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
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Mathematical Trick
This mathematical trick is used in computation involving functions of a variable that can be expressed as a power series in that variable. The crux of this technique is to reduce the function of a variable, say
A particular case which is of great use in physics is in averaging the free energy, or
which reduces the task of averaging to solving a relatively simpler Gaussian integral. The replica trick involves extending this argument to the case where
Clearly, such an argument poses many mathematical questions, and the resulting formalism for performing the limit
Physical applications
The replica trick is used in determining ground states of statistical mechanical systems, in the mean field approximation. Typically, for systems in which the determination of ground state is easy, one can analyze fluctuations near the ground state. But in cases where, for some reason the determination of ground state is hard, one uses the replica method. An example is the case of a quenched disorder in a spin system like spin glass with different types of magnetic links between spin sites, thereby causing many configurations to have the same energy. Hence finding a particular ground state is hard.
In statistical physics of quenched disorder systems, any two states (set of configurations) with the same realization of the disorder, or in case of Spin glasses, with the same distribution of ferromagnetic and antiferromagnetic bonds, are called replicas of each other. 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. Hence replicas are introduced for integrating out the disorder in a system.
In the case of a spin glass, we expect the free energy per spin (or any self averaging quantity) in the thermodynamic limit to be independent of the particular values of ferromagnetic and antiferromagnetic couplings between individual sites, across the lattice. So, we explicitly find the free energy as a function of the disorder parameter (in this case, parameters of the distribution of ferromagnetic and antiferromagnetic bonds) and average the free energy over all realizations of the disorder (all values of the coupling between sites, each with its corresponding probability, given by the distribution function). As free energy takes the form:
REM: The easiest Replica problem
The Random Energy Model (REM) is one of the simplest models of statistical mechanics of disordered systems, and probably the simplest model to show the meaning and power of the Replica Trick to the level 1 of Replica Symmetry Breaking. 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.