally referred to as Bosonization only works in 1 spatial dimension. The technique is a mapping between the the Fock space of fermions (with linear dispersion) and the Fock space of Bosons.

The proof that the mapping is an isomorphism was given by Haldane who computed the grandcanonic partition function in the two cases (and obtained the same result).

In practice (part of) the success of this technique is that some Hamiltonians that look interacting in the fermionic picture become non-interacting after Bosonization (i.e. in the bosonic picture) and so can be solved exactly. This was done by Tomonaga, Luttinger and others.