The McCoy-Wu Model in the Mean-field Approximation

TL;DR

This paper analyzes the critical behavior of the McCoy-Wu model with layered randomness using mean-field theory, revealing specific critical exponents and the absence of Griffiths singularities.

Contribution

It provides a mean-field analysis of the McCoy-Wu model, determining critical exponents and the distribution of reduced critical temperatures in the presence of randomness.

Findings

Critical exponents: β ≈ 3.6 (bulk), β₁ ≈ 4.1 (surface)

Specific heat exponent: α ≈ -3.1

Power law distribution of reduced critical temperature: P(t_c) ∼ t_c^ω with ω ≈ 3.1

Abstract

We consider a system with randomly layered ferromagnetic bonds (McCoy-Wu model) and study its critical properties in the frame of mean-field theory. In the low-temperature phase there is an average spontaneous magnetization in the system, which vanishes as a power law at the critical point with the critical exponents $\beta \approx 3.6$ and $\beta_1 \approx 4.1$ in the bulk and at the surface of the system, respectively. The singularity of the specific heat is characterized by an exponent $\alpha \approx -3.1$. The samples reduced critical temperature $t_c=T_c^{av}-T_c$ has a power law distribution $P(t_c) \sim t_c^{\omega}$ and we show that the difference between the values of the critical exponents in the pure and in the random system is just $\omega \approx 3.1$. Above the critical temperature the thermodynamic quantities behave analytically, thus the system does not exhibit Griffiths singularities.