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arXiv:2006.11361 · Physical review. E

Perturbative study of the one-dimensional quantum clock model

Bingnan Zhang

Published Oct 9, 2020• 1 citation
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Abstract

We calculate the ground-state energy density $\ensuremath{\epsilon}(g)$ 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 Pad\'e approximation, we extract the singular structure of ${\ensuremath{\epsilon}}^{\ensuremath{''}}(g)$ or $\ensuremath{\epsilon}(g)$. 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 $\ensuremath{\alpha}=0.34\ifmmode\pm\else\textpm\fi{}0.01$ for $N=3$, and $\ensuremath{\alpha}=\ensuremath{-}0.01\ifmmode\pm\else\textpm\fi{}0.01$ for $N=4$. For $N>4$, there are two exponential singularities related by ${g}_{c1}=1/{g}_{c2}$, and $\ensuremath{\epsilon}(g)$ behaves as $A{e}^{\ensuremath{-}\frac{c}{|{g}_{c}{\ensuremath{-}g|}^{\ensuremath{\sigma}}}}+\text{analytic}\phantom{\rule{4.pt}{0ex}}\text{terms}$ near ${g}_{c}$. The exponent $\ensuremath{\sigma}$ 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 $\ensuremath{\sigma}=0.5$.

DOI
10.1103/physreve.102.042110

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