Editing Spin glass (section)

==Sherrington–Kirkpatrick model==
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 [[replica trick|replicas]] of the [[partition function (statistical mechanics)|partition function]] of the system.

An important, exactly solvable model of a spin glass was introduced by [[David Sherrington (physicist)|David Sherrington]] and [[Scott Kirkpatrick]] in 1975. It is an [[Ising model]] with long range frustrated ferro- as well as antiferromagnetic couplings. It corresponds to a [[mean-field theory|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-spin 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 linked 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:

: <math>
H = - \frac 1\sqrt N \sum_{i<j} J_{ij} S_i S_j
</math>

where <math>J_{ij}, S_i, S_j</math> 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 with the replica method. The subsequent work of interpretation of the Parisi solution—by [[Marc Mézard|M. Mezard]], [[Giorgio Parisi|G. Parisi]], [[Miguel Ángel Virasoro (physicist)|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]].<ref>{{cite book |last1=Talagrand |first1=Michel |title=Mean Field Models for Spin Glasses |date=10 November 2010 |publisher=Springer Berlin |location=Heidelberg |isbn=978-3-642-15202-3 |url=https://doi.org/10.1007/978-3-642-15202-3 |access-date=14 January 2025}}</ref>

=== Phase diagram ===
[[File:De Almeida-Thouless line.svg|thumb|de Almeida-Thouless curve.]]
When there is a uniform external magnetic field of magnitude <math>
M
</math>, the energy function becomes<math display="block">
H = - \frac 1\sqrt N \sum_{i<j} J_{ij} S_i S_j - M \sum_i S_i
</math>Let all couplings <math>
J_{ij}
</math> are IID samples from the gaussian distribution of mean 0 and variance <math>
J^2
</math>. In 1979, J.R.L. de Almeida and [[David J. Thouless|David Thouless]]<ref name=":2" /> found that, as in the case of the Ising model, the mean-field solution to the SK model becomes unstable when under low-temperature, low-magnetic field state.

The stability region on the phase diagram of the SK model is determined by two dimensionless parameters <math>
x := \frac{kT}{J}, \quad y := \frac{M}{J}
</math>. Its phase diagram has two parts, divided by the ''de Almeida-Thouless curve'', The curve is the solution set to the equations<ref name=":2">{{Cite journal |last1=Almeida |first1=J R L de |last2=Thouless |first2=D J |date=May 1978 |title=Stability of the Sherrington-Kirkpatrick solution of a spin glass model |url=https://iopscience.iop.org/article/10.1088/0305-4470/11/5/028 |journal=Journal of Physics A: Mathematical and General |volume=11 |issue=5 |pages=983–990 |doi=10.1088/0305-4470/11/5/028 |bibcode=1978JPhA...11..983D |issn=0305-4470}}</ref><math display="block">
\begin{aligned}
& x^2 = \frac{1}{(2 \pi)^{1 / 2}} \int \mathrm{d} z\; \mathrm{e}^{-\frac 12 z^2} \operatorname{sech}^4\left(\frac{q^{1 / 2} z + y}{x}\right), \\
& q=\frac{1}{(2 \pi)^{1 / 2}} \int \mathrm{d} z\; \mathrm{e}^{-\frac{1}{2} z^2} \tanh ^2\left(\frac{q^{1 / 2} z + y}{x}\right) .
\end{aligned}
</math>The phase transition occurs at <math>x = 1</math>. Just below it, we have<math display="block">
y^2 \approx \frac 43 ( 1-x)^3.
</math>At low temperature, high magnetic field limit, the line is<math display="block">
x \approx \frac{4}{3\sqrt{2\pi}} e^{-\frac 12 y^2}
</math>