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

Abstract

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.

Kardar-Parisi-Zhang equation with short-range correlated noise: Emergent symmetries and nonuniversal observables

“…As for the behaviour of D at intermediate values of ξ, although no exact expression is available yet, two independent predictions have been obtained in the form of an implicit equation, D γ ∝ (4/ξ) γ/3 (1 − D), with γ ∈ {3/2, 6} and numerical pre-factors of order 10 [62], invoking in particular a GVM computation with a full replica-symmetry-breaking [62,68]. These analytical predictions are qualitatively consistent with numerical measurements either on a directed polymer on a discrete lattice [65] or in a direct numerical integration of the continuous KPZ equation [66]. At last, we mention that D corresponds to the 'fudging' parameter f in [62,67], which coincides in fact with the full-replica-symmetrybreaking parameter in the GVM computations presented in [63].…”

“…In particular, the induced modification of the flow equation ofĝ k cannot be computed within NLO approximation, such that the RG flow of the theory is constrained to preserve the identity exponent χ + z = 2 at the IR fixed point. However, the presence of temporal correlations do not affect the results found in [62,63,65,66] (where the full time evolution is considered) in any noticeable way when compared to results obtained in a Galilean invariant set-up. This strongly suggests that the IR physics is indeed Galilean invariant.…”

“…5. The behaviour of the amplitude D was previously addressed in [62,[65][66][67], fixing ξ and varying the diffusion coefficient ν within the original KPZ equation Eq. (1) (see Sec.…”

“…The role of a finite ξ has been addressed in a series of previous studies in the language of the 1+1 DP endpoint [62][63][64][65][66][67] and recently in [68,69], combining analytical and numerical approaches to characterise the complete time dependence of the KPZ fluctuations, starting from the socalled 'sharp-wedge' initial condition. It was predicted in particular (within the Gaussian Variational Method (GVM) and a Bethe ansatz analysis as well as with a direct numerical integration of the correlated KPZ equation) that the amplitude of the KPZ fluctuations must display a crossover, as ξ is tuned, from the exact solution of the uncorrelated case (ξ = 0) [4,5,33] to a regime of large-scale correlations.…”

Nanoscale studies of ferroelectric domain walls as pinned elastic interfaces

“…2(d). With finite temperatures, the effective thickness of the interface increases ξ ef f,T >0 > ξ T =0 as a result of the induced thermal jitter [30], allowing continual exploration of the pinning potential landscape at energy scales ∼ k B T . Meanwhile, due to dipolar interactions, the stiffness of the interface, and thus its elasticity under deformations perpendicular to the polar orientation is higher than that for deformations along the polar orientation.…”