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
Xi and
Ri 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 $\ensuremath{\eta}$.
The two-dimensional Q-state Potts model with real couplings has a first-order transition for Q>4. We study a loop-model realization in which Q is a continuous parameter. This model allows for the collision of a critical and a tricritical fixed point at Q=4, which then emerge as complex conformally invariant theories at Q>4, or even complex Q, for suitable complex coupling constants. All critical exponents can be obtained as analytic continuation of known exact results for Q≤4. We verify this scenario in detail for Q=5 using transfer-matrix computations.
We compute the combined two and three loop order correction to the spin-spin correlation functions for the 2D Ising and q-states Potts model with random bonds at the critical point. The procedure employed is the renormalisation group approach for the perturbation series around the conformal field theories representing the pure models. We obtain corrections for the correlations functions which produce crossover in the amplitude but don't change the critical exponent in the case of the Ising model and which produce a shift in the critical exponent, due to randomness, in the case of the Potts model. Comparison with numerical data is discussed briefly.