Perturbative study of the one-dimensional quantum clock model

We calculate the ground-state energy density epsilon(g) for the one-dimensional N-state quantum clock model up to order 18, where g is the coupling and N = 3, 4, 5, . . . , 10, 20. Using methods based on the Pade approximation, we extract the singular structure of epsilon "(g) or epsilon(g). They correspond to the specific heat and free energy of the classical two-dimensional (2D) clock model. We find that, for N = 3, 4, there is a single critical point at g(c) = 1. The heat capacity exponent of the corresponding 2D classical model is alpha = 0.34 +/- 0.01 for N = 3, and alpha = -0.01 +/- 0.01 for N = 4. For N > 4, there are two exponential singularities related by g(c1) = 1/g(c2), and epsilon(g) behaves as Ae(-c/vertical bar gc-g vertical bar sigma) + analytic terms near g(c). The exponent sigma gradually grows from 0.2 to 0.5 as N increases from 5 to 9, and it stabilizes at 0.5 when N > 9. The phase transitions exhibited in these models should be generalizations of the Kosterlitz-Thouless transition, which has sigma = 0.5.