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Journal of Physics A Mathematical and General

Yang-Lee edge singularity in the one-dimensional long-range Ising model

Zvonko Glumac, Katarina Uzelac

Published Jan 21, 1991• 14 citations
Publisher

Abstract

The finite-range scaling method has been used to study the Yang-Lee edge singularity problem for the one-dimensional ferromagnetic Ising model with long-range interactions decaying with distance r, as r -d- sigma . The phase diagram and the critical exponent nu have been calculated for different values of sigma by using finite ranges L<or=9. The critical value sigma c dividing the long-range from the short-range critical regime has been discussed.

DOI
10.1088/0305-4470/24/2/022

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The finite-range scaling method has been used to study the Yang-Lee edge singularity problem for the one-dimensional ferromagnetic Ising model with long-range interactions decaying with distance r, as r -d- sigma . The phase diagram and the critical exponent nu have been calculated for different values of sigma by using finite ranges L<or=9. The critical value sigma c dividing the long-range from the short-range critical regime has been discussed.
The critical behaviour of the one-dimensional Ising model with long-range ferromagnetic interactions decaying with distance r as 1/r 1+ sigma has been studied by scaling the range of interactions. Exact calculations have been done for a system with finite ranges up to the tenth neighbour for 0< sigma <or=1. Applying the range scaling, the critical temperature, critical exponent nu and the anomalous dimension of the order parameter have been calculated. Additional analysis of the convergence of the method has been performed by applying the Vanden Broeck and Schwartz extrapolation procedure in addition to the simple least-squares approximation and by evaluating the convergence exponents.
We apply renormalization-group methods to study the density of Yang-Lee zeros at the edges of the gap in Ising systems with correlated random-Tc{T}_{c}Tc​ impurities. The impurity correlations are assumed to fall off at large distances as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}\ensuremath{\theta})}$ (\ensuremath{\theta}>0). Both short- and long-range spin interactions decaying as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d+\ensuremath{\sigma})}$ (0\ensuremath{\sigma}2) are considered. We find that the edge singularities are described by a new fixed point in the underlying ${\ensuremath{\varphi}}^{3}$-field model with imaginary coupling and imaginary correlated random fields. We obtain the edge exponents to leading order in the \ensuremath{\epsilon}\ifmmode \tilde{}\else \~{}\fi{} expansion, where \ensuremath{\epsilon}\ifmmode \tilde{}\else \~{}\fi{}=dc{d}_{c}dc​-d with dc{d}_{c}dc​=8+\ensuremath{\theta} for short- and dc{d}_{c}dc​=4\ensuremath{\sigma}+\ensuremath{\theta} for long-range interactions which do not obey any apparent dimensional reduction rule.
Critical exponents for a one-dimensional general continuous Ising model with long-range ferromagnetic interactions decaying as 1/r1+σ are calculated using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between −1 and 1. The critical point behavior is investigated. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents; the singularities in the specific heat and susceptibility depend on σ. For σ=1, there are extremely weak finite size effects: the first-order and the second-order cumulants of the order parameter yield ν=2.42(1). The susceptibility exponent γ=2.259(9). Results for σ=0.7 will be shown and discussed.
The Ising ferromagnet, consisting of magnetic spins, is the simplest system showing phase transitions and critical phenomena at finite temperatures. The Ising ferromagnet has played a central role in our understanding of phase transitions and critical phenomena. Also, the Ising ferromagnet explains the gas-liquid phase transitions accurately. In particular, the Ising ferromagnet in a nonzero magnetic field has been one of the most intriguing and outstanding unsolved problems. We study analytically the partition function zeros in the complex magnetic-field plane and the Yang-Lee edge singularity of the infinite-range Ising ferromagnet in an external magnetic field. In addition, we compare the Yang-Lee edge singularity of the infinite-range Ising ferromagnet with that of the square-lattice Ising ferromagnet in an external magnetic field.
The quantum phase transition of a one-dimensional transverse field Ising model in an imaginary longitudinal field is studied. A new order parameter M is introduced to describe the critical behaviors in the Yang-Lee edge singularity (YLES). The M does not diverge at the YLES point, a behavior different from other usual parameters. We term this unusual critical behavior around YLES as the pseudo-YLES. To investigate the static and driven dynamics of M, the (1+1) dimensional ferromagnetic-paramagnetic phase transition ((1+1) D FPPT) critical region, (0+1) D YLES critical region and the (1+1) D YLES critical region of the model are selected. Our numerical study shows that the (1+1) D FPPT scaling theory, the (0+1) D YLES scaling theory and (1+1) D YLES scaling theory are applicable to describe the critical behaviors of M, demonstrating that M could be a good indicator to detect the phase transition around YLES. Since M has finite value around YLES, it is expected that M could be quantitatively measured in experiments.