Abstract
In quantum systems described by covariant families of 1-particle Schrödinger operators on half-spaces the pressure on the boundary per unit energy is topologically quantised if the Fermi energy lies in a gap of the bulk spectrum. Its relation with the integrated density of states can be expressed in an integrated version of Streda’s formula. This leads also to a gap labelling theorem for systems with constant magnetic field. The proof uses non-commutative topology.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bellissard, J.: K-theory of C*-algebras in solid state physics. In: Statistical Mechanics and Field Theory: Mathematical Aspects, Lecture Notes in Physics 257, edited by Dorlas, T., Hugenholtz, M., Winnink, M., Berlin: Springer-Verlag, 1986, pp. 99–156
Bellissard, J.: Gap labelling theorems for Schrödinger operators. In: From Number Theory to Physics, Berlin: Springer, 1993, pp. 538–630
Bellissard, J., van Elst, A., Schulz-Baldes, H.: The Non-Commutative Geometry of the Quantum Hall Effect. J. Math. Physics 35, 5373–5451 (1994)
Bellissard, J., Benedetti, R., Gambaudo, J.: Spaces of tilings, finite telescopic approximations and gap-labelling. To appear in Commun. Math. Phys.
Benameur, M., Oyono-Oyono, H.: Gap-labelling for quasi-crystals (proving a conjecture by Bellissard, J.). In: Proc. Operator-Algebras and Mathematical Physics (Constanta 2001), eds. Combes, J.M. et al., Bucharest: Theta Foundation 2003
Connes, A.: An analogue of the Thom isomorphism. Adv. in Math. 39, 31–55 (1981)
Connes, A.: Non-Commutative Geometry. San Diego: Acad. Press 1994
Elliott, G., Natsume, T., Nest, R.: Cyclic cohomology for one-parameter smooth crossed products. Acta Math. 160, 285–305 (1988)
Elliott, G., Natsume, T., Nest, R.: The Heisenberg group and K-theory. K-Theory 7, 409–428 (1993)
Kaminker, J., Putnam, I.F.: A proof of the Gap Labeling Conjecture. Michigan J. Math. 51, 537–547 (2003)
Kellendonk, J., Schulz-Baldes, H.: Quantization of edge currents for continuous magnetic operators. J. Funct. Anal. 209, 388–413 (2004)
Kellendonk, J., Schulz-Baldes, H.: Boundary maps for C*-crossed products with ℝ with an application to the Quantum Hall Effect. Commun. Math. Phys. 249, 611–637 (2004)
Kellendonk, J.: Topological quantization of boundary forces and the integrated density of states. J. Phys. A. 37, L161–L166 (2004)
Kellendonk, J., Zois, I.P.: Rotation numbers, Gap Labelling and Boundary forces. http://www.ma.utexas.edu/mp_arc/c/04/04-217.pdf, 2004. J. Phys. A: Math. Gen. 38, 3937–3946 (2005)
Pastur, L.: Spectral properties of disordered systems in the one-body approximation. Commun. Math. Phys. 75, 179–196 (1980)
Rieffel, M. A.: Connes’ analogue for crossed products of the Thom isomorphism. Contemp. Math. 10, 143–154 (1982)
Streda, P.: Theory of quantised Hall conductivity in two dimensions. J. Phys. C. 15, L717–L721 (1982)
Author information
Authors and Affiliations
Additional information
Communicated by A. Connes
Rights and permissions
About this article
Cite this article
Kellendonk, J. Gap Labelling and the Pressure on the Boundary. Commun. Math. Phys. 258, 751–768 (2005). https://doi.org/10.1007/s00220-005-1338-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-005-1338-1