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 Pad\'e approximation,
we extract the singular structure of ϵ′′(g) or ϵ(g). They
correspond to the specific heat and free energy of the classical 2D clock
model\cite{Suzu}. We find that, for N=3,4, there is a single critical point
at gc​=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 gc1​=1/gc2​,
and ϵ(g) behaves as Ae−∣gc​−g∣σc​+analytic terms
near gc​. The exponent σ gradually grows from 0.2 to 0.5 as N
increases from 5 to 9. These findings partially agree with those in\cite{Elit},
and these models are thus generalizations of Kosterlitz-Thouless transition,
which has $\sigma=0.5