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Complex-temperature singularities of the susceptibility in the d = 2 Ising model: I. Square lattice

V. Matveev, Robert E. Shrock

Published Jan 1, 2015• 30 citations
OpenAlex

Abstract

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.

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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.
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 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 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 ${\ensuremath{\beta}}_{e}=\ensuremath{-}0.125(1)$, 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=5q=5q=5, our finding is consistent with the exact value ae{a}_{e}ae​, ${a}_{e}^{*}=2(\ensuremath{-}1\ensuremath{\mp}i)$. Further discussions of complex-temperature phase diagrams are given.
We derive low-temperature series (in the variable ) for the spontaneous magnetization, susceptibility, and specific heat of the spin-S Ising model on the square lattice for , 2, , and 3. We determine the location of the physical critical point and non-physical singularities. The number of non-physical singularities closer to the origin than the physical critical point grows quite rapidly with S. The critical exponents at the singularities which are closest to the origin and for which we have reasonably accurate estimates are independent of S. Due to the many non-physical singularities, the estimates for the physical critical point and exponents are poor for higher values of S, though consistent with universality.
Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K; for the triangular lattice to v 16 , for the square lattice to v 21 and for the honeycomb lattice to v 32 , inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.
By combining all available information it is possible to extend the numerical knowledge of the high- and low-temperature zero-field isothermal susceptibility of the Ising model. Estimates of the third most singular term of the ferromagnetic singularity are obtained, using both direct estimates and an extension of the Generalized Law of Corresponding States. A recent conjecture of Barouch, McCoy and Wu (1973) concerning the equality of the constant term in the asymptotic expansion of the high- and low-temperature ferromagnetic susceptibility is confirmed numerically. Finally, a non-physical critical exponent of the low-temperature susceptibility of the triangular lattice is identified.