• Agoritsas, Elisabeth

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 Tc(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 F̅ (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 C̅ (t,y) and R̅ (t,y). We compute the exact solution of its linearized evolution R̅ 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/Tc(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 D̃t. 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.

Topics

• impedance spectroscopy
• elasticity
• random
• one-dimensional
• magnetic domain wall
• liquid crystal