Scaling of the critical Hopfield capacity p_c(N)

Determine how the critical Hopfield capacity p_c(N) scales with the system size N in the transverse-field quantum Hopfield model with dilute memories, specifically deciding whether p_c(N) grows logarithmically as O(log_2 N) or algebraically as O(N^σ). Here p_c(N) denotes the largest number of stored patterns for which the finite-size Landau–Zener gap scaling Δ ∝ N^{-1/3} and the critical behavior with exponent a = 1/2 remain valid for p < p_c(N).

Background

The paper studies the transverse-field quantum Hopfield model with dilute memories and analyzes its adiabatic spectra and annealing dynamics near a second-order quantum phase transition. For p = 1, the model maps to the Lipkin–Meshkov–Glick (LMG) model, yielding critical exponents a = 1/2 and b = 1/3 for the closing of the macroscopic gap and the finite-size scaling of the critical gap, respectively.

For 1 < p < log_2 N, the system can be treated as a collection of large spins. After a Holstein–Primakoff transformation and mode decomposition via the matrix T constructed from pattern structure, the authors show that for p below some threshold p_c the low-energy sector consists of p nearly critical modes with successive avoided crossings whose finite-size gaps each scale as N^{-1/3}, similar to the p = 1 case. Intermode interactions are argued to be subleading in 1/N and do not change the scaling exponents.

However, the analysis does not determine how the threshold number of patterns p_c depends on N. The authors note that p_c could scale logarithmically with N or algebraically with N, and resolving this scaling is left open.