Analysis of the Hopfield Model Incorporating the Effects of Unlearning

We analyze a variant of the Hopfield model that incorporates an unlearning mechanism based on spin correlations in the high-temperature regime. In the large system limit where extensively many patterns are stored, we employ the replica method under the replica symmetric ansatz to characterize the model analytically. Our analysis provides a systematic and self-consistent framework that yields order-parameter equations and stability conditions at finite temperatures over a wide range of parameter settings. The resulting theory accurately captures the behavior of the signal-to-noise ratio, the memory capacity, and the criteria for selecting optimal hyperparameters, in agreement with the qualitative findings of Nokura (1996 \textit{J. Phys. A: Math. Gen.} \textbf{29} 3871). Moreover, the theoretical predictions show good agreement with numerical simulations, supporting the conclusion that unlearning enhances memory capacity by suppressing spurious memories.