Replica trick

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In statistical physics of spin glasses and other systems with quenched disorder, the replica trick is a mathematical technique based on the application of the formula

$\lim_{n\to 0} {Z^n-1\over n}=\log Z$

The significance of this formula in statistical physics arises when the quantity $Z$ represents the partition function of a system, from which most macroscopic statistical quantities such as entropy, free-energy, heat-capacity, etc, can be calculated. For example, in a spin glass, the pseudo-random interaction between different magnetic dipoles means that the orientation of these dipoles that has lowest energy is very difficult to predict, and is far richer than in an analogous ferromagnetic system.

For systems with quenched disorder, one typically expects that macroscopic quantities will be self-averaging, whereby any macroscopic quantity for a specific realization of the disorder will be indistinguishable from the same quantity calculated by averaging over all possible realizations of the disorder. When trying to calculate these macroscopic properties mathematically, it is typically desirable to introduce this average directly into one's calculations by calculating quantities such as

$\left\langle \log Z \right\rangle$

where the angle-brackets represents an average over all realizations of the disorder. Although this average is often extremely difficult to perform directly, an average of the form

$\left\langle Z^n \right\rangle$

is much easier to perform, provided that $n$ is an integer. In this case, the quantity $Z n$ represents the joint partition function of $n$ identical systems. The replica trick involves extending this argument to the case where $n$ is no longer constrained to be an integer, by positing that if $Z n$ can be calculated for all positive integers $n$ then this may be sufficient to allow the limiting behaviour as $n\to0$ to be calculated.

Clearly, such an argument poses many mathematical questions, and the resulting formalism for performing the limit $n\to0$ typically introduces many subtleties (see Mezard et al). When using mean field theory to perform one's calculations, taking this limit often requires introducing extra order parameters, in consequence of 'replica symmetry breaking' which is closely related to ergodicity breaking and slow dynamics within disorder systems.

[edit] REM: The easiest Replica problem

The Random Energy Model (REM) is one of the simplest models of statisical mechanics of disordered systems, and probably the simplest model to show the meaning and power of the Replica Trick to the level of 1 Replica Symmetry Break. The model is especially suitable for this introduction because an exact result by a different procedure is known, and the Replica Trick can be proved to work by crosschecking of results.