2019/05/31 by Timo Aspelmeier, T. Aspelmeier, M. A. Moore
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Applied mathematics #Complex Network Analysis Techniques #Complex Systems and Time Series Analysis #Condensed matter physics #Convergence (economics) #Eigenvalues and eigenvectors #Geometry #Hessian matrix #Ising model #Mathematical physics #Mathematics #Physics #Quantum mechanics #Replica #Saddle #Saddle point #Spin glass #Spins #State (computer science) #Theoretical and Computational Physics #cond-mat.dis-nn
paper · pdf · doi:10.1103/physreve.100.032127
published as Phys. Rev. E 100, 032127 (2019) · 8 pages, 6 figures, modified text and figures, references added
arxiv created 2019/09/06 · openalex publication_date 2019/09/18 · arxiv updated 2019/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We show that the only solutions of the Thouless-Anderson-Palmer (TAP) equations for the Sherrington-Kirkpatrick model of Ising spin glasses which can be found by iteration are those whose free energy lies on the border between replica-symmetric and broken-replica-symmetric states, when the number of spins N is large. Convergence to this same borderline also happens in quenches from a high-temperature initial state to a locally stable state where each spin is parallel to its local field; both are examples of self-organized criticality. At this borderline the band of eigenvalues of the Hessian associated with a solution extends to zero, so the states reached have marginal stability. We have also investigated the factors which determine the free-energy difference between a stationary solution corresponding to a saddle point and its associated minimum, which is the barrier which has to be surmounted to escape from the vicinity of a TAP minimum or pure state.