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arXiv:2309.05576 · Physical review. E

Reentrant ferromagnetic ordering of the random-field Heisenberg model in d>2 dimensions: Fourier-Legendre renormalization-group theory

Alpar Türkoğlu, A. Nihat Berker

Published Jan 16, 2024• 1 citation
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Abstract

The random-magnetic-field classical Heisenberg model is solved in spatial dimensions d≥2 using the recently developed Fourier-Legendre renormalization-group theory for 4π steradians continuously orientable spins, with renormalization-group flows of 12 500 variables. The random-magnetic-field Heisenberg model is exactly solved in 10 hierarchical models, for d=2,2.26,2.46,2.58,2.63,2.77,2.89,3. For nonzero random fields, ferromagnetic order is seen for d>2. This ordering, at d=2.46,2.58,2.63,2.77,2.89,3, shows reentrance as a function of temperature.

DOI
10.1103/physreve.109.014114

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The random-magnetic-field classical Heisenberg model is solved in spatial dimensions d≥2 using the recently developed Fourier-Legendre renormalization-group theory for 4π steradians continuously orientable spins, with renormalization-group flows of 12 500 variables. The random-magnetic-field Heisenberg model is exactly solved in 10 hierarchical models, for d=2,2.26,2.46,2.58,2.63,2.77,2.89,3. For nonzero random fields, ferromagnetic order is seen for d>2. This ordering, at d=2.46,2.58,2.63,2.77,2.89,3, shows reentrance as a function of temperature.
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The three-dimensional random-field Ising model has been approximately solved by pursuing the global renormalization-group trajectories of the full, coupled probability distributions of local fields and bonds. This study, which in effect is the global analysis of the renormalization-group flows of 220 physical quantities (based on a generalization of the Migdal–Kadanoff approximation), yields several results that were previously unsuspected. The boundary between the ferromagnetic and paramagnetic phases is hybrid order, in the sense that the magnetization is discontinuous (as in a first-order transition) and the specific heat has a power-law singularity (as in a second-order transition). The magnetic properties of the d=3 random-field Ising model are presented in the form of equimagnetization lines. The study has been repeated for a variety of dimensions d. The lower critical dimension dl=2 is correctly obtained. Moreover, this study indicates the existence of an intermediate-critical dimension dt (between the lower- and upper-critical dimensions) where a singularity occurs in the critical properties as a function of dimension.
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