  Within the Projected Entangled Pair State (PEPS) tensor network formalism, a
simple update (SU) method has been used to investigate the time evolution of a
two-dimensional U(1) critical spin-1/2 spin liquid under Hamiltonian quench
[Phys. Rev. B 106, 195132 (2022)]. Here we introduce two different variational
frameworks to describe the time dynamics of SU(2)-symmetric
translationally-invariant PEPS, aiming to improve the accuracy. In one
approach, after using a Trotter-Suzuki decomposition of the time evolution
operator in term of two-site elementary gates, one considers a single bond
embedded in an environment approximated by a Corner Transfer Matrix
Renormalization Group (CTMRG). A variational update of the two tensors on the
bond is performed under the application of the elementary gate and then, after
symmetrization of the site tensors, the environment is updated. In the second
approach, a cluster optimization is performed on a finite (periodic) cluster,
maximizing the overlap of the exact time-evolved state with a symmetric
finite-size PEPS ansatz. Observables are then computed on the infinite lattice
contracting the infinite-PEPS (iPEPS) by CTMRG. We show that the variational
schemes outperform the SU method and remain accurate over a significant time
interval before hitting the entanglement barrier. Studying the spectrum of the
transfer matrix, we find that the asymptotic correlations are very well
preserved under time evolution, including the critical nature of the singlet
correlations, as expected from the Lieb-Robinson (LR) bound theorem.
Consistently, the system (asymptotic) boundary is found to bedescribed by the
same Conformal Field Theory of central charge c = 1 during time evolution. We
also compute the time-evolution of the short distance spin-spin correlations
and estimate the LR velocity.
