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 energy F[over ¯](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 correlators C[over ¯](t,y) and R[over ¯](t,y). We compute the exact solution of its linearized evolution R[over ¯](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 D̃(∞)(T,ξ). We present an extended scaling analysis of the roughness, supported by saddle-point arguments on its path-integral representation, which predicts D̃(∞)~1/T at high temperatures and D̃(∞)~1/T(c)(ξ) at low temperatures. We identify the connection between the temperature-induced crossover of D̃(∞)(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 D̃(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.