Percolation of Zero-Weight Paths and the Shape of the Phase Boundary in the Two-Dimensional Random-Bond Ising Model

Abstract

We explore the connection between the low-temperature boundary of the ferromagnetic phase in the two-dimensional $\pm J$ random-bond Ising model, where antiferromagnetic bonds occur with probability $p$ and a geometric transition dubbed ``zero-weight percolation''. We argue that the onset of this percolation characterized by the emergence of a percolating path containing an equal number of $+J$ and $-J$ bonds is incompatible with ferromagnetic ordering. Due to its purely geometrical nature, this percolation criterion is a property of a disorder realization and is independent of the temperature, which in turn suggests that the ferromagnetic phase boundary is vertical below the Nishimori point in the $(p,T)$ plane. Using a dynamic-programming algorithm combined with finite-size scaling, we identify the critical disorder at which zero-weight paths first percolate as $p_c=0.1000(2)$, and we extract the associated critical exponents $\nu =1.26(1)$, $\beta /\nu =0.85(1)$, $\gamma /\nu =0.264(5)$, and fractal dimension $d_f\approx 1.11$. The value of $p_c$ is below the previously reported values of the critical disorder strength corresponding to the loss of the ferromagnetic order, both at zero temperature and the Nishimori point. Nevertheless, we argue that the percolation transition studied in this paper is behind the loss of ferromagnetism and thus provides a new, purely geometrical perspective on the stability of ferromagnetic order in disordered spin systems.