Replica trick
Encyclopedia
In statistical physics
of spin glass
es and other systems with quenched disorder, the replica trick is a mathematical technique based on the application of the formula
, into powers of 
or, in other words, replicas of 
, and perform the same computation which is to be done on 
, using the powers of 
.
A particular case which is of great use in physics, is the averaging of a the free energy or
, over values of 
given along with a certain probability distribution, which is typically taken to be a gaussian. and the function 
. Notice that if it was 
(or more generally, any power of 
) and not its log which we wanted to average, the resulting integral (assuming a gaussian distribution) would be of the form 
, which can be performed by completing squares and carrying out the standard gaussian integration
. But we have the special property or the limit form expression for the logarithm function, given by:
which clearly reduces the task of averaging to solving a relatively simpler gaussian integral.
The replica trick involves extending this argument to the case where
is no longer constrained to be an integer, by positing that if 
can be calculated for all positive integers 
then this may be sufficient to allow the limiting behaviour as 
to be calculated.
Clearly, such an argument poses many mathematical questions, and the resulting formalism for performing the limit
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.
s 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 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, on 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 \emph{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:
where
describes the disorder (for spin glasses, it describes the nature of magnetic interaction between each of the individual sites 
and 
) and 
denotes the average over all values of the couplings described in 
, weighted with a given distribution. To perform the averaging over the logarithm function, the replica trick come in handy, in replacing the logarithm with its limit form mentioned above. In this case, the quantity 
represents the joint partition function of 
identical systems.
(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.
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
Statistical physics
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...
of spin glass
Spin glass
A spin glass is a magnet with frustrated interactions, augmented by stochastic disorder, where usually ferromagnetic and antiferromagnetic bonds are randomly distributed...
es and other systems with quenched disorder, the replica trick is a mathematical technique based on the application of the formula
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, into powers of or, in other words, replicas of , and perform the same computation which is to be done on , using the powers of .A particular case which is of great use in physics, is the averaging of a the free energy or
, over values of given along with a certain probability distribution, which is typically taken to be a gaussian. and the function . Notice that if it was (or more generally, any power of ) and not its log which we wanted to average, the resulting integral (assuming a gaussian distribution) would be of the form , which can be performed by completing squares and carrying out the standard gaussian integrationGaussian integral
The Gaussian integral, also known as the Euler-Poisson integral or Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.It is named after the German mathematician and...
. But we have the special property or the limit form expression for the logarithm function, given by:
which clearly reduces the task of averaging to solving a relatively simpler gaussian integral.
The replica trick involves extending this argument to the case where
is no longer constrained to be an integer, by positing that if can be calculated for all positive integers then this may be sufficient to allow the limiting behaviour as to be calculated.Clearly, such an argument poses many mathematical questions, and the resulting formalism for performing the limit
typically introduces many subtleties (see Mezard et al.).When using mean field theory
Mean field theory
Mean field theory is a method to analyse physical systems with multiple bodies. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases . The n-body system is replaced by a 1-body problem with a chosen good external field...
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.
Physical Applications
The replica trick is used in determining ground stateGround state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
s 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 Spin glass
Spin glass
A spin glass is a magnet with frustrated interactions, augmented by stochastic disorder, where usually ferromagnetic and antiferromagnetic bonds are randomly distributed...
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, on 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
Self-averaging
A self-averaging physical property of a disordered system is one that can be described by averaging over a sufficiently large sample. The concept was introduced by Ilya Mikhailovich Lifshitz.- Definition :...
, 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 \emph{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:
where
describes the disorder (for spin glasses, it describes the nature of magnetic interaction between each of the individual sites and ) and denotes the average over all values of the couplings described in , weighted with a given distribution. To perform the averaging over the logarithm function, the replica trick come in handy, in replacing the logarithm with its limit form mentioned above. In this case, the quantity represents the joint partition function of identical systems.REM: The easiest Replica problem
The Random Energy ModelRandom Energy Model
In statistical physics of disordered systems, the random energy model is a toy model of a system with quenched disorder. It concerns the statistics of a system of N particles, such that the number of possible states for the systems grow as 2^N, while the energy of such states is a Gaussian...
(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 methodCavity method
The Cavity method is a mathematical method due to M. Mezard, Giorgio Parisi and Miguel Angel Virasoro in 1985 to solve some mean field type of models in statistical physics, specially adapted to disordered systems. It has been used to compute properties of ground states in many condensed matter and...
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
Tree (graph theory)
In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree...