Improved perturbation theory for the Kardar-Parisi-Zhang equation.

We apply a number of schemes which variationally improve perturbation theory for the Kardar-Parisi-Zhang equation in order to extract estimates for the dynamic exponent z. The results for the various schemes show the same broad features, giving closer agreement with numerical simulations in low dimensions than self-consistent methods. They do, however, continue to predict that z = 2 in some critical dimension dc in disagreement with the findings of simulations. PACS numbers: 05.40.+j, 05.70.Ln, 68.35.Ja Typeset using REVTEX 1 The Kardar-Parisi-Zhang (KPZ) equation [1] is perhaps the simplest nonlinear stochastic equation of the diffusion type. Nevertheless, the large-distance scaling behavior and anomalous dimensions have not yet been understood on the basis of systematic calculational schemes, such as the renormalization group. To date the most reliable estimates for the anomalous dimensions, obtained by direct analytic means from the KPZ equation, have come from self-consistent or pseudo-variational procedures [2–4]. These approaches are, in general, rather ad hoc and it would be useful to have a method of improving estimates and perhaps gaining some idea of their accuracy. The relative merits of procedures such as these, genuine variational procedures (by which we mean those with an associated bound) and “improvement” methods such as “the principle of minimal sensitivity” (PMS) [5], have recently been compared for stochastic processes described by simple Langevin equations (or equivalently Fokker-Planck equations) [6]. In this Letter we extend these considerations to the KPZ equation — which can be formulated as a functional Fokker-Planck equation. It turns out that, while genuine variational techniques are not straightforward to use in this case, the application of the PMS yields improved values for the dynamic exponent z which agree better with the results of numerical simulation of models believed to be in the KPZ universality class. The KPZ equation for surface growth in (d+ 1) dimensions with random deposition is ḣ(~x, t) = ν0∇ h + g(∇h) + η(~x, t), (1) where the single-valued function h(~x, t) represents the interface and ~x is a d-dimensional vector. The subscript “0” on surface tension ν0 and noise strength D0 (below) is used to distinguish these bare parameters from the effective (renormalized) ones to be introduced later. The noise η(~x, t) is Gaussian-distributed with zero mean 〈η(~x, t)〉 = 0 and deltafunction correlations 〈η(~x, t)η(~x′, t′)〉 = 2D0δ(~x − ~x ′)δ(t− t′). (2) 2 Equivalently the noise may be specified in terms of the probability density functional: P[η] ∼ exp {