Spin glass
In ferromagnetic solids, magnetic spins all align in the same direction; this is analogous to a crystal's lattice-based structure.
On the physical side, spin glasses are real materials with distinctive properties, a review of which was published in 1982.
[2] On the mathematical side, simple statistical mechanics models, inspired by real spin glasses, are widely studied and applied.
Upon reaching Tc, the sample becomes a spin glass, and further cooling results in little change in magnetization.
Magnetization then decays slowly as it 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.
Experimental measurements on the order of days have shown continual changes above the noise level of instrumentation.
Surprisingly, the sum of the two complicated 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,[5] at least in the limit of very small external fields.
representing the magnetic nature of the spin-spin interactions are called bond or link variables.
In order to determine the partition function for this system, one needs to average the free energy
Under the assumption of replica symmetry, the mean-field free energy is given by the expression:[6] 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 David Sherrington and Scott Kirkpatrick in 1975.
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 with the replica method.
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 stability region on the phase diagram of the SK model is determined by two dimensionless parameters
The ergodic aspect of spin glass was instrumental in the awarding of half the 2021 Nobel Prize in Physics to Giorgio Parisi.
[9][10][11] For physical systems, such as dilute manganese in copper, the freezing temperature is typically as low as 30 kelvins (−240 °C), and so the spin-glass magnetism appears to be practically without applications in daily life.
Elemental crystalline neodymium is paramagnetic at room temperature and becomes an antiferromagnet with incommensurate order upon cooling below 19.9 K.[12] Below this transition temperature it exhibits a complex set of magnetic phases[13][14] that have long spin relaxation times and spin-glass behavior that does not rely on structural disorder.
[15] A detailed account of the history of spin glasses from the early 1960s to the late 1980s can be found in a series of popular articles by Philip W. Anderson in Physics Today.
[16][17][18][19][20][21][22][23] In 1930s, material scientists discovered the Kondo effect, where the resistivity of nominally pure gold reaches a minimum at 10 K, and similarly for nominally pure Cu at 2 K. It was later understood that the Kondo effect occurs when a nonmagnetic metal contains a very small fraction of magnetic atoms (i.e., at high dilution).
This term fell out of favor as the theoretical understanding of spin glasses evolved, recognizing that the magnetic frustration arises not just from a simple mixture of ferro- and antiferromagnetic interactions, but from their randomness and frustration in the system.
It was then discovered that the replica method was correct, but the problem lies in that the low-temperature broken symmetry in the SK model cannot be purely characterized by the Edwards-Anderson order parameter.
At the full replica breaking ansatz, infinitely many order parameters are required to characterize a stable solution.
[27] 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.
[28] 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.
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.
