# Spin glass

Schematic representation of the random spin structure of a spin glass (top) and the ordered one of a ferromagnet (bottom)
Glass (amorphous SiO2)
Quartz (crystalline SiO2)
The magnetic disorder of spin glass compared to a ferromagnet is analogous to the positional disorder of glass (left) compared to quartz (right).

A spin glass is a disordered magnet with frustrated interactions, augmented by stochastic positions of the spins, where conflicting interactions, namely both ferromagnetic and also antiferromagnetic bonds, are randomly distributed with comparable frequency. The term "glass" comes from an analogy between the magnetic disorder in a spin glass and the positional disorder of a conventional, chemical glass, e.g., a window 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.

Above the spin glass transition temperature, Tc,[note 1] 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.[1] 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.{{[J. Phys.: Condens. Matter 10 (1998) 11049–11054. Printed in the UK. BB]}}

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.[citation needed]

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 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,[2] at least in the limit of very small external fields.

## Edwards–Anderson model

In this model, we have spins arranged on a $d$-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.[3] The Hamiltonian for this spin system is given by:

$H = -\sum_{\langle ij\rangle}J_{ij}S_{i}S_{j},$

where $S_{i}$ refers to the Pauli spin matrix for the spin-half particle at lattice point $i$. A negative value of $J_{ij}$ denotes an antiferromagnetic type interaction between spins at points $i$ and $j$. The sum runs over all nearest neighbor positions on a lattice, of any dimension. The variables $J_{ij}$ 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 $f[J_{ij}] = -\dfrac{1}{\beta}\ln \mathcal{Z}\left[J_{ij}\right]$ where $\mathcal{Z}\left[J_{ij}\right] = \operatorname{Tr}_{S}e^{\left(-\beta H\right)}$, over all possible values of $J_{ij}$. The distribution of values of $J_{ij}$ is taken to be a gaussian with a mean $J_{0}$ and a variance $J^{2}$:

$P(J_{ij}) = \dfrac{1}{\sqrt{2\pi J^2}}\exp\left\{-\dfrac{N}{2J^2}\left(J_{ij} - \dfrac{J_0}{N}\right)^2\right\}.$

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 $m = 0$ along with a non-vanishing value of the two point correlation function between spins at the same lattice point but at two different replicas: $q = \sum_{i=1}^N S^\alpha_i S^\beta_i \neq 0$, where $\alpha, \beta$ are replica indices. The order parameter for the ferromagnetic to spin glass phase transition is therefore $q$, and that for paramagnetic to spin glass is again $q$. Hence the new set of order parameters describing the three magnetic phases consists of both $m$ and $q$. 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:

\begin{align} \beta f = &- \dfrac{\beta^2 J^2}{4}(1-q)^2 + \dfrac{\beta J_0 r m^r}{2} \\ &- \int \exp\left(-\frac{z^2}{2}\right)\log \left(2\cosh\left(\beta J z + \beta J_{0}m\right)\right) \, \mathrm{d}z. \end{align}

## 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.

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:

$H = -\sum_{i

where $J_{ij}, S_{i}, S_{j}$ 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 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.[citation needed]

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.

## Infinite-range model

The infinite-range model is a generalization of the Sherrington–Kirkpatrick model where we not only consider two spin interactions but $r$-spin interactions, where $r \leq N$ and $N$ 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:

$H = -\sum_{i_1 < i_2 < \cdots < i_r} J_{i_1 \dots i_r} S_{i_1}\cdots S_{i_r}$

where $J_{i_1\dots i_r}, S_{i_1},\dots, S_{i_r}$ have similar meanings as in the EA model. The $r\to \infty$ limit of this model is known as the Random energy model. In this limit, it can be seen that the probability of the spin glass existing in a particular state, 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 $\dfrac{J_0}{N}$ and variance $\dfrac{J^2}{N}$, is given as:

$P(J_{i_1\cdots i_r}) = \sqrt{\dfrac{N^{r-1}}{J^2 \pi r!}} \exp\left\{-\dfrac{N^{r-1}}{J^2 r!}\left(J_{i_1\cdots i_r} - \dfrac{J_0 r!}{2N^{r-1}}\right)\right\}$

The order parameters for this system are given by the magnetization $m$ and the two point spin correlation between spins at the same site $q$, 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[3] in terms of $m$ and $q$, under the assumption of replica symmetry as well as 1-Replica Symmetry Breaking.[3]

\begin{align} \beta f &= \dfrac{\beta^2 J^2 q^r}{4} - \dfrac{r\beta^2 J^2 q^r}{2} - \dfrac{\beta^2 J^2}{4} + \dfrac{\beta J_0 r m^r}{2} + \dfrac{r\beta^2 J^2 q^{r-1}}{4\sqrt{2\pi}} \\ &\qquad + \int \exp\left(-\frac{z^2}{2}\right)\log \left(2\cosh\left(\beta Jz\sqrt{\dfrac{rq^{r-1}}{2}} + \dfrac{\beta J_0 r m^{r-1}}{2}\right)\right) \, \mathrm{d}z \end{align}

## Non-ergodic behavior and applications

A so-called non-ergodic behavior happens in spin glasses below the freezing temperature $T_f$, since below that temperature the system cannot escape from the ultradeep minima of the hierarchically-disordered energy landscape.[note 2] Although the freezing temperature is typically as low as 30 kelvin (−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 networks, i.e. in theoretical brain research, and in the mathematical-economical theory of optimization.

1. ^ $T_c$ is identical with the so-called "freezing temperature" $T_f$