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Physical Review Letters

Yang-Lee Edge Singularity and ϕ 3 Field Theory

Michael Ellis Fisher

Published Jun 19, 1978• 485 citations
Publisher

Abstract

The edge of the gap in the distribution of Yang-Lee zeros at $H=i{H}_{0}(T)$ on the imaginary magnetic field axis in ferromagnets above ${T}_{c}$ 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 $d=2$ and $d=3$ and with exact results for $d=1$ ($\ensuremath{\eta}=\ensuremath{-}1$).

DOI
10.1103/physrevlett.40.1610

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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 \ensuremathδ\ensuremath{\delta}\ensuremathδ and \ensuremathη\ensuremath{\eta}\ensuremathη, the density of zeros obeys G(H\ensuremath′\ensuremath′)\ensuremath∼[H\ensuremath′\ensuremath′\ensuremath−H0(T)]\ensuremathσ\mathcal{G}({H}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\sim}{[{H}^{\ensuremath{'}\ensuremath{'}}\ensuremath{-}{H}_{0}(T)]}^{\ensuremath{\sigma}}G(H\ensuremath′\ensuremath′)\ensuremath∼[H\ensuremath′\ensuremath′\ensuremath−H0​(T)]\ensuremathσ, with \ensuremathσ=1\ensuremathδ=(d\ensuremath−2+\ensuremathη)(d2\ensuremath−\ensuremathη)\ensuremath{\sigma}=\frac{1}{\ensuremath{\delta}}=\frac{(d\ensuremath{-}2+\ensuremath{\eta})}{({d}_{2}\ensuremath{-}\ensuremath{\eta})}\ensuremathσ=\ensuremathδ1​=(d2​\ensuremath−\ensuremathη)(d\ensuremath−2+\ensuremathη)​. Classical behavior (\ensuremathσ=12\ensuremath{\sigma}=\frac{1}{2}\ensuremathσ=21​) occurs for d>d\ifmmode×\else\texttimes\fi=6d>{d}^{\ifmmode\times\else\texttimes\fi{}}=6d>d\ifmmode×\else\texttimes\fi=6. The appropriate field-theoretic renormalization group entails a w\ensuremathϕ3w{\ensuremath{\phi}}^{3}w\ensuremathϕ3 coupling and, with \ensuremathϵ=6\ensuremath−d> 0\ensuremath{\epsilon}=6\ensuremath{-}d>~0\ensuremathϵ=6\ensuremath−d> 0, yields \ensuremathη\ensuremath≈\ensuremath−\ensuremathϵ9\ensuremath{\eta}\ensuremath{\approx}\ensuremath{-}\frac{\ensuremath{\epsilon}}{9}\ensuremathη\ensuremath≈\ensuremath−9\ensuremathϵ​ for all n<\ensuremath∞n<\ensuremath{\infty}n<\ensuremath∞. 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 (\ensuremathη=\ensuremath−1\ensuremath{\eta}=\ensuremath{-}1\ensuremathη=\ensuremath−1).
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 density of Lee-Yang zeros, in the thermodynamic limit, for classical nnn-vector models and for the quantum Heisenberg model is studied in the asymptotic high-temperature limit. It is shown that the high-temperature series expansions for these models reduce, in this limit, to the corresponding low-density expansions for the monomer-dimer problem with negative dimer activity. If the density of zeros, $g({h}^{\ensuremath{'}\ensuremath{'}})$, on the imaginary axis of the complex reduced-magnetic-field plane, $h=\frac{H}{{k}_{B}T}={h}^{\ensuremath{'}}+i{h}^{\ensuremath{'}\ensuremath{'}}$, has an algebraic singularity at the edge of the gap in the zero distribution, $g({h}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\sim}{[|{h}^{\ensuremath{'}\ensuremath{'}}|\ensuremath{-}{h}_{0}(T)]}^{\ensuremath{\sigma}}$, then $\ensuremath{\sigma}$ is independent of nnn in this limit. Analyzing dimer density series on various lattices by means of the ratio test, Dlog Pad\'e, the recursion-relation method, and inhomogeneous differential approximants, we obtain the estimates $\ensuremath{\sigma}=\ensuremath{-}0.163\ifmmode\pm\else\textpm\fi{}3$ for d=2d=2d=2 dimensions and $\ensuremath{\sigma}=0.086\ifmmode\pm\else\textpm\fi{}15$ for d=3d=3d=3.
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.
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.
Yang-Lee edge singularities are the branch point of free energy on the complex plane of physical parameters and have been shown to be the simplest universality class of phase transitions.However, Yang-Lee edge singularities have not been regarded as experimentally observable since they occur at complex physical parameters which are unphysical.A recent discovery regarding the relation between partition functions and probe spin coherence makes it experimentally feasible to access the complex plane of physical parameters.However, how to extract the critical point and the critical exponent of Yang-Lee edge singularities in many-body systems, which occurs only at the thermodynamic limit, has still been elusive.Here we show that the quantum coherence of a probe spin coupled to finite-size Ising-type spin systems presents universal scaling behavior near the Yang-Lee edge singularity.The finite-size scaling behavior of the quantum coherence of the probe spin predicts that one can extract the critical point and the critical exponent of the Yang-Lee edge singularity of an Ising-type spin system in the thermodynamic limit from the spin coherence measurement of the probe spin coupled to finite-size Ising-type spin systems.This finding provides a practical approach to studying the nature of Yang-Lee edge singularities of many-body systems.In 1952, Yang and Lee [1, 2] initiated the examination of phase transitions by studying the partition function zeros, termed Lee-Yang zeros, on the complex plane of a magnetic field h (or fugacity z for fluids).They found that for a system above its critical temperature T c , the partition function must be nonzero throughout some neighborhood of the real axis of the magnetic field within a gap determined by two edges at  ( ) h T YL .Therefore the free energy is an analytic function of the real magnetic field.On the other hand, below the critical temperature T c , the Lee-Yang zeros will come arbitrarily to the real axis as the thermodynamic limit is taken, destroying the analyticity of the free energy in h for those real fields at which the Lee-Yang zeros accumulate.Kortman and Griffiths [3] pointed out that the edges of Lee-Yang zeros of a many-body system in the thermodynamic limit are singularity points, termed Yang-Lee edge singularities [4].Fisher [4] proposed that the Yang-Lee edge singularity could be considered as a new second order phase transition point with associated critical exponents.Renormalization group analyses [4] show that the Yang-Lee edge singularity is associated with an j i 3 theory and the crossover dimensionality of the Yang-Lee edge singularity is d c = 6.Later, the investigation of the Yang-Lee edge singularity has been extended to many other models, such as the classical nvector model [5], the quantum Heisenberg model [5], the spherical model [6], the quantum one-dimensional (1D) transverse Ising model [7], the hierarchical model [8], branched polymers [9], the directed-site animalsenumeration problem [10], Ising models on fractal lattices [11], Ising systems with correlated disorder [12], fluid models with repulsive-core interactions [13], the antiferromagnetic Ising model [14] etc.A comprehensive review of the study of Yang-Lee edge singularities can be found in [15, 16].Recently, Kibble-Zurek scaling was shown to appear at the Yang-Lee edge singularity [17], which supports that the Yang-Lee edge singularity is similar to conventional second order phase transitions.Yang-Lee edge singularity occurs in the ferromagnetic Ising models above its critical temperatures in a purely imaginary magnetic field  h i (figure 1).For  > ( ) h h T YL , the partition function acquires zeros, which becomes dense on the line with > ( ) Ih h T YL .The density of Lee-Yang zeros  ( ) g T h , in the thermodynamic limit behaves as  ~ -s ( ) | ( )| g T h h h T , YL when   ( ) h h T YL from above, with σ being the universal critical
The densities of Yang-Lee zeros for the Ising ferromagnet on the L x L square lattice are evaluated from the exact grand partition functions (L=3 approximately 16). The properties of the density of Yang-Lee zeros are discussed as a function of temperature T and system size L. The three different classes of phase transitions for the Ising ferromagnet--first-order phase transition, second-order phase transition, and Yang-Lee edge singularity--are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite-size systems. The divergence of the density of zeros at Yang-Lee edge in high temperatures (Yang-Lee edge singularity), which has been detected only by the series expansion until now for the square-lattice Ising ferromagnet, is obtained from the finite-size data. The identification of the orders of phase transitions in small systems is also discussed using the density of Yang-Lee zeros.