Elastic manifolds in disordered environments: energy statis- tics

– The energy of an elastic manifold in a random landscape at T = 0 is shown numerically to obey a probability distribution that depends on size of the box it is put into. If the extent of the spatial fluctuations of the manifold is much less than that of the system, a crossover takes place to the Gumbel-distribution of extreme statistics. If they are comparable, the distributions have non-Gaussian, stretched exponential tails. The low-energy and high-energy stretching exponents are roughly independent of the internal dimension and the fluctuation degrees of freedom. The statistical mechanics of elastic objects or manifolds changes in the presence of disorder, since temperature becomes irrelevant as a scaling variable. The statistical properties are determined by the competition of elasticity and randomness. Examples of systems where this happens are domain walls (DW) in random magnets and flux lines in superconductors [1, 2, 3]. At the zero temperature “fixed point” in the renormalization group language, the geometry of the object becomes critical. The fluctuations are self-affine below the upper critical dimension of the Hamiltonian. This upper critical dimension may or may not have a finite value depending on the dimensionality of the possible fluctuations. Examples are provided by the one-dimensional directed polymer (DP) in n ≥ 1 fluctuation dimensions, which has a low-temperature phase for which nc = ∞ [2]. Another case is given by D internal-dimensional random manifolds in d = (D + n) dimensions, where n = 1, which are known to have the upper critical dimension Dc = 4 [3]. The criticality of elastic manifolds is manifested by the distribution of energies, P (E), which is not a Gaussian. This was shown by extensive simulations of directed polymers in d = (D + n) = (1 + 1) and (1+2) dimensions (d is the total, embedding space dimension of a system) [4]. In these simulations a DP was let to minimize its energy by keeping one end fixed, and letting the other one wander freely. The outcome is that for energies smaller than the average, E − ≪ 〈E〉, the distribution P (E) ∼ exp[−|E|− ], i.e., stretched exponential. Note that the distribution is normalized in such a way that the average of E is zero and its variance