Interface roughening due to random impurities at low temperatures

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

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.

Finite-temperature and finite-time scaling of the directed polymer free energy with respect to its geometrical fluctuations

We study the fluctuations of the directed polymer in 1+1 dimensions in a Gaussian random environment with a finite correlation length ξ and at finite temperature. We address the correspondence between the geometrical transverse fluctuations of the directed polymer, described by its roughness, and the fluctuations of its free energy, characterized by its two-point correlator. Analytical arguments are provided in favor of a generic scaling law between those quantities, at finite time, nonvanishing ξ, and explicit temperature dependence. Numerical results are in good agreement both for simulations on the discrete directed polymer and on a continuous directed polymer (with short-range correlated disorder). Applications to recent experiments on liquid crystals are discussed.

Thermodynamic fingerprints of disorder in flux line lattices and other glassy mesoscopic systems

We examine probability distributions for thermodynamic quantities in finite-sized random systems close to criticality. Guided by available exact results, a general ansatz is proposed for replicated free energies, which leads to scaling forms for cumulants of various macroscopic observables. For the specific example of a planar flux line lattice in a two-dimensional superconducting film near H(c1), we provide detailed scaling results for the statistics of the magnetic flux density, susceptibility, heat capacity, and their cross correlations, which can be tested in a recently used experimental setup [Bolle et al., Nature (London) 399, 43 (1999)].