Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation

Abstract

Experimental realizations of a 1D interface always exhibit a finite microscopic width \(\xi>0\); 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 \(T_c(\xi)\). 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 \(\xi\). We derive the exact `time'-evolution equations of the disorder free-energy \(\bar{F}(t,y)\), its derivative \(\eta (t,y)\), and their respective two-point correlators \(\bar{C}(t,y)\) and \(\bar{R}(t,y)\). We compute the exact solution of its linearized evolution \(\bar{R}^{lin}(t,y)\), and we combine its qualitative behavior and the asymptotic properties known for an uncorrelated disorder (\(\xi=0\)), to construct a `toymodel' leading to a simple description of the DP. This model is characterized by Brownian-like free-energy fluctuations, correlated at small \(|y|<\xi\), of amplitude \(\tilde{D}_{\infty}(T,\xi)\). We present an extended scaling analysis of the roughness predicting \(\tilde{D}_{\infty} \sim 1/T\) at high-temperatures and \(\tilde{D}_{\infty} \sim 1/T_c(\xi)\) at low-temperatures. We identify the connection between the temperature-induced crossover and the full replica-symmetry breaking in previous Gaussian Variational Method computations. Finally we discuss the consequences of the low-temperature regime for two experimental realizations of KPZ interfaces, namely the static and quasistatic behavior of magnetic domain walls and the high-velocity steady-state dynamics of interfaces in liquid crystals.