Entanglement entropy and massless phase in the antiferromagnetic three-state quantum chiral clock model

The von Neumann entanglement entropy is used to estimate the critical point ${h}_c/J\ensuremath{\simeq}0.143(3)$ of the mixed ferro-antiferromagnetic three-state quantum Potts model $H={\ensuremath{\sum}}_i[J({X}_i{X}_i+1^{\phantom{\rule{0.16em}{0ex}}2}+{X}_i^{\phantom{\rule{0.16em}{0ex}}2}{X}_i+1)\ensuremath{-}h\phantom{\rule{0.16em}{0ex}}{R}_i]$, where ${X}_i$ and ${R}_i$ are standard three-state Potts spin operators and $J>0$ is the antiferromagnetic coupling parameter. This critical point value gives improved estimates for two Kosterlitz-Thouless transition points in the antiferromagnetic ($\ensuremath{\beta}<0$) region of the $\mathrm{\ensuremath{\Delta}}\ensuremath{-}\ensuremath{\beta}$ phase diagram of the three-state quantum chiral clock model, where $\mathrm{\ensuremath{\Delta}}$ and $\ensuremath{\beta}$ are, respectively, the chirality and coupling parameters in the clock model. These are the transition points ${\ensuremath{\beta}}_c\ensuremath{\simeq}\ensuremath{-}0.143(3)$ at $\mathrm{\ensuremath{\Delta}}=\frac{1}{2}$ between incommensurate and commensurate phases and ${\ensuremath{\beta}}_c\ensuremath{\simeq}\ensuremath{-}7.0(1)$ at $\mathrm{\ensuremath{\Delta}}=0$ between disordered and incommensurate phases. The von Neumann entropy is also used to calculate the central charge $c$ of the underlying conformal field theory in the massless phase $h\ensuremath{\le}{h}_c$. The estimate $c\ensuremath{\simeq}1$ in this phase is consistent with the known exact value at the particular point $h/J=\ensuremath{-}1$ corresponding to the purely antiferromagnetic three-state quantum Potts model. The algebraic decay of the Potts spin-spin correlation in the massless phase is used to estimate the continuously varying critical exponent $\ensuremath{\eta}$.