Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation

Abstract

Experimental realizations of a 1D interface always exhibit a finite microscopic width $\xi >0$; 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 $T_c(\xi )$. 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 $\xi$. We derive the exact `time'-evolution equations of the disorder free-energy $\bar{F}(t,y)$, its derivative $\eta (t,y)$, and their respective two-point correlators $\bar{C}(t,y)$ and $\bar{R}(t,y)$. We compute the exact solution of its linearized evolution ${\bar{R}}^{lin}(t,y)$, and we combine its qualitative behavior and the asymptotic properties known for an uncorrelated disorder ($\xi =0$), to construct a `toymodel' leading to a simple description of the DP. This model is characterized by Brownian-like free-energy fluctuations, correlated at small $|y|<\xi$, of amplitude ${\tilde{D}}_{\infty }(T,\xi )$. We present an extended scaling analysis of the roughness predicting ${\tilde{D}}_{\infty }\sim 1/T$ at high-temperatures and ${\tilde{D}}_{\infty }\sim 1/T_c(\xi )$ at low-temperatures. We identify the connection between the temperature-induced crossover and the full replica-symmetry breaking in previous Gaussian Variational Method computations. 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.

33 pages, 6 figures. The initial preprint arXiv:1209.0567v1 has been split into two parts upon refereeing process. The first part gathers the analytical results and is published (see reference below). It corresponds to the current version of arXiv:1209.0567. The second part gathers the numerical results and corresponds the other arXiv preprint arXiv:1305.2364

References in corpus (12)

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• Creep and flow regimes of magnetic domain wall motion in ultrathin Pt/Co/Pt films with perpendicular anisotropy
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• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
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• Thermal rounding of the depinning transition in ultrathin Pt/Co/Pt films
• Temperature-induced crossovers in the static roughness of a one-dimensional interface
• The universal high temperature regime of pinned elastic objects
• Finite-temperature and finite-time scaling of the directed polymer free-energy with respect to its geometrical fluctuations
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Cited by in corpus (20)

• Pinning dependent field driven domain wall dynamics and thermal scaling in an ultrathin Pt/Co/Pt magnetic film
• Pinning of Domain Walls in thin Ferromagnetic Films
• Numerical Approaches on Driven Elastic Interfaces in Random Media
• KPZ equation with short-range correlated noise: emergent symmetries and non-universal observables
• Strength and length-scale of the interaction between domain walls and pinning disorder in thin ferromagnetic films
• Driven interfaces: from flow to creep through model reduction
• From bulk descriptions to emergent interfaces: connecting the Ginzburg-Landau and elastic line models
• Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation: numerical study
• Effect of disorder geometry on the critical force in disordered elastic systems
• A numerical study of the statistics of roughness parameters for fluctuating interfaces
• "Sinc"-Noise for the KPZ Equation
• Tuning Ginzburg-Landau theory to quantitatively study thin ferromagnetic materials
• Degradation of domains with sequential field application
• Power countings versus physical scalings in disordered elastic systems - Case study of the one-dimensional interface
• Zero temperature limit for (1+1) directed polymers with correlated random potential
• Loss of memory of an elastic line on its way to limit cycles
• Velocity distribution functions and intermittency in one-dimensional randomly forced Burgers turbulence
• Finite temperature crossovers in periodic disordered systems
• Microscopic interplay of temperature and disorder of a one-dimensional elastic interface
• Field-dependent roughness of moving domain walls in a Pt/Co/Pt magnetic thin film