Accurate Gauge-Invariant Tensor Network Simulations for Abelian Lattice Gauge Theory in (2+1)D: ground state and real-time dynamics

We propose a novel tensor network method to achieve accurate and efficient simulations of Abelian lattice gauge theories (LGTs) in (2+1)D for both ground state and real-time dynamics. The first key is to identify a gauge canonical form (GCF) of gauge-invariant tensor network states, which already simplifies existing algorithms for (1+1)D LGTs. The second key is to employ the GCF of projected entangled-pair state (PEPS) combining with variational Monte Carlo (VMC), enabling efficient computations for (2+1)D LGTs. We demonstrate the versatile capability of this approach for accurate ground state simulation of pure $Z_2$, $Z_3$ and $Z_4$ gauge theory, odd-$Z_2$ gauge theories, and $Z_2$ gauge theory coupled to hard-core bosons, on square lattices up to $32 \times 32$. Furthermore, we demonstrate that it allows for accurate simulations of real-time dynamics up to long-time, exemplified by the dynamics of elementary excitations of the deconfined $Z_2$ gauge field on a $10\times10$ lattice. This is also the first example of using VMC to simulate the real-time dynamics of PEPS, whose impact may extend beyond gauge theory.