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Pierre Le Doussal (LPTENS) Kay J. Wiese (LPTENS)Florian Kühnel (Universität Bielefeld )
Long-range correlated random field Long-range correlated random field and random anisotropy and random anisotropy O(N)O(N) models models
Andrei A. Fedorenko
CNRS-Laboratoire de Physique Theorique de l'Ecole Normale Superieure, Paris, France
AAF, P. Le Doussal, and K.J. Wiese, Phys. Rev. E 74, 061109 (2006).
AAF and F. Kühnel, Phys. Rev. B 75, 174206 (2007)
CompPhys07, 30th November 2007, Leipzig
• Random field and random anisotropy O(N) modelsRandom field and random anisotropy O(N) models
• Long-range correlated disorderLong-range correlated disorder
• Functional renormalization group Functional renormalization group
• Phase diagramsPhase diagrams and critical exponents and critical exponents
OutlineOutline
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
• vortex phases in impure superconductors
• disordered liquid crystals • amorphous magnets • He-3 in aerogels
RFIM,
A. A. Abrikosov, 1957
Bragg glass: no translational order, but no dislocations
Decoration
T. Giamarchi, P.Le Doussal, 1995
P.Kim et al, 1999 Disorder destroys the true long-range order A.I. Larkin, 1970
RF,
Klein et al , 1999 Bragg peaks
RA,
Random field and random anisotropy O(N) symmetric models
- component spin
Hamiltonian
- quenched random field
Random field model
Random anisotropy model
- strength of uniaxial anisotropy
- random unit vector
Dimensional reduction.
Perturbation theory suggests that the critical behavior of both models is that of the pure models in .
Dimensional reduction is wrong!
Correlated disorder
Correlation function of disorder potential
• Long-range correlated disorder
dimensional extended defects with random orientation
Probablity that both points belong to the same extended defect
Real systems often contains extended defects in the form of linear dislocations, planar grain boundaries, three-dimensional cavities, fractal structures, etc.
Another example: systems confined in fractal-like porous media (yesterday talk by Christian von Ferber)
A. Weinrib, B.I Halperin, 1983
Phase diagram and critical exponents
The ferromagnetic-paramagnetic transition is described by three independent critical exponents
Connected two-point function
The true long-range order can exist only above the lower critical dimension (A.J. Bray, 1986)
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
Disconnected two-point function
The divergence of the correlation length is described by
Schwartz-Soffer inequality: (RF)
Generalized Schwartz – Soffer inequality
T.Vojta, M.Schreiber, 1995
The “minimal ” model
Hamiltonian
- random potential
There is infinite number of relevant operators (D.S. Fisher, 1985)
Replicated Hamiltonian
SR disorder LR disorder
are arbitrary in the RF case and even in the RA case
FRG for uncorrelated RF and RA models
We have to look for a non-analytic fixed point!
D.S. Fisher, 1985
FRG equation in terms of periodic for RF and for RA
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)
M.Tissier, G.Tarjus, 2006and with
........
such that
TT FP gives exponents corresponding to dimensional reduction
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
RA model has a similar behavior with the main difference that and
M.Tissier, G.Tarjus, 2006
P. Le Doussal, K.Wiese, 2006
FRG to two-loop order
M. Tissier, G. Tarjus, 2006
M.Tissier, G.Tarjus, 2006
Long-range correlated random field O(N) model above
Stability regions of various FPs
Positive eigenvalue
LR disorder modifies the critical behavior for
Critical exponents:
There is no true long-range order.
Long-range correlated random field O(N) model below
However, there are two quasi-long-range ordered phases
Phase diagram
Generalized Schwartz – Soffer inequality
T.Vojta, M.Schreiber, 1995
is satisfied at equality
There is no true long-range order.
Long-range correlated random anisotropy O(N) model below
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
Open questionsOpen 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
SummarySummary
• 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 .