Statistical Physics and Condensed Matter
NB: former research topic, project expired 1999
Flux-lines, Directed Polymers, Surface Growth: The KPZ-equation
Harald Kinzelbach
Contact:
kibach@tphys.uni-heidelberg.de (Harald Kinzelbach)
Introduction
Many properties of disordered Type-II superconductors are determined
by the interaction between magnetic vortex lines penetrating
the specimen, and randomly distributed crystal defects. On a mesoscopic
scale, the problem can be described by a (seemingly) simple model of
directed lines in a random potential. By an elementary transformation, this
model can be mapped onto the so called Kardar-Parisi-Zhang-equation
(KPZ-equation). This equation originally has been introduced as a simple
nonlinear evolution equation describing a growing surface.
But in various disguises, it appears ubiquitously in a number of
non-equilibrium statistical problems such as stochastically driven fluid
dynamics or dissipative transport.
The large scale behaviour of these systems typically shows a characteristic
non-trivial disorder-dominated regime. It is a notorious difficulty of this so
called "strong-coupling regime" that its properties are more or less inaccessible
to any known systematic treatment, let alone to an exact solution.
In particular, all standard renormalization group treatments fail to produce
the corresponding fixed point which determines the selfsimilar scaling behavior
of the systems.
This feature, which the model shares with other driven systems such as fully
developed turbulence, and also with complicated disorder-dominated systems
like spin glasses, is one major open problem in statistical physics.
Abstracts of recent publications on the subject
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Depinning in a random medium
by Harald Kinzelbach and Michael Lässig,
Journal of Physics A28, 6535, (1995)
We develop a renormalized continuum field theory for a directed polymer
interacting with a random medium and a single extended defect.
The renormalization group is based on the operator algebra of the
pinning potential; it has novel features due to the breakdown of hyperscaling in a random system. There is a second-order transition between a localized and a delocalized phase of the polymer; we obtain analytic results on its critical pinning strength and scaling exponents. Our results are directly related to spatially inhomogeneous Kardar-Parisi-Zhang surface growth.
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Interacting Flux Lines in a Random Medium
by Harald Kinzelbach and Michael Lässig, Phys.Rev.Lett. 75, 2208, (1995)
We study the continuum field theory for an ensemble of directed lines
in 1+d' dimensions that live in a medium with quenched
point disorder and interact via short-ranged pair forces.
In the strong-disorder (or low-temperature) regime,
attractive forces generate a bound state with a localization length
that diverges algebraically for vanishing interactions.
Repulsive forces lead to mutual avoidance with a pair distribution
function reminiscent of fermions.
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Upper Critical Dimension of the Kardar-Parisi-Zhang Equation
by Michael Lässig and Harald Kinzelbach, Phys.Rev.Lett. 78, 903, (1997)
The strong-coupling regime of Kardar-Parisi-Zhang surface
growth driven by short-ranged noise has an upper critical dimension
d> less or equal to four (where the dynamic exponent z takes the
value z(d>) = 2). To derive this, we use the mapping onto
directed polymers with quenched disorder. Two such polymers coupled by
a small contact attraction of strength u are shown to form a bound
state at all temperatures T<Tc, the
roughening temperature of a single polymer. Comparing the
singularities of the localization length below Tc and
at Tc yields that d> is less than or
equal to 4.
-
reply to comments by T. Ala-Nissila and by J.M. Kim
by Michael Lässig and Harald Kinzelbach, Phys.Rev.Lett. 80, 889, (1998)
[Statistical
Physics Group]
[Many Body Physics Division]
[Institute for
Theoretical Physics]
12/1998