Universal high-temperature regime of pinned elastic objects

Abstract

We study the high temperature regime within the glass phase of an elastic object with short ranged disorder. In that regime we argue that the scaling functions of any observable are described by a continuum model with a δ-correlated disorder and that they are universal up to only two parameters that can be explicitly computed. This is shown numerically on the roughness of directed polymer models and by dimensional and functional renormalization group arguments. We discuss experimental consequences such as non-monotonous behaviour with temperature. PACS numbers:Elastic objects pinned by quenched disorder are ubiquitous in nature, e.g. vortex lattices in superconductors [1], magnetic domain walls [2]. They are modeled by elastic manifolds, of internal dimension d, parameterized by a N -component displacement field u(x), submitted to random potentials. These systems have been described using the collective pinning theory [3], in terms of the characteristic Larkin length R c , and more recently using functional renormalization group (FRG) [4,5] in terms of a RG fixed point at zero temperature, with only few universality classes depending on N, d and the nature of elasticity and disorder -short range (SR) or long range (LR). For scales larger than R c these objects exhibit glass phases with statistically self-similar ground states of roughness scaling as u ∼ x ζ and self-similar energy landscape with free energy exponent θ. Whenever θ > 0 the T = 0 fixed point is attractive and thermal fluctuations are irrelevant at low temperature for large systems. In some cases (θ F > 0 see below) the glass phase extends to all temperatures, with, however, a crossover at T = T dep . At high temperature T > T dep , the system unbinds from individual pins but remains collectively pinned, and the Larkin length increases with T [1, 6, 7]. This has interesting consequences, e.g. a reentrant region in the phase diagram of high T c superconductors [9]. It was generally argued that, due to this effect, the roughness amplitude decays with temperature for SR disorder [6,8,10].Besides its interest to experiments, the high temperature regime is also at the center of recent works on the directed polymer (DP) d = 1, a problem in close connection to KPZ growth and Burgers turbulence [11][12][13][14]. In two dimension (N = 1) it is amenable to the Bethe Ansatz method using a continuum model with δ-function correlation in the random potential [15]. Since the collective pinning theory and the FRG show that a finite correlation range in the disorder is an important ingredient to describe pinning, an outstanding question has been the domain of validity of this model. Recently the distribution of the free energy F of polymers of length x was computed within the δ-correlated model [8,13,16,17].At large scale x one recovers the Tracy Widom distribution [19], previously proved to hold [18] for a discrete DP model, but at T = 0. At first sight, it suggests that the δ-function model also captures the universality at low temperature, in agreement with the i...

Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation

“…It is well known in the DP context [46] that in the weak noise limit in d = 1, 2, the KPZ equation in presence of a SR correlated noise becomes equivalent, beyond some scale, to a model with delta correlations in space and that the only information that is retained about the noise at large scale is the integral over space of the noise correlations. That shows once again that the backward and forward equations lead to equivalent descriptions.…”

Directed polymer near a hard wall and KPZ equation in the half-space

“…It gives ln z as an output with z = Z/Z. As was established in [19,41,42] in the high T limit at fixed λ, where λ = (t/(2T 4 ))…”

“…Here we callt the (integer) polymer length. At high temperature, we follow [19,41,42] and define the partition sum (PS) Z(t) = γt e − 1 T (x,τ )∈γt V (x,τ ) of paths γt directed along the diagonal of a square lattice from (0, 0) to (t/2,t/2) with only (1, 0) or (0, 1) moves. We denote space x = (i − j)/2 and time τ = i + j.…”

Midpoint Distribution of Directed Polymers in the Stationary Regime: Exact Result Through Linear Response

“…In the case of the 1 + 1-dimensional continuum DP, the model that is studied in the present paper, the same endpoint distribution was also obtained in [49] through a replica Bethe Ansatz calculation for polymers of arbitrary length. Let us mention here that small-length properties of the continuum DP are also interesting from the point-of-view of universality because of the special role that it plays in the universality class: the continuum DP is the universal weak noise limit of DP-models on the square lattice [50,51]. That means that small-scale properties of the continuum DP are related to those of arbitrary DPs in a weakly disordered environment.…”