The standard picture of spin glasses on square lattices has two phases: paramagnetic at high temperature and spin-glass at low temperature. But modified Ising models on the Nishimori line — where the temperature matches the disorder distribution — suggest a richer structure. The question is whether a Mattis-like intermediate phase exists between the ferromagnetic and spin-glass phases.
This paper finds it, using tensor-network-based sampling that handles both bond and spin configurations hierarchically. The method reaches system sizes up to L=256, far beyond what Monte Carlo typically manages for these frustrated systems. The results show that spin-glass and ferromagnetic transitions occur separately on the Nishimori line, with an intermediate Mattis phase between them.
The critical exponents place these transitions in a different universality class than the standard Edwards-Anderson model. The reentrant transition — entering and exiting the spin-glass phase as temperature varies — is confirmed, but the nature of the transitions is distinct. The intermediate phase is not a curiosity: it's a separate thermodynamic state, identified by the divergence of different order parameters at different temperatures. The tensor-network approach succeeds where Monte Carlo struggles because it directly computes the partition function rather than sampling from it.