Localization transition of stiff directed lines in random media

We investigate the localization of stiff directed lines with bending energy by a short-range random potential. Using perturbative arguments, Flory arguments, and a replica calculation, we show that a stiff directed line in $1+d$ dimensions undergoes a localization transition with increasing disorder for $d>2/3$. We demonstrate that this transition is accessible by numerical transfer matrix calculations in $1+1$ dimensions and analyze the properties of the disorder-dominated phase. On the basis of the two-replica problem, we propose a relation between the localization of stiff directed lines in $1+d$ dimensions and of directed lines under tension in $1+3d$ dimensions, which is strongly supported by identical free energy distributions. This shows that pair interactions in the replicated Hamiltonian determine the nature of directed line localization transitions with consequences for the critical behavior of the Kardar-Parisi-Zhang (KPZ) equation. Furthermore, we quantify how the persistence length of the stiff directed line is reduced by disorder.

Figure 1

Paths with lowest energy for given ending states (top, globally optimal path thicker) for the (a) stiff and (b) tense directed lines (right). The dots mark favorable regions of the Gaussian random potential $V$ [realizations of the quenched disorder are not identical in (a) and (b)].

Figure 2

Sketch of the phase diagram as implied by Flory-type arguments. The arrows indicate the flow of the disorder “strength” $g$ under renormalization.

Figure 3

(a) Rescaled $G_F\left(X\right)=P[(F-\overline{F})/\Delta F]$ free energy distribution for a stiff directed line (SDL) in $1+1$ dimensions. We show distributions for $T=0$ [light (blue) squares, ground state energy] as well as for three finite temperatures $T<T_c$ (black thin solid line), $T\approx T_c$ (red, thicker solid line), and $T>T_c$ (green, light thin solid line). Results for a directed line (DL) in $1+3$ dimensions are shown (dark circles). (b) Local (see text) roughness exponent $2{\zeta }_{\kappa }$ as a function of $T$. Deviations from the analytical value $2{\zeta }_{\kappa }=3$ at high temperatures indicate numerical problems, nonetheless there is a clear dip at $T\approx 1.4$, which we identify as the critical temperature. For low temperatures, we find values consistent with $\omega \approx 0.11$. Inset: Reduced free energy $\delta F=\overline{F}-{\overline{F}}_{\text{ann}}$ rescaled by ${\ln }^{1/2}L$ for lengths $L=50,60,...,100$ as a function of $T$. There is a pseudocrossing around $T\approx 1.38$. (c) The overlap order parameter $q$, as a function of $T$. We estimated $q$ from finite lengths using a fit $q_T\left(L\right)=a\left(T\right)/L+q_{\infty }\left(T\right)$. Inset: Double-logarithmic plot of the overlap $q$ versus $T_c-T$ (with $T_c=1.44$); the solid line is given by $q\sim {(T_c-T)}^{-{\beta }^{\prime }}$ with ${\beta }^{\prime }\approx -1.36$.

Figure 4

Disorder-reduced persistence length $L_p$ of the SDL for $g=1$. $L_p$ matches its thermal value (solid line) for $T>T_c$ and is reduced and approximately constant for $T<T_c$. Inset: $L_p$ at $T=0$ versus the potential strength $g$. The Flory-result $L_p\sim g^{-1}$ (dotted line) matches the data.