Gauge Neural Network with Z(2) Synaptic Variables: Phase Structure and Simulation of Learning and Recalling Patterns

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We study the Z(2) gauge-invariant neural network which is defined on a partially connected random network and involves Z(2) neuron variables $$S_i$$ ( $$=\pm $$ 1) and Z(2) synaptic connection (gauge) variables $$J_{ij}$$ ( $$=\pm $$ 1). Its energy consists of the Hopfield term $$-c_1S_iJ_{ij}S_j$$ , double Hopfield term $$-c_2 S_iJ_{ij}J_{jk} S_k$$ , and the reverberation (triple Hopfield) term $$-c_3 J_{ij}J_{jk}J_{ki}$$ of synaptic self interactions. For the case $$c_2=0$$ , its phase diagram in the $$c_3-c_1$$ plane has been studied both for the symmetric couplings $$J_{ij}=J_{ji}$$ and asymmetric couplings ( $$J_{ij}$$ and $$J_{ji}$$ are independent); it consists of the Higgs, Coulomb and confinement phases, each of which is characterized by the ability of learning and/or recalling patterns. In this paper, we consider the phase diagram for the case of nonvanishing $$c_2$$ , and examine its effect. We find that the $$c_2$$ term enlarges the region of Higgs phase and generates a new second-order transition. We also simulate the dynamical process of learning patterns of $$S_i$$ and recalling them and measure the performance directly by overlaps of $$S_i$$ . We discuss the difference in performance for the cases of Z(2) variables and real variables for synaptic connections.