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:

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:

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

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

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

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:

Let $$f_n$$ be the mean over $$J$$ defined by:

As:

we have:

Using $$\sum_JP[J]=1$$ and $$\ln(1+x)=x+O(x)$$ one has:

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.

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 \index{travelling salesman problem}).