Perturbative study of the one-dimensional quantum clock model

We calculate the ground-state energy density ɛ (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 ɛ^″(g ) or ɛ (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_{c 1}=1 /g_{c 2} , and ɛ (g ) behaves as A e^{-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 .