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Physical review. B, Condensed matter

Yang-Lee edge singularity in systems with correlated disorder

Bosiljka Tadić, R. Pirc

Published Mar 1, 1988• 3 citations
Publisher

Abstract

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-${T}_{c}$ 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{}=${d}_{c}$-d with ${d}_{c}$=8+\ensuremath{\theta} for short- and ${d}_{c}$=4\ensuremath{\sigma}+\ensuremath{\theta} for long-range interactions which do not obey any apparent dimensional reduction rule.

DOI
10.1103/physrevb.37.3569

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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\ensuremath−(d\ensuremath−\ensuremathθ){R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}\ensuremath{\theta})}R\ensuremath−(d\ensuremath−\ensuremathθ) (\ensuremath{\theta}>0). Both short- and long-range spin interactions decaying as \ensuremath{\sim}R\ensuremath−(d+\ensuremathσ){R}^{\mathrm{\ensuremath{-}}(d+\ensuremath{\sigma})}R\ensuremath−(d+\ensuremathσ) (0\ensuremath{\sigma}2) are considered. We find that the edge singularities are described by a new fixed point in the underlying \ensuremathφ3{\ensuremath{\varphi}}^{3}\ensuremathφ3-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.
The edge of the gap in the distribution of Yang-Lee zeros at H=iH0(T)H=i{H}_{0}(T)H=iH0​(T) on the imaginary magnetic field axis in ferromagnets above Tc{T}_{c}Tc​ is essentially a critical point. In terms of the edge exponents $\ensuremath{\delta}$ and $\ensuremath{\eta}$, the density of zeros obeys $\mathcal{G}({H}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\sim}{[{H}^{\ensuremath{'}\ensuremath{'}}\ensuremath{-}{H}_{0}(T)]}^{\ensuremath{\sigma}}$, with $\ensuremath{\sigma}=\frac{1}{\ensuremath{\delta}}=\frac{(d\ensuremath{-}2+\ensuremath{\eta})}{({d}_{2}\ensuremath{-}\ensuremath{\eta})}$. Classical behavior ($\ensuremath{\sigma}=\frac{1}{2}$) occurs for $d>{d}^{\ifmmode\times\else\texttimes\fi{}}=6$. The appropriate field-theoretic renormalization group entails a $w{\ensuremath{\phi}}^{3}$ coupling and, with $\ensuremath{\epsilon}=6\ensuremath{-}d>~0$, yields $\ensuremath{\eta}\ensuremath{\approx}\ensuremath{-}\frac{\ensuremath{\epsilon}}{9}$ for all $n<\ensuremath{\infty}$. This correlates well with refined series estimates for d=2d=2d=2 and d=3d=3d=3 and with exact results for d=1d=1d=1 ($\ensuremath{\eta}=\ensuremath{-}1$).
The quantum one-dimensional transverse Ising model with an imaginary longitudinal field is studied at T=0T=0T=0 using a real-space renormalization-group method. Evaluation is made of the real part of the ground-state longitudinal magnetization which is related to the Yang-Lee distribution of zeros of the partition function for the two-dimensional classical Ising model. The edge exponents are found as $\ensuremath{\sigma}=\ensuremath{-}0.25\ifmmode\pm\else\textpm\fi{}0.05$ and $\ensuremath{\eta}=\ensuremath{-}1.0\ifmmode\pm\else\textpm\fi{}0.02$, in agreement with series expansion estimates.
A T=0 real-space renormalization-group method is presented and applied to the quantum 1D transverse field Ising model in the presence of a longitudinal field. This is an indirect way to study the equation of state of the equivalent 2D classical Ising model at finite temperature. When the longitudinal field is purely imaginary we obtain a new critical behavior which is related to the Yang-Lee edge singularity of the corresponding classical model. We extract the edge exponent σ in agreement with previous series expansion estimates.
The edge of the gap in the distribution of the Yang-Lee zeros is concluded to be a singular point of square-root type in all hierarchical models, in distinction to expectations based on renormalization-group arguments for short-range models in dimensions d<6d<6d<6.