Spin glass

A spin glass is a disordered material exhibiting high magnetic frustration. The origin of the behavior can be either a disordered structure (such as that of a conventional, chemical glass) or a disordered magnetic doping in an otherwise regular structure. Frustration is the inability of the system to remain in a single lowest energy state (the ground state). Spin glasses have many ground states which are never explored on experimental time scales.

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It is the time dependence which distinguishes spin glasses from other magnetic systems. Beginning above the spin glass transition temperature, T_c, 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 T_c 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 remnant 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 single function can fit the curve of magnetization versus time adequately. 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 remnant 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 change is rapid and, if carefully examined, is found to be exponential decay with a very small time constant.

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

Surprisingly, the sum of the two complex functions of time (the zero-field-cooled and remnant 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.

In addition to unusual experimental properties, spin glasses are the subject of extensive theoretical and computational investigation. A great deal of early theoretical work on spin-glasses uses a form of mean field theory based on a set of replicas of the partition function of the system. An influential exactly-solvable model of a spin-glass was introduced by D. Sherrington and S. Kirkpatrick, and led to considerable theoretical extensions of mean field theory to model the slow dynamics of the magnetisation, and the complex non-ergodic equilibrium state. The equilibrium solution of the model, after some early 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, non-selfaverageness. Further developments led to the creation of the cavity method, allowing to study the low temperature phase without replica. A rigorous proof of the Parisi solution has been provided by the work of Francesco Guerra and Michel Talagrand in the 2000.

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

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.

The cavity method has been successfully employed to analyze the typical case of Constraint Satisfaction Problems like the random KSAT and the coloring of random graphs.

See also

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

References

• 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)
• P. Nordblad, L. Lundgren and L. Sandlund, J. Mag. and Mag. Mater. 54, pp. 185 (1986)
• S. Boettcher, Emory University
• Spin glass on arxiv.org