Replica symmetry breaking is the loss of permutation symmetry among copies of a system used in the replica method. For a random partition function 
, that method formally uses
![E[lnZ]=lim_(n->0)(E[Z^n]-1)/n,](/images/equations/ReplicaSymmetryBreaking/NumberedEquation1.svg) |
interpreting the positive integer 
as the number of copies and then applying analytic
continuation. For 
replicas, an order parameter is commonly represented by a
symmetric matrix
 |
where 
and 
label replicas. A replica-symmetric ansatz, or assumed
trial form, takes all off-diagonal 
to be equal. In a replica-symmetry-broken ansatz,
the off-diagonal entries instead have a hierarchical block structure, which becomes
a function 
in the limit 
.
Parisi (1978) introduced the construction in the mathematical analysis of mean-field spin glasses. Its hierarchy encodes a nested organization of many equilibrium states rather than a preferred physical replica.
