Depinning in a Random Medium

We develop a renormalized continuum field theory for a directed polymer interacting with a random medium and a single extended defect. The renormalization group is based on the operator algebra of the pinning potential; it has novel features due to the breakdown of hyperscaling in a random system. There is a second-order transition between a localized and a delocalized phase of the polymer; we obtain analytic results on its critical pinning strength and scaling exponents. Our results are directly related to spatially inhomogeneous Kardar-Parisi-Zhang surface growth. PACS numbers: 74.40 Ge, 5.40 +j, 64.60 Ak. ∗ Electronic mail: [email protected] † Electronic mail: [email protected] Low-dimensional manifolds in media with quenched disorder are objects encountered in a large variety of different physical realizations. Obvious examples are interfaces in disordered bulk media and random field systems [1, 2] or magnetic flux lines in dirty superconductors [3], but there is also a deep connection to the problem of nonequilibrium surface growth [4, 5] and randomly driven hydrodynamics [6]. Furthermore, the theory serves as a simple paradigm for more complicated, fully frustrated random systems such as spin glasses [7]. A one dimensional manifold in a random medium is a phenomenological continuum model for a (single) magnetic flux line in type-II superconductors with impurities [3, 15], where the flux line interacts with an ensemble of quenched point defects (represented by a random potential). In addition to these point impurities there may also be extended (e.g columnar or planar) defects in the system. Experiments that systematically probe the effect of this kind of impurities have recently become possible in high temperature superconductors [8]. The statistics of the line configurations is governed by an energetic competition: point defects tend to roughen the flux line; it performs large transversal excursions in order to take advantage of locally favourable regions. An attractive extended defect, on the other hand, suppresses these excursions and, if it is sufficiently strong, localizes the line to within a finite transversal distance ξ⊥. The two regimes are separated by a second order phase transition where the localization length ξ⊥ diverges. In contrast to temperature-driven transitions, it involves the competition of two different configuration energies rather than energy and entropy and is hence governed by a zero-temperature renormalization group fixed point. The system is described by an effective Hamiltonian H = ∫ dt    1 2 ( dr dt )2 − V (r, t) + ρ0 Φ(t)    . (1) Here r(t) denotes the displacement vector of the flux line (also called directed polymer) in d transversal dimensions, as a function of the longitudinal “timelike” co-