  Realizations of gauge theories in setups of quantum synthetic matter open up
the possibility of probing salient exotic phenomena in condensed matter and
high-energy physics, along with potential applications in quantum information
and science technologies. In light of the impressive ongoing efforts to achieve
such realizations, a fundamental question regarding quantum link model
regularizations of lattice gauge theories is how faithfully they capture the
quantum field theory limit of gauge theories. Recent work [Zache, Van Damme,
Halimeh, Hauke, and Banerjee, at
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.106.L091502 has shown
through analytic derivations, exact diagonalization, and infinite matrix
product state calculations that the low-energy physics of $1+1$D
$\mathrm{U}(1)$ quantum link models approaches the quantum field theory limit
already at small link spin length $S$. Here, we show that the approach to this
limit also lends itself to the far-from-equilibrium quench dynamics of lattice
gauge theories, as demonstrated by our numerical simulations of the Loschmidt
return rate and the chiral condensate in infinite matrix product states, which
work directly in the thermodynamic limit. Similar to our findings in
equilibrium that show a distinct behavior between half-integer and integer link
spin lengths, we find that criticality emerging in the Loschmidt return rate is
fundamentally different between half-integer and integer spin quantum link
models in the regime of strong electric-field coupling. Our results further
affirm that state-of-the-art finite-size ultracold-atom and NISQ-device
implementations of quantum link lattice gauge theories have the real potential
to simulate their quantum field theory limit even in the far-from-equilibrium
regime.
