Z(2) gauge neural network and its phase structure

We study general phase structures of neural-network models that have Z(2) local gauge symmetry. The Z(2) spin variable _Si=±1$S_i=\pm 1$ on the i$i$-th site describes a neuron state as in the Hopfield model, and the Z(2) gauge variable _Jij=±1$J_{ij}=\pm 1$ describes a state of the synaptic connection between j$j$-th and i$i$-th neurons. The gauge symmetry allows for a self-coupling energy among _Jij$J_{ij}$’s such as _JijJjkJki$J_{ij}{J}_{jk}{J}_{ki}$, which describes reverberation of signals. Explicitly, we consider the three models; (I) an annealed model with full and partial connections of _Jij$J_{ij}$, (II) a quenched model with full connections where _Jij$J_{ij}$ is treated as a slow quenched variable, and (III) a quenched three-dimensional lattice model with the nearest-neighbor connections. By numerical simulations, we examine their phase structures paying attention to the effect of the reverberation term, and compare them with each other and with the annealed 3D lattice model which has been studied beforehand. By noting the dependence of thermodynamic quantities upon the total number of sites and the connectivity among sites, we obtain a coherent interpretation to understand these results. Among other things, we find that the Higgs phase of the annealed model is separated into two stable spin-glass phases in the quenched models (II) and (III).