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SUMMARY:Energy levels and wave functions of Bloch electrons in rational an
 d irrational magnetic fields - Pascal Bugnion (TCM\, Physics\, Cambridge)
DTSTART:20120309T143000Z
DTEND:20120309T150000Z
UID:TALK35987@talks.cam.ac.uk
CONTACT:Daniel Cole
DESCRIPTION:"D. R. Hofstadter Phys. Rev. B 14\, 2239 (1976) ":http://prb.a
 ps.org/abstract/PRB/v14/i6/p2239_1\n\nAn effective single-band Hamiltonian
  representing a crystal electron in a uniform magnetic field is constructe
 d from the tight-binding form of a Bloch band by replacing ℏk⃗ by the 
 operator p⃗-eA⃗/c. The resultant Schrödinger equation becomes a finit
 e-difference equation whose eigenvalues can be computed by a matrix method
 . The magnetic flux which passes through a lattice cell\, divided by a flu
 x quantum\, yields a dimensionless parameter whose rationality or irration
 ality highly influences the nature of the computed spectrum. The graph of 
 the spectrum over a wide range of "rational" fields is plotted. A recursiv
 e structure is discovered in the graph\, which enables a number of theorem
 s to be proven\, bearing particularly on the question of continuity. The r
 ecursive structure is not unlike that predicted by Azbel'\, using a contin
 ued fraction for the dimensionless parameter. An iterative algorithm for d
 eriving the clustering pattern of the magnetic subbands is given\, which f
 ollows from the recursive structure. From this algorithm\, the nature of t
 he spectrum at an "irrational" field can be deduced\; it is seen to be an 
 uncountable but measure-zero set of points (a Cantor set). Despite these-f
 eatures\, it is shown that the graph is continuous as the magnetic field v
 aries. It is also shown how a spectrum with simplified properties can be d
 erived from the rigorously derived spectrum\, by introducing a spread in t
 he field values. This spectrum satisfies all the intuitively desirable pro
 perties of a spectrum. The spectrum here presented is shown to agree with 
 that predicted by A. Rauh in a completely different model for crystal elec
 trons in a magnetic field. A new type of magnetic "superlattice" is introd
 uced\, constructed so that its unit cell intercepts precisely one quantum 
 of flux. It is shown that this cell represents the periodicity of solution
 s of the difference equation. It is also shown how this superlattice allow
 s the determination of the wave function at nonlattice sites. Evidence is 
 offered that the wave functions belonging to irrational fields are everywh
 ere defined and are continuous in this model\, whereas those belonging to 
 rational fields are only defined on a discrete set of points. A method for
  investigating these predictions experimentally is sketched.
LOCATION:TCM Seminar Room\, Cavendish Laboratory
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