Replica trick

In the <a href="/facts/Statistical_physics/SnlJE78a">statistical physics</a> of <a href="/facts/Spin_glass/v2Cxng9Y">spin glasses</a> and other systems with <a href="/facts/Quenched_disorder/VAqeaphv">quenched disorder</a>, 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 <a href="/facts/Partition_function_(statistical_mechanics)/q65sJrrn">partition function</a>, or a similar thermodynamic function.
It is typically used to simplify the calculation of 
 
 
 
 
 
 
 ln
 ⁡
 Z
 
 ¯
 
 
 
 
 {\displaystyle {\overline {\ln Z}}}
 
, the <a href="/facts/Expected_value/1XV0JKL8">expected value</a> 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 <a href="/facts/Carlson%2527s_theorem/p7524I9a">Carlson's theorem</a> hold, especially that the ratio 
 
 
 
 (
 
 Z
 
 n
 
 
 −
 1
 )
 
 /
 
 n
 
 
 {\displaystyle (Z^{n}-1)/n}
 
 is of <a href="/facts/Exponential_type/2v8QrUTm">exponential type</a> less than <a href="/facts/Pi/vC0UBj1q">π</a>.)
It is occasionally necessary to require the additional property of replica <a href="/facts/Symmetry_breaking/q3yqVlNQ">symmetry breaking</a> (RSB) in order to obtain physical results, which is associated with the breakdown of <a href="/facts/Ergodicity/CKY3Gr9v">ergodicity</a>.

Replica trick open-in-new

Replica trick