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

Abstract

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 $\ensuremath{\eta}$.