Projected Entangled Pair States for Lattice Gauge Theories with Dynamical Fermions

Abstract

Lattice gauge theory is an important framework for studying gauge theories that arise in the Standard Model and condensed matter physics. Yet many systems (or regimes of those systems) are difficult to study using conventional techniques, such as action-based Monte Carlo sampling. In this paper, we demonstrate the use of gauged Gaussian projected entangled pair states as an ansatz for a lattice gauge theory involving dynamical physical matter. We study a $\mathbb{Z}_2$ gauge theory on a two dimensional lattice with a single flavor of fermionic matter on each lattice site. For small systems, our results show agreement with results computed by exactly diagonalizing the Hamiltonian, and demonstrate that the approach is computationally feasible for larger system sizes where exact results are unavailable. This is a further step on the road to studying higher dimensions and other gauge groups with manageable computational costs while avoiding the sign problem.

Figures (8)

• Figure 1: A diagram of the 2D lattice. Matter is shown on lattice sites in yellow and gauge fields on links in blue. The blue box shows the labeling convention for a plaquette.
• Figure 1: The ground state energies, normalized by lattice size, for the free fermion Hamiltonian. The dashed lines are simply guides for the eye. Even for moderate lattice sizes, the normalized ground state energy is relatively close to its value in the thermodynamic limit.
• Figure 2: A graphical representation of the construction of a gauged gaussian projected entangled pair state (GGPEPS) for a fixed gauge field configuration. We start on the left with the vacuum of the physical modes on each site (shown in yellow), create 4 virtual modes on each link around each site (turquoise), and couple them to each other and to the physical modes. We then couple the appropriate virtual modes to the gauge fields (blue). Finally, we project the virtual modes from neighboring sites onto a maximally entangled state, and trace out the virtual modes. The bottom row illustrates each operation locally; it is in fact applied to the entire lattice at each stage.
• Figure 3: A graphical representation of the state which shows that an entanglement area law is automatically obeyed. The physical matter on the lattice sites is shown in yellow, the virtual modes are shown in turquoise, and the gauge fields are not shown. The entanglement between the region highlighted in purple and the rest of the lattice is proportional to the number of links through which the boundary cuts, since the only source of entanglement is the projection onto maximally entangled states of the virtual modes on the same link (enclosed by the turquoise dashed lines).
• Figure 4: The various energy observables for the $2\times2$ ground states. The dots show results given by our ansatz, showing their close match to the exact diagonalization results (solid lines) for each term in the Hamiltonian. The interaction coupling $g_\text{I}$ was fixed at $g_\text{I} = 1.0$, and the mass coupling $g_\text{M}$ was fixed at $g_\text{M} = 1.0$. The horizontal axis is the electric coupling $g_\text{E} = \lambda$; the magnetic coupling is defined as $g_\text{B} = 1/\lambda$. The results shown here do not rely on Monte Carlo, but rather on exact contraction --- for a $2 \times 2$ system, it is possible to simply iterate over all $2^{2L^2} = 256$ gauge configurations of the entire lattice. No error bars are shown as no sampling error is present and we have not quantified numerical error due to the minimization procedure.