Appendix 7: Determination of Correlation Lengths

(i) Introduction

We are interested in two kinds of correlation length: the percolation correlation length, denoted by ξ_p, and the thermal correlation length, denoted by ξ_T. Both concepts are sometimes said to concern "average cluster size", or "average distance between occupied sites (or spins) in a cluster".¹

These concepts may be explicated intuitively in various ways, but for these lengths to be measured (and for the dynamics of dilute spin systems to be studied by means of such measurements) it is necessary to give operational definitions of them, i.e., to specify procedures whereby measurements may be made.

The percolation correlation length is conceptually simpler than the thermal correlation length, but if we first define the thermal correlation length then we can define the percolation correlation length as a special case.

Let Γ₁(r,T) denote the spatial correlation function (type 1), which is the probability that a spin at a distance r from a given spin in a spin system at equilibrium at temperature T has the same value as that spin.

Let Γ₂(r,T) denote the spatial correlation function (type 2), which is the probability that a spin at a distance r from a given spin in a spin system at equilibrium at temperature T belongs to the same spin cluster as that spin.²

These probabilities must be obtained by averaging over many states of the system (using importance sampling, made possibly by the choice of dynamics algorithm).

Clearly Γ₁(r,T) ≥ Γ₂(r,T), because all spins in a single spin cluster have the same spin value.

Barber (1983, pp.157-8) says that in an infinite spin system at temperature T, Γ₂(r,T) is expected to vary as e^-r/ξ_T as r → ∞, where ξ_T is a parameter which is called the thermal correlation length.³  This is not true of Γ₁(r,T), which can be shown as follows: Suppose we have an Ising spin system at high temperature, then the spins will be more-or-less randomly up or down. Thus in such a system Γ₁(r,T) will be about ½ for large r, and so will not approach zero for large r as e^-r/ξ_T does.

The assumption that Γ₂(r,T) ~ e^-r/ξ_T as r → ∞ implies that

1/ξ_T = - lim _{r → ∞} [(Γ₂(r,T)/r]

Thus by making this assumption and by measuring Γ₂(r,T) we can extract a value for ξ_T.

Barber (1983, p.158) says that this definition of ξ_T is "rather difficult to apply if the limit is restricted by the finite size of the system." Following Fisher & Burford (1967) he suggests rather that "we can consider the second moment of Γ(r,T)" and define

ξ₂(T) = [ Σ_r r².Γ(r,T) / Σ_r r.Γ(r,T) ] ^½

Since the sums are over all r, it is not obvious that this definition is any easier to apply in a finite system than the simpler one above.

As we shall see in the next section, Stauffer and Aharony (1991) suggest an analogous definition for the percolation correlation length.

From a definition of the thermal correlation length we can derive a definition of the percolation correlation length by restricting the spin system to one in which the spins can have only one possible value. Then, in the terminology of Appendix 6, every open-bond cluster is a spin cluster, and every nearest-neighbour cluster is a single-value cluster. Under the assumption of just one spin value the hierarchy of spin cluster types given in Appendix 6 reduces to:

Nearest neighbour cluster = Single-value cluster
                          |    
        Open-bond cluster = Spin cluster

A single-value cluster is not the same as an open-bond cluster, as is shown by the following:

+1(a) --- +1(b) --- +1(c)

+1(d)       .         .      

 .        +1(e) --- +1(f)

Here { a, b, c, d } is a single-value cluster but is not an open-bond cluster.

We are now in a position to define the percolation correlation length for a lattice (of a particular geometry) with a particular concentration of sites and a particular concentration of open bonds between nearest neighbour spins. Assume that there is only one possible spin value, then the spin system cannot change, so the temperature is irrelevant, and there is no need for the system to attain equilibrium. The initial state of the system is the only state relevant to the percolation correlation length. Under this assumption the percolation correlation length is the same as the thermal correlation length, however the latter is defined.

(ii) The Percolation Correlation Length

ξ_p is, intuitively, "some average distance of two sites belonging to the same cluster" (Stauffer and Aharony 1991, p.60), and it "is proportional to a typical cluster diameter" (ibid., p. 22). Stauffer and Aharony (1991, p.60) define it as

ξ² = Σ_r r² g(r) / Σ_r g(r)

where g(r) is the correlation function, which they define as "the probability that a site at a distance r from an occupied site is also occupied and belongs to the same cluster" (ibid., p. 59). With the proviso that "all sums over cluster sizes ... are understood to exclude the infinite cluster, if one is present" (ibid., p.37).

This allows us to give an operational definition of ξ_p provided we have an operational definition of g(r). Such an operational definition is given in the remainder of this section.

First, however, we must note that in studies of dilute spin systems it is usual to consider systems which are either site-diluted or bond-diluted but not both. Nevertheless it is easy to conceive of a spin system which is simultaneously site-diluted and bond-diluted. In such a system the site concentration is (as usual) the proportion of lattice sites occupied by spins, and the bond concentration is the proportion of occupied nearest neighbour sites between which there is an open bond.

For example, in the following spin system (in which up-spins are marked by ‘X’, down-spins by ‘O’ and empty sites by ‘.’) 7 of the 9 lattice sites are occupied, so the site concentration is 7/9, and of the 6 pairs of nearest neighbour spins (occupied sites) 3 of them have open bonds, so the bond concentration is 3/6 = 1/2.⁴

X --- O     X

.     X     .
      |
X     X --- O 

The considerations below apply to lattices of all three types: (i) site-diluted, (ii) bond-diluted and (iii) simultaneously site- and bond-diluted.

Consider any spin system of size L in each dimension incorporating a lattice geometry with a coordination number cn, and using boundary conditions. Each lattice site has cn nearest neighbour sites, and these are in cn different directions from that site. Suppose we have generated a particular configuration of occupied sites and open bonds with a particular site concentration and a particular bond concentration. (For the measurement of the percolation correlation length we are not concerned with the values of the spins at those sites, which may be assumed to be all the same.)

Since the above definition of the percolation correlation length uses the concept of a cluster we must now clarify what exactly is meant.

In Appendix 6 we distinguished several types of cluster. The type which is relevant here is the open-bond cluster, i.e., a set of nearest neighbour sites each of which is connected directly or indirectly to all other sites in the set by means of open bonds (spin values are ignored).

We also must define the concept of a spanning cluster, which corresponds in a finite lattice to an infinite cluster in an infinite lattice (i.e., a cluster which is unbounded).

A spanning cluster is a cluster which connects one side of the lattice with the opposite side of the lattice, as illustrated by the cluster of up-spins (‘X’) shown below:

O --- O     O

.     X     .
      |
X --- X --- X 

In studies of spin model dynamics it is common to use periodic boundary conditions, according to which sites on the boundaries of a finite lattice are connected to sites on the opposite sides of the lattice as if they were nearest neighbour sites. (With free boundary conditions there are no such connections.) This does not go well with the above definition of a spanning cluster, since then the following spin system must be said to have a spanning cluster because the bottom left-hand up-spin on the left side of the lattice is connected to an up-spin on the opposite side of the lattice.

   O --- O     X     0

   .     X     .     O
         |
   .     X     .     O
         
-- X     .     X     X --

Thus, unless we wish to involve ourselves in unnecessary complexity, we can speak of a spanning cluster only in the context of free boundary conditions.

Suppose then that we have a means for identifying the open-bond clusters and that we have assigned a unique number to each open-bond cluster and that we know which open-bond cluster any particular spin belongs to. Suppose also that we can detect if a spin belongs to a spanning cluster (in the sense defined above).

Then for any distance n, with 1 ≤ n < L/2, we define the percolation correlation function gp(n) operationally as follows:

X = 0  
Y = 0
For each spin which does not belong to a spanning open-bond cluster:
    Increment Y.
    For each direction r:
        Increment X if the lattice site at a distance n in direction r
            is occupied and belongs to the same open-bond cluster.
    End For
End For
gp(n) = X/(Y*cn)

Here, for each spin which is not a member of a spanning open-bond cluster, we consider cn sites in each of the cn directions, as to whether or not they are occupied sites in the same open-bond cluster. Thus X/(Y*cn) is the probability that a site at a distance n from a spin (not in a spanning cluster) is occupied and belongs to the same open bond cluster, which accords with the definition of the correlation function given above.

Note that this definition is independent of the particular lattice geometry.

Then (for a lattice which is either site-diluted, bond-diluted or both) we define the percolation correlation length ξ_p as:

  ______________________________________
√ ( Σ_n=1^n=L/2-1 n² gp(n) ) / ( Σ_n=1^n=L/2-1 gp(n) )

This is ξ_p for a particular configuration of occupied and open bonds. To obtain a mean percolation correlation length corresponding to a particular site concentration and a particular bond concentration we must, of course, generate a large number of configurations and take the mean of the ξ_p values.

(iii) The Thermal Correlation Length, Version A

If we regard the thermal correlation length ξ_T as exactly analogous to the percolation correlation length except that it is a measure of the average size of spin clusters (at thermal equilibrium at temperature T), rather than of clusters of occupied sites, then we may use an analogous operational definition to measure ξ_T as follows:

Suppose we have a spin system as above and that we have generated a particular configuration of occupied sites and open bonds in this lattice with a given site concentration and a given bond concentration. We assign some initial spin values and allow the system to equilibrate. We then use our cluster identification procedure, suitably modified for spin values rather than just occupied sites, to identify all the spin clusters⁵  and to identify any spanning spin clusters. Then for any distance n, with 1 ≤ n < L/2, we define the thermal correlation function gt(n) operationally as follows:

X = 0  
Y = 0
For each spin which does not belong to a spanning spin cluster:
    Increment Y.
    For each direction r:
        Increment X if the lattice site at a distance n in direction r
            is occupied by a spin which belongs to the same spin cluster.
    End For
End For
gt(n) = X/(Y*cn)

Here, for each spin which is not in a spanning spin cluster, we consider cn sites in each of the cn directions, as to whether or not it is occupied by a spin in the same spin cluster. Thus X/(Y*cn) is the probability that a site at a distance n from a given spin is occupied by a spin which belongs to the same spin cluster.

Then we can define the thermal correlation length (version A) ξ_T as:

  _____________________________________
√ ( Σ_n=1^n=L/2-1 n² gt(n) ) / ( Σ_n=1^n=L/2-1 gt(n) )

This, however, is a measurement of the thermal correlation length for one particular state of the system. The thermal correlation length, however, is a thermal average, and so we must consider many states of the system at thermal equilibrium. Thus to obtain ξ_T properly we must allow the system to evolve further, after equilibration, and take the mean of the ξ_T values of many states.

Moreover, this is a measure of the thermal correlation length for a particular initial configuration of occupied sites and open bonds. To obtain a mean thermal correlation length corresponding to a particular site concentration, a particular bond concentration and a particular temperature we must generate a large number of initial site and bond configurations, allow them to equilibrate, and take the mean of the ξ_T values arrived at as described above.

(iv) Observations on this Definition

A spin cluster is always a subset of an open-bond cluster, because spins in the same cluster must be connected by a chain of open bonds. Hence it might be thought that for any n in the procedures above the value of the counter X in the case of site correlation must be ≤ the value of X in the case of spin correlation, implying that gt(n) ≤ gp(n), and hence ξ_T ≤ ξ_p, which we might not want.

However, we cannot immediately conclude this, because the above procedures ignore both spanning open-bond clusters (in the case of ξ_p) and spanning spin clusters (in the case of ξ_T). It is possible to have a spanning open-bond cluster which does not contain a spanning spin cluster, in which case the numbers of spins in the (non-spanning) spin clusters contribute to ξ_T but the number of sites in the spanning open-bond cluster does not contribute toward ξ_p, allowing for the possibility of ξ_T > ξ_p.

(v) The Thermal Correlation Length, Version B

If we wish to have a definition of ξ_T which is more likely to allow ξ_T > ξ_p then we may modify the operational definition of ξ_T, in such a way as to ignore spin clusters entirely, as follows:

X = 0
For each spin:
    For each direction r:
        Increment X if the lattice site at a distance n in direction r
            is occupied by a spin which has the same value.
     End For
End For
gt(n) = X/(N*cn)

where N is the number of spins.

Operationally, these alternative definitions of ξ_T are equally valid (although the second is computationally easier). One might object to the second on the grounds that spins in distinct open-bond clusters cannot interact, since spins can interact only by means of a chain of open bonds. If two spins cannot interact then it is questionable to speak of "correlation" between them (except in a merely statistical sense). Or one could accept this consequence and instead abandon the intuitive definition that ξ_T measures the average size of a spin cluster.

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