No-passing Rule in the Ground State Evolution of the Random-Field Ising Model

We exactly prove the no-passing rule in the ground state evolution of the random-field Ising model (RFIM) with monotonically varying external field. In particular, we show that the application of the no-passing rule can speed up the calculation of the zero-temperature equilibrium $M(H)$ curve dramatically.Comment: 7 pages, 4 figure