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arXiv:cond-mat/9605176 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Complex-temperature singularities in Potts models on the square lattice

V. Matveev, Robert E. Shrock

Published Dec 1, 1996• 48 citations
arXivHTMLPDF

Abstract

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 $\mathrm{Re}(a)<0$ (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 $3<~q<~8$, 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 ${a}_{e}$,${a}_{e}^{*}$ 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 ${a}_{e}$, ${a}_{e}^{*}=2(\ensuremath{-}1\ensuremath{\mp}i)$. Further discussions of complex-temperature phase diagrams are given.

DOI
10.1103/physreve.54.6174

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We report some results on the complex-temperature (CT) singularities of qqq-state Potts models on the square lattice. We concentrate on the problematic region Re(a)<0\mathrm{Re}(a)<0Re(a)<0 (where a=eKa={e}^{K}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< 83<~q<~83< 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{a}_{e}ae​,ae∗{a}_{e}^{*}ae∗​ of arcs protruding into the (complex-temperature extension of the) ferromagnetic phase. Exponents for these singularities are determined; e.g., for q=3q=3q=3, we find \ensuremathβe=\ensuremath−0.125(1){\ensuremath{\beta}}_{e}=\ensuremath{-}0.125(1)\ensuremathβe​=\ensuremath−0.125(1), consistent with \ensuremathβe=\ensuremath−18{\ensuremath{\beta}}_{e}=\ensuremath{-}\frac{1}{8}\ensuremathβe​=\ensuremath−81​. 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=5q=5q=5, our finding is consistent with the exact value ae{a}_{e}ae​, ae∗=2(\ensuremath−1\ensuremath∓i){a}_{e}^{*}=2(\ensuremath{-}1\ensuremath{\mp}i)ae∗​=2(\ensuremath−1\ensuremath∓i). Further discussions of complex-temperature phase diagrams are given.
We calculate complex-temperature (CT) zeros of the partition function for the qqq-state Potts model on the honeycomb and kagom\'e lattices for several values of qqq. These give information on the CT phase diagrams. A comparison of results obtained for different boundary conditions and a discussion of some CT singularities are given. Among other results, our findings show that the Potts antiferromagnet with q=4q=4q=4 and q=5q=5q=5 on the kagom\'e lattice has no phase transition at either finite or zero temperature.
We study complex-temperature singularities of the Ising model on the triangular and honeycomb lattices. We first discuss the complex-T phases and their boundaries. From exact results, we determine the complex-T singularities in the specific heat and magnetization. For the triangular lattice we discuss the implications of the divergence of the magnetization at the point (where ) and extend a previous study by Guttmann of the susceptibility at this point with the use of differential approximants. For the honeycomb lattice, from an analysis of low-temperature series expansions, we have found evidence that the uniform and staggered susceptibilities and both have divergent singularities at , and our numerical values for the exponents are consistent with the hypothesis that the exact values are . The critical amplitudes at this singularity were calculated. Using our exact results for and together with numerical values for from series analyses, we find that the exponent relation is violated at z = -1 on the honeycomb lattice; the right-hand side is consistent with being equal to 4 rather than 2. The connections of the critical exponents at these two singularities on the triangular and honeycomb lattice are discussed.
We study the complex-temperature phase diagram of the square-lattice Ising model for nonzero external magnetic field H, i.e., for 0\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}\ensuremath{\infty}, where \ensuremath{\mu}=${\mathit{e}}^{\mathrm{\ensuremath{-}}2\mathrm{\ensuremath{\beta}}\mathit{H}}$. We also carry out a similar analysis for -\ensuremath{\infty}\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}0. The results for the interval -1\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1 provide a way of continuously connecting the two known exact solutions of this model, viz., at \ensuremath{\mu}=1 (Onsager, Yang) and \ensuremath{\mu}=-1 (Lee and Yang). Our methods include numerical calculations of complex-temperature zeros of the partition function and an analysis of low-temperature series expansions. For real nonzero H, the inner branch of a lima\ifmmode \mbox{\c{c}}\else \c{c}\fi{}on bounding the FM phase breaks and forms two complex-conjugate arcs. We study the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs. For \ensuremath{\mu}0, there are two line segments of singularities on the negative and positive u axis, and we carry out a similar study of the behavior at the inner endpoints of these arcs, which constitute the nearest singularities to the origin in this case. Finally, we also determine the exact complex-temperature phase diagrams at \ensuremath{\mu}=-1 on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero-field phase diagrams. \textcopyright{} 1996 The American Physical Society.
We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that, as one approaches the point u=u s =-1 (where u=e '4K ) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form X approximately A s '(1+u)(- gamma s ') with exponent gamma s '=3/2. The critical amplitude A s ' is calculated. Other critical exponents are found to be alpha s '= sigma s =0 and beta s = 1/4 , so that the scaling relation alpha s '+2 beta s + gamma s ' is satisfied. However, using exact results for beta s on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: beta s is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u=-1 from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.
We present calculations of the complex-temperature zeros of the partition functions for 2D Ising models on the square lattice with spin s=1s=1s=1, 3/2, and 2. These give insight into complex-temperature phase diagrams of these models in the thermodynamic limit. Support is adduced for a conjecture that all divergences of the magnetisation occur at endpoints of arcs of zeros protruding into the FM phase. We conjecture that there are 4[s2]−24[s^2]-24[s2]−2 such arcs for s≥1s \ge 1s≥1, where [x][x][x] denotes the integral part of xxx.
king model on a square lattice. From an analysis of low-temperature series expansions. we find evidence that, as one approaches the point U = us =- I (where U = e-") from within the complex extensions of the FM or AFM phases. the susceptibility has 3. divergent singularity of the farm,y- A:(l + u)-y: with exponent r: = i. The critical amplitude A: is calculated. Other critical exponents are found to be ci = cs = 0 and ps = $, so thY the scaling relation U: + 26, + y: = 2 is satisfied. However, using exact results for ps on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: ps is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length nnd hence susceptibility are finite GS one approaches the point I ( =-1 from within the symmetric phase. This is m n h e d by an explicit study of high-tempemture series expansions. We investigate the complex-temperature singularities of the susceptibility of the 1.