  Although gauge invariance is a postulate in fundamental theories of nature
such as quantum electrodynamics, in quantum-simulation implementations of gauge
theories it is compromised by experimental imperfections. In a recent work
[Halimeh and Hauke,
\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.125.030503}{Phys.
Rev. Lett. \textbf{125}, 030503 (2020)}], it has been shown in finite-size
spin-$1/2$ quantum link lattice gauge theories that upon introducing an
energy-penalty term of sufficiently large strength $V$, unitary gauge-breaking
errors at strength $\lambda$ are suppressed $\propto\lambda^2/V^2$ up to all
accessible evolution times. Here, we show numerically that this result extends
to quantum link models in the thermodynamic limit and with larger spin-$S$. As
we show analytically, the dynamics at short times is described by an
\textit{adjusted} gauge theory up to a timescale that is at earliest
$\tau_\text{adj}\propto\sqrt{V/V_0^3}$, with $V_0$ an energy factor. Moreover,
our analytics predicts that a renormalized gauge theory dominates at
intermediate times up to a timescale $\tau_\text{ren}\propto\exp(V/V_0)/V_0$.
In both emergent gauge theories, $V$ is volume-independent and scales at worst
$\sim S^2$. Furthermore, we numerically demonstrate that robust gauge
invariance is also retained through a single-body gauge-protection term, which
is experimentally straightforward to implement in ultracold-atom setups and
NISQ devices.

Quantum Gases;Strongly Correlated Electrons;High Energy Physics - Lattice;High Energy Physics - Phenomenology;Quantum Physics