Reliability of lattice gauge theories in the thermodynamic limit

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 paper [Halimeh and Hauke, Phys. Rev. Lett. 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 λ are suppressed ∝λ2/V2 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 adjusted gauge theory up to a timescale that is at earliest τadj∝√V/V30, with V0 an energy factor. Moreover, our analytics predicts that a renormalized gauge theory dominates at intermediate times up to a timescale τren∝exp(V/V0)/V0. In both emergent gauge theories, V is volume independent and scales at worst ∼S2. 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