Criticality of compact and noncompact $(1+1)D$ quantum dissipative $Z_4$-models

We study two versions of a $(1+1)D$ $Z_4$-symmetric model with Ohmic bond dissipation. In one version the phase variable is restricted to the interval $[0,2\pi\rangle$, while the domain is unrestricted in the other. The compact model features a completely ordered phase with a broken $Z_4$-symmetry and a disordered phase, separated by a critical line. The non-compact model features three phases. In addition to the two phases exhibited by the compact model, there is also an intermediate phase, characterized by isotropic power-law phase correlations. We calculate the dynamical critical exponent $z$ along the critical lines of both models to see if the compactness of the variable is relevant to the critical scaling between space and imaginary time. We find $z\approx1$ for the single phase transition in the compact model as well as for both transitions in the non- compact model.