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On subgroups in division rings of type
Let be a division ring with center . We say that is a {\em
division ring of type } if for every two elements the division
subring is a finite dimensional vector space over . In this paper
we investigate multiplicative subgroups in such a ring.Comment: 10 pages, 0 figure
Weak localization of bulk channels in topological insulator thin film
Weak antilocalization (WAL) is expected whenever strong spin-orbit coupling
or scattering comes into play. Spin-orbit coupling in the bulk states of a
topological insulator is very strong, enough to result in the topological phase
transition. However, the recently observed WAL in topological insulators seems
to have an ambiguous origin from the bulk states. Starting from the effective
model for three-dimensional topological insulators, we find that the lowest
two-dimensional (2D) bulk subbands of a topological insulator thin film can be
described by the modified massive Dirac model. We derive the
magnetoconductivity formula for both the 2D bulk subbands and surface bands.
Because with Relatively large gap, the 2D bulk subbands may lie in the regimes
where the unitary behavior or even weak localization (WL) is also expected,
instead of always WAL. As a result, the bulk states may contribute small
magnetoconductivity or even compensate the WAL from the surface states.
Inflection in magnetoconductivity curves may appear when the bulk WL channels
outnumber the surface WAL channels, providing a signature of the weak
localization from the bulk states.Comment: 9 pages, 5 figure
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