Realizable solutions of the Thouless-Anderson-Palmer equations
1Institute for Theoretical Physics, Georg-August-Universität Göttingen, D37077, Göttingen, Germany.
Physical Review. E
|October 24, 2019
Summary
Iterative solutions for Ising spin glasses converge to a critical state with marginal stability. This finding applies to both replica-symmetric and broken-replica-symmetric states in large systems.
Area of Science:
- Statistical Mechanics
- Condensed Matter Physics
- Computational Physics
Background:
- The Sherrington-Kirkpatrick (SK) model is a fundamental model for understanding spin glasses.
- Thouless-Anderson-Palmer (TAP) equations provide a framework for analyzing the properties of such systems.
- Understanding the nature of solutions and their stability is crucial for characterizing complex magnetic materials.
Purpose of the Study:
- To identify the characteristics of solutions to the TAP equations for the SK model that are discoverable through iterative methods.
- To investigate the phenomenon of self-organized criticality in the context of spin glass models.
- To analyze the free-energy landscape and stability of stationary states in Ising spin glasses.
Main Methods:
- Analysis of the Thouless-Anderson-Palmer (TAP) equations for the Sherrington-Kirkpatrick (SK) model.
- Investigation of iterative solution methods and their convergence properties.
- Examination of the Hessian matrix eigenvalues to determine state stability.
- Study of free-energy differences between saddle points and minima.
Main Results:
- Iterative solutions for the SK model's TAP equations converge to states on the border between replica-symmetric and broken-replica-symmetric phases for large N.
- Quenches from high-temperature states to locally stable states also exhibit convergence to this borderline, demonstrating self-organized criticality.
- States reached at this borderline possess marginal stability, indicated by the Hessian eigenvalue band extending to zero.
- Factors influencing the free-energy barrier between saddle points and minima were investigated.
Conclusions:
- The iterative approach to solving TAP equations for Ising spin glasses yields solutions with marginal stability at the replica-symmetry breaking transition.
- Self-organized criticality plays a role in reaching these marginally stable states through iterative processes and quenches.
- Understanding these stability properties is key to characterizing the complex free-energy landscape of spin glasses.
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