Speaker
Benoit Estienne
(Princeton University)
Description
In the quantum Hall effect, the density operators at
different wave-vectors generally do not commute and give
rise to the Girvin MacDonald Plazmann (GMP) algebra with
important consequences such as ground-state center of
mass degeneracy at fractional filling fraction, and W_{1 +
\infty} symmetry of the filled Landau levels. We show that
the natural generalization of the GMP algebra to higher
dimensional topological insulators involves the concept of a
D-algebra formed by using the fully anti-symmetric tensor
in D-dimensions. For insulators in even dimensional space,
the D-algebra is isotropic and closes for the case of constant
non-Abelian F(k) ^ F(k) ... ^ F(k) connection (D-Berry
curvature), and its structure factors are proportional to the
D/2-Chern number. In odd dimensions, the algebra is not
isotropic, contains the weak topological insulator index
(layers of the topological insulator in one less dimension)
and does not contain the Chern-Simons \theta form (F ^ A
- 1/3 A ^ A ^ A in 3 dimensions). The possible relation to D-
dimensional volume preserving diffeomorphisms and parallel
transport of extended objects is also discussed.
