Perturbative study of the one-dimensional quantum clock model.

Abstract

We calculate the ground-state energy density ε(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 Padé approximation, we extract the singular structure of ε^{″}(g) or ε(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 α=0.34±0.01 for N=3, and α=-0.01±0.01 for N=4. For N>4, there are two exponential singularities related by g_{c1}=1/g_{c2}, and ε(g) behaves as Ae^{-c/|g_{c}-g|^{σ}}+analyticterms near g_{c}. The exponent σ 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 σ=0.5.