Exact Replica Symmetric solution for transverse field Hopfield model under finite Trotter size

TL;DR

This paper provides an exact replica symmetric solution for the quantum Hopfield model with finite Trotter slices, revealing quantitative differences from static approximation results by analyzing the model's phase diagram.

Contribution

It introduces an exact solution for the quantum Hopfield model with finite Trotter size using the replica method, and compares it to the static approximation.

Findings

Quantitative differences from static approximation results

Phase diagram with respect to transverse field and pattern number

Introduction of a quasi-static relation for analysis

Abstract

We analyze the quantum Hopfield model in which an extensive number of patterns are embedded in the presence of a uniform transverse field. This analysis employs the replica method under the replica symmetric ansatz on the Suzuki-Trotter representation of the model, while keeping the number of Trotter slices $M$ finite. The statistical properties of the quantum Hopfield model in imaginary time are reduced to an effective $M$-spin long-range classical Ising model, which can be extensively studied using a dedicated Monte Carlo algorithm. This approach contrasts with the commonly applied static approximation, which ignores the imaginary time dependency of the order parameters, but allows $M \to \infty$ to be taken analytically. During the analysis, we introduce an exact but fundamentally weaker static relation, referred to as the quasi-static relation. We present the phase diagram of the model with respect to the transverse field strength and the number of embedded patterns, indicating a small but quantitative difference from previous results obtained using the static approximation.