For a link describing the experimental fractional quantum Hall effect and its discoverers, cf. the 1998 Nobel Prize in Physics press release. In the paper [MM] below, we propose a noncommutative geometry model on the hyperbolic plane for the fractional quantum Hall effect, extending earlier work done in [CHMM] and also building on fundamental work mainly by Bellissard and collaborators, who established a noncommutative geometry model on the Euclidean plane for the integer quantum Hall effect. The hyperbolic metric plays the role of the effective interaction of the charge carriers in the Hamiltonian. The Hall conductance is derived to be a cyclic 2-cocycle on the algebra of observables and its expression resembles a generalized Kubo formula. Under the assumption that the Fermi level is in a spectral gap of the Hamiltonian, we establish that the Hall conductance is an integer mutiple of orbifold Euler characteristics of cocompact Fuchsian groups, and is therefore topological in character as well as fractional valued. As a consequence, the Hall conductance is stable under small deformations of the Hamiltonian.Thus this model can be generalized to systems with disorder as in [CHM], where the hypothesis that the Fermi level is in a spectral gap of the Hamiltonian can be relaxed to the assumption that it is in a gap of extended states, which is a necessary step in order to establish the presence of Hall plateaux. A table below compares the experimentally observed fractions of the Hall conductance, to the fractions arising from our model, whereas another table below lists some new fractional Hall conductances predicted from our model. For another interesting link on the role of hyperbolic geometry in describing structure and properties of crystalline materials, cf. some current research at ANU.
[MM] M. Marcolli and V. Mathai,
Twisted index theory on good orbifolds, II: fractional quantum numbers,
Communications in Mathematical Physics,
217 (2001) 55-87.
[CHM] A. Carey, K. Hannabuss and V. Mathai,
Quantum Hall Effect on the Hyperbolic Plane in the presence of disorder,
Letters in Mathematical Physics,
47 (1999) 215-236.
[CHMM] A. Carey, K. Hannabuss, V. Mathai and P. McCann,
Quantum Hall Effect on the Hyperbolic Plane,
Communications in Mathematical Physics,
190 (1998) 629-673.
The table below compares the experimentally observed fractions
of the Hall conductance, to the fractions arising from our model.
| Experimental values for the
Hall conductance |
Fractions realized as orbifold Euler characteristics
of orbifolds with g = 1 or g = 0 in their signature |
| 5/3 | S(1;6,6) |
| 4/3 | S(1;3,3) |
| 7/5 | S(0;5,5,10,10) |
| 4/5 | S(1;5) |
| 5/7 | S(0;7,14,14) |
| 2/3 | S(1;3) |
| 3/5 | S(0;5,10,10) |
| 4/7 | S(0;7,7,7) |
| 5/9 | ??? |
| 4/9 | S(0;3,9,9) |
| 3/7 | ??? |
| 2/5 | S(0;5,5,5) |
| 1/3 | S(0;3,6,6) |
| 5/2 | S(1;6,6,6) |
The table below lists some new fractional Hall conductances predicted
from our model.
| Some fractions for the Hall conductance
predicted by our model |
These fractions realized as orbifold Euler characteristics
of orbifolds with g = 1 or g = 0 in their signature |
| 4/3 | S(0;3,3,3,3,3), S(1;3,3) |
| 2/3 | S(0;2,2,2,2,3), S(1;3) |
| 4/7 | S(0;7,7,7) |
| 1/2 | S(0;4,8,8), S(1;2) |
| 4/9 | S(0;3,9,9) |
| 2/5 | S(0;5,5,5) |
| 1/3 | S(0;4,4,6), S(0;2,2,2,6) |
| 1/4 | S(0;2,8,8) S(0;4,4,4) |
| 1/5 | S(0;2,5,10) |
| 4/21 | S(0;3,7,7) |
| 1/6 | S(0;2,4,12), S(0;3,3,6) |
| 1/8 | S(0;2,4,8) |
| 1/12 | S(0;2,4,6), S(0;3,3,4) |
| 1/24 | S(0;2,3,8) |
| 1/42 | S(0;2,3,7) |