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 $\xi >0$; 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 $T_c\left(\xi \right)$. 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 $\overline{F}(t,y)$, which encodes the microscopic disorder integrated by the DP up to a growing time $t$ and an endpoint position $y$, its derivative $\eta (t,y)$, and their respective two-point correlators $\overline{C}(t,y)$ and $\overline{R}(t,y)$. We compute the exact solution of its linearized evolution ${\overline{R}}^{\mathrm{lin}}(t,y)$ and we combine its qualitative behavior and the asymptotic properties known for an uncorrelated disorder ($\xi =0$) 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 $\left|y\right|\lesssim \xi$, and of amplitude ${\tilde{D}}_{\infty }(T,\xi )$. We present an extended scaling analysis of the roughness, supported by saddle-point arguments on its path-integral representation, which predicts ${\tilde{D}}_{\infty }\sim 1/T$ at high temperatures and ${\tilde{D}}_{\infty }\sim 1/T_c\left(\xi \right)$ at low temperatures. We identify the connection between the temperature-induced crossover of ${\tilde{D}}_{\infty }(T,\xi )$ 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 ${\tilde{D}}_t$. 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.

Figure 1

Left: 1D interface configuration of displacement field $u\left(z\right)$ with respect to the $z$ axis; definition of its relative displacement $\Delta u_z\left(r\right)$ at a length scale $r$ with respect to the internal coordinate $z$. Right: Focus on all the segments of the 1D interface starting from $(0,0)$ and ending at $[t,y(t\left)\right]$; definition of the DP's endpoint position $y\left(t\right)$ after a growing time $t$. The translation table between those two representations is given by $(z,x)\leftrightarrow (t^{\prime },y^{\prime })$ for the coordinates, $u\left(z\right)\leftrightarrow y\left(t^{\prime }\right)$ for the trajectory, $\mathcal{P}\left[\Delta u\left(r\right)\right]\leftrightarrow \mathcal{P}(t,y)$ for the geometrical PDF, and $B\left(r\right)=\overline{\langle \Delta u{\left(r\right)}^2\rangle }\leftrightarrow \overline{\langle y{\left(t\right)}^2\rangle }=B\left(t\right)$ for the roughness function.

Figure 2

Free-energy landscape seen by the DP endpoint as a function of time or length scale $t$. Top: Graph of $F_{\mathrm{th}}(t,y)+{\overline{F}}_V(t,y)$ [imposing $\overline{{\overline{F}}_V(t,y)}\equiv 0$ for simplification]: the thermal parabola $F_{\mathrm{th}}(t,y)=\frac{c{y}^2}{2t}$ flattens and unveils the disorder fluctuations ${\overline{F}}_V(t,y)$, which sketches a KPZ surface in its steady state at asymptotically large times. Bottom: Alternative point of view with the graph of ${\partial }_y{F}_V(t,y)=\frac{cy}{t}+{\eta }_V(t,y)$, where the random phase is progressively revealed with increasing length scale.

Figure 3

Top: Schematic graphs of the two-point correlators $\overline{C}(t,y)$ and $\overline{R}(t,y)$ (respectively in full and dashed curves) at fixed time $t>t_{\mathrm{sat}}$, which suggest the generic decomposition (44, 45, 46); they display the two characteristic length scales $\tilde{\xi }$ and ${\ell }_t$ in the $y$ direction; the dashed area below the central peak of $\overline{R}$ corresponds roughly to the saturation amplitude ${\tilde{D}}_{\infty }$ and translates into the slope of the intermediate linear behavior of $\overline{C}$ [since ${\partial }_y^2\overline{C}(t,y)=2\overline{R}(t,y)$ by (21)]. Bottom: Corresponding PDF $\mathcal{P}(t,y)$, whose variance is the roughness $B\left(t\right)$; the dashed area emphasizes the most probable positions of the DP endpoint, which exclude the large $y$ and thus the saturation wings of $\overline{C}$ or the negative bumps of $\overline{R}$.

Figure 4

${\tilde{D}}_t$ as a function of $t$ (left: linear scale, right: logarithmic scale). In green (bottom curve), the solution of the full differential equation (95); in red (top curve), the solution of the equation without the nonlinear KPZ term; the dashed horizontal lines are the corresponding large-time asymptotics; in purple dotted, the short-time asymptotics (98). Chosen parameters: $c=D=T=\xi =1$, $c_0=c_2=\frac{1}{\sqrt{4\pi }}$, $c_1=c_3=\frac{1}{2}$, $c_4=\frac{1}{4}$.

Figure 5

Kernel $\widehat{K}(\overline{y},\overline{w})$ defined in (E5) linking the linearized free-energy correlator ${\overline{C}}^{\mathrm{lin}}(t,y)$ and the disorder correlator $R_{\xi }\left(y\right)$ with the combination of (E3) and (E4).

Figure 6

The finite-time correlator ${\overline{R}}^{\mathrm{lin}}(t,y)$ (thin purple lines) for a Gaussian disorder correlator $R_{\xi }\left(y\right)=e^{-y^2/\left(4{\xi }^2\right)}/\sqrt{4\pi {\xi }^2}$, plotted as a function of $y$ for different times, compared to its infinite-time limit $R_{\xi }\left(y\right)$ (thick red line), with $\xi =1$, $c=1$, $D=1$, and $T=1$. The central peak develops with increasing times from the flat initial condition $\overline{R}(0,y)\equiv 0$. Larger times correspond to lighter colors. Left inset: same behavior for ${\overline{C}}^{\mathrm{lin}}(t,y)$. Right inset: same behavior for $\frac{1}{2}{\partial }_y{\overline{C}}^{\mathrm{lin}}(t,y)={\int }_0^{y}d{y}^{\prime }{\overline{R}}^{\mathrm{lin}}(t,y^{\prime })$.