Strong-Coupling Fixed Point of the KPZ Equation

– We present a new approach to the Kardar-Parisi-Zhang (KPZ) equation based on the non-perturbative renormalisation group (NPRG). The NPRG flow equations derived here, while embedding all the known analytical results, moreover allow to follow the strong-coupling fixed point describing the rough phase in all dimensions, and thus yield the qualitatively-robust complete phase diagram of the problem. The Kardar-Parisi-Zhang (KPZ) equation, though originally introduced as a coarse-grained description of non-equilibrium interface growth [1], has acquired a broader significance over the past decades as a simple model for generic scale invariance and non-equilibrium phase transitions [2]. It is indeed intimately related to many other important physical systems, such as randomly stirred fluid (Burgers equation) [3], directed polymers in random media [4], dissipative transport [5, 6] or magnetic flux lines in superconductors [8]. The KPZ equation has thus emerged as one of the fundamental theoretical models to investigate universality classes in non-equilibrium scaling phenomena and phase transitions [2]. It is a non-linear Langevin equation which describes the large-distance, long-time dynamics of the growth process specified by a single-valued height function h(x, t) on a d-dimensional substrate x ∈ IR: ∂th(x, t) = ν∇h(x, t) + λ/2 ( ∇h(x, t) 2 + η(x, t), (1) where η(x, t) is a zero mean uncorrelated noise with variance 〈η(x, t)η(x, t′)〉 = 2Dδ(x − x) δ(t − t). This equation reflects the competition between the surface tension smoothing force ν∇2h, the preferential growth along the local normal to the surface represented by the non-linear term and the Langevin noise η which tends to roughen the interface and mimics the stochastic nature of the growth. The objective is to determine the profile of the stationary interface, characterised by the two-point correlation function C(|x−x′|, t− t) ≡ 〈[h(x, t)− h(x, t′)]2〉 and, in particular, its large-scale properties where C is expected to assume the scaling form C(L, τ) = L f( τ L ), where χ and z are the roughness and dynamical exponents respectively. These two exponents are not independent since the Galilean symmetry [3] — the invariance of Eq. (1) under an infinitesimal tilting of the interface — enforces the scaling relation z + χ = 2.