Replica cluster move in condensed matter physics refers to a family of non-local cluster algorithms used to simulate spin glasses.
[1][2][3] It is an extension of the Swendsen-Wang algorithm in that it generates non-trivial spin clusters informed by the interaction states on two (or more) replicas instead of just one.
It is different from the replica exchange method (or parallel tempering), as it performs a non-local update on a fraction of the sites between the two replicas at the same temperature, while parallel tempering directly exchanges all the spins between two replicas at different temperature.
However, the two are often used alongside to achieve state-of-the-art efficiency in simulating spin-glass models.
It is based on the observation that the total Hamiltonian of two independent Ising replicas α and β,
is the orientation of the 4-state clock,[5] then the total Hamiltonian can be represented as
In the graphical representation of this model, there are two types of bonds that can be open, referred to as blue and red.
[6][7] A cluster formed with blue bonds is referred to as a blue cluster, and a super-cluster formed together with both blue and red bonds is referred to as a grey cluster.
Once the clusters are generated, there are two types of non-local updates that can be made to the clock states independently in the clock clusters (and thus the spin states in both replicas).
Following this, for every grey cluster (blue clusters connected with red bonds), we can rotate all the clock states simultaneously by a random angle.
It can be shown that both updates are consistent with the bond-formation rules, and satisfy detailed balance.
However, the algorithm is not necessarily efficient, as a giant grey cluster will tend to span the entire lattice at sufficiently low temperatures (e.g. even at paramagnetic phases of spin-glass models).
The Houdayer cluster move is a simpler cluster algorithm based on a site percolation process on sites with negative spin overlaps.
[8] For two independent Ising replicas, we can define the spin overlap as
(with a percolation ratio of 1) until the maximal cluster is formed.
This gives an acceptance ratio of 1 as calculated from the Metropolis-Hastings rule.
The efficiency of this algorithm is highly sensitive to the site percolation threshold of the underlying lattice.
If the percolation threshold is too small, then a giant cluster will likely span the entire lattice, resulting in the trivial update of exchanging nearly all the spins between the replicas.
This is why the original algorithm only performs well in low dimensional settings[8][9] (where the site percolation ratio is sufficiently high).
For instance, one can restrict the cluster moves to low-temperature replicas where one expects only a few number of negative-overlap sites to appear[1] (such that the algorithm does not percolate supercritically).
In addition, one can perform a global spin-flip in one of the two replicas when the number of negative-overlap sites exceeds half the lattice size, in order to further suppress the percolation process.
In each Houdayer cluster, the algorithm forms open bonds with probability
This will form sub-clusters that are smaller than the Houdayer clusters, and the spins in these sub-clusters can then be exchange between replicas in a similar fashion as a Houdayer cluster move.