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Spin glass
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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.
s the time dependence which distinguishes spin glasses from other magnetic systems. Beginning above the spin glass transition temperature, Tc, 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).
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Encyclopedia
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, 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 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 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 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 Tc 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 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, 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 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. Indeed, such
models have been found to display spin glass phases, bordered by a sharp phase transition.
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.
See also
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