Spin glass
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 simulation
s.
Above the spin glass transition temperature
, Tc,
is identical with the so-called "freezing temperature" 
the spin glass exhibits typical magnetic behaviour (such as paramagnetism
).
If a magnetic field is applied as the sample is cooled to the transition temperature, magnetization of the sample increases as described by the Curie law. Upon reaching Tc, the sample becomes a spin glass and further cooling results in little change in magnetization. This is referred to as the field-cooled magnetization.
When the external magnetic field is removed, the magnetization of the spin glass falls rapidly to a lower value known as the remanent 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. 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.
Spin glasses differ from ferromagnetic materials by the fact that after the external magnetic field is removed from a ferromagnetic substance, the magnetization remains indefinitely at the remanent value. Paramagnetic materials differ from spin glasses by the fact that, after the external magnetic field is removed, the magnetization rapidly falls to zero, with no remanent magnetization. In each case the decay is rapid and exponential.
If the sample is cooled below Tc in the absence of an external magnetic field and a magnetic field is applied after the transition to the spin glass phase, there is a rapid initial increase to a value called the zero-field-cooled magnetization. A slow upward drift then occurs 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.
-dimensional lattice with only nearest neighbor interactions similar to the Ising model
. This model can be solved exactly for the critical temperatures and a glassy phase is observed to exist at low temperatures. The Hamiltonian for this spin system is given by:
Magnet
A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, and attracts or repels other magnets.A permanent magnet is an object...
with frustrated interactions
Geometrical frustration
In condensed matter physics, the term geometrical frustration means a phenomenon in which the geometrical properties of the crystal lattice or the presence of conflicting atomic forces forbid simultaneous minimization of the interaction energies acting at a given site.This may lead to highly...
, augmented by stochastic
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
disorder, where usually ferromagnetic and antiferromagnetic bonds are randomly distributed. Its magnetic ordering resembles the positional ordering of a conventional, chemical glass
Glass
Glass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is soda-lime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...
.
Spin glasses display many metastable structures leading to a plenitude of time scales which are difficult to explore experimentally or in simulation
Simulation
Simulation is the imitation of some real thing available, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system....
s.
Magnetic behavior
It is the time dependence which distinguishes spin glasses from other magnetic systems.Above the spin glass transition temperature
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
, Tc,
is identical with the so-called "freezing temperature" the spin glass exhibits typical magnetic behaviour (such as paramagnetismParamagnetism
Paramagnetism is a form of magnetism whereby the paramagnetic material is only attracted when in the presence of an externally applied magnetic field. In contrast with this, diamagnetic materials are repulsive when placed in a magnetic field...
).
If a magnetic field is applied as the sample is cooled to the transition temperature, magnetization of the sample increases as described by the Curie law. Upon reaching Tc, the sample becomes a spin glass and further cooling results in little change in magnetization. This is referred to as the field-cooled magnetization.
When the external magnetic field is removed, the magnetization of the spin glass falls rapidly to a lower value known as the remanent 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. 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.
Spin glasses differ from ferromagnetic materials by the fact that after the external magnetic field is removed from a ferromagnetic substance, the magnetization remains indefinitely at the remanent value. Paramagnetic materials differ from spin glasses by the fact that, after the external magnetic field is removed, the magnetization rapidly falls to zero, with no remanent magnetization. In each case the decay is rapid and exponential.
If the sample is cooled below Tc in the absence of an external magnetic field and a magnetic field is applied after the transition to the spin glass phase, there is a rapid initial increase to a value called the zero-field-cooled magnetization. A slow upward drift then occurs 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.
Edwards–Anderson model
In this model, we have spins arranged on a-dimensional lattice with only nearest neighbor interactions similar to the Ising modelIsing model
The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
. This model can be solved exactly for the critical temperatures and a glassy phase is observed to exist at low temperatures. The Hamiltonian for this spin system is given by:
-
where
refers to the Pauli spin matrix for the spin-half particle at lattice point 
. A negative value of 
denotes an antiferromagnetic type interaction between spins at points 
and 
. The sum runs over all nearest neighbor positions on a lattice, of any dimension.
The variables
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 
where 
, over all possible values of 
. The distribution of values of 
is taken to be a gaussian with a mean 
and a variance 
:
-
Solving for the free energy using the replica method, below a certain temperature, a new magnetic phase called the spin glass phase (or glassy phase) of the system is found to exist which is characterized by a vanishing magnetization
along with a non-vanishing value of the two point correlation function between spins at the same lattice point but at two different replicas: 
, where 
are replica incides. The order parameter for the ferromagnetic to spin glass phase transition is therefore 
, and that for paramagnetic to spin glass is again 
. Hence the new set of order parameters describing the three magnetic phases constitutes of both 
and 
.
Free energy of this system can be found, both under assumptions of replica symmetry as well as considering replica symmetry breaking. Under the assumption of replica symmetry, the free energy is given by the expression:
-
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 theoryMean field theoryMean field theory is a method to analyse physical systems with multiple bodies. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases . The n-body system is replaced by a 1-body problem with a chosen good external field...
based on a set of replicasReplica trickIn statistical physics of spin glasses and other systems with quenched disorder, the replica trick is a mathematical technique based on the application of the formula- Mathematical Trick :...
of the partition functionPartition function (statistical mechanics)Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
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 modelIsing modelThe Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
with long range frustrated ferro- as well as antiferromagnetic couplings. It corresponds to a mean field approximationMean field theoryMean field theory is a method to analyse physical systems with multiple bodies. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases . The n-body system is replaced by a 1-body problem with a chosen good external field...
of spin glasses describing the slow dynamics of the magnetization and the complex non-ergodic equilibrium state.
Unlike the Edwards–Anderson (EA) model, in the system though only two spins interactions are considered, the range of each interaction can be potentially infinite (of the order of the size of the lattice). Therefore we see that any two spins can be lined with a ferromagnetic or an antiferromagnetic bond and the distribution of these is given exactly as in the case of Edwards–Anderson model. The Hamiltonian for SK model is very similar to the EA model:
-
where
have same meanings as in the EA model. The equilibrium solution of the model, after some initial attempts by Sherrington, Kirkpatrick and others, was found by Giorgio ParisiGiorgio ParisiGiorgio Parisi is an Italian theoretical physicist. He is best known for his works concerning statistical mechanics, quantum field theory and various aspects of physics, mathematics and science in general....
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 methodCavity methodThe Cavity method is a mathematical method due to M. Mezard, Giorgio Parisi and Miguel Angel Virasoro in 1985 to solve some mean field type of models in statistical physics, specially adapted to disordered systems. It has been used to compute properties of ground states in many condensed matter and...
, 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 TalagrandMichel TalagrandMichel Pierre Talagrand is a French mathematician. Docteur ès sciences since 1977, he has been, since 1985, Directeur de Recherches at CNRS and a member of the Functional Analysis Team of the Institut de Mathématique of Paris...
.
The formalism of replica mean field theory has also been applied in the study of neural networksNeural NetworksNeural Networks is the official journal of the three oldest societies dedicated to research in neural networks: International Neural Network Society, European Neural Network Society and Japanese Neural Network Society, published by Elsevier...
, where it has enabled calculations of properties such as the storage capacity of simple neural network architectures without requiring a training algorithm (such as backpropagationBackpropagationBackpropagation is a common method of teaching artificial neural networks how to perform a given task. Arthur E. Bryson and Yu-Chi Ho described it as a multi-stage dynamic system optimization method in 1969 . It wasn't until 1974 and later, when applied in the context of neural networks and...
) 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 transitionPhase transitionA phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
s.
Besides its relevance in condensed matter physics, spin glass theory has acquired a strongly interdisciplinary character, with applications to neural networkNeural networkThe term neural network was traditionally used to refer to a network or circuit of biological neurons. The modern usage of the term often refers to artificial neural networks, which are composed of artificial neurons or nodes...
theory, computer
science, theoretical biology, econophysicsEconophysicsEconophysics is an interdisciplinary research field, applying theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic processes and nonlinear dynamics...
etc.
Infinite-range model
The infinite-range model is a generalization of the Sherrington–Kirkpatrik model where we not only consider two spin interactions but
-spin interactions, where 
and 
is the total number of spins. Unlike the Edwards–Anderson model, similar to the SK model, the interaction range is still infinite. The Hamiltonian for this model is described by:
-
where
have same meanings as in the EA model. The 
limit of this model is known as the Random energy modelRandom Energy ModelIn statistical physics of disordered systems, the random energy model is a toy model of a system with quenched disorder. It concerns the statistics of a system of N particles, such that the number of possible states for the systems grow as 2^N, while the energy of such states is a Gaussian...
. In this limit, it can be seen that the probability of the spin glass existing in a particular state, does not depends only on the energy of that state and not on the individual spin configurations in it.
A gaussian distribution of magnetic bonds across the lattice is assumed usually to solve this model. Any other distribution is expected to give the same result, as a consequence of the central limit theorem. The gaussian distribution function, with mean
and variance 
, is given as:
-
The order parameters for this system are given by the magnetization
and the two point spin correlation between spins at the same site 
, in two different replicas, which are the same as for the SK model. This infinite range model can be solved explicitly for the free energy in terms of 
and 
, under the assumption of replica symmetry as well as 1-Replica Symmetry Breaking.
-
Non-ergodic behavior and applications
A so-called non-ergodicErgodicityIn mathematics, the term ergodic is used to describe a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over space. In physics the term is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.-Etymology:The word ergodic is...
behavior happens in spin glasses below the freezing temperature
, since below that temperature the system cannot escape from the ultradeep minima of the hierarchically-disordered energy landscape.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," Although the freezing temperature is typically as low as 30 kelvinKelvinThe kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
(−240 degrees Celsius), so that the spin glass 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 networksNeural NetworksNeural Networks is the official journal of the three oldest societies dedicated to research in neural networks: International Neural Network Society, European Neural Network Society and Japanese Neural Network Society, published by Elsevier...
, i.e. in theoretical brainBrainThe brain is the center of the nervous system in all vertebrate and most invertebrate animals—only a few primitive invertebrates such as sponges, jellyfish, sea squirts and starfishes do not have one. It is located in the head, usually close to primary sensory apparatus such as vision, hearing,...
research, and in the mathematical-economical theory of optimizationOptimization (mathematics)In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
.
See also
- Quenched disorder
- Replica trickReplica trickIn statistical physics of spin glasses and other systems with quenched disorder, the replica trick is a mathematical technique based on the application of the formula- Mathematical Trick :...
- Cavity methodCavity methodThe Cavity method is a mathematical method due to M. Mezard, Giorgio Parisi and Miguel Angel Virasoro in 1985 to solve some mean field type of models in statistical physics, specially adapted to disordered systems. It has been used to compute properties of ground states in many condensed matter and...
- Geometrical frustrationGeometrical frustrationIn condensed matter physics, the term geometrical frustration means a phenomenon in which the geometrical properties of the crystal lattice or the presence of conflicting atomic forces forbid simultaneous minimization of the interaction energies acting at a given site.This may lead to highly...
- Phase transitionPhase transitionA phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
- Antiferromagnetic interaction
- Crystal structureCrystal structureIn mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...
- Spin iceSpin iceA spin ice is a substance that is similar to water ice in that it can never be completely frozen. This is because it does not have a single minimal-energy state. A spin ice has "spin" degrees of freedom , with frustrated interactions which prevent it freezing...
- Orientational glassOrientational glassIn solid-state physics, an orientational glass is a molecular solid in which crystalline long-range order coexists with quenched disorder in some rotational degree of freedom....
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