B. Estienne, N. Regnault, B. A. Bernevig
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 - 2/3 A ^ A ^ A in 3
dimensions). The Chern-Simons form appears in a certain combination of the
parallel transport and simple translation operator which is not an algebra. The
possible relation to D-dimensional volume preserving diffeomorphisms and
parallel transport of extended objects is also discussed.
View original:
http://arxiv.org/abs/1202.5543
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