Given the fundamental difference between bosons and fermions, it is remarkable that a mapping between a fermionic system and a bosonic system, known as Jordan-Wigner transformation, exists in one spatial dimension. While a direct generalization to higher dimensional systems is possible, it is not quite satisfactory, as the short-range hoppings may be mapped to terms with non-local Jordan-Wigner strings.

In this talk, we describe a 2d analog of the Jordan–Wigner transformation which maps an arbitrary fermionic system on a 2d lattice to a lattice gauge theory while preserving the locality of the Hamiltonian [1]. We will discuss the bosonization procedure in detail for systems on 2d square lattices as well as arbitrary triangularized lattices. Then we discuss the generalization of this method to 3d [2] and even arbitrary dimensions [3].

[1] Yu-An Chen et al, Exact bosonization in two spatial dimensions and a new class of lattice gauge theories, Annals of Physics 393 (2018) 234–253.

[2] Yu-An Chen et al, Bosonization in three spatial dimensions and a 2-form gauge theory, Phys. Rev. B. 100.245127 (2019).

[3] Yu-An Chen, Exact bosonization in arbitrary dimensions, Phys. Rev. Research. 2.033527 (2020).