Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation
Elisabeth Agoritsas, Vivien Lecomte, Thierry Giamarchi
备注：
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
Experimental realizations of a 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 $\bar{F}(t,y)$, its derivative $η(t,y)$, 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 ($ξ=0$), 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 $|y|<ξ$, of amplitude $\tilde{D}_{\infty}(T,ξ)$. We present an extended scaling analysis of the roughness predicting $\tilde{D}_{\infty} \sim 1/T$ at high-temperatures and $\tilde{D}_{\infty} \sim 1/T_c(ξ)$ 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.
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；School of Science, Constructor University（康斯坦茨大学理学院）；Department of Physics, Massachusetts Institute of Technology（麻省理工学院物理系）
