Ward's identity in critical dynamics

Abstract

We consider the relaxation of an order-parameter fluctuation of wave numberk in a system undergoing a second-order phase transition. In general, close to the critical point, wherek ⁻¹ ≪κ⁻¹ (the correlation length) the relaxation rate has a linear dependence onκ/k of the form γ(k, κ) = γ(k, 0)x(1−aκ/k). In analogy with the use of Ward's identity in elementary particle physics, we show that the numerical coefficienta is readily calculated by means of a “mass insertion.” We demonstrate, furthermore, that this initial linear drop is the main feature of the fullκ/k dependence of the scaling functionR ^−x γ(k,κ), wherex is the dynamic critical exponent andR=(k²+κ ²)^1/2 is the “distance” variable.