Eliminating fermionic matter fields in lattice gauge theories

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 $G=U\left(N\right)$ and $SU\left(2N\right)$, where $N\in \mathbb{N}$. For $SU(2N+1)$, one can also carry out the transformation after introducing an extra idle ${\mathbb{Z}}_2$ gauge field, so that the resulting symmetry group trivially contains ${\mathbb{Z}}_2$ as a normal subgroup. Those results have implications in the field of quantum simulations of high-energy physics models.

Figure 1

General lattice gauge theory settings. (a) The fermionic fields reside on vertices, while the gauge fields on the links of a square lattice. Notation conventions for (b) the plaquette interaction of $H_{\text{B}}$ as used in Eq. (5b); (c),(d) the link interactions of $H_{\text{I}}$ Eq. (6); (e) gauss law Eq. (7).

Figure 2

Notation conventions for the transformation and its results. (a) Auxiliary fermions of I—$c_x$, and of type II—${\alpha }_x,{\beta }_x,{\gamma }_x,{\delta }_x$ and $E$ conventions around a single vertex $x$, for Eq. (20). (b) The fermionic modes along links $f_{h,v}$ are built out of the Majorana modes associated with the link's endpoints, (c) the sign factors ${\xi }_{h,v}$ Eq. (21), (d) sign factors for ${\tilde{H}}_{\text{B}}$ Eq. (27), (e) sign factors ${\xi }_{\mathcal{M}}$ for the transformed horizontal meson operator $\tilde{\mathcal{M}}$ Eq. (30)—links with circles contribute to the sign factor.

Figure 3

(a) System with open boundary conditions. Apart from the original gauge field degrees of freedom (blue dots), there are auxiliary gauge degrees of freedom (green) on the links; the three kinds of plaquettes are indicated: bulk (blue), boundary (orange), and corner (green). (b) Loop configuration. Highlighted are the spins in $\left|1\right.〉$. All blue loops are closed and thus they respect that gauss law Eq. (40), whereas the one in red is open, and thus violates that law at the edges, i.e., in the plaquettes marked in red.