Some aspects of infinite range models of spin glasses: theory and numerical simulations

The expression " spin glasses " was originally coined to describe metallic alloys of a non-magnetic metal with few, randomly substituted, magnetic impurities. Experimental evidences where obtained for a low temperature " spin-glass " phase characterized by a non-periodic freezing of the magnetic moments (the spins) with a very slow, and strongly history dependent, response to external perturbations (this later aspect lead more recently to many fascinating developments). The theoretical analysis of this phenomenon lead to the celebrated Edwards–Anderson model [1] of spin-glasses: classical spins on the sites of a regular lattice with random interactions between nearest neighbor spins. However, after more than thirty years of intense studies, the very nature of the low temperature phase of the Edwards–Anderson model in three dimensions is still debated, even in the simple case of Ising spins. Two main competing theories exist: the mean field approach originating from the work of Sherrington and Kirkpatrick [2], and the so-called " droplet " [3] or scaling theory of spin glasses. The mean field approach is the application to this problem of the conventional approach to phase transitions in statistical physics: one first builds a mean field theory after identifying the proper order parameter, solve it (usually a straightforward task) and then study the fluctuations around the mean field solution. Usually, fluctuations turn out to have mild effects for space dimensions above the so-called upper critical dimension (up to infinite space dimension, where mean field is exact). Below the upper critical dimension fluctuations have major effects and non-perturbative techniques are needed to handle them. The second item of this agenda (solving the mean field equations) led, with spin glasses, to severe unexpected difficulties, and revealed a variety of new fascinating phenomena. The last step is the subject of the so called " replica field theory " , which is still facing formidable difficulties. These notes are an introduction to the physics of the infinite range version of the Edwards–Anderson model, the so-called Sherrington–Kirkpatrick model, namely a model of classical spins that are not embedded in Euclidean space, with all pairs of spins interacting with a random interaction. If there is 2 Alain Billoire no more debate whether Parisi famous solution of the Sherrington–Kirkpatrick model in the infinite volume limit is correct, much less is known, as mentioned before, about the Edwards–Anderson model in three dimensions, with numerical simulations as one of our main sources of knowledge. …