Talk at CompPhys04 in Leipzig

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CompPhys07, 30th November 2007, Leipzig
Long-range correlated random field and random
anisotropy O(N) models
Andrei A. Fedorenko CNRS-Laboratoire de Physique
Theorique de l'Ecole Normale Superieure, Paris,
France
Pierre Le Doussal (LPTENS) Kay J. Wiese
(LPTENS) Florian Kühnel (Universität Bielefeld )
AAF and F. Kühnel, Phys. Rev. B 75, 174206 (2007)
AAF, P. Le Doussal, and K.J. Wiese, Phys. Rev. E
74, 061109 (2006).
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Outline

• Random field and random anisotropy O(N) models
• Long-range correlated disorder
• Functional renormalization group
• Phase diagrams and critical exponents

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Examples of random field and random anisotropy
systems
Def. N-component order parameter is coupled to
a random field. Random Field (RF)
linear coupling Random
Anisotropy (RA) bilinear coupling

• diluted antiferromagnets in uniform magnetic
  field

RFIM,

• vortex phases in impure superconductors

RF,
A. A. Abrikosov,   1957
Decoration
Bragg peaks
Disorder destroys the true long-range order
P.Kim et al, 1999
Klein et al , 1999
A.I. Larkin, 1970
Bragg glass no translational order, but no
dislocations
T. Giamarchi, P.Le Doussal, 1995

• disordered liquid crystals
• amorphous magnets
• He-3 in aerogels

RA,
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Random field and random anisotropy O(N) symmetric
models
Random field model
Hamiltonian
- component spin
- quenched random field
Random anisotropy model
Dimensional reduction. Perturbation theory
suggests that the critical behavior of both
models is that of the pure models in
. Dimensional reduction is wrong!
- strength of uniaxial anisotropy
- random unit vector
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Correlated disorder
Real systems often contains extended defects in
the form of linear dislocations, planar grain
boundaries, three-dimensional cavities, fractal
structures, etc.

• Long-range correlated disorder
• 
  dimensional extended defects with random
  orientation

Correlation function of disorder potential
Probablity that both points belong to the same
extended defect
A. Weinrib, B.I Halperin, 1983
Another example systems confined in fractal-like
porous media
(yesterday talk by Christian von Ferber)
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Phase diagram and critical exponents
The true long-range order can exist only above
the lower critical dimension (A.J.
Bray, 1986)
The ferromagnetic-paramagnetic transition is
described by three independent critical exponents

Connected two-point function
Disconnected two-point function
Schwartz-Soffer inequality
(RF)
Generalized Schwartz Soffer inequality
T.Vojta, M.Schreiber, 1995
The divergence of the correlation length is
described by

• Below the lower critical dimension only a
  quasi-long-range order is possible
• order parameter is zero
• infinite correlation length, i.e., power law
  decay of correlations

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The minimal model
There is infinite number of relevant operators
(D.S. Fisher, 1985)
Hamiltonian
- random potential
SR disorder
LR disorder
Replicated Hamiltonian
are arbitrary in the RF case and even in the RA
case
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FRG for uncorrelated RF and RA models
FRG equation in terms of
periodic for RF and for RA
D.S. Fisher, 1985
We have to look for a non-analytic fixed point!
D.E. Feldman, 2002
RF model above the lower critical dimension
Singly unstable FP exists for
........
For there
is a crossover to a weaker non-analytic FP
(TT-phenomen)
such that
and
with
M.Tissier, G.Tarjus, 2006
TT FP gives exponents corresponding to
dimensional reduction
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FRG for uncorrelated RF and RA O(N) models
RF model below the lower critical dimension (
) for
There is a stable FP which describes a
quasi-long-range ordered (QLRO) phase

FRG to two-loop order
P. Le Doussal, K.Wiese, 2006
M. Tissier, G. Tarjus, 2006
M.Tissier, G.Tarjus, 2006
RA model has a similar behavior with the main
difference that and
M.Tissier, G.Tarjus, 2006
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FRG for long-range correlated RF and RA O(N)
models
One-loop flow equations
Critical exponents
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Long-range correlated random field O(N) model
above
Stability regions of various FPs
Critical exponents
Positive eigenvalue
LR disorder modifies the critical behavior for
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Long-range correlated random field O(N) model
below
There is no true long-range order.
However, there are two quasi-long-range ordered
phases
Phase diagram
Generalized Schwartz Soffer inequality
T.Vojta, M.Schreiber, 1995
is satisfied at equality
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Long-range correlated random anisotropy O(N)
model below
There is no true long-range order.
There are two quasi-long-range ordered phases
Two different QLRO has been observed in NMR
experiments with He-3 in aerogel???
V.V. Dmitriev, et al, 2006
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Summary

• Correlation of RF changes the critical behavior
  above the lower critical dimension and modifies
  the critical exponents. Below the lower critical
  dimension LR correlated RF creates a new LR QLRO
  phase.
• LR RA does not change the critical behavior
  above the lower critical dimension
• for , but creates a
  new LR QLRO phase below the lower critical
  dimension .

Open questions

• Metastability in TT region with subcusp
  non-analyticity corrections to scaling,
  distributions of observables
• Equilibrium and nonequilibrium dynamics of the RF
  and RA models, aging