Some New Results on Complex-Temperature Singularities in Potts Models on the Square Lattice

TL;DR

This paper investigates complex-temperature singularities in q-state Potts models on the square lattice, revealing new singularities and their properties through series expansions and zero analyses, with implications for phase diagram structures.

Contribution

It provides new evidence and analytic insights into the complex-temperature singularities of Potts models, especially in the challenging Re(a)<0 region, including their locations and critical exponents.

Findings

Existence of complex conjugate CT singularities at specific endpoints.

Determination of critical exponents, e.g., β_e ≈ -1/8 for q=3.

Analytic expressions for singularity positions, e.g., for q=5, at 2(-1 ± i).

Abstract

We report some new 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 $a=e^K$) in which CT zeros of the partition function are sensitive to finite lattice artifacts. From analyses of low-temperature series expansions for $3 \le q \le 8$, 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 $a_e, \ a_e^*$ of arcs protruding into the (complex-temperature extension of the) FM phase. Exponents for these singularities are determined; e.g., for $q=3$, we find $\beta_e=-0.125(1)$, consistent with $\beta_e=-1/8$. By duality, these results also imply associated arcs extending to the (CT extension of the) symmetric PM 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 $a_e,a_e^*=2(-1 \mp i)$. Further discussions of complex-temperature phase diagrams are given.