Some comments on the Sherrington-Kirkpatrick model of spin glasses

AbstractWe determine explicitly and discuss in detail the effects of the joint presence of a longitudinal and a transversal (random) magnetic field on the phases of the Random Energy Model and its hierarchical generalization, the GREM. Our results extent known results both in the classical case of vanishing transversal field and in the quantum case for vanishing longitudinal field. Following Derrida and Gardner, we argue that the longitudinal field has to be implemented hierarchically also in the Quantum GREM. We show that this ensures the shrinking of the spin glass phase in the presence of the magnetic fields as is also expected for the Quantum Sherrington–Kirkpatrick model.

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FINDING MINIMA IN COMPLEX LANDSCAPES: ANNEALED, GREEDY AND RELUCTANT ALGORITHMS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202505000765 ◽

2005 ◽

Vol 15 (09) ◽

pp. 1349-1369 ◽

Cited By ~ 1

Author(s):

PIERLUIGI CONTUCCI ◽

CRISTIAN GIARDINÀ ◽

CLAUDIO GIBERTI ◽

CECILIA VERNIA

Keyword(s):

Complex Systems ◽

Cost Function ◽

Spin Glasses ◽

Optimization Problems ◽

New Class ◽

Kirkpatrick Model ◽

Annealing Procedure ◽

The Cost ◽

Complex Landscapes

We consider optimization problems for complex systems in which the cost function has a multivalleyed landscape. We introduce a new class of dynamical algorithms which, using a suitable annealing procedure coupled with a balanced greedy-reluctant strategy drive the systems towards the deepest minimum of the cost function. Results are presented for the Sherrington–Kirkpatrick model of spin-glasses.

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The Infinite-Range Spin Glass

Spin Glasses and Complexity ◽

10.23943/princeton/9780691147338.003.0006 ◽

2013 ◽

Author(s):

Daniel L. Stein ◽

Charles M. Newman

Keyword(s):

Symmetry Breaking ◽

Spin Glass ◽

Spin Glasses ◽

Mean Field Theory ◽

Mean Field ◽

Replica Symmetry ◽

Replica Symmetry Breaking ◽

Glass Model ◽

Kirkpatrick Model ◽

Technical Details

This chapter introduces mean field theory, both as a general class of models and in its specific incarnation in spin glasses, the Sherrington–Kirkpatrick model. This is undoubtedly the most theoretically studied spin glass model by far, and the best understood. For the nonphysicist the going may get a little heavy in this chapter once replica symmetry breaking is introduced, with its attendant features of many states, non-self-averaging, and ultrametricity—but an attempt is made to define and explain what all of these things mean and why replica symmetry breaking represents such a radical departure from more conventional and familiar modes of symmetry breaking. While this is a central part of the story of spin glasses proper, the nonphysicist who wants to skip the technical details can safely omit certain sections in the chapter and continue on without losing the essential thread of the discussion that follows.

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DISTRIBUTION OF PURE STATES IN SHORT-RANGE SPIN GLASSES

International Journal of Modern Physics B ◽

10.1142/s0217979210055779 ◽

2010 ◽

Vol 24 (14) ◽

pp. 2091-2106 ◽

Cited By ~ 3

Author(s):

C. M. NEWMAN ◽

D. L. STEIN

Keyword(s):

Spin Glasses ◽

Short Range ◽

Glass Phase ◽

Single Pair ◽

Mixed States ◽

Spin Glass Phase ◽

Kirkpatrick Model ◽

Pure States ◽

Mathematical Construct ◽

Infinite Range

We review the structure of the spin glass phase in the infinite-range Sherrington–Kirkpatrick model and the short-range Edwards–Anderson (EA) model. While the former is now believed to be understood, the nature of the latter remains unresolved. However, considerable insight can be gained through the use of the metastate, a mathematical construct that provides a probability measure on the space of all thermodynamic states. Using tools provided by the metastate construct, possibilities for the nature of the organization of pure states in short-range spin glasses can be considerably narrowed. We review the concept of the "ordinary" metastate, and also newer ideas on the excitation metastate, which has been recently used to prove existence of only a single pair of ground states in the EA Ising model in the half-plane. We close by presenting a new result, using metastate methods, on the number of mixed states allowed in the EA model.

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