30 research outputs found
The Loewner equation: maps and shapes
In the last few years, new insights have permitted unexpected progress in the
study of fractal shapes in two dimensions. A new approach, called
Schramm-Loewner evolution, or SLE, has arisen through analytic function theory
and probability theory, and given a new way of calculating fractal shapes in
critical phenomena, the theory of random walks, and of percolation. We present
a non-technical discussion of this development aimed to attract the attention
of condensed matter community to this fascinating subject
Finite Size Effects and Irrelevant Corrections to Scaling near the Integer Quantum Hall Transition
We present a numerical finite size scaling study of the localization length
in long cylinders near the integer quantum Hall transition (IQHT) employing the
Chalker-Coddington network model. Corrections to scaling that decay slowly with
increasing system size make this analysis a very challenging numerical problem.
In this work we develop a novel method of stability analysis that allows for a
better estimate of error bars. Applying the new method we find consistent
results when keeping second (or higher) order terms of the leading irrelevant
scaling field. The knowledge of the associated (negative) irrelevant exponent
is crucial for a precise determination of other critical exponents,
including multifractal spectra of wave functions. We estimate ,
which is considerably larger than most recently reported values. Within this
approach we obtain the localization length exponent confirming
recent results. Our stability analysis has broad applicability to other
observables at IQHT, as well as other critical points where corrections to
scaling are present.Comment: 6 pages and 3 figures, plus supplemental material
Conformal invariance and multifractality at Anderson transitions in arbitrary dimensions
Electronic wave functions at Anderson transitions exhibit multifractal
scaling characterized by a continuum of generalized multifractal exponents
with vector indices . In a field
theory description of the transitions, there are corresponding multifractal
operators with scaling dimensions .
Assuming conformal invariance and using the conformal bootstrap framework, we
derive a constraint that implies that the generalized multifractal spectrum
must be quadratic in all in any dimension . As
several numerical studies have shown deviations from parabolicity, we argue
that conformal invariance is likely absent at Anderson transitions in
dimensions
Wave function correlations and the AC conductivity of disordered wires beyond the Mott-Berezinskii law
In one-dimensional disordered wires electronic states are localized at any
energy. Correlations of the states at close positive energies and the AC
conductivity in the limit of small frequency are described by
the Mott-Berezinskii theory. We revisit the instanton approach to the
statistics of wave functions and AC transport valid in the tails of the
spectrum (large negative energies). Applying our recent results on functional
determinants, we calculate exactly the integral over gaussian fluctuations
around the exact two-instanton saddle point. We derive correlators of wave
functions at different energies beyond the leading order in the energy
difference. This allows us to calculate corrections to the Mott-Berezinskii law
(the leading small frequency asymptotic behavior of ) which
approximate the exact result in a broad range of . We compare our
results with the ones obtained for positive energies.Comment: 7 pages, 3 figure
Statistics of Conductances and Subleading Corrections to Scaling near the Integer Quantum Hall Plateau Transition
We study the critical behavior near the integer quantum Hall plateau
transition by focusing on the multifractal (MF) exponents describing the
scaling of the disorder-average moments of the point contact conductance
between two points of the sample, within the Chalker-Coddington network model.
Past analytical work has related the exponents to the MF exponents
of the local density of states (LDOS). To verify this relation, we
numerically determine the exponents with high accuracy. We thereby
provide, at the same time, independent numerical results for the MF exponents
for the LDOS. The presence of subleading corrections to scaling
makes such determination directly from scaling of the moments of virtually
impossible. We overcome this difficulty by using two recent advances. First, we
construct pure scaling operators for the moments of which have precisely
the same leading scaling behavior, but no subleading contributions. Secondly,
we take into account corrections to scaling from irrelevant (in the
renormalization group sense) scaling fields by employing a numerical technique
("stability map") recently developed by us. We thereby numerically confirm the
relation between the two sets of exponents, (point contact conductances)
and (LDOS), and also determine the leading irrelevant (corrections
to scaling) exponent as well as other subleading exponents. Our results
suggest a way to access multifractality in an experimental setting.Comment: 7 pages and 4 figures, plus Supplemental materia
Metal-insulator transition in a 2D system of chiral unitary class
We perform a numerical investigation of Anderson metal-insulator transition
(MIT) in a twodimensional system of chiral symmetry class AIII by combining
finite-size scaling, transport, density of states, and multifractality studies.
The results are in agreement with the sigma-model renormalization-group theory,
where MIT is driven by proliferation of vortices. We determine the phase
diagram and find an apparent non-universality of several parameters on the
critical line of MIT, which is consistent with the analytically predicted slow
renormalization towards the ultimate fixed point of the MIT. The
localization-length exponent is estimated as .Comment: 11 pages, 10 figure
Generalized multifractality at metal-insulator transitions and in metallic phases of 2D disordered systems
We study generalized multifractality characterizing fluctuations and
correlations of eigenstates in disordered systems of symmetry classes AII, D,
and DIII. Both metallic phases and Andersonlocalization transitions are
considered. By using the non-linear sigma-model approach, we construct
pure-scaling eigenfunction observables. The construction is verified by
numerical simulations of appropriate microscopic models, which also yield
numerical values of the corresponding exponents. In the metallic phases, the
numerically obtained exponents satisfy Weyl symmetry relations as well as
generalized parabolicity (proportionality to eigenvalues of the quadratic
Casimir operator). At the same time, the generalized parabolicity is strongly
violated at critical points of metal-insulator transitions, signalling
violation of local conformal invariance. Moreover, in classes D and DIII, even
the Weyl symmetry breaks down at critical points of metal-insulator
transitions. This last feature is related with a peculiarity of the sigma-model
manifolds in these symmetry classes: they consist of two disjoint components.
Domain walls associated with these additional degrees of freedom are crucial
for ensuring Anderson localization and, at the same time, lead to the violation
of the Weyl symmetry.Comment: 36 pages, 14 figure