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:

ln

⁡

Z

=

lim

n

→

0

Z

n

−

1

n

{\displaystyle \ln Z=\lim _{n\to 0}{Z^{n}-1 \over n}}

or:

ln

⁡

Z

=

lim

n

→

0

∂

Z

n

∂

n

{\displaystyle \ln Z=\lim _{n\to 0}{\frac {\partial Z^{n}}{\partial n}}}

where

Z

{\displaystyle Z}

is most commonly the partition function, or a similar thermodynamic function.

It is typically used to simplify the calculation of

ln

⁡

Z

¯

{\displaystyle {\overline {\ln Z}}}

, the expected value of

ln

⁡

Z

{\displaystyle \ln Z}

, reducing the problem to calculating the disorder average

Z

n

¯

{\displaystyle {\overline {Z^{n}}}}

where

n

{\displaystyle n}

is assumed to be an integer. This is physically equivalent to averaging over

n

{\displaystyle n}

copies or replicas of the system, hence the name.

The crux of the replica trick is that while the disorder averaging is done assuming

n

{\displaystyle n}

to be an integer, to recover the disorder-averaged logarithm one must send

n

{\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. (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

(

Z

n

−

1

)

/

n

{\displaystyle (Z^{n}-1)/n}

is of exponential type less than π.)

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

f

(

z

)

{\displaystyle f(z)}

using its power series: into powers of

z

{\displaystyle z}

or in other words replicas of

z

{\displaystyle z}

, and perform the same computation which is to be done on

f

(

z

)

{\displaystyle f(z)}

, using the powers of

z

{\displaystyle z}

.

A particular case which is of great use in physics is in averaging the thermodynamic free energy,

F

=

−

k

B

T

ln

⁡

Z

[

J

i

j

]

,

{\displaystyle F=-k_{\rm {B}}T\ln Z[J_{ij}],}

over values of

J

i

j

{\displaystyle J_{ij}}

with a certain probability distribution, typically Gaussian.

The partition function is then given by

Z

[

J

i

j

]

∼

e

−

β

J

i

j

.