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CATEGORIES:TCM Journal Club
SUMMARY:Energy levels and wave functions of Bloch electron
 s in rational and irrational magnetic fields - Pas
 cal Bugnion (TCM\, Physics\, Cambridge)
DTSTART;TZID=Europe/London:20120309T143000
DTEND;TZID=Europe/London:20120309T150000
UID:TALK35987AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/35987
DESCRIPTION:"D. R. Hofstadter Phys. Rev. B 14\, 2239 (1976) ":
 http://prb.aps.org/abstract/PRB/v14/i6/p2239_1\n\n
 An effective single-band Hamiltonian representing 
 a crystal electron in a uniform magnetic field is 
 constructed from the tight-binding form of a Bloch
  band by replacing ℏk⃗ by the operator p⃗-eA⃗/c. T
 he resultant Schrödinger equation becomes a finite
 -difference equation whose eigenvalues can be comp
 uted by a matrix method. The magnetic flux which p
 asses through a lattice cell\, divided by a flux q
 uantum\, yields a dimensionless parameter whose ra
 tionality or irrationality highly influences the n
 ature of the computed spectrum. The graph of the s
 pectrum over a wide range of "rational" fields is 
 plotted. A recursive structure is discovered in th
 e graph\, which enables a number of theorems to be
  proven\, bearing particularly on the question of 
 continuity. The recursive structure is not unlike 
 that predicted by Azbel'\, using a continued fract
 ion for the dimensionless parameter. An iterative 
 algorithm for deriving the clustering pattern of t
 he magnetic subbands is given\, which follows from
  the recursive structure. From this algorithm\, th
 e nature of the spectrum at an "irrational" field 
 can be deduced\; it is seen to be an uncountable b
 ut measure-zero set of points (a Cantor set). Desp
 ite these-features\, it is shown that the graph is
  continuous as the magnetic field varies. It is al
 so shown how a spectrum with simplified properties
  can be derived from the rigorously derived spectr
 um\, by introducing a spread in the field values. 
 This spectrum satisfies all the intuitively desira
 ble properties of a spectrum. The spectrum here pr
 esented is shown to agree with that predicted by A
 . Rauh in a completely different model for crystal
  electrons in a magnetic field. A new type of magn
 etic "superlattice" is introduced\, constructed so
  that its unit cell intercepts precisely one quant
 um of flux. It is shown that this cell represents 
 the periodicity of solutions of the difference equ
 ation. It is also shown how this superlattice allo
 ws the determination of the wave function at nonla
 ttice sites. Evidence is offered that the wave fun
 ctions belonging to irrational fields are everywhe
 re defined and are continuous in this model\, wher
 eas those belonging to rational fields are only de
 fined on a discrete set of points. A method for in
 vestigating these predictions experimentally is sk
 etched.
LOCATION:TCM Seminar Room\, Cavendish Laboratory
CONTACT:Daniel Cole
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