Variational method for $\mathbb{Z}_K$ wavefunctions in spin-$J$ PXP model

We investigate the approach of time-dependent variational principle (TDVP) for the one-dimensional spin-$J$ PXP model with detuning, which is relevant for programmable Rydberg atom arrays. The variational manifold is chosen as the minimally entangled $\mathbb{Z}_K$ matrix-product-states (MPS). We demonstrate that variational dynamics and variational error can be expressed as rapidly convergent series in the thermodynamic limit. In particular, for $J=1/2$ and the limiting case $J\rightarrow \infty$, the TDVP results become exact and significantly simplified.