Reliability of Lattice Gauge Theories

Currently, there are intense experimental efforts to realize lattice gauge theories in quantum simulators. Except for specific models, however, practical quantum simulators can never be fine-tuned to perfect local gauge invariance. There is thus a strong need for a rigorous understanding of gauge-invariance violation and how to reliably protect against it. As we show through analytic and numerical evidence, in the presence of a gauge invariance-breaking term the gauge violation accumulates only perturbatively at short times before proliferating only at very long times. This proliferation can be suppressed up to infinite times by energetically penalizing processes that drive the dynamics away from the initial gauge-invariant sector. Our results provide a theoretical basis that highlights a surprising robustness of gauge-theory quantum simulators.

Figure 1

Right: given a gauge theory with Hamiltonian $H_0$ on $L$ matter sites, undesired processes $\propto \lambda H_1$ that break gauge symmetry drive the dynamics away from the initial gauge-invariant sector (green bubble) given by the analog of Gauss’s law, $G_j|\psi ⟩=0\forall j$ to other sectors in the total Hilbert space (blue bubble). At late times, the average gauge violation assumes a random value. Left: through the introduction of a protection term $\propto V{H}_G$ that energetically penalizes gauge-violating processes, gauge invariance is retained up to infinite times. Two sharply distinct regimes emerge: an uncontrolled-violation regime where $V$ is too small to energetically isolate the initial gauge-invariant sector, and a controlled-violation regime for sufficiently large $V$ where the infinite-time violation scales as $\sim (\lambda /V{)}^2$. The axes are both log scale. See Fig. 4 for a detailed quantitative presentation on ${\mathrm{Z}}_2$ ($p=1$) and U(1) ($p=2$) gauge theories.

Figure 2

Effect of gauge invariance-breaking terms on the dynamics of a ${\text{Z}}_2$ gauge theory. Spatiotemporal averages of (a) the gauge-invariance violation in Eq. (4), (b) the electric field in Eq. (5), and (c) the staggered boson number in Eq. (6), deviate only gradually from the ideal dynamics, before gauge-noninvariant effects begin to dominate after a timescale of $t_{\mathrm{dev}}\propto 1/\lambda$. Insets: rescaled deviations from ideal dynamics, showing the perturbative growth with $\lambda$. Data for $L=6$ matter sites. See the Supplemental Material [21] for further information.

Figure 3

Dynamics of the spatiotemporal averages of (a) the gauge-invariance violation in Eq. (4), (b) the electric field in Eq. (5), and (c) the staggered boson number in Eq. (6) at error strength $\lambda =0.05$ and at various values of the protection strength $V$ for $L=6$ matter sites. We see two clear regimes: at small $V$ the gauge-noninvariant behavior dominates after a timescale $\propto 1/\lambda$. When $V$ is sufficiently large (controlled-violation regime), the gauge-invariance violation in (a) is indefinitely suppressed by $(\lambda /V{)}^2$, while deviations in the electric field in (c) are suppressed by $\lambda /V$ up to a timescale $\propto V/{\lambda }^2$. The deviation in the staggered boson number becomes uncontrolled only at times beyond $t\propto V/{\lambda }^2$. See the Supplemental Material [21] for corresponding results at $\lambda =0.005$ and 0.5.

Figure 4

Infinite-time violation ${\sum }_j⟨G_j(t\to \infty )⟩/L$ in the ${\mathrm{Z}}_2$ gauge theory (top panel) and ${\sum }_j⟨G_j^2(t\to \infty )⟩/L$ in the U(1) gauge theory (bottom panel) as a function of $\lambda /V$ for $\lambda =0.05$, computed for several system sizes $L$. We see two separate regimes where at large $V$ the violation is controlled and scales as $(\lambda /V{)}^2$, while at small $V$ the violation is uncontrolled. The dotted lines indicate the largest $V$ at which there is an energy overlap between the initial gauge-invariant sector, and other symmetry sectors, which we protect against. In the controlled-violation regime, the violation is system size-independent. The vertical dotted lines correspond to the largest value of $V$ (for a given system size $L$) at which the initial gauge-invariant sector still has spectral overlap with other sectors.