We report some results on the complex-temperature (CT) singularities of
q-state Potts models on the square lattice. We concentrate on the problematic region
Re(a)<0 (where
a=eK) in which CT zeros of the partition function are sensitive to finite lattice artifacts. From analyses of low-temperature series expansions for
3< q< 8, we establish the existence, in this region, of complex-conjugate CT singularities at which the magnetization and susceptibility diverge. From calculations of zeros of the partition function, we obtain evidence consistent with the inference that these singularities occur at endpoints
ae,
ae∗ of arcs protruding into the (complex-temperature extension of the) ferromagnetic phase. Exponents for these singularities are determined; e.g., for
q=3, we find ${\ensuremath{\beta}}_{e}=\ensuremath{-}0.125(1)$, consistent with ${\ensuremath{\beta}}_{e}=\ensuremath{-}\frac{1}{8}$. By duality, these results also imply associated arcs extending into the (CT extension of the) symmetric paramagnetic phase. Analytic expressions are suggested for the positions of some of these singularities; e.g., for
q=5, our finding is consistent with the exact value
ae, ${a}_{e}^{*}=2(\ensuremath{-}1\ensuremath{\mp}i)$. Further discussions of complex-temperature phase diagrams are given.
Within a real space renormalisation group (RG) scheme, the authors study the criticality of the ferromagnetic Z(4) model on an anisotropic square lattice. They use an RG cluster which has already proved to be very efficient for the Potts model on the same lattice. The establishment of the RG recurrence relations is greatly simplified through the break-collapse method. The phase diagram (exhibiting ferromagnetic, paramagnetic and nematic-like phases) recovers all the available exact results, and is believed to be of high precision everywhere. If the model is alternatively thought of as being associated with a particular hierarchical lattice rather than with the square lattice, then it is exact everywhere.
Identifying endpoint criticality is challenging because pseudo-first-order signatures can persist over accessible system sizes and obscure the underlying scaling behavior. This difficulty is especially acute in the frustrated square-lattice
J1--
J2 Ising model, where the location and even the nature of the stripe-ordering endpoint have remained controversial for decades. Here we treat this endpoint as a relevant--marginal two-field scaling problem and develop a generalized field-mixing construction to resolve the endpoint ambiguity. Mixed-histogram symmetry defines a finite-size pseudotransition manifold, transforming pseudo-first-order finite-size structure into a scaling trajectory on which the normal relevant detuning is removed and the residual marginal drift is exposed. Along this manifold, Binder analysis locates the endpoint as a distinct singular point, while mixed-histogram scaling, comparison with the Ashkin--Teller Potts point, and complementary Binder scaling establish four-state Potts endpoint criticality with logarithmic corrections.