Appendix 8: Relaxation Time and Singular Dynamic Scaling
(i) Relaxation Time
Hohenberg and Halperin (1977) write:
When a system is at or close to the critical point, anomalies occur in a wide variety of dynamic properties ... such as transport coefficients and relaxation rates ... [and] multi-time correlation functions ...
Dynamic properties may be measured by a variety of experiments. For example, time-dependent correlation functions are determined by inelastic scattering of neutrons ... Relaxation rates may be measured by changing the temperature or some other parameter, and then monitoring the rate at which a system relaxes toward equilibrium. Relaxation rates may also be obtained indirectly from ... magnetic resonance experiments. (Hohenberg and Halperin, 1977, p.436)
In a theoretical study Harris and Stinchcombe (1986, p.869) identify the "characteristic time τc for the decay of magnetic correlations ... [with] the average time a domain wall takes to diffuse a correlation length." In a similar study Henley (1985, p.2030) defines a relaxation time τ(L,T) as "the typical (median-among-realizations) lifetime of the slowest relaxing mode of a piece P of PC [percolation cluster] with diameter L. The characteristic time of the infinite system is τ(T) = limL→∞τ(L,T) ~ τ(ξT,T)."
When we consider how to actually measure the relaxation time in a spin system (at a given temperature and degree of site- or bond-dilution) a review of the literature reveals that there are at least three approaches (see, e.g., Jain and Lage 1993, Chowdhury and Stauffer 1986, Biswal and Chowdhury 1991 and Binder 1997, p.512), the details of which need not detain us here.
(ii) Dynamic Scaling
At the critical temperature various properties (e.g., magnetic susceptibility) are found to vary as a power of the difference of the temperature from the critical temperature, so that these properties diverge or vanish at the critical point. Such powers are called "critical exponents". The exponent which describes the behaviour of the relaxation time in the vicinity of the critical temperature is called "the dynamic critical exponent".
The hypothesis of "dynamic scaling" was first introduced by Ferrell et al. (1967) and subsequently generalized by Halperin and Hohenberg (1967, 1969). This hypothesis implies that as the temperature T of a system approaches the critical temperature Tc at which a phase transition occurs, the relaxation time τ varies as (ξT)z where ξT is the thermal correlation length (a quantity which describes the tendency of spins separated by a certain distance to be in the same state). In other words, as the temperature of a system approaches the critical temperature the log of the relaxation time, lnτ, depends linearly on the log of the correlation length, lnξ, i.e., lnτ ~ Z.lnξ and thus τ ~ ξZ, where Z is the dynamic critical exponent.
This hypothesis has been confirmed for pure systems (i.e., those with no impurities, or "disorder"), and for such systems the dynamic critical exponent Z has been found to be c. 1.67. In contrast with critical dynamic behaviour discovered subsequently, this is now known as "conventional" or "standard" dynamic scaling.
(iii) Singular Dynamic Scaling
Aeppli, Guggenheim and Uemura (1984, 1985) conducted a neutron-scattering experiment using Rb2CopMg1-pF4, a magnetic material in which a certain proportion p of the magnetic Co ions have been replaced by non-magnetic Mg ions. This is effectively a 2d site-diluted antiferromagnet, with magnetic ion concentration p = 0.58 close to the percolation threshold pc = 0.5927 for a 2d square lattice.
Under the assumption of conventional dynamic scaling Aeppli et al. found that the dynamic exponent Z for the relaxation time had a value close to 2.4, considerably higher than what was expected. This result was considered anomalous and generated considerable interest.
Although there were some attempts to fit this result with the hypothesis of conventional dynamic scaling Henley (1985) and Harris and Stinchcombe (1986) proposed an alternative theoretical view, namely, that at pc the "standard" form of dynamic scaling breaks down and is replaced by a "singular" form, wherein the log of the relaxation time depends on the log of the correlation length not in a linear way but in a quadratic way, i.e., ln(τ) = f(lnξT) where f(x) = Ax2 + Bx + C, so that the relaxation time ~ ξZeff where Zeff (the "effective dynamic exponent") = A(ln ξT) + B, with constant A and B.
According to this view Zeff itself diverges as we approach the bicritical point, T = Tc and p = pc. This would explain the "anomalously" large value obtained by Aeppli et al. This value would have been larger if the concentration of magnetic Co ions had been closer to the percolation threshold.
In recent years numerous computational studies have been conducted using Monte Carlo simulations based upon the dilute Ising model and the q-state Potts model. Singular dynamic scaling has been confirmed by Jain (1986) at pc and by Biswal and Chowdhury (1991) below pc for ξT of the order of ξp (where ξp is the percolation correlation length).
A more complete study of singular dynamic scaling in dilute spin models with various lattice geometries awaits a researcher who finds the subject worthy of investigation and who has access to the computing power required for the many simulation runs required.