Abstract
The conjecture is verified that the optimum, energy minimizing, magnetic flux for a half-filled band of electrons hopping on a planar, bipartite graph is π per square plaquette. We require only that the graph has periodicity in one direction and the result includes the hexagonal lattice (with flux 0 per hexagon) as a special case. The theorem goes beyond previous conjectures in several ways: (1) It does not assume, a priori, that all plaquettes have the same flux (as in Hofstadter’s model). (2) A Hubbard-type on-site interaction of any sign, as well as certain longer range interactions, can be included. (3) The conclusion holds for positive temperature as well as the ground state. (4) The results hold in D ≥ 2 dimensions if there is periodicity in D — 1 directions (e.g., the cubic lattice has the lowest energy if there is flux π in each square face).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Y. Hasegawa, P. Lederer, T. M. Rice, and P. B. Wiegmann, Phys. Rev. Lett. 63, 907 (1989).
G. Kotliar, Phys. Rev. B 37, 3664 (1988).
P.S. Rokhsar, Phys. Rev. B 42, 2526 (1990); Phys. Rev. Lett 65, 1506 (1990).
A. Barelli, J. Bellissard, and R. Rammal, J. Phys. (France) 51, 2167 (1990).
J. Bellissard and R. Rammal, J. Phys. (France) 51, 1803 (1990); 51, 2153 (1990); Europhys. Lett. (Switzerland) 13, 205 (1990).
P.R. Hofstadter, Phys. Rev. B 14, 2239 (1976).
P. G. Harper, Proc. Phys. Soc. London A 68, 874 (1955); 68, 879 (1955).
E. H. Lieb, HeIv. Phys. Acta 65, 247 (1992).
E. H. Lieb and M. Loss, Duke Math. J. 71, 337 (1993).
P.B. Wiegmann, Physica (Amsterdam) 153C, 103 (1988).
X. G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B 39, 11413 (1989).
See also F. Nori, B. Douçot, and R. Rammal, Phys. Rev. B 44, 7637 (1991); F. Nori, E. Abrahams, and G.T. Zimanyi, Phys. Rev. B 41, 7277 (1990).
J. Fröhlich, R. Israel, B. Simon, and E. Lieb, Commun. Math. Phys. 62, 1 (1978).
D. Brydges, J. Fröhlich, and E. Seiler, Ann. Phys. (N.Y.) 121, 227 (1979).
E. H. Lieb and B. Nachtergaele, The Stability of the Peierls Instability for Ring-Shaped Molecules (to be published).
T. Kennedy and E. H. Lieb, Physica (Amsterdam) 138A, 320 (1986).
E.H. Lieb, Phys. Rev. Lett 62, 1201 (1989); 62, 1927 (E) (1989).
E. H. Lieb, in Proceedings of 1993 Conference in honor of G. F. Dell'Antonio, "Advances in Dynamical Systems and Quantum Physics" (World Scientific, Singapore, to be published); in Proceedings of 1993 NATO ASW `The Physics and Mathematical Physics of the Hubbard Model" (Plenum, New York, to be published ).
Y. Meir, Y. liefen, and O. Entin-Wohlman, Phys. Rev. Lett. 63, 798 (1989).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lieb, E.H. (2004). Flux Phase of the Half-Filled Band. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds) Condensed Matter Physics and Exactly Soluble Models. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06390-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-06390-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-06093-9
Online ISBN: 978-3-662-06390-3
eBook Packages: Springer Book Archive