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In condensed matter physics, a **spin glass** is a magnet that can randomly exhibit both ferromagnetic and antiferromagnetic properties due to the probabilistic distribution of the internal elements that produce the magnetic (spin) effects. The internal spin orientation, whose vector sum gives the macroscopic effect, is stochastic and varies with temperature but never reaches equilibrium, so the internal configuration is said to be “frustrated”.

They can be defined as sets of elementary (spin) magnets whose interactions are, at random, ferromagnetic or antiferromagnetic according to a well-defined probability law. The first such materials were constructed by L. Neel around 1932 by diluting a magnetically active transition metal (e.g., iron) in a noble matrix (e.g., gold or silver) with concentrations around 1%. The interaction between the spin of the former is transmitted through the free electrons of the latter producing a strong interaction of variable sign.

This irregular internal structure of spin orientation is analogous to that of the positions of atoms in an amorphous solid, such as glass is and from which the name comes. As for the glass, which solidifying does not reach an ordered crystalline structure, also the spin glasses slowly change their internal state, with speeds that decelerate with the passage of time so that the internal structures can be considered metastable.

The study of mathematical models to explain such complex structures is also useful for other applications in physics, chemistry, material science, neural networks, and other complex systems.

At high temperature the behavior of spin glass does not differ from that of any other magnetic material. On the contrary, when the temperature drops below a certain specific value, new glass-like effects appear: the response to an external magnetic field becomes very slow and the state of the system depends on the previous history (hysteresis effects). The equilibrium state seems unattainable, as confirmed by the so called aging experiments: observations show that the speed with which the system moves away from a certain starting configuration is inversely proportional to the preparation time of the initial state. The longer one waits for the system to equilibrate, the slower the relaxation processes become.

Thus, it is not possible to characterize the regime change with temperature as a true phase transition corresponding to a change in the equilibrium state. Only when the external magnetic field is zero does a phase transition exist, however, corresponding to the appearance of local spontaneous magnetization. Because of the disorder, this magnetization has zero average in a finite volume and can only be observed by neutron scattering experiments.

The phenomenology of glassy behavior, particularly aging, points to the presence of many long-lived average states relative to the observation time scale, but it is difficult to conclude whether, in the infinite volume limit, the average life of some of these actually tends to infinity. The energy, as a function of spin configurations, describes a very irregular surface. It has numerous valleys separated by barriers of varying heights. Almost literally, the barriers surrounding the valley where the system is located hide the higher but more distant ones. Slow relaxation processes prevent, both experimentally and in numerical simulations, the true state of equilibrium from being known.

## The spin glass as an archetype of a complex system

The spin glass is the simplest of a new class of systems where disorder appears structural. Its behavior is so rich that it has become the preferred model of statistical mechanics in the study of complex systems. In particular, a certain approximation to it, the so-called Sherrington-Kirkpatrick (SK) model, can be solved completely within a new analytical scheme. The solution, derived by G. Parisi in the years 1979-80, for the innovations that involves, represents one of the most important advances in the study of complexity. In particular it shows that in the mean-field approximation there is a phase transition corresponding to ergodicity breaking: at sufficiently low temperatures the system has a large number of equilibrium states (infinite when N, the number of spins, tends to infinity) that correspond to deep and wide valleys in the energy surface.

In the behavior of this model plays a predominant role the concept of frustration introduced by G. Toulouse. Any spin triplet is said frustrated if the pairwise interactions are such that some of them tend to put two spin parallel, while the others would like them antiparallel and no choice for the three spin s_{A}, s_{B}, s_{C} can satisfy them all.

Paradoxically it follows that more than one configuration satisfies the largest possible number of interactions, and therefore it results in greater richness and diversity among the equilibrium states. In the SK model typically half of all triplets are frustrated. If you modify the model so that the frustration rate decreases, the number of equilibrium states also decreases.

## The method of the replicas and the solution of Parisi

Since there are an infinite number of equilibrium states, the SK system will evolve within one of them. So the time average of any observable will not be equal to the average made on the canonical ensemble. In these cases the usual methods of statistical mechanics require knowledge of the external magnetic field (or boundary condition) that projects the system into a particular equilibrium state. In disordered systems such knowledge is not there and, on the other hand, this methodology would not allow to study the global structure of the energy surface.

The method of replications solves these problems. In Parisi’s formulation, the appearance of an infinite number of equilibrium states is encoded as a breaking of a symmetry: the group of permutations of n elements where n → 0. The method provides detailed information about the relative distances between valleys. It follows that there are valleys at all possible mutual distances within a certain interval. By computing predictions for more than two replications, multiple correlations between different locations can be studied.

In particular from the study of three replicas we deduce a surprising result: chosen three valleys at random, with probability given by the canonical distribution, the one that is farthest from the other two is at the same distance of these: all triangles are equilateral or isosceles with acute angles. This particular correlation is called ultrametricity following the mathematician M. Krasner who studied its properties. Numerical simulations have, in general, confirmed the general description of Parisi’s solution. Similar energy surfaces have been found in other theories and/or models. The corrections to the mean-field approximation are too complex and, for the moment, do not allow us to deduce what happens when the magnitude of the interaction becomes finite.

## Applications of glass concepts to complex systems

The SK model is a prototype of many-state equilibrium systems. Therefore it becomes a possible choice when modeling complex systems. In particular, P.W. Anderson has underlined how the two properties, richness in the number of states and relative stability of each of them with respect to small perturbations are fundamental to model biosystems. The neural networks of J.J. Hopfield, suggested as models of brain functioning, are the best known example although not the only one. In these models, magnetic spins are interpreted as formal neurons, while couplings are translated into synaptic intensities.

Two different dynamic processes are distinguished. In the first, called *recognition*, the couplings remain fixed while the network starts from any configuration of coactive neurons and converges to one of the possible equilibrium states. The mutual organization of the latter reproduces many of the properties discussed above. The second process, called *learning*, is instead specific to neural networks. During learning, the system interacts with the external world, where structural disorder takes refuge, and synaptic intensities evolve towards values that allow to model the equilibrium states of recognition. The replication method was successfully applied to the understanding of both dynamic processes (“recognition,” Amit et al. 1985; “learning,” Gardner 1988).

Another important application is in the field of complex optimization. It can be shown that the problem of finding the algorithm leading to the minimum energy configuration in the SK model in a reasonable time (proportional to a power of the number of spins) belongs to the class of the most difficult problems, i.e., the so-called NP-complete class, to which also the problem of partitioning a graph belongs. Mathematically finding the absolute minimum of the energy is hampered by the existence of many local minima. A detailed knowledge of the topography of the energy surface as provided by the method of replicas is important to design an algorithm that takes these obstacles into account.

## References

- M. Mezard, G. Parisi, M.A. Virasoro,
*Spin glass theory and beyond*, Singapore 1987; D. Amit,*Modeling brain function*, Cambridge 1989.