Z2 symmetry: Ising spin ±1

Spins with one components: S=±1
Energy=-S.hlocal=-S.h
P(x)=e-Energy/T
       =eh.S
The probability of the Ising states (±1) are shown on the left. On the right the Walker probability.

The fastest Algorithm is the Restricted Metropolis Algorithm for the Ising ±1 spin. Restricted means that only the opposite state of the present state is tested. It is the only case where the Metropolis is better than the Heat-Bath (see performances below). We have also tested a multi-spins heat bath algorithm. We did not found that the performances increase at the critical temperature.
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Probabilities and Performances :
  • Figure: comparison of the time consumption to simulate a two dimensional ferromagnetic square lattice for various algorithms (Metropolis,Me, Restricted metropolis, Med, Direct Heat Bath, DHB, Walter Algorithm for 4 spins on a plaquette, WA4, Restricted Walter Algorithm Hasting for 4 spins on a plaquette, WAH4d)
    The critical temperatures are shown by the squares
    The restricted Metropolis Algorithm is the fastest at the critical temperature. We can show more generally that this conclusion holds whatever the system is. It is mainly due to the presence of zero local field (see article). The Ising ±1 case is the only one where the Metropolis is better than the Heat Bath.
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Algorithms in C