Editing Spin glass (section)

=== 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>