25 research outputs found
Applicability of layered sine-Gordon models to layered superconductors: II. The case of magnetic coupling
In this paper, we propose a quantum field theoretical renormalization group
approach to the vortex dynamics of magnetically coupled layered
superconductors, to supplement our earlier investigations on the
Josephson-coupled case. We construct a two-dimensional multi-layer sine-Gordon
type model which we map onto a gas of topological excitations. With a special
choice of the mass matrix for our field theoretical model, vortex dominated
properties of magnetically coupled layered superconductors can be described.
The well known interaction potentials of fractional flux vortices are
consistently obtained from our field-theoretical analysis, and the physical
parameters (vortex fugacity and temperature parameter) are also identified. We
analyse the phase structure of the multi-layer sine--Gordon model by a
differential renormalization group method for the magnetically coupled case
from first principles. The dependence of the transition temperature on the
number of layers is found to be in agreement with known results based on other
methods.Comment: 7 pages, 1 figure, published in J. Phys.: Condens. Matte
Functional renormalization group with a compactly supported smooth regulator function
The functional renormalization group equation with a compactly supported
smooth (CSS) regulator function is considered. It is demonstrated that in an
appropriate limit the CSS regulator recovers the optimized one and it has
derivatives of all orders. The more generalized form of the CSS regulator is
shown to reduce to all major type of regulator functions (exponential,
power-law) in appropriate limits. The CSS regulator function is tested by
studying the critical behavior of the bosonized two-dimensional quantum
electrodynamics in the local potential approximation and the sine-Gordon scalar
theory for d<2 dimensions beyond the local potential approximation. It is shown
that a similar smoothing problem in nuclear physics has already been solved by
introducing the so called Salamon-Vertse potential which can be related to the
CSS regulator.Comment: JHEP style, 11 pages, 2 figures, proofs corrected, accepted for
publication by JHE
Some results and problems for anisotropic random walks on the plane
This is an expository paper on the asymptotic results concerning path
behaviour of the anisotropic random walk on the two-dimensional square lattice
Z^2. In recent years Mikl\'os and the authors of the present paper investigated
the properties of this random walk concerning strong approximations, local
times and range. We give a survey of these results together with some further
problems.Comment: 20 page
A central limit theorem for time-dependent dynamical systems
The work [8] established memory loss in the time-dependent (non-random) case
of uniformly expanding maps of the interval. Here we find conditions under
which we have convergence to the normal distribution of the appropriately
scaled Birkhoff-like partial sums of appropriate test functions. A substantial
part of the problem is to ensure that the variances of the partial sums tend to
infinity (cf. the zero-cohomology condition in the autonomous case). In fact,
the present paper is the first one where non-random, i. e. specific examples
are also found, which are not small perturbations of a given map. Our approach
uses martingale approximation technique in the form of [9]
Truncation Effects in the Functional Renormalization Group Study of Spontaneous Symmetry Breaking
We study the occurrence of spontaneous symmetry breaking (SSB) for O (N) models using functional renormalization group techniques. We show that even the local potential approximation (LPA) when treated exactly is sufficient to give qualitatively correct results for systems with continuous symmetry, in agreement with the Mermin-Wagner theorem and its extension to systems with fractional dimensions. For general N (including the Ising model N = 1) we study the solutions of the LPA equations for various truncations around the zero field using a finite number of terms (and different regulators), showing that SSB always occurs even where it should not. The SSB is signalled by Wilson-Fisher fixed points which for any truncation are shown to stay on the line defined by vanishing mass beta functions
Lectures on the functional renormalization group method
These introductory notes are about functional renormalization group equations
and some of their applications. It is emphasised that the applicability of this
method extends well beyond critical systems, it actually provides us a general
purpose algorithm to solve strongly coupled quantum field theories. The
renormalization group equation of F. Wegner and A. Houghton is shown to resum
the loop-expansion. Another version, due to J. Polchinski, is obtained by the
method of collective coordinates and can be used for the resummation of the
perturbation series. The genuinely non-perturbative evolution equation is
obtained in a manner reminiscent of the Schwinger-Dyson equations. Two variants
of this scheme are presented where the scale which determines the order of the
successive elimination of the modes is extracted from external and internal
spaces. The renormalization of composite operators is discussed briefly as an
alternative way to arrive at the renormalization group equation. The scaling
laws and fixed points are considered from local and global points of view.
Instability induced renormalization and new scaling laws are shown to occur in
the symmetry broken phase of the scalar theory. The flattening of the effective
potential of a compact variable is demonstrated in case of the sine-Gordon
model. Finally, a manifestly gauge invariant evolution equation is given for
QED.Comment: 47 pages, 11 figures, final versio