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{{Short description|Mathematical limit applied in statistical physics}}
In the [[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:
<math display="block">\ln Z=\lim_{n\to 0} {Z^n-1\over n}</math> or where
It is typically used to simplify the calculation of <math>\overline{\ln Z}</math>, the [[expected value]] of <math>\ln Z</math>, reducing the problem to calculating the disorder average <math>\overline{Z^n}</math> where <math>n</math> is assumed to be an integer. This is physically equivalent to averaging over <math>n</math> copies or '''''replicas''''' of the system, hence the name.
▲:<math>\ln Z = \lim_{n\to 0} \frac{\partial Z^n}{\partial n}</math>
The crux of the replica trick is that while the disorder averaging is done assuming <math>n</math> to be an integer, to recover the disorder-averaged logarithm one must send <math>n</math> 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. (
▲where <math>Z</math> is most commonly the [[Partition function (statistical mechanics)|partition function]], or a similar thermodynamic function.
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 typically used to simplify the calculation of <math>\overline{\ln Z}</math>, reducing the problem to calculating the disorder average <math>\overline{Z^n}</math> where <math>n</math> is assumed to be an integer. This is physically equivalent to averaging over <math>n</math> copies or '''''replicas''''' of the system, hence the name.
▲The crux of the replica trick is that while the disorder averaging is done assuming <math>n</math> to be an integer, to recover the disorder-averaged logarithm one must send <math>n</math> 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 <math>(Z^n-1)/n</math> 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 [[
Expand <math>f(z)</math> using its [[power series]]: into powers of <math>z</math> or in other words replicas of <math>z</math>, and perform the same computation which is to be done on <math>f(z)</math>, using the powers of <math>z</math>.
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A particular case which is of great use in physics is in averaging the [[thermodynamic free energy]],
:<math>F = -k_{\rm B} T \ln Z[J_{ij}],</math>
over values of <math>J_{ij}</math> with a certain probability distribution, typically Gaussian.<ref name="nishimori_book">{{cite book
The [[Partition function (statistical mechanics)|partition function]] is then given by
:<math>Z[J_{ij}] \sim e^{-\beta J_{ij}}.</math>
Notice that if we were calculating just <math>Z[J_{ij}]</math> (or more generally, any power of <math>J_{ij}</math>) and not its logarithm which we wanted to average, the resulting integral (assuming a Gaussian distribution) is just
:<math>\int dJ_{ij} \, e^{-\beta J - \alpha J^
a standard [[Gaussian integral]] which can be easily computed (e.g. completing the square).
To calculate the free energy, we use the replica trick:<math display="block">\ln Z = \lim_{n\to 0}\dfrac{Z^{n}-1}{n}</math>which reduces the complicated task of averaging the logarithm to solving a relatively simple Gaussian integral, provided <math>n</math> is an integer.<ref>{{cite journal|last=Hertz|first=John|title=Spin Glass Physics|date=March–April 1998}}</ref>
The replica trick postulates that if <math>Z^n</math> can be calculated for all positive integers <math>n</math> then this may be sufficient to allow the limiting behavior as <math>n\to0</math> to be calculated.
Clearly, such an argument poses many mathematical questions, and the resulting formalism for performing the limit <math>n\to0</math> typically introduces many subtleties.<ref>{{Cite book|title=Spin Glass Theory and Beyond|last1=Mezard|first1=M|last2=Parisi|first2=G|last3=Virasoro|first3=M|date=1986-11-01|publisher=WORLD SCIENTIFIC|isbn=9789971501167|series=World Scientific Lecture Notes in Physics|volume=
When using [[mean
== Physical applications ==
The replica trick is used in determining [[ground state]]s of statistical mechanical systems, in the [[mean
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.<ref name="spin glass pedestrians" group="papers on spin glasses">{{cite journal|last=Tommaso Castellani|first=Andrea Cavagna|title=Spin-glass theory for pedestrians|journal=Journal of Statistical Mechanics: Theory and Experiment|volume=2005|issue=5|pages=P05012|date=May 2005|doi=10.1088/1742-5468/2005/05/P05012|arxiv = cond-mat/0505032 |bibcode = 2005JSMTE..05..012C |s2cid=118903982}}</ref> 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.
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:<math>
F = \overline{F[J_{ij}]} = -
</math>
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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
▲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 (graph theory)|tree-like graphs]].
Another alternative method is the [[Supersymmetry#Supersymmetry in condensed matter physics|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: <ref group="other approaches">''Supersymmetry in Disorder and Chaos'', Konstantin Efetov, Cambridge university press, 1997.</ref>
Also, it has been demonstrated <ref group="other approaches">A. Kamenev and A. Andreev, cond-mat/9810191; C. Chamon, A. W. W. Ludwig, and C. Nayak, cond-mat/9810282.</ref> that the [[Keldysh
==Remarks==
{{reflist|group=nb}}
The first of the above
:<math>\begin{align}\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 + \
For the second identity, one simply uses the definition of the derivative
:<math>\begin{align}
\lim_{n \rightarrow 0} \dfrac{\partial Z^n}{\partial n} &= \lim_{n \rightarrow 0} \dfrac{\partial e^{n\ln Z}}{\partial n}\\[5pt]
&= \lim_{n \rightarrow 0} Z^n\ln Z\\[5pt]
&=\lim_{n \rightarrow 0} (1 + n\ln Z +\cdots )\ln Z\\[5pt]
&= \ln Z ~~.\end{align}</math>
==References==
* S Edwards (1971), "Statistical mechanics of rubber". In ''Polymer networks: structural and mechanical properties'', (eds A. J. Chompff & S. Newman). New York: Plenum Press, ISBN 978-1-4757-6210-5.
* {{Cite book |last1=Mézard |first1=Marc |title=Spin glass theory and beyond: an introduction to the replica method and its applications |last2=Parisi |first2=Giorgio |last3=Virasoro |first3=Miguel Ángel |date=1987 |publisher=World scientific |isbn=978-9971-5-0116-7 |series=World Scientific lecture notes in physics |location=Teaneck, NJ, USA}}
* {{cite arXiv |last=Charbonneau |first=Patrick |title=From the replica trick to the replica symmetry breaking technique |date=2022-11-03 |class=physics.hist-ph |eprint=2211.01802}}
===Papers on spin glasses===
{{Reflist|group=papers on spin glasses}}
===Books on
{{Reflist|group=books on spin glasses}}
===References to other approaches===
{{Reflist|group=other approaches}}
{{Reflist}}
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