Static fluctuations of a thick one-dimensional interface in the 1+1 directed polymer formulation
Elisabeth Agoritsas, Vivien Lecomte, and Thierry Giamarchi
Accepted
Experimental realizations of a 1D interface always exhibit a finite microscopic width x > 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 Tc(x). 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 x. We derive the exact `time'-evolution equations of the disorder free-energy [`F](t,y) - which encodes the microscopic disorder integrated by the DP up to a growing `time' t and an endpoint position y - its derivative h(t,y), and their respective two-point correlators [`C](t,y) and [`R](t,y). We compute the exact solution of its linearized evolution [`R]\textlin(t,y), and we combine its qualitative behavior and the asymptotic properties known for an uncorrelated disorder (x = 0), to justify the construction of a `toymodel' leading to a simple description of the DP properties. This model is characterized by Gaussian Brownian-like free-energy fluctuations, correlated at small \valabsy <~x, and of amplitude [D\tilde](T,x). We present an extended scaling analysis of the roughness, supported by saddle-point arguments on its path-integral representation, which predicts [D\tilde] ~ 1/T at high-temperatures and [D\tilde] ~ 1/Tc(x) at low-temperatures. We identify the connection between the temperature-induced crossover of [D\tilde](T,x) and the full replica-symmetry breaking (full-RSB) in previous Gaussian Variational Method (GVM) computations. In order to refine our toymodel with respect to finite-`time' geometrical fluctuations, we propose an effective `time'-dependent amplitude [D\tilde]t. 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.