Geometry of rare regions behind Griffiths singularities in random quantum magnets

István A Kovács,Ferenc Iglói

doi:10.1038/s41598-022-05096-z

Abstract

In many-body systems with quenched disorder, dynamical observables can be singular not only at the critical point, but in an extended region of the paramagnetic phase as well. These Griffiths singularities are due to rare regions, which are locally in the ordered phase and contribute to a large susceptibility. Here, we study the geometrical properties of rare regions in the transverse Ising model with dilution or with random couplings and transverse fields. In diluted models, the rare regions are percolation clusters, while in random models the ground state consists of a set of spin clusters, which are calculated by the strong disorder renormalization method. We consider the so called energy cluster, which has the smallest excitation energy and calculate its mass and linear extension in one-, two- and three-dimensions. Both average quantities are found to grow logarithmically with the linear size of the sample. Consequently, the energy clusters are not compact: for the diluted model they are isotropic and tree-like, while for the random model they are quasi-one-dimensional.

Highlights

In many-body systems with quenched disorder, dynamical observables can be singular at the critical point, but in an extended region of the paramagnetic phase as well
One peculiarity of random many-body systems is that dynamical observables can be singular at the critical point, but in extended regions of the paramagnetic and ferromagnetic phases as well
This singular behavior is due to rare regions, which occur with very low probability, but have a very large relaxation time, which diverges in the thermodynamic limit

Summary

Introduction

In many-body systems with quenched disorder, dynamical observables can be singular at the critical point, but in an extended region of the paramagnetic phase as well. In a random ferromagnet, the linear susceptibility, χ , can be divergent in an extended part of this region, which is called the Griffiths-phase[5] This type of singular behaviour is due to rare regions[6], in which there are extreme fluctuations of strong couplings. The relaxation time, τ , associated with turning the spins in such domains can be extremely large and it has no upper limit in the thermodynamic limit This type of Griffiths singularities are responsible for non-analytic behaviour of several average physical quantities, besides the susceptibility one can mention the specific heat and the auto-correlation function. In the Griffiths-phase, the average susceptibility, χ and that of the specific heat, cV , show power-law singularities at low temperatures, T: χ (T) ∼ T−1+d/z and cV (T) ∼ Td/z , at zero temperature χ (0) is divergent for z > d6,10

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