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 $h_c/J \simeq 0.143(3)$ of the mixed ferro-antiferromagnetic three-state quantum Potts model $H = \sum_i [ J ( X_i X_{i+1}^{\,2} + X_i^{\,2} X_{i+1} ) - h\, R_i ]$, 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 ($\beta < 0$) region of the $\Delta$--$\beta$ phase diagram of the three-state quantum chiral clock model, where $\Delta$ and $\beta$ are, respectively, the chirality and coupling parameters in the clock model. These are the transition points $\beta_c \simeq - 0.143(3)$ at $\Delta = \frac12$ between incommensurate and commensurate phases and $\beta_c \simeq - 7.0(1)$ at $\Delta = 0$ 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 $h \le h_c$. The estimate $c \simeq 1$ in this phase is consistent with the known exact value at the particular point $h/J = -1$ 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 $\eta$. 
Yan-Wei Dai, Sam Young Cho, Murray T. Batchelor and Huan-Qiang Zhou
cond-mat.stat-mech