Complex-Temperature Singularities of the Susceptibility in the $d=2$ Ising Model. I. Square Lattice

Abstract

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 $\chi \sim A_s^{\prime }(1+u{)}^{-{\gamma }_s^{\prime }}$ with exponent ${\gamma }_s^{\prime }=3/2$. The critical amplitude $A_s^{\prime }$ is calculated. Other critical exponents are found to be ${\alpha }_s^{\prime }={\alpha }_s=0$ and ${\beta }_s=1/4$, so that the scaling relation ${\alpha }_s^{\prime }+2{\beta }_s+{\gamma }_s^{\prime }=2$ 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.

arXiv commentLatex file, 27 pages of text plus figures appended to file. ITP-SB-94-37. (further results added to sections 4, 7)

References in corpus (3)

• Series studies of the Potts model. I: The simple cubic Ising model
• Low-Temperature Series Expansions for the Spin-1 Ising Model
• Series studies of the Potts model. II: Bulk series for the square lattice

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