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arXiv:2110.02928 · Journal of Statistical Mechanics Theory and Experiment

Finite size scaling functions of the phase transition in the ferromagnetic Ising model on random regular graphs

S. R. Kulkarni, Deepak Dhar

Published Feb 1, 2022• 1 citation
arXivHTMLPDF

Abstract

Abstract We discuss the finite-size scaling of the ferromagnetic Ising model on random regular graphs. These graphs are locally tree-like, and in the limit of large graphs, the Bethe approximation gives the exact free energy per site. In the thermodynamic limit, the Ising model on these graphs show a phase transition. This transition is rounded off for finite graphs. We verify the scaling theory prediction that this rounding off is described in terms of the scaling variable [ T / T c − 1] S 1/2 (where T and T c are the temperature and the critical temperature respectively, and S is the number of sites in the graph), and not in terms of a power of the diameter of the graph, which varies as log S . We determine the theoretical scaling functions for the specific heat capacity and the magnetic susceptibility of the absolute value of the magnetization in closed form and compare them to Monte Carlo simulations.

DOI
10.1088/1742-5468/ac4c3e

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The presence of a random magnetic field in ferromagnetic systems leads, in the broken phase, to an anomalous convergence of some thermodynamic quantities to their asymptotic limits. Here we show a general method, based on the replica trick, to compute analytically the finite size correction to the average free energy. We apply this method to two mean field Ising models, fully connected and random regular graphs, and compare the results to exact numerical algorithms. We argue that this behaviour is present in finite dimensional models as well.
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