2,186 research outputs found
Convergence of multiple ergodic averages along cubes for several commuting transformations
We prove the norm convergence of multiple ergodic averages along cubes for
several commuting transformations, and derive corresponding combinatorial
results. The method we use relies primarily on the "magic extension"
established recently by B. ~Host
Multiple recurrence for two commuting transformations
This paper is devoted to a study of the multiple recurrence of two commuting
transformations. We derive a result which is similar but not identical to that
of one single transformation established by Bergelson, Host and Kra. We will
use the machinery of "magic systems" established recently by B. Host for the
proof
Convergence of weighted polynomial multiple ergodic averages
We study here weighted polynomial multiple ergodic averages. A sequence of
weights is called universally good if any polynomial multiple ergodic average
with this sequence of weights converges in . We find a necessary
condition and show that for any bounded measurable function on an
ergodic system, the sequence is universally good for almost
every . The linear case was understood by Host and Kra
Lower bound in the Roth theorem for amenable groups
Let , be two commuting probability measure-preserving actions of a
countable amenable group such that the group spanned by these actions acts
ergodically. We show that on a syndetic set for any measurable set and any
. The proof uses the concept of a sated system introduced by
Austin.Comment: v2: added a counterexample showing that the exponent in the main
result cannot be improved to 3. To appear in Ergodic Theory and Dynamical
System
Surface-edge state and half quantized Hall conductance in topological insulators
We propose a surface-edge state theory for half quantized Hall conductance of
surface states in topological insulators. The gap opening of a single Dirac
cone for the surface states in a weak magnetic field is demonstrated. We find a
new surface state resides on the surface edges and carries chiral edge current,
resulting in a half-quantized Hall conductance in a four-terminal setup. We
also give a physical interpretation of the half quantized conductance by
showing that this state is the product of splitting of a boundary bound state
of massive Dirac fermions which carries a conductance quantum
Quantum percolation in quantum spin Hall antidot systems
We study the influences of antidot-induced bound states on transport
properties of two- dimensional quantum spin Hall insulators. The bound
statesare found able to induce quantum percolation in the originally insulating
bulk. At some critical antidot densities, the quantum spin Hall phase can be
completely destroyed due to the maximum quantum percolation. For systems with
periodic boundaries, the maximum quantum percolationbetween the bound states
creates intermediate extended states in the bulk which is originally gapped and
insulating. The antidot in- duced bound states plays the same role as the
magnetic field inthe quantum Hall effect, both makes electrons go into
cyclotron motions. We also draw an analogy between the quantum percolation
phenomena in this system and that in the network models of quantum Hall effect
Ergodic averages of commuting transformations with distinct degree polynomial iterates
We prove mean convergence, as , for the multiple ergodic averages
,
where are integer polynomials with distinct degrees, and
are commuting, invertible measure preserving transformations,
acting on the same probability space. This establishes several cases of a
conjecture of Bergelson and Leibman, that complement the case of linear
polynomials, recently established by Tao. Furthermore, we show that, unlike the
case of linear polynomials, for polynomials of distinct degrees, the
corresponding characteristic factors are mixtures of inverse limits of
nilsystems. We use this particular structure, together with some
equidistribution results on nilmanifolds, to give an application to multiple
recurrence and a corresponding one to combinatorics.Comment: 44 pages, small correction in the proof of Lemma 7.5, appeared in the
Proceedings of the London Mathematical Societ
Local diffusion theory of localized waves in open media
We report a first-principles study of static transport of localized waves in
quasi-one-dimensional open media. We found that such transport, dominated by
disorder-induced resonant transmissions, displays novel diffusive behavior. Our
analytical predictions are entirely confirmed by numerical simulations. We
showed that the prevailing self-consistent localization theory [van Tiggelen,
{\it et. al.}, Phys. Rev. Lett. \textbf{84}, 4333 (2000)] is valid only if
disorder-induced resonant transmissions are negligible. Our findings open a new
direction in the study of Anderson localization in open media.Comment: 4 pages, 2 figure
Electric field modulation of topological order in thin film semiconductors
We propose a method that can consecutively modulate the topological orders or
the number of helical edge states in ultrathin film semiconductors without a
magnetic field. By applying a staggered periodic potential, the system
undergoes a transition from a topological trivial insulating state into a
non-trivial one with helical edge states emerging in the band gap. Further
study demonstrates that the number of helical edge state can be modulated by
the amplitude and the geometry of the electric potential in a step-wise
fashion, which is analogous to tuning the integer quantum Hall conductance by a
megntic field. We address the feasibility of experimental measurement of this
topological transition.Comment: 4 pages, 4 figure
Surface and Edge States in Topological Semi-metals
We study the topologically non-trivial semi-metals by means of the 6-band
Kane model. Existence of surface states is explicitly demonstrated by
calculating the LDOS on the material surface. In the strain free condition,
surface states are divided into two parts in the energy spectrum, one part is
in the direct gap, the other part including the crossing point of surface state
Dirac cone is submerged in the valence band. We also show how uni-axial strain
induces an insulating band gap and raises the crossing point from the valence
band into the band gap, making the system a true topological insulator. We
predict existence of helical edge states and spin Hall effect in the thin film
topological semi-metals, which could be tested with future experiment. Disorder
is found to significantly enhance the spin Hall effect in the valence band of
the thin films
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