Static fluctuations of a thick one-dimensional interface in the 1+1 directed polymer formulation

Abstract

Experimental realizations of a one-dimensional (1D) interface always exhibit a finite microscopic width ξ > 0; its influence is erased by thermal fluctuations at sufficiently high temperatures, but turns out to be a crucial ingredient for the description of the interface fluctuations below a characteristic temperature T c (ξ ). Exploiting the exact mapping between the static 1D interface and a 1 + 1 directed polymer (DP) growing in a continuous space, we study analytically both the free-energy and geometrical fluctuations of a DP, at finite temperature T , with a short-range elasticity and submitted to a quenched random-bond Gaussian disorder of finite correlation length ξ . We derive the exact time-evolution equations of the disorder free energyF (t,y), which encodes the microscopic disorder integrated by the DP up to a growing time t and an endpoint position y, its derivative η(t,y), and their respective two-point correlatorsC(t,y) andR(t,y). We compute the exact solution of its linearized evolution R lin (t,y) and we combine its qualitative behavior and the asymptotic properties known for an uncorrelated disorder (ξ = 0) to justify the construction of a "toy model" leading to a simple description of the DP properties. This model is characterized by Gaussian Brownian-type free-energy fluctuations, correlated at small |y| ξ , and of amplitude D ∞ (T ,ξ). We present an extended scaling analysis of the roughness, supported by saddle-point arguments on its path-integral representation, which predicts D ∞ ∼ 1/T at high temperatures and D ∞ ∼ 1/T c (ξ ) at low temperatures. We identify the connection between the temperature-induced crossover of D ∞ (T ,ξ) and the full replica symmetry breaking in previous Gaussian variational method (GVM) computations. In order to refine our toy model with respect to finite-time geometrical fluctuations, we propose an effective time-dependent amplitude D t . Finally, we discuss the consequences of the low-temperature regime for two experimental realizations of Kardar-Parisi-Zhang interfaces, namely, the static and quasistatic behavior of magnetic domain walls and the high-velocity steady-state dynamics of interfaces in liquid crystals.

Velocity distribution functions and intermittency in one-dimensional randomly forced Burgers turbulence

“…However, at low temperatures, T uR 1/3 , the typical distance between particles (defined by the wave function Ψ(x)) becomes comparable with the size R of the interaction potential U (x) and its approximation by the δ-function is no longer valid. The zero temperature limit of the considered system has been studied in [17,23] In the limit of low temperatures it is convenient to redefine the parameters of the system in the following way:…”

Kardar-Parisi-Zhang equation with short-range correlated noise: Emergent symmetries and nonuniversal observables

“…This function is precisely the quantity that has been studied analytically and numerically in [62,66]. In these studies, the complete time-evolution ofR(t, r) was investigated, starting from the 'sharp-wedge' initial condition for the 1d KPZ equation.R(r) can be normalised asR(r) := [ D(ξ)/ξ] E ξ (r/ξ) with D(ξ) defined through rR (r) = D. The normalisation and the shape of the correlator are characterised by D(ξ) and E ξ (x) respectively.…”

“…VII. of [62] for overviews and further references. In this article we discuss in particular the non-universal amplitude of the interface roughness [see Eq.…”

“…The role of a finite ξ has been addressed in a series of previous studies in the language of the 1+1 DP endpoint [62][63][64][65][66][67] and recently in [68,69], combining analytical and numerical approaches to characterise the complete time dependence of the KPZ fluctuations, starting from the socalled 'sharp-wedge' initial condition. It was predicted in particular (within the Gaussian Variational Method (GVM) and a Bethe ansatz analysis as well as with a direct numerical integration of the correlated KPZ equation) that the amplitude of the KPZ fluctuations must display a crossover, as ξ is tuned, from the exact solution of the uncorrelated case (ξ = 0) [4,5,33] to a regime of large-scale correlations.…”

Finite-temperature crossovers in periodic disordered systems

“…In particular, for interfaces this question has recently been the focus of several studies (see, e.g., Refs. [14][15][16][17] and references therein).…”