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

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


H=\sum J_{ij}S_iS_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_iS_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_JP[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_JP[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}).