Some Aspects of Infinite-Range Models of Spin Glasses: Theory and Numerical Simulations

Abstract

The expression “spin glasses” was originally coined to describe metallic alloys of a nonmagnetic metal with few, randomly substituted, magnetic impurities. Experimental evidences were obtained for a low-temperature “spin-glass” gly history dependent, response to external perturbations (this later aspect leads, more recently, to many fascinating developments). The theoretical analysis of this phenomenon leads 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 30 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 nonperturbative 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 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. It is accordingly important to test the various methods of analysis proposed for the Edwards-Anderson model, in the Sherrington–Kirkpatrick model case first.In a first part, I motivate and introduce the Edwards–Anderson and Sherrington–Kirkpatrick models. In the second part, I sketch the analytical solution of the Sherrington–Kirkpatrick model, following Parisi. I next give the physical interpretation of this solution. This is a vast subject, and I concentrate on the major points and give references for more developments. The third part presents the numerical simulation approach and compares some numerical results to theoretical expectations. The last part, more detailed, is about the specific problem of finite size effects for the free energy, which is interesting for both theoretical and practical point of views. I have left aside several very interesting aspects, such as the problem of chaos [4,5] the TAP approach [6] (see [7] for numerical results) and the computation of the complexity [8].There are many books and review articles about spin glasses and related phenomena: One may start with the text book by Fischer and Hertz [10], the review by Binder and Young [9], which gives a very complete account of the situation in 1986 both experimental, theoretical and numerical with many detailed analytical computations, and (at a higher level) the book by Mézard, Parisi and Virasoro [11]. More recent references include [12] and [13]. The recent book by de Dominicis and Giardina [14] gives a very compete exposition of the replica field theory. Reviews on various aspects of the physics of spin glasses can be found in [15,16,17]. The nonequilibrium behavior of spin glasses has been the subject of intense work during the recent years, see [18,19,20] for reviews.