Stiff directed lines in random media

We investigate the localization of stiff directed lines with bending energy by a short-range random potential. We apply perturbative arguments, Flory scaling arguments, a variational replica calculation, and functional renormalization to 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 in detail. 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 equation. We support the proposed relation to directed lines via multifractal analysis, revealing an analogous Anderson transition-like scenario and a matching correlation length exponent. 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 (a) the stiff and (b) the tense directed line (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

Schematic summary of our findings for the relation between DLs and SDLs. Our results for the energy fluctuation exponent $\omega$ [introduced below in Eq. (8)] and for the free-energy distribution $G_F\left(x\right)$ are presented in Secs. 7b1 and 7c. The shaded area marks the range of the short-ranged (binding) potential. The effective binding interaction becomes apparent in the replica formalism (Sec. 5); the importance of two-replica interactions for SDLs is one essential result of this work; see Secs. 7d and 7e.

Figure 3

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

Figure 4

The roughness $\overline{\langle z^2\rangle }$ as a function of the temperature. There are two distinct regimes where the roughness scales as $\overline{\langle z^2\rangle }\sim T^0$ and $\overline{\langle z^2\rangle }\sim T$, respectively. We plotted these asymptotics as well. We show lengths $L=50,60,...,100$, so the ``scattering'' of the symbols for one temperature indicate the quality of the rescaling with $L^{-3}$ (see also Fig. 5).

Figure 5

Roughness exponent $2{\zeta }_{\kappa }$ for various temperatures, computed via (37). The deviation for high temperatures from the analytical value $2\zeta =3$ indicates numerical shortcomings; 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 $\omega \approx 0.11$.

Figure 6

The generalized persistence length ${\tilde{L}}_p$ according to Eqs. (39) and (43) as a function of the temperature $T$ (43). ${\tilde{L}}_p$ matches its thermal value (solid line) for $T>T_c$ and is reduced and approximately constant for $T<T_c$. Inset: ${\tilde{L}}_p$ at $T=0$ versus the potential strength $g$. The Flory result ${\tilde{L}}_p\sim g^{-1}$ (dotted line), see Eq. (40), matches the data.

Figure 7

Parameters $a\left(T\right)$ (blue plus signs) and ${\omega }^{\prime }\left(T\right)$ (red $X$s) used in fitting $\overline{{\langle v\rangle }^2}$ [cf. Eq. (42)]. The latter vanishes around, but not exactly at, $T=T_c$. The horizontal line corresponds to the literature value for DLs in 1+3 dimensions, $\omega =0.186$.

Figure 8

Distribution of the largest ($w_1$, light green solid line) and second-largest ($w_2$, blue dashed line) statistical weight $w\left(z\right)=Z\left(z\right)/Z$ and the information entropy $s=-{\sum }_{z}w\left(z\right)\ln w\left(z\right)$ (black solid line).

Figure 9

Fluctuation exponent $\omega$ determined by the free-energy fluctuations.

Figure 10

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).

Figure 11

Double logarithmic plot of the negative logarithm of the rescaled free-energy distribution $G_F\left(X\right)$ [$X=(F-\overline{F})/\Delta F$] at low temperatures $T<T_c$ for the DL (light blue stars for $X<0$ and yellow squares for $X>0$) and the SDL (red plus signs for $X<0$ and dark blue $X$s for $X>0$). We see identical behavior for SDL and DL consistent with exponents $\eta \approx 1.23$ and ${\eta }^{\prime }\approx 1.84$; see text.

Figure 12

Rescaled potential energy distribution for a SDL in 1+1 dimensions. Plotted are distributions for three finite temperatures, $T<T_c$ (black thin solid line), $T\approx T_c$ (red thick solid line), and $T>T_c$ (light green solid line). Results for a DL in 1+3 dimensions are shown in blue (medium solid line). Here the brown (dark dashed) curve is an approximation using a one-parameter Gumbel distribution [cf. Eq. (52a) with $m=1.7$], whereas the yellow (light dashed) curve is the normal distribution.

Figure 13

The 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 the temperature $T$. There is a pseudocrossing of the lines around $T\approx 1.38$.

Figure 14

Finite-size scaling for the SDL adsorption problem that corresponds to the two-replica binding; see text for a detailed explanation. The scaling uses $T_c=1.44$ and $\nu =2$. As only one sample is needed for the calculation we used larger system lengths $L=100$ (red plus signs), 200 (green $X$s), and 500 (light blue solid squares).

Figure 15

Overlap order parameter $q$ as a function of $T$. We estimate $q$ 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 16

Finite-size scaling ($L=15,...,30$) for the overlap order parameter. Our best results are $\Sigma \approx 0.75$ for the exponent related to the decay of the order parameter and $\nu \approx 2$ for the correlation length exponent. As before, we used $T_c=1.44$.

Figure 17

Generalized dimensions $D\left(q\right)$ and $\tilde{D}\left(q\right)$ at criticality $T=T_{c,\kappa }$. The dashed line is $D_{T>T_c}=3/2$. Solid lines are monomial fits for large $q$, see text.

Figure 18

Finite-size scaling $\overline{Y_2}{L}^{\tilde{\tau }\left(2\right)}[(T_c-T)/T_c{L}^{1/\nu }]$ giving $\nu =2$ and $T_c=1.41429$ for lengths $L=20,30,...,100$, the points for $L=100$ are interconnected by a line as a visual guidance. This does not work for the alternatively proposed value $\nu =4$. For $T<T_c$, $\overline{Y_2}$ remains finite for large $L$ and, therefore, the finite-size scaling observable diverges as $L^{\tilde{\tau }\left(2\right)}$.

Figure 19

The singularity spectrum $f\left(\alpha \right)$; see Eq. (60). For a better comparison with the expectations the plot also shows lines corresponding to $f\left(\alpha \right)=D\left(2\right)=2$, $f\left(\alpha \right)=D\left(1\right)$, and $f\left(\alpha \right)=\alpha$; see text. The bisector and $f\left(\alpha \right)$ touch at an $\alpha$ that is slightly smaller than $\alpha =1.5$. We consider this to be wrong, but it is consistent with the determined generalized dimensions; see also Fig. 20.

Figure 20

The directly measurable $\alpha \left(q\right)$ (green) and $f\left(q\right)$ (red) from which Fig. 19 was created. The contact point is at $\alpha =f\left(\alpha \right)\approx 1.4$.

Figure 21

Schematic summary of our results supporting a relation between DLs in $1+3d$ and SDLs in $1+d$ dimensions and the importance of replica pair interactions for the localization transition in a random potential.