Magnetization-resolved density of states and mixed-order transition in the two-dimensional random bond Ising model: an entropic sampling study

Systems with quenched disorder possess complex energy landscapes that are challenging to explore under conventional Monte Carlo methods. In this work, we implement an efficient entropy sampling scheme for accurate computation of the entropy function in low-energy regions. The method is applied to the two-dimensional ±J random-bond Ising model, where frustration is controlled by the fraction p of ferromagnetic bonds. We investigate the low-temperature paramagnetic-ferromagnetic phase boundary below the multicritical point at T_N = 0.9530(4), P_N = 0.89078(8), as well as the zero-temperature ferromagnetic-spin-glass transition. Finite-size scaling analysis reveals that the phase boundary for T < T_N exhibits reentrant behavior. By analyzing the evolution of the magnetization-resolved density of states g(E, M) and ground-state spin configurations against increasing frustration, we provide strong evidence that the zero-temperature transition is a mixed-order. Finite-size scaling conducted on the spin-glass side supports the validity of β = 0, where β is the magnetization exponent, with a correlation length exponent ν = 1.50(8). Our results provide new insights into the nature of the ferromagnetic-to-spin-glass phase transition in an extensively degenerate ground state.