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arXiv:2006.12541 · Physical Review Letters

Universal Location of the Yang-Lee Edge Singularity in O(N) Theories

Andrew Connelly, Gregory Johnson, Fabian Rennecke, Vladimir V. Skokov

Published Nov 3, 2020• 53 citations
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Abstract

We determine a previously unknown universal quantity, the location of the Yang-Lee edge singularity for the O(N) theories in a wide range of N and various dimensions. At large N, we reproduce the N→∞ analytical result on the location of the singularity and, additionally, we obtain the mean-field result for the location in d=4 dimensions. In order to capture the nonperturbative physics for arbitrary N, d and complex-valued external fields, we use the functional renormalization group approach.

DOI
10.1103/physrevlett.125.191602

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We determine a previously unknown universal quantity, the location of the Yang-Lee edge singularity for the O(N) theories in a wide range of N and various dimensions. At large N, we reproduce the N→∞ analytical result on the location of the singularity and, additionally, we obtain the mean-field result for the location in d=4 dimensions. In order to capture the nonperturbative physics for arbitrary N, d and complex-valued external fields, we use the functional renormalization group approach.
We discuss the analytic continuation of scaling function in the three-dimensional Z ( 2 ) , O ( 2 ) and O ( 4 ) universality classes using the Schofield representation of the magnetic equation of state. We show that a determination of the location of Lee-Yang edge singularities and, in the case of Z ( 2 ) , also the Langer edge singularity yields stable results. Results for the former are in good agreement with functional renormalization group calculations. We also present results for the location of the Langer edge singularity in the 3 d , Z ( 2 ) universality class. We find that in terms of the complex scaling variable z the distance of the Langer edge singularity to the critical point agrees within errors with that of the Lee-Yang edge singularity. Furthermore the magnitude of the discontinuity along the Langer branch cut is an order of magnitude smaller than that along the Lee-Yang branch cut. Published by the American Physical Society 2024
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.
We show that a recently proposed field-theoretic model of Anderson localization has the same critical behaviour as that found in the problem of branched polymers. Thus this model in D dimensions is in the same universality class as that studied by Parisi and Sourlas, which includes the Yang-Lee edge singularity in D — 2 dimensions. We stress that critical properties can be studied either at constant order parameter or at constant field and that critical exponents in the two cases are related by a Fisher renormalization. We note that constant order parameter exponents diverge at a critical dimension Dc.
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 ($\ensuremath{\eta}=\ensuremath{-}1$).