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Physical review. B, Condensed matter

Complex-temperature-plane zeros: Scaling theory and multicritical mean-field models

M. L. Glasser, Vladimir Privman A, Lawrence S. Schulman

Published Feb 1, 1987• 33 citations
Publisher

Abstract

We formulate a finite-size scaling theory for the partition function. This leads to a scaling description of the complex-temperature-plane zeros. The asymptotic form of the scaling function for large complex arguments is conjectured and used to calculate explicitly the location of an unbounded number of zeros. Scaling predictions are checked against the exactly solvable generalized infinite-range models which exhibit multicritical mean-field behavior. New mathematical results are reported for the asymptotic behavior of integrals representing finite-size scaling functions of infinite-range models.

DOI
10.1103/physrevb.35.1841

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We formulate a finite-size scaling theory for the partition function. This leads to a scaling description of the complex-temperature-plane zeros. The asymptotic form of the scaling function for large complex arguments is conjectured and used to calculate explicitly the location of an unbounded number of zeros. Scaling predictions are checked against the exactly solvable generalized infinite-range models which exhibit multicritical mean-field behavior. New mathematical results are reported for the asymptotic behavior of integrals representing finite-size scaling functions of infinite-range models.
In contrast to an infinite chain, the low-temperature expansion of a one-dimensional free-field Ising model has a strong dependence on boundary conditions. I derive an explicit formula for the leading term of the expansion both under open and periodic boundary conditions and show that they are related to different distributions of the partition function zeros on the complex temperature plane. In particular, when a periodic boundary condition is imposed, the leading coefficient of the expansion grows with increasing size of the chain, due to the zeros approaching the origin.
All of the information on the behavior of the four-dimensional Ising model is contained in the distribution and density of its partition function zeros. This model is believed to belong to the same universality class as the ${\mathrm{\ensuremath{\varphi}}}_{4}^{4}$ model which plays a central role in relativistic quantum field theory. Here the scaling behavior of the edge of the distribution of zeros and the asymptotic form for the density of zeros are determined. The finite-size dependency of the density of zeros---or the distance between zeros---at the infinite volume critical point is found using both analytic and numerical approaches. As with a previous analysis of the lowest lying zero, emphasis is laid on the multiplicative logarithmic corrections to mean field scaling behavior which are related to the triviality of the Ising and ${\mathrm{\ensuremath{\varphi}}}^{4}$ models in four dimensions.
We extend the finite-size scaling theory into the complex temperature plane and present an alternative scenario for the second-order phase transition where the second-order derivatives of the partition function diverge at the transition point with the same exponent but different (discontinuous) amplitudes ${\mathit{A}}_{\ifmmode\pm\else\textpm\fi{}}$. The complex scaling partition function Zs{\mathit{Z}}_{\mathit{s}}Zs​ for the d=3 Ising model on a simple-cubic lattice is calculated using microcanonical Monte Carlo technique and is used to verify the scenario. The method of complex Zs{\mathit{Z}}_{\mathit{s}}Zs​ can be used as an alternative to the series method for the study of the second-order phase transition.