Statics and dynamics of elastic manifolds in media with long-range correlated disorder

1. CompPhys06, 1st December 2006, Leipzig 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 properties • Functional renormalization group • Fixed points and critical exponents • Response to tilting force • Summary AAF, P. Le Doussal, and K.J. Wiese, cond-mat/0609234

2. 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. In all cases the configuration of manifold can be descibed by a displecment field 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) Elastic Manifolds in the Nature

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

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

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

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

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

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

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

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

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

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).