Replica trick
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: formula_1 or: formula_2 where formula_3 is most commonly the partition function, or a similar thermodynamic function. It is typically used to simplify the calculation of formula_4, reducing the problem to calculating the disorder average formula_5 where formula_6 is assumed to be an integer. This is physically equivalent to averaging over formula_6 copies or replicas of the system, hence the name. The crux of the replica trick is that while the disorder  averaging is done assuming formula_6 to be an integer, to recover the disorder-averaged logarithm one must send formula_6 continuously to zero. 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. (To prove that the replica trick works, one would have to prove that Carlson's theorem holds, that is, that the ratio formula_10 is of exponential type less than pi.) 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. General formulation. It is generally used for computations involving analytic functions (can be expanded in power series). Expand formula_11 using its power series: into powers of formula_12 or in other words replicas of formula_12, and perform the same computation which is to be done on formula_11, using the powers of formula_12. A particular case which is of great use in physics is in averaging the thermodynamic free energy, over values of formula_17 with a certain probability distribution, typically Gaussian. The partition function is  then given by Notice that if we were calculating just formula_19 (or more generally, any power of formula_17) 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:formula_22which reduces the complicated task of averaging the logarithm to solving a relatively simple Gaussian integral, provided formula_6 is an integer. The replica trick postulates that if formula_24 can be calculated for all positive integers formula_6 then this may be sufficient  to allow the limiting behavior as formula_26 to be calculated. Clearly, such an argument poses many mathematical questions, and the resulting formalism for performing the limit formula_26 typically introduces many subtleties. 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. 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. Otherwise one uses the replica method. An example is the case of a quenched disorder in a system like a spin glass with different types of magnetic links between spins, leading to many different configurations of spins having the same energy. 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. 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 formula_17 describes the disorder (for spin glasses, it describes the nature of magnetic interaction between each of the individual sites formula_30 and formula_31) and we are taking the average over all values of the couplings described in formula_32, weighted with a given distribution. To perform the averaging over the logarithm function, the  replica trick comes in handy, in replacing the logarithm with its limit form mentioned above. In this case, the quantity formula_24 represents the joint partition function of formula_6 identical systems. 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. See also. The cavity method is an alternative method, often of simpler use than the replica method, for studying disordered mean-field problems. It has been devised to deal with models on locally tree-like graphs. Another alternative method is the supersymmetric method. The use of the supersymmetry method provides a mathematical rigorous alternative to the replica trick, but only in non-interacting systems. See for example the book: Also, it has been demonstrated that the Keldysh technique provides a viable alternative to the replica approach. Remarks. The above identity is easily understood via  Taylor expansion: Papers on Spin Glasses References. Books on Spin Glasses References to other approaches