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Von Neumann Lattices of Wannier Functions for Bloch Electrons in a Magnetic Field

Wilkinson, M. (1986) Von Neumann Lattices of Wannier Functions for Bloch Electrons in a Magnetic Field. Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences, 403 (1824). pp. 135-166. ISSN 1364-5021. doi:10.1098/rspa.1986.0006.

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The problem of Bloch electrons in a magnetic field in two dimensions can be reduced to a one-dimensional problem with a Hamiltonian H that is a periodic function of x and p. Wannier functions can be defined for the sub-bands of the spectrum of this effective Hamiltonian. When the Chern class (quantized Hall conductance integer) of the sub-band is zero, the Weyl-Wigner formalism can be used to represent these Wannier functions by a von Neumann lattice. It is shown how this von Neumann lattice of Wannier functions can be defined for irrational as well as rational magnetic fields. An important benefit from using the Weyl-Wigner formalism is that symmetries of the periodic potential are reflected by symmetries of the effective Hamiltonian in phase space. It is shown how the Wannier functions can be defined so that their Wigner functions have the point symmetries of the effective Hamiltonian. An example of how these results can prove useful is given: if we take matrix elements of the Hamiltonian between the Wannier states of a sub-band, we derive a new effective Hamiltonian describing this sub-band, which is again a periodic function of coordinate and momentum operators. Since, by projecting onto a sub-band, we have also reduced the number of degrees of freedom, this operation is a renormalization group transformation. It is shown that the symmetry of the new effective Hamiltonian in phase space is the same as that of the original one. This preservation of symmetry helps to explain some unusual properties of the spectrum when the Hamiltonian has fourfold symmetry.

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Additional Information:© 1986 The Royal Society. Received 24 June 1985. I wish to thank Professor B. Simon, Professor D. J. Thouless and Professor M. V. Berry, F.R.S., for useful discussions. Some of the results of section 3.3 overlap with a joint paper to be written by B. Simon and myself. A major part of this paper forms part of a Ph.D thesis of the University of Bristol, where I was supported by the S.E.R.C. I also wish to acknowledge the award of a Weingart Fellowship from the California Institute of Technology.
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Issue or Number:1824
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Official Citation:Von Neumann Lattices of Wannier Functions for Bloch Electrons in a Magnetic Field Author(s): M. Wilkinson Source: Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 403, No. 1824 (Jan. 8, 1986), pp. 135-166 Published by: The Royal Society Stable URL:
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ID Code:54194
Deposited On:29 Jan 2015 04:08
Last Modified:10 Nov 2021 20:30

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