5,473 research outputs found
Dihedral symmetries of multiple logarithms
This paper finds relationships between multiple logarithms with a dihedral
group action on the arguments. I generalize the combinatorics developed in
Gangl, Goncharov and Levin's R-deco polygon representation of multiple
logarithms to find these relations. By writing multiple logarithms as iterated
integrals, my arguments are valid for iterated integrals as over an arbitrary
field
Killing the Hofstadter butterfly, one bond at a time
Electronic bands in a square lattice when subjected to a perpendicular
magnetic field form the Hofstadter butterfly pattern. We study the evolution of
this pattern as a function of bond percolation disorder (removal or dilution of
lattice bonds). With increasing concentration of the bonds removed, the
butterfly pattern gets smoothly decimated. However, in this process of
decimation, bands develop interesting characteristics and features. For
example, in the high disorder limit, some butterfly-like pattern still persists
even as most of the states are localized. We also analyze, in the low disorder
limit, the effect of percolation on wavefunctions (using inverse participation
ratios) and on band gaps in the spectrum. We explain and provide the reasons
behind many of the key features in our results by analyzing small clusters and
finite size rings. Furthermore, we study the effect of bond dilution on
transverse conductivity(). We show that starting from the clean
limit, increasing disorder reduces to zero, even though the
strength of percolation is smaller than the classical percolation threshold.
This shows that the system undergoes a direct transition from a integer quantum
Hall state to a localized Anderson insulator beyond a critical value of bond
dilution. We further find that the energy bands close to the band edge are more
stable to disorder than at the band center. To arrive at these results we use
the coupling matrix approach to calculate Chern numbers for disordered systems.
We point out the relevance of these results to signatures in
magneto-oscillations.Comment: minor typos fixe
Effects of local periodic driving on transport and generation of bound states
We periodically kick a local region in a one-dimensional lattice and
demonstrate, by studying wave packet dynamics, that the strength and the time
period of the kicking can be used as tuning parameters to control the
transmission probability across the region. Interestingly, we can tune the
transmission to zero which is otherwise impossible to do in a time-independent
system. We adapt the non-equilibrium Green's function method to take into
account the effects of periodic driving; the results obtained by this method
agree with those found by wave packet dynamics if the time period is small. We
discover that Floquet bound states can exist in certain ranges of parameters;
when the driving frequency is decreased, these states get delocalized and turn
into resonances by mixing with the Floquet bulk states. We extend these results
to incorporate the effects of local interactions at the driven site, and we
find some interesting features in the transmission and the bound states.Comment: 14 pages, 12 figures; added several references and corrected some
typo
Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees
This paper defines a generalization of the Connes-Moscovici Hopf algebra,
that contains the entire Hopf algebra of rooted trees. A
relationship between the former, a much studied object in non-commutative
geometry, and the later, a much studied object in perturbative Quantum Field
Theory, has been established by Connes and Kreimer. The results of this paper
open the door to study the cohomology of the Hopf algebra of rooted trees
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