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Physical Review Letters

Yang-Lee Edge Singularity from a Real-Space Renormalization-Group Method

Katarina Uzelac, Pierre M. Pfeuty, Rémi Jullien

Published Sep 17, 1979• 44 citations
Publisher

Abstract

The quantum one-dimensional transverse Ising model with an imaginary longitudinal field is studied at $T=0$ 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.

DOI
10.1103/physrevlett.43.805

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Physical Review Letters•Jun 19, 1978•485 citations
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 \ensuremathσ=\ensuremath−0.25\ifmmode±\else\textpm\fi0.05\ensuremath{\sigma}=\ensuremath{-}0.25\ifmmode\pm\else\textpm\fi{}0.05\ensuremathσ=\ensuremath−0.25\ifmmode±\else\textpm\fi0.05 and \ensuremathη=\ensuremath−1.0\ifmmode±\else\textpm\fi0.02\ensuremath{\eta}=\ensuremath{-}1.0\ifmmode\pm\else\textpm\fi{}0.02\ensuremathη=\ensuremath−1.0\ifmmode±\else\textpm\fi0.02, 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 ground state of the quantum one-dimensional transverse-field Ising model in a longitudinal field is studied using a real-space renormalization-group method. The longitudinal magnetization in the ground state is calculated. Considering a real longitudinal field, this study provides an indirect investigation of the equation of state of the equivalent classical two-dimensional Ising model. Considering then the case of a purely imaginary longitudinal field, a new critical behavior is found which corresponds to the Yang-Lee edge singularity of the classical equivalent. In this last case the details of calculations and results are given, published earlier in a letter form.
A quantum renormalization-group technique is applied to the one-dimensional transverse field Ising model in a complex longitudinal field. The Yang-Lee edge singularity is analysed in this model in view of its analogy with the 2d classical Ising model. Previous two-level calculations are improved by retaining four levels in the blocking procedure. A more precise edge exponent σ is found in agreement with the infinite block extrapolation of the two-level calculation, but still larger in absolute value than the latest series analysis results. The occurrence of oscillations in the real part of the magnetization appears to be an artifact of the real-space method, which introduces finite size effects.
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.
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 ($\ensuremath{\eta}=\ensuremath{-}1$).