Transcript Talk at CompPhys04 in Leipzig

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
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.
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 Disordered Media: Models
Hamiltonian
elasticity constant
random potential with zero mean and correlator
SR disorder
LR disorder
Universality classes
Random Bond (RB):
are short-range functions
Domain wall (DW) in random-bond magnets
Random Field (RF) :
for large
for extended defects
Interface in a medium with planes
of disorder with random orientation
DW in random-field magnets, depinning
Random Periodic (RP):
are periodic
CDW, vortex lattice (Bragg glass)
Quantity of interest
Roughness exponent
Periodic systems
(LR)
Driven dynamics
The equation of motion (overdamped dynamics):
friction,
The typical force-velocity characteristics
driving force density
pinning force correlator (
):
Flow
Depinning transition (
Depinning
Creep
velocity:
depinning transition
dynamic exponent:
Creep (
thermal rounding
,
velocity:
,
)
)
Perturbation theory
Action
Observabales
Diagramatic rules
propagator
SR disorder vertex
LR disorder vertex
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
Interfaces
RF
FRG to two-loop (P. Chauve, PLD, KJW, 2001)
Periodic systems
RB
Depinning
(depinning)
FRG for system with LR correlated disorder
Correction to disorder
a
b
c
d
Correction to mobility and elasticity
dot line - either SR disorder or LR disorder.
a , b , and c contribute to SR disorder,
d to LR disorder.
New fixed points
Flow equations in statics:
Flow equations in dynamics:
Critical
exponents:
Double expansion in
new universality classes
and
Random Bond Disorder
Fixed point
Stability analysis
Eigenfunctions computed at the LR RB FP
corresponding eigenvalue is
LR RB Fixed point for
LR disorder at the LR RB FP
analytic function, while SR disorder
has a cusp, i.e.
LR RB FP is stable for
SR RB FP controls the behavior for
is an
Universal amplitude:
Roughness exponent
In constrast to SR disorder
(Exact to all orders!!!)
is preserved along RG flow
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 (
):
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 disorder at the LR RF FP
analytic function, while SR disorder
has a cusp, i.e.
is an
Universal amplitude (Bragg glass):
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
Depinning transition
)
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)
point-like disorder
Bragg glass
Weak Bose glass
columnar disorder
Bose glass
No response to a weak transverse force (L. Balents, 1993)
Tilting force:
(transverse Meissner effect)
SR disorder:
LR disorder:
Two-loop order:
Localized
columnar disorder:
(
-finite )
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).