Introduction to Mathematical Physics/N body problems and statistical equilibrium/Spin glasses

Assume that a spin glass system \index{spin glass}(see section{secglassyspin}) has the energy:

$H=\sum J_{ij}S_{i}S_{j}$

Values of variable $S_{i}$ are $+1$ if the spin is up or $-1$ if the spin is down. Coefficient $J_{ij}$ is $+1$ if spins $i$ and $j$ tend to be oriented in the same direction or $-1$ if spins $i$ and $j$ tend to be oriented in opposite directions (according to the random position of the atoms carrying the spins). Energy is noted:

$H_{J}=\sum J_{ij}S_{i}S_{j}$

where $J$ in $H_{J}$ denotes the $J_{ij}$ distribution. Partitions function is:

$Z_{J}=\sum _{[s]}e^{-\beta H_{J}[s]}$

where $[s]$ is a spin configuration. We look for the mean ${\bar {f}}$ over $J_{ij}$ distributions of the energy:

${\bar {f}}=\sum _{J}P[J]f_{J}$

where $P[j]$ is the probability density function of configurations $[J]$, and where $f_{J}$ is:

$f_{J}=-\ln Z_{J}.$

This way to calculate means is not usual in statistical physics. Mean is done on the "chilled" $J$ variables, that is that they vary slowly with respect to the $S_{i}$'s. A more classical mean would consist to $\sum _{J}P[J]\sum _{[s]}e^{-\beta H_{J}[s]}$ (the $J$'s are then "annealed" variables). Consider a system $S_{j}^{n}$ compound by $n$ replicas\index{replica} of the same system $S_{J}$. Its partition function $Z_{J}^{n}$ is simply:

$Z_{J}^{n}=(Z_{J})^{n}$

Let $f_{n}$ be the mean over $J$ defined by:

$f_{n}=-{\frac {1}{n}}\ln \sum _{J}P[J](Z_{J})^{n}$

As:

$\ln Z=\lim _{n\rightarrow 0}{\frac {Z^{n}-1}{n}}$

we have:

$\lim _{n\rightarrow 0}f_{n}=\lim _{n\rightarrow 0}\ln(\sum _{J}P[J][1+n\ln Z_{J}])$

Using $\sum _{J}P[J]=1$ and $\ln(1+x)=x+O(x)$ one has:

${\bar {f}}=\lim _{n\rightarrow 0}f_{n}.$

By using this trick we have replaced a mean over $\ln Z$ by a mean over $Z^{n}$; price to pay is an analytic prolongation in zero. Calculations are then greatly simplified [ph:sping:Mezard87].

Calculation of the equilibrium state of a frustrated system can be made by simulated annealing method .\index{simulated annealing} An numerical implementation can be done using the Metropolis algorithm\index{Metropolis}. This method can be applied to the travelling salesman problem (see [ma:compu:Press92] \index{travelling salesman problem}).