# Calculating the ground state of an Ising model with $\sigma_i = (0,1)$ spin state assignments (do Barahona & Istrail's NP-hardness results hold?)

In a typical Ising model, one has possible spin assignments of $\sigma_i = \pm 1$. However, one can also imagine a $q = 2$ Potts model generalization with spin assignments $\sigma_i = (0,1)$. Is there a special name for this variant of an Ising model?

I'm asking this question because I'm curious if the results of (Barahona -- http://yaroslavvb.com/papers/barahona-on.pdf) & (Istrail 2000 -- http://www.brown.edu/Research/Istrail_Lab/papers/p87-istrail.pdf), showing the that calculation of the ground state is NP-hard for three-dimensional integer lattices (Barahona) and more generally non-planar lattices (Istrail), hold for an Ising model with $\sigma_i = (0,1)$ spin assignments?

It seems that it should from their proof arguments, though I do wonder if I'm missing something?

Generally speaking, what sort of rules can I "get away with" for spin state assignments while allowing for the results of (Barahona 1982) & (Istrail 2000) to hold?