Editing Spin glass (section)

==Edwards–Anderson model==
This is similar to the [[Ising model]]. In this model, we have spins arranged on a <math>d</math>-dimensional lattice with only nearest neighbor interactions. This model can be solved exactly for the critical temperatures and a glassy phase is observed to exist at low temperatures.<ref name=nishimori>{{cite book|last=Nishimori|first=Hidetoshi|title=Statistical Physics of Spin Glasses and Information Processing: An Introduction|year=2001|publisher=Oxford University Press|location=Oxford|isbn=9780198509400|pages=243}}</ref> The [[Hamiltonian mechanics|Hamiltonian]] for this spin system is given by:

: <math>H = -\sum_{\langle ij\rangle} J_{ij} S_i S_j,</math>

where <math>S_i</math> refers to the [[Pauli spin matrix]] for the spin-half particle at lattice point <math>i</math>, and the sum over <math>\langle ij\rangle</math> refers to summing over neighboring lattice points <math>i</math> and <math>j</math>. A negative value of <math>J_{ij}</math> denotes an antiferromagnetic type interaction between spins at points <math>i</math> and <math>j</math>. The sum runs over all nearest neighbor positions on a lattice, of any dimension. The variables <math>J_{ij}</math> representing the magnetic nature of the spin-spin interactions are called bond or link variables.

In order to determine the [[Partition function (statistical mechanics)|partition function]] for this system, one needs to average the [[Helmholtz free energy|free energy]] <math>f\left[J_{ij}\right] = -\frac{1}{\beta} \ln\mathcal{Z}\left[J_{ij}\right]</math> where <math>\mathcal{Z}\left[J_{ij}\right] = \operatorname{Tr}_S \left(e^{-\beta H}\right)</math>, over all possible values of <math>J_{ij}</math>. The distribution of values of <math>J_{ij}</math> is taken to be a Gaussian with a mean <math>J_0</math> and a variance <math>J^2</math>:

: <math>P(J_{ij}) = \sqrt{\frac{N}{2\pi J^2}} \exp\left\{-\frac N {2J^2} \left(J_{ij} - \frac{J_0}{N}\right)^2\right\}.</math>

Solving for the free energy using the [[replica trick|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 <math>m = 0</math> along with a non-vanishing value of the two point correlation function between spins at the same lattice point but at two different replicas:

: <math>q = \sum_{i=1}^N S^\alpha_i S^\beta_i \neq 0,</math>

where <math>\alpha, \beta</math> are replica indices. The [[order parameter]] for the ferromagnetic to spin glass phase transition is therefore <math>q</math>, and that for paramagnetic to spin glass is again <math>q</math>. Hence the new set of order parameters describing the three magnetic phases consists of both <math>m</math> and <math>q</math>.

Under the assumption of replica symmetry, the mean-field free energy is given by the expression:{{r|nishimori}}

: <math>\begin{align}
  \beta f ={} - \frac{\beta^2 J^2}{4}(1 - q)^2 + \frac{\beta J_0 m^2}{2}
              - \int \exp\left( -\frac{z^2} 2 \right) \log \left(2\cosh\left(\beta Jz + \beta J_0 m\right)\right) \, \mathrm{d}z.
\end{align}</math>