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. For a link describing the experimental integer quantum Hall effect and its discoverers, cf. the 1985 Nobel Prize in Physics press release. 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.

The latest development is the result in [KMS] that the low lying energy bands of the effective Hamiltonian in the fractional quantum Hall effect, do not contribute to the Hall conductance, in the semiclassical limit as the electro-magnetic coupling constant goes to zero. This was previously known only for the integer quantum Hall effect. The crude physical explanation for this semiclassical vanishing theorem for the Hall conductance is as follows. The Hall conductance for the model operator vanishes, since it is the Hamiltonian of a crystal with perfectly isolated atoms, therefore there can be no current flowing through it - this remains valid for small perturbations of the model operator and therefore for the effective Hamiltonian.

A recent survey of the orbifold and noncommutative geometry approach to the fractional quantum Hall effect can be found in [MM2].

February 2008 - January 2016, Editor in charge of "Global Analysis, Noncommutative Geometry, and the Mathematics of String Theory", Proceedings of the American Mathematical Society.

I am Managing Editor of the J. Geometry and Physics special issue on String geometries, Dualities and Topological Matter,

I have co-organized multiple international IGA workshops on noncommutative geometry and the fractional quantum Hall effect at Adelaide.

References

[MT18] V. Mathai and G. C. Thiang,
Topological phases on the hyperbolic plane,
26 pp., [1712.02952]

[MT172] V. Mathai and G. C. Thiang,
Differential topology of semimetals,
Communications in Mathematical Physics,
355, no. 2,(2017) 561-602.

[MT171] V. Mathai and Guo Chuan Thiang,
Global topology of Weyl semimetals and Fermi arcs,
J. Physics A: Mathematical and Theoretical (Letter),
50 no. 11 (2017) 11LT01, 11pp [1607.02242]
publicity at JPhys+

[HMT17] K. Hannabuss, V. Mathai and Guo Chuan Thiang,
T-duality trivializes bulk-boundary correspondence: the noncommutative case,
Letters in Mathematical Physics,
(published online) 39 pp.

[HMT] K. Hannabuss, V. Mathai and Guo Chuan Thiang,
T-duality trivializes bulk-boundary correspondence: the parametrised case,
Advances in Theoretical and Mathematical Physics,
20 no. 5 (2016) 1193-1226, [1510.04785]

[MT3] V. Mathai and Guo Chuan Thiang,
T-duality trivializes bulk-boundary correspondence: some higher dimensional cases
Annales Henri Poincare,
17 no. 12 (2016) 3399--3424,, [1506.04492]

[MT2] V. Mathai and Guo Chuan Thiang,
T-duality trivializes bulk-boundary correspondence
Communications in Mathematical Physics,
345 no. 2, (2016) 675-701, [1505.05250]

[MT] V. Mathai and Guo Chuan Thiang,
T-duality and topological insulators
Journal of Physics A: Mathematical and Theoretical (Fast Track Communication)
48 (2015) no. 42, 42FT02, 10pp, [1503.01206]
publicity at IOPSCIENCE

[CHM2] A. Carey, K. Hannabuss and V. Mathai,
Quantum Hall Effect and Noncommutative Geometry,
Journal of Geometry and Symmetry in Physics,
6 (2006) 16-37, [math.OA/0008115]

[DMY] J. Dodziuk, V. Mathai and S. Yates,
Arithmetic properties of eigenvalues of generalized Harper operators on graphs,
Communications in Mathematical Physics,
262 no. 2 (2006) 269-297. [math.SP/0311315]

[MM2] M. Marcolli and V. Mathai,
Towards the fractional quantum Hall effect:
a noncommutative geometry perspective,
"Noncommutative Geometry and Number Theory".
Editors C.Consani, M. Marcolli, Aspects of Mathematics,
Vieweg Verlag, Wiesbaden, 2006, pages 235-261.
[cond-mat/0502356]

[KMS] Y. Kordyukov, V. Mathai and M. Shubin,
Equivalence of spectral projections in semiclassical limit
and a vanishing theorem for higher traces in K-theory,
J.Reine Angew.Math.(Crelle Journal)
581 (2005) 193 - 236. [math.DG/0305189]

[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)