# 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:

${\displaystyle \ln Z=\lim _{n\to 0}{Z^{n}-1 \over n}}$
where ${\displaystyle Z}$ is most commonly the partition function, or a similar thermodynamic function. In this form, the expression is formally exact. It is typically used in calculating the disorder-averaged value of ${\displaystyle {\overline {\ln Z}}}$, where the complicated problem of calculating the average of a logarithm of a disordered quantity can be simplified by applying the above identity, reducing the problem to calculating the disorder average ${\displaystyle {\overline {Z^{n}}}}$ where ${\displaystyle n}$ is assumed to be an integer. This is physically equivalent to averaging the disorder across ${\displaystyle n}$ copies of the system in question, or replicas, hence the name.

The crux of the replica trick is that while the disorder averaging is done assuming ${\displaystyle n}$ to be an integer, to recover the disorder-averaged logarithm one must send ${\displaystyle n}$ 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.

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. –

## 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 ${\displaystyle f(z)}$, into powers of ${\displaystyle z}$ or, in other words, replicas of ${\displaystyle z}$, and perform the same computation which is to be done on ${\displaystyle f(z)}$, using the powers of ${\displaystyle z}$.

A particular case which is of great use in physics is in averaging the free energy ${\textstyle F=-k_{b}T\ln Z[J_{ij}]}$, over values of ${\displaystyle J_{ij}}$ with a certain probability distribution, typically Gaussian,[books on spin glasses 1] and the function ${\displaystyle Z[J_{ij}]\sim e^{-\beta J_{ij}}}$. Notice that if it were ${\displaystyle Z[J_{ij}]}$ (or more generally, any power of ${\displaystyle J_{ij}}$) and not its logarithm which we wanted to average, the resulting integral (assuming a Gaussian distribution) would be of the form ${\displaystyle \int dJ_{ij}e^{-\beta J-\alpha J^{2}}}$, which can be performed by completing squares and carrying out the standard Gaussian integration. Instead, here we employ the following identity for the logarithm function:

${\displaystyle \ln Z=\lim _{n\to 0}{\dfrac {Z^{n}-1}{n}}}$
which reduces the complicated task of averaging the logarithm to solving a relatively simple Gaussian integral, provided ${\displaystyle n}$ is an integer.[1] The replica trick postulates that if ${\displaystyle Z^{n}}$ can be calculated for all positive integers ${\displaystyle n}$ then this may be sufficient to allow the limiting behaviour as ${\displaystyle n\to 0}$ to be calculated.

Clearly, such an argument poses many mathematical questions, and the resulting formalism for performing the limit ${\displaystyle n\to 0}$ 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, 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. But in cases where, for some reason, the determination of ground state is hard, one uses the replica method.[papers on spin glasses 1] 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.[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: ${\displaystyle F={\overline {F[J_{ij}]}}=-k_{B}T{\overline {\ln Z[J]}}}$ where ${\displaystyle J_{ij}}$ describes the disorder (for spin glasses, it describes the nature of magnetic interaction between each of the individual sites ${\displaystyle i}$ and ${\displaystyle j}$) and ${\displaystyle [\cdots ]}$ denotes the average over all values of the couplings described in ${\displaystyle J}$, 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 ${\displaystyle Z^{n}}$ represents the joint partition function of ${\displaystyle n}$ 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.

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: [other approaches 1]

Also, it has been demonstrated [other approaches 2] that the Keldysh technique provides a viable alternative to the replica approach.

## Remarks

The above identity is easily understood:

{\displaystyle {\begin{aligned}\lim _{n\rightarrow 0}{\dfrac {Z^{n}-1}{n}}&=\lim _{n\rightarrow 0}{\dfrac {e^{n\ln Z}-1}{n}}\\&=\lim _{n\rightarrow 0}{\dfrac {n\ln Z+{1 \over 2!}(n\ln Z)^{2}+\dots }{n}}\\&=\ln Z~~.\end{aligned}}}

## References

• M Mezard, G Parisi & M Virasoro, "Spin Glass Theory and Beyond", World Scientific, 1987

Papers on Spin Glasses

1. ^ Parisi, Giorgio (17 January 1997). "On the replica approach to spin glasses".
2. ^ Tommaso Castellani, Andrea Cavagna (May 2005). "Spin-glass theory for pedestrians". arXiv:cond-mat/0505032. Bibcode:2005JSMTE..05..012C. doi:10.1088/1742-5468/2005/05/P05012. Retrieved 3 April 2011.

Books on Spin Glasses

1. ^ Nishimori, Hidetoshi (2001). "2". Statistical physics of spin glasses and information processing : an introduction (PDF). Oxford [u.a.]: Oxford Univ. Press. p. 13. ISBN 0-19-850940-5.

References to other approaches

1. ^ Supersymmetry in Disorder and Chaos, Konstantin Efetov, Cambridge university press, 1997.
2. ^ A. Kamenev and A. Andreev, cond-mat/9810191; C. Chamon, A. W. W. Ludwig, and C. Nayak, cond-mat/9810282.
1. ^ Hertz, John (March–April 1998). "Spin Glass Physics".