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Spin glass

A spin glass is a magnet with frustrated interactions, augmented by stochastic disorder, where usually ferromagnetic and antiferromagnetic bonds are randomly distributed. Its magnetic ordering resembles the positional ordering of a conventional, chemical glass.

Spin glasses display many metastable structures leading to a plenitude of time scales which are difficult to explore experimentally or in simulations.

Magnetic behavior

It is the time dependence which distinguishes spin glasses from other magnetic systems. Beginning above the spin glass transition temperature, Tc,[1] where the spin glass exhibits more typical magnetic behavior (such as paramagnetism as discussed here but other kinds of magnetism are possible), if an external magnetic field is applied and the magnetization is plotted versus temperature, it follows the typical Curie law (in which magnetization is inversely proportional to temperature) until Tc is reached, at which point the magnetization becomes virtually constant (this value is called the field-cooled magnetization). This is the onset of the spin glass phase. When the external field is removed, the spin glass has a rapid decrease of magnetization to a value called the remanent magnetization, and then a slow decay as the magnetization approaches zero (or some small fraction of the original value—this remains unknown). This decay is non-exponential and no simple function can fit the curve of magnetization versus time adequately.[citation needed] This slow decay is particular to spin glasses. Experimental measurements on the order of days have shown continual changes above the noise level of instrumentation.

If a similar test is run on a ferromagnetic substance, when the external field is removed there is a rapid change to a remanent value that then stays constant in time. For a paramagnet, when the external field is removed the magnetization rapidly goes to zero and stays there. In each case the decay is rapid and exponential.

If instead, the spin glass is cooled below Tc in the absence of an external field and then a field is applied, there is a rapid initial increase to a value called the zero-field-cooled magnetization followed by a slow upward drift toward the field cooled magnetization.

Surprisingly, the sum of the two complex functions of time (the zero-field-cooled and remanent magnetizations) is a constant, namely the field-cooled value, and thus both share identical functional forms with time (Nordblad et al.), at least in the limit of very small external fields.

The model of Sherrington and Kirkpatrick

In addition to unusual experimental properties, spin glasses are the subject of extensive theoretical and computational investigations. A substantial part of early theoretical work on spin-glasses dealt with a form of mean field theory based on a set of replicas of the partition function of the system.

An important exactly-solvable model of a spin-glass was introduced by D. Sherrington and S. Kirkpatrick in 1975. It is an Ising model with long range frustrated ferro- as well as antiferromagnetic couplings. It corresponds to a mean field approximation of spin glasses describing the slow dynamics of the magnetization and the complex non-ergodic equilibrium state.

The equilibrium solution of the model, after some initial attempts by Sherrington, Kirkpatrick and others, was found by Giorgio Parisi in 1979 within the replica method. The subsequent work of interpretation of the Parisi solution—by M. Mezard, G. Parisi, M.A. Virasoro and many others—revealed the complex nature of a glassy low temperature phase characterized by ergodicity breaking, ultrametricity and non-selfaverageness. Further developments led to the creation of the cavity method, which allowed study of the low temperature phase without replicas. A rigorous proof of the Parisi solution has been provided in the work of Francesco Guerra and Michel Talagrand.

The formalism of replica mean field theory has also been applied in the study of neural networks, where it has enabled calculations of properties such as the storage capacity of simple neural network architectures without requiring a training algorithm (such as backpropagation) to be designed or implemented.

More realistic spin glass models with short range frustrated interactions and disorder, like the Gaussian model where the couplings between neighboring spins follow a Gaussian distribution, have been studied extensively as well, especially using Monte Carlo simulations. These models display spin glass phases bordered by sharp phase transitions.

Besides its relevance in condensed matter physics, spin glass theory has acquired a strongly interdisciplinary character, with applications to neural network theory, computer science, theoretical biology, econophysics etc.

Non-ergodic behavior, and applications

A so-called non-ergodic behavior happens in spin-glasses below the freezing temperature Tf, since below that temperature the system cannot escape from the ultradeep minima of the hierarchically-disordered energy landscape [2] . Although the freezing temperature is typically as low as 30 kelvin (−240 degrees Celsius) so that the spinglass-magnetism appears to be practically without applications in daily life, there are applications in different contexts, e.g. in the already mentioned theory of neural networks, i.e. in theoretical brain research, and in the mathematical-economical theory of optimization.

See also

* Quenched disorder
* Replica trick
* Cavity method
* Geometrical frustration
* Phase transition
* Antiferromagnetic interaction
* Crystal structure
* Spin ice


References

1. ^ Tc is identical with the so-called "freezing temperature" Tf
2. ^ The hierarchical disorder of the energy landscape may be verbally characterized by a single sentence: in this landscape there are "(random) valleys within still deeper (random) valleys within still deeper (random) valleys, ..., etc,"


Literature

* D. Sherrington, S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975)
* P. Nordblad, L. Lundgren and L. Sandlund, J. Mag. and Mag. Mater. 54, pp. 185 (1986)
* K. Binder, A. P. Young, Rev. Mod. Phys. 58, 801 (1986)
* Bryngelson, Joseph D. and Peter G. Wolynes, "Spin glasses and the statistical mechanics of protein folding", Proc. Natl. Acad. Sci. USA. Vol. 84, pp. 7524-7528 (1987).
* K.H. Fischer and J.A. Hertz, Spin Glasses, Cambridge University Press (1991)
* Mezard, Marc; Giorgio Parisi, Miguel Angel Virasoro (1987). Spin glass theory and beyond. Singapore: World Scientific. ISBN 9971-5-0115-5.
* J. A. Mydosh, Spin Glasses, Taylor & Francis (1995)
* M. Talagrand, Annals of Probability 28, 1018 (2000)
* F. Guerra and F.L. Toninelli, Comm. in Math. Physics 230, 71 (2002)


External links

* Statistics of frequency of the term "Spin glass" in arxiv.org

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