CERN Accelerating science

The smallest non-trivial system in two spatial dimensions, which four lattice sites and boundaries that are periodically identified. The theory can be formulated either in terms of electric fields and plaquettes or in terms of rotors and plaquettes. One plaquette can always be removed.

Dependence of the non-compact theory on the value of $b_{\rm max}$ for two values of $\ell$. The dashed and dotted lines show the commutator expectation value for the first and second/third lattice plaquette as a function of $b_{\rm max} / b^{\rm NC}_{\rm max}$. In the solid lines we show the results for energy difference compared to the analytical result. All curves are minimized for values of $b_{\rm max} \approx b^{\rm NC}_{\rm max}$. As these plots are for the non-compact theory, they are independent of the value of $g$.

Expectation value of the plaquette operator for $\ell = 2$ (red), $\ell = 3$ (green), $\ell = 4$ (blue). The solid lines show the results of this work, while the dashed lines with circles and dotted lines with cross marks denote the results of~\cite{Haase:2020kaj} in the magnetic and electric basis, respectively. The ratios to the result of our work with $\ell = 6$ are shown below. In the bottom plot we also show the analytical solution of the non-compact theory, which should give the correct result at low values of $g$.

Dependence on the value of $L_{\rm opt}$ as defined via the sequence fidelity given in Eq.~\eqref{eq:sequenceFid} on the coupling constant $g$. We show the results for values $\ell = 3, 4, 5$ in green, blue and magenta, respectively. For values above about $g \lesssim 0.37$ the value of $L_{\rm opt}$ starts to rise proportional to $g$. The plot looks similar to the Fig. 3(d) of~\cite{Haase:2020kaj}, but a detailed comparison is not possible due to the log-log nature of the plots.

Comparison of $x_\text{max}$ to $x_\text{max}^{\rm C}$ for $\ell = 3, 4$, $g = 0.1$ and $\beta_x, \beta_p =1$. The solid lines show the fractional difference between the energies of the digitizied compact theory and the analytic result; the dashed line shows the deviation from the canonical commutation relation, as defined in Eq.~\eqref{eq:canoncialComm}.

Same as Fig.~\ref{fig:ToyModelCompareSmall}, but with $g = 0.7$. Note that for $\ell = 4$, the value of $x_\text{max}^{\rm C}$ freezes out at $g \sim 0.83$ and the theory becomes equivalent to the KS formulation.