Fig. 2: Conjectured ground-state phase diagrams.
We show two qualitative sketches of phase diagrams for the effective model (4). In (a), we consider U(1) matter (Δ1 = Δ2 = 0) coupled to a dynamical \({{\mathbb{Z}}}_{2}\) gauge field as discussed in the main text. Along the vertical direction the filling is tuned, which yields an even (odd) \({{\mathbb{Z}}}_{2}\) pure gauge theory in the vacuum (Mott insulator) illustrated by the gray regions. In between the matter and gauge degrees-of-freedom interplay, for which we examined the limiting cases. Above the deconfined region, we expect a superfluid regime (yellow), while above the confined region composite mesons of \({{\mathbb{Z}}}_{2}\) charges may condense (red). In (b), we show the phase diagram for an Ising \({{\mathbb{Z}}}_{2}\) LGT as proposed by Fradkin and Shenker6. The 2D quantum Hamiltonian of the Ising \({{\mathbb{Z}}}_{2}\) mLGT has equal hopping t and pairing Δ1 strength and can thus be mapped on a classical 3D Ising theory. Because our model with quantum \({{\mathbb{Z}}}_{2}\) matter coupled to dynamical \({{\mathbb{Z}}}_{2}\) gauge fields has slight anisotropy between hopping and pairing, t ≠ Δ1, as well as additional anomalous pairing terms Δ2, the classical mapping can only work approximately. We anticipate that the phase diagram should be qualitatively very similar to (b).
