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 (KPZ) 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.
随机介质中的受驱动弹性流形在临界驱动力作用下表现出一个脱钉过渡到一个速度非零的状态。我们研究了刚性定向线的脱钉，这是由弯曲刚度，而不是线张力。他们的运动方程是（淬火）赫林-穆林斯方程，它也描述了表面扩散控制的表面生长。刚性定向线特别有趣，因为在有限温度下的静态问题中存在局部化转变，并且通常利用米德尔顿定理（A. Middleton，Phys. Rev. Lett. 68，670（1992））不适用。我们采用分析参数和数值模拟来确定临界指数，并将我们的研究结果与以前的工作和功能重整化群的结果，我们扩展到不同的线弹性。我们看到两个不同的相关长度指数的证据。
We present the general theory of Ising transitions in isotropic elastic media with vanishing thermal expansion. By constructing a minimal model with appropriate spin-lattice couplings, we show that in two dimensions near a continuous transition the elasticity is anomalous in unusual ways: the system either significantly stiffens with a hitherto unknown unique positional order logarithmically stronger than quasi-long range order, or, as the inversion-asymmetry of the order parameter in its coupling with strain increases, it destabilizes when system size $L$ exceeds a finite threshold. At three …
The one-point distribution of the height for the continuum Kardar-Parisi-Zhang (KPZ) equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature. Using an importance sampling approach, the distribution is obtained over a large range of values, down to a probability density as small as $10^{-1000}$ in the tails. The short time behavior is investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations, showing a spectacular agreement with the analytical expressions. The …
We consider field theory formulation for directed polymers and interfaces in the presence of quenched disorder. We write a series representation for the averaged free energy, where all the integer moments of the partition function of the model contribute. The structure of field space is analysed for polymers and interfaces at finite temperature using the saddle-point equations derived from each integer moments of the partition function. For the case of an interface we obtain the wandering exponent $ξ= (4-d)/2$, also obtained by the conventional replica method for the replica symmetric scenario…
Using the non-interacting Anderson tight-binding model on the Bethe lattice as a toy model for the many-body quantum dynamics, we propose a novel and transparent theoretical explanation of the anomalously slow dynamics that emerges in the bad metal phase preceding the Many-Body Localization transition. By mapping the time-decorrelation of many-body wave-functions onto Directed Polymers in Random Media, we show the existence of a glass transition within the extended regime separating a metallic-like phase at small disorder, where delocalization occurs on an exponential number of paths, from a b…
Unified theory for the scaling of the crossover between strong and weak disorder behaviors of optimal paths and directed/undirected polymers in disordered media
Consider the short-time probability distribution $\mathcal{P}(H,t)$ of the one-point interface height difference $h(x=0,τ=t)-h(x=0,τ=0)=H$ of the stationary interface $h(x,τ)$ described by the Kardar-Parisi-Zhang equation. It was previously shown that the optimal path -- the most probable history of the interface $h(x,τ)$ which dominates the upper tail of $\mathcal{P}(H,t)$ -- is described by any of \emph{two} ramp-like structures of $h(x,τ)$ traveling either to the left, or to the right. These two solutions emerge, at a critical value of $H$, via a spontaneous breaking of the mirror symmetry …
