Fermionic Gaussian projected entangled pair states in $3+1\mathrm{D}$: Rotations and relativistic limits

Fermionic Gaussian projected entangled pair states (PEPS) are fermionic tensor network state constructions that describe the physics of ground states of noninteracting fermionic Hamiltonians. As noninteracting states, one may study and analyze them very efficiently, in both analytical and numerical means. Recently it was shown that they may be used as the starting point—after applying so-called PEPS gauging mechanisms—for variational study of nontrivial, interacting states of lattice gauge theories. This is done using sign-problem free variational Monte Carlo techniques. In this work we show how to generalize such states from two to three spatial dimensions, focusing on spin representations and requirements of lattice rotations. We present constructions that are crucial for the application of the above-mentioned variational Monte Carlo techniques for studying nonperturbative lattice gauge theory physics, with fermionic matter, in $2+1\mathrm{D}$ and $3+1\mathrm{D}$ models. Thus, the constructions presented here are crucial for the study of nontrivial lattice gauge theories with fermionic tensor network states.

Figure 1

The spatial rotation: The labeling conventions of the legs of a single site ($m$), with respect to the unit vectors ${\widehat{\mathbf{e}}}_i$. When a rotation around ${\widehat{\mathbf{e}}}_i$ is carried out, the coordinate $\mathbf{x}$ is mapped to ${\mathrm{\Lambda }}_i\mathbf{x}$, and the virtual fermionic modes $c$ and $d$ associated with the legs are permuted using the matrices $R^{(i)}$ defined in the text (and can be constructed intuitively from this figure).

Figure 2

Graphical illustration of the PEPS construction in (a) two space dimensions and (b) three space dimensions. The operators $A(\mathbf{x})$ (in blue) act on the sites and the virtual legs are connected via $w_i(\mathbf{x})$ (green). Virtual legs are drawn in black and physical legs of the PEPS are drawn in yellow. The dotted lines illustrate the possibly larger lattice and the periodic boundary conditions.