2,283 research outputs found
On quantum mechanics with a magnetic field on R^n and on a torus T^n, and their relation
We show in elementary terms the equivalence in a general gauge of a
U(1)-gauge theory of a scalar charged particle on a torus T^n = R^n/L to the
analogous theory on R^n constrained by quasiperiodicity under translations in
the lattice L. The latter theory provides a global description of the former:
the quasiperiodic wavefunctions defined on R^n play the role of sections of the
associated hermitean line bundle E on T^n, since also E admits a global
description as a quotient. The components of the covariant derivatives
corresponding to a constant (necessarily integral) magnetic field B = dA
generate a Lie algebra g_Q and together with the periodic functions the algebra
of observables O_Q . The non-abelian part of g_Q is a Heisenberg Lie algebra
with the electric charge operator Q as the central generator; the corresponding
Lie group G_Q acts on the Hilbert space as the translation group up to phase
factors. Also the space of sections of E is mapped into itself by g in G_Q . We
identify the socalled magnetic translation group as a subgroup of the
observables' group Y_Q . We determine the unitary irreducible representations
of O_Q, Y_Q corresponding to integer charges and for each of them an associated
orthonormal basis explicitly in configuration space. We also clarify how in the
n = 2m case a holomorphic structure and Theta functions arise on the associated
complex torus. These results apply equally well to the physics of charged
scalar particles on R^n and on T^n in the presence of periodic magnetic field B
and scalar potential. They are also necessary preliminary steps for the
application to these theories of the deformation procedure induced by Drinfel'd
twists.Comment: Latex2e file, 22 pages. Final version appeared in IJT
Hofstadter Problem on the Honeycomb and Triangular Lattices: Bethe Ansatz Solution
We consider Bloch electrons on the honeycomb lattice under a uniform magnetic
field with flux per cell. It is shown that the problem factorizes
to two triangular lattices. Treating magnetic translations as Heisenberg-Weyl
group and by the use of its irreducible representation on the space of theta
functions, we find a nested set of Bethe equations, which determine the
eigenstates and energy spectrum. The Bethe equations have simple form which
allows to consider them further in the limit by the technique
of Thermodynamic Bethe Ansatz and analyze Hofstadter problem for the irrational
flux.Comment: 7 pages, 2 figures, Revte
Diffractive energy spreading and its semiclassical limit
We consider driven systems where the driving induces jumps in energy space:
(1) particles pulsed by a step potential; (2) particles in a box with a moving
wall; (3) particles in a ring driven by an electro-motive-force. In all these
cases the route towards quantum-classical correspondence is highly non-trivial.
Some insight is gained by observing that the dynamics in energy space, where
is the level index, is essentially the same as that of Bloch electrons in a
tight binding model, where is the site index. The mean level spacing is
like a constant electric field and the driving induces long range hopping
1/(n-m).Comment: 19 pages, 11 figs, published version with some improved figure
Simultaneous measurement of coordinate and momentum on a von Neumann lattice
It is shown that on a finite phase plane the -coordinates and the sites
of a von Neumann lattice are conjugate to one another. This elementary result
holds when the number defining the size of the phase plane can be expressed
as a product, , with and being relatively prime.
As a consequence of this result a hitherto unknown wave function is defined
giving the probability of simultaneously measuring the momentum and coordinate
on the von Neumann lattice.Comment: Published in EPL 83 (2008) 1000
Theory of a magnetic microscope with nanometer resolution
We propose a theory for a type of apertureless scanning near field microscopy
that is intended to allow the measurement of magnetism on a nanometer length
scale. A scanning probe, for example a scanning tunneling microscope (STM) tip,
is used to scan a magnetic substrate while a laser is focused on it. The
electric field between the tip and substrate is enhanced in such a way that the
circular polarization due to the Kerr effect, which is normally of order 0.1%
is increased by up to two orders of magnitude for the case of a Ag or W tip and
an Fe sample. Apart from this there is a large background of circular
polarization which is non-magnetic in origin. This circular polarization is
produced by light scattered from the STM tip and substrate. A detailed retarded
calculation for this light-in-light-out experiment is presented.Comment: 17 pages, 8 figure
Algebraic Geometry Approach to the Bethe Equation for Hofstadter Type Models
We study the diagonalization problem of certain Hofstadter-type models
through the algebraic Bethe ansatz equation by the algebraic geometry method.
When the spectral variables lie on a rational curve, we obtain the complete and
explicit solutions for models with the rational magnetic flux, and discuss the
Bethe equation of their thermodynamic flux limit. The algebraic geometry
properties of the Bethe equation on high genus algebraic curves are
investigated in cooperationComment: 28 pages, Latex ; Some improvement of presentations, Revised version
with minor changes for journal publicatio
Factorizations and Physical Representations
A Hilbert space in M dimensions is shown explicitly to accommodate
representations that reflect the prime numbers decomposition of M.
Representations that exhibit the factorization of M into two relatively prime
numbers: the kq representation (J. Zak, Phys. Today, {\bf 23} (2), 51 (1970)),
and related representations termed representations (together with
their conjugates) are analysed, as well as a representation that exhibits the
complete factorization of M. In this latter representation each quantum number
varies in a subspace that is associated with one of the prime numbers that make
up M
Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects
We present a unified theory for wave-packet dynamics of electrons in crystals
subject to perturbations varying slowly in space and time. We derive the
wave-packet energy up to the first order gradient correction and obtain all
kinds of Berry-phase terms for the semiclassical dynamics and the quantization
rule. For electromagnetic perturbations, we recover the orbital magnetization
energy and the anomalous velocity purely within a single-band picture without
invoking inter-band couplings. For deformations in crystals, besides a
deformation potential, we obtain a Berry-phase term in the Lagrangian due to
lattice tracking, which gives rise to new terms in the expressions for the
wave-packet velocity and the semiclassical force. For multiple-valued
displacement fields surrounding dislocations, this term manifests as a Berry
phase, which we show to be proportional to the Burgers vector around each
dislocation.Comment: 12 pages, RevTe
Quantum Hall effect in a p-type heterojunction with a lateral surface quantum dot superlattice
The quantization of Hall conductance in a p-type heterojunction with lateral
surface quantum dot superlattice is investigated. The topological properties of
the four-component hole wavefunction are studied both in r- and k-spaces. New
method of calculation of the Hall conductance in a 2D hole gas described by the
Luttinger Hamiltonian and affected by lateral periodic potential is proposed,
based on the investigation of four-component wavefunction singularities in
k-space. The deviations from the quantization rules for Hofstadter "butterfly"
for electrons are found, and the explanation of this effect is proposed. For
the case of strong periodic potential the mixing of magnetic subbands is taken
into account, and the exchange of the Chern numbers between magnetic subands is
discussed.Comment: 12 pages, 5 figures; reported at the 15th Int. Conf. on High Magnetic
Fields in Semicond. Phys. (Oxford, UK, 2002
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