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We introduce a class of variational states to study ground state propertiesand real-time dynamics in (2+1)-dimensional compact QED. These are based oncomplex Gaussian states which are made periodic in order to account for thecompact nature of the $U(1)$ gauge field. Since the evaluation of expectationvalues involves infinite sums, we present an approximation scheme for the wholevariational manifold. We calculate the ground state energy density for latticesizes up to $20 \times 20$ and extrapolate to the thermodynamic limit for thewhole coupling region. Additionally, we study the string tension both byfitting the potential between two static charges and by fitting the exponentialdecay of spatial Wilson loops. As the ansatz does not require a truncation inthe local Hilbert spaces, we analyze truncation effects which are present inother approaches. The variational states are benchmarked against exactsolutions known for the one plaquette case and exact diagonalization resultsfor a $\mathbb{Z}_3$ lattice gauge theory. Using the time-dependent variationalprinciple, we study real-time dynamics after various global quenches, e.g. thetime evolution of a strongly confined electric field between two charges aftera quench to the weak-coupling regime. Up to the points where finite sizeeffects start to play a role, we observe equilibrating behavior.

Real-time dynamics in 2+1d compact QED using complex periodic Gaussian states