1. Statics and dynamics of elastic manifolds in media with long-range correlated disorder Andrei A. Fedorenko, Pierre Le Doussal and Kay J. Wiese CNRS-Laboratoire de Physique Theorique de l'Ecole Normale Superieure, Paris, France Outline: Elastic manifolds in the nature Models and their basic propertiesModels and their basic properties Functional renormalization groupFunctional renormalization group Fixed points and critical exponentsFixed points and critical exponents Response to tilting forceResponse to tilting force SummarySummary AAF, P. Le Doussal, and K.J. Wiese, cond-mat/0609234 CompPhys06, 1st December 2006, Leipzig

2. Elastic Manifolds in the Nature Domain wall (DW) in an Ising ferromagnet with either Random Bond (RB) or Random Field (RF) disorder. An experiment on a thin Cobalt film (left) ( S. Lemerle, et al 1998 ) Cartoon of vortex lattice deformed by disorder. A contact line for the wetting of a disordered substrate by Glycerine. Experimental setup (left). The disorder consists of randomly deposited islands of Chromium, appearing as bright spots (top right). Temporal evolution of the retreating contact-line (bottom right). ( S. Moulinet, et al 2002 ) In all cases the configuration of manifold can be descibed by a displecment field

3. Elastic Manifolds in Disordered Media: Models elasticity constant Hamiltonian random potential with zero mean and correlator Universality classes Random Bond (RB): are short-range functions Random Field (RF) : for large Random Periodic (RP): are periodic CDW, vortex lattice (Bragg glass) Domain wall (DW) in random-bond magnets DW in random-field magnets, depinning Roughness exponent Quantity of interest SR disorder Periodic systems LR disorder for extended defects Interface in a medium with planes of disorder with random orientation (LR)

4. Driven dynamics The typical force-velocity characteristics Depinning transition (, ) Creep (, ) The equation of motion (overdamped dynamics): driving force density friction, pinning force correlator ( ) : velocity: dynamic exponent: velocity: Creep Depinning Flow depinning transition thermal rounding

5. Perturbation theory Action Observabales Diagramatic rules propagator SR disorder vertex LR disorder vertex

6. FRG for short-range correlated disorder Fixed-point solution Depinning transition (T. Nattermann, S. Stepanow, et al 1992) FRG equation to one-loop ( D.S. Fisher, 1986 ) has cusp above Larkin scale Perturbation theory to all orders gives dimensional reduction (incorrect) Imry – Ma gives FRG to two-loop (P. Chauve, PLD, KJW, 2001) Exponents RFRB Depinning InterfacesPeriodic systems (depinning)

7. FRG for system with LR correlated disorder Correction to disorder Flow equations in statics: Flow equations in dynamics: Critical exponents: dot line - either SR disorder or LR disorder. a, b, and c contribute to SR disorder, d to LR disorder. Correction to mobility and elasticity New fixed points new universality classes Double expansion in and

8. Random Bond Disorder LR RB Fixed point for Roughness exponent Eigenfunctions computed at the LR RB FP LR disorder at the LR RB FP is an analytic function, while SR disorder has a cusp, i.e. LR RB FP is stable for SR RB FP controls the behavior for Universal amplitude: In constrast to SR disorder is preserved along RG flow Stability analysisFixed point corresponding eigenvalue is (Exact to all orders!!!)

9. Random Field Disorder LR RF Fixed point for Depinning transition Roughness exponent: Universal amplitude ( ): LR RF FP is stable for SR RF FP controls the behavior for NOTE: that in fact this is a FP of mixed type: SR disorder is effectively RB and LR – RF !!! Fixed point Stability analysis Eigenfunctions computed at the LR RF FP corresponding eigenvalue is

10. Random periodic LR RP Fixed point Universal amplitude (Bragg glass): Depinning transition Two first eigenvectors computed at the LR RP FP (only SR disorder is shown, LR ) corresponding eigenvalue is, LR disorder SR disorder for different LR disorder at the LR RF FP is an analytic function, while SR disorder has a cusp, i.e. LR RP FP is unstable with respect to non-potential perturbation corresponding to : LR RP FP is stable for SR RP FP controls the behavior for Fixed point Stability analysis

11. Tilting field: from linear response to transverse Meissner effect Flux lines in the presence of disorder (neglecting disclocations in flux lattice) point-like disorder columnar disorder LR disorder (extended defects with random orientation) Bragg glassBose glassWeak Bose glass Tilting force: No response to a weak transverse force SR disorder: LR disorder: columnar disorder: ( -finite ) (L. Balents, 1993) (transverse Meissner effect) Localized Two-loop order:

12. Summary We have derived the FRG equations which describe the large scale behavior of elastic manifolds in statics and near depinning transition in the presence of long-range correlated disorder. We have found 3 new fixed points which control the scaling behavior of Random Bond, Random Field and Periodic systems and identified the regions of their stability. In contrast to systems with only SR correlated random filed a mixed type of fixed point appears in systems with LR correlations. The static and dynamic critical exponents are computed to one-loop order. We have study the response of elastic manifold subjected to the tilting force in the presence of long-range correlated disorder. We argue existence of a new glass phase with properties interpolating between properties of the Bragg glass (point-like disorder) and Bose glass (columnar disorder).