Number of solutions of the TAP equations for p--spin interaction spin glasses

TL;DR

This paper calculates the number of solutions to TAP equations in p-spin spin glass models, revealing a discontinuous jump at the critical temperature for p>2, and connects these results to known models like SK and REM.

Contribution

It provides a detailed calculation of the solution count for TAP equations in p-spin models, highlighting a discontinuous change at the critical temperature for p>2.

Findings

Number of solutions becomes exponentially large below T_c.

Discontinuous jump in solution count at T_c for p>2.

Results connect to SK model at p=2 and REM at p→∞.

Abstract

The number $\langle N_s\rangle$ of solutions of the equations of Thouless, Anderson and Palmer for p--spin interaction spin glass models is calculated. Below a critical temperature $T_c$ this number becomes exponentially large, as it is in the SK--model ($p=2$). But in contrast to this, for any $p>2$ the factor $\alpha(T)=N^{-1} \ln\langle N_s\rangle$ jumps discontinuously at $T_c(p)$, which is consistent with the discontinuity occuring within the mean--field theory for these models. For zero temperature the results obtained by Gross and M\'ezard are reproduced, and for $p\rightarrow\infty$ one gets the result for the random energy model.