In the 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:
, reducing the problem to calculating the disorder average
The crux of the replica trick is that while the disorder averaging is done assuming
to be an integer, to recover the disorder-averaged logarithm one must send
This apparent contradiction at the heart of the replica trick has never been formally resolved, however in all cases where the replica method can be compared with other exact solutions, the methods lead to the same results.
(A natural sufficient rigorous proof that the replica trick works would be to check that the assumptions of Carlson's theorem hold, especially that the ratio
It is occasionally necessary to require the additional property of replica symmetry breaking (RSB) in order to obtain physical results, which is associated with the breakdown of ergodicity.
It is generally used for computations involving analytic functions (can be expanded in power series).
A particular case which is of great use in physics is in averaging the thermodynamic free energy, over values of
[1] The partition function is then given by Notice that if we were calculating just
) and not its logarithm which we wanted to average, the resulting integral (assuming a Gaussian distribution) is just a standard Gaussian integral which can be easily computed (e.g. completing the square).
To calculate the free energy, we use the replica trick:
which reduces the complicated task of averaging the logarithm to solving a relatively simple Gaussian integral, provided
Clearly, such an argument poses many mathematical questions, and the resulting formalism for performing the limit
[3] When using mean-field theory to perform one's calculations, taking this limit often requires introducing extra order parameters, a property known as "replica symmetry breaking" which is closely related to ergodicity breaking and slow dynamics within disorder systems.
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.
In the statistical physics of systems with quenched disorder, any two states 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.
[papers on spin glasses 2] 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.
Introducing replicas allows one to perform this average over different disorder realizations.
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: where
describes the disorder (for spin glasses, it describes the nature of magnetic interaction between each of the individual sites
) and we are taking the average over all values of the couplings described in
To perform the averaging over the logarithm function, the replica trick comes in handy, in replacing the logarithm with its limit form mentioned above.
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.
It has been devised to deal with models on locally tree-like graphs.
The use of the supersymmetry method provides a mathematical rigorous alternative to the replica trick, but only in non-interacting systems.