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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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