Finite-lattice Hamiltonian results for the phase structure of theZ(q)models and theq-state Potts models

We use our finite-lattice approach to study the phase transitions of the Hamiltonian formulation for the (1 + 1)-dimensional $Z\mathrm{}\left(q\right)\mathrm{}$ models. We confirm the known result that these models possess two phases for $q<\mathrm{}{q}_c$ and three phases for $q>~q_c$. Our calculation, however, gives $q_c=6\mathrm{}$, while the perturbative calculation predicts $q_c=5\mathrm{}$. Similar calculations for the (1 + 1)-dimensional $q$-state Potts models for $q=4, 5, \mathrm{and} 8$ failed to differentiate the second-order transition expected for $q<~4$ from the first-order transition expected for $q>~5$.