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ANDERSON LOCALIZATION, BRANCHED POLYMERS AND THE YANG-LEE EDGE SINGULARITY.

Alan J. McKane, T. C. Lubensky, Alan J. McKane

Published Jan 1, 1981• 0 citations
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Abstract

It is shown that a recently proposed field-theoretic model of Anderson localization has the same critical behavior as that found in te 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 minus 2 dimensions. It is stressed 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. It is noted that constant order parameter exponents diverge at a critical dimension D//c.

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Anderson localization, branched polymers and the Yang-Lee edge singularity

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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.

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Andrew Connelly, Gregory Johnson, Fabian Rennecke +1 more

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.

arXiv:2006.12541•Physical Review Letters•Nov 3, 2020•53 citations
It is shown that a recently proposed field-theoretic model of Anderson localization has the same critical behavior as that found in te 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 minus 2 dimensions. It is stressed 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. It is noted that constant order parameter exponents diverge at a critical dimension D//c.
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 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.
We show here for the one-dimensional spin-1/2 axial-next-to-nearest-neighbor Ising model in an external magnetic field that the linear density of Yang-Lee zeros may diverge with critical exponent σ=-2/3 at the Yang-Lee edge singularity. The necessary condition for this unusual behavior is the triple degeneracy of the transfer-matrix eigenvalues. If this condition is absent we have the usual value σ=-1/2 . Analogous results have been found in the literature in the spin-1 Blume-Emery-Griffths model and in the three-state Potts model in a magnetic field with two complex components. Our results support the universality of σ=-2/3 which might be a one-dimensional footprint of a tricritical version of the Yang-Lee edge singularity possibly present also in higher-dimensional spin models.
The isothermal magnetization m(H) of the metamagnet FeCl2 is measured in axial magnetic fields 0≤µ0Ha≤12 T at temperatures 34≤T≤53 K above the Néel temperature, where the system is essentially a two-dimensional Ising ferromagnet. The analysis of the data indicates experimental accessibility of the critical exponent µ of the Yang-Lee edge singularities. They manifest themselves in divergences of the density functions g(θ), which quantify the distribution of the zeros of the partition function on the Lee-Yang unit circle in the complex plane. In accordance with the hypothesis of universality, a critical exponent close to the theoretical prediction for the two-dimensional Ising ferromagnet, µ = -1/6, is found.
We investigate the critical properties of the Lee-Yang model in less than six spacetime dimensions using truncations of the functional renormalization group flow. We give estimates for the critical exponents, study the dependence on the regularization scheme, and show the convergence of our results for increasing size of the truncations in four and five dimensions. While with our truncations it is numerically challenging to approach the three-dimensional case, we provide a simple approximation which allows us to qualitatively study the Lee-Yang model in two and three dimensions and use it to argue the existence of further nonunitary multicritical theories including one which is relevant for the universality class of the Blume-Capel model.