29 research outputs found
Symmetric space description of carbon nanotubes
Using an innovative technique arising from the theory of symmetric spaces, we
obtain an approximate analytic solution of the Dorokhov-Mello-Pereyra-Kumar
(DMPK) equation in the insulating regime of a metallic carbon nanotube with
symplectic symmetry and an odd number of conducting channels. This symmetry
class is characterized by the presence of a perfectly conducting channel in the
limit of infinite length of the nanotube. The derivation of the DMPK equation
for this system has recently been performed by Takane, who also obtained the
average conductance both analytically and numerically. Using the Jacobian
corresponding to the transformation to radial coordinates and the
parameterization of the transfer matrix given by Takane, we identify the
ensemble of transfer matrices as the symmetric space of negative curvature
SO^*(4m+2)/[SU(2m+1)xU(1)] belonging to the DIII-odd Cartan class. We rederive
the leading-order correction to the conductance of the perfectly conducting
channel and its variance Var(log(delta g)). Our results are in
complete agreement with Takane's. In addition, our approach based on the
mapping to a symmetric space enables us to obtain new universal quantities: a
universal group theoretical expression for the ratio Var(log(delta
g)/ and as a byproduct, a novel expression for the localization
length for the most general case of a symmetric space with BC_m root system, in
which all three types of roots are present.Comment: 23 pages. Text concerning symmetric space description augmented,
table and references added. Version to be published on JSTA
Nonuniversality in quantum wires with off-diagonal disorder: a geometric point of view
It is shown that, in the scaling regime, transport properties of quantum
wires with off-diagonal disorder are described by a family of scaling equations
that depend on two parameters: the mean free path and an additional continuous
parameter. The existing scaling equation for quantum wires with off-diagonal
disorder [Brouwer et al., Phys. Rev. Lett. 81, 862 (1998)] is a special point
in this family. Both parameters depend on the details of the microscopic model.
Since there are two parameters involved, instead of only one, localization in a
wire with off-diagonal disorder is not universal. We take a geometric point of
view and show that this nonuniversality follows from the fact that the group of
transfer matrices is not semi-simple. Our results are illustrated with
numerical simulations for a tight-binding model with random hopping amplitudes.Comment: 12 pages, RevTeX; 3 figures included with eps
Quantum Lie algebras associated to and
Quantum Lie algebras \qlie{g} are non-associative algebras which are
embedded into the quantized enveloping algebras of Drinfeld and Jimbo
in the same way as ordinary Lie algebras are embedded into their enveloping
algebras. The quantum Lie product on \qlie{g} is induced by the quantum
adjoint action of . We construct the quantum Lie algebras associated to
and . We determine the structure constants and the
quantum root systems, which are now functions of the quantum parameter .
They exhibit an interesting duality symmetry under .Comment: Latex 9 page
Calogero-Sutherland techniques in the physics of disorderd wires
We discuss the connection between the random matrix approach to disordered
wires and the Calogero-Sutherland models. We show that different choices of
random matrix ensembles correspond to different classes of CS models. In
particular, the standard transfer matrix ensembles correspond to CS model with
sinh-type interaction, constructed according to the root lattice pattern.
By exploiting this relation, and by using some known properties of the zonal
spherical functions on symmetric spaces we can obtain several properties of the
Dorokhov-Mello-Pereyra-Kumar equation, which describes the evolution of an
ensemble of quasi one-dimensional disordered wires of increasing length .
These results are in complete agreement with all known properties of disordered
wires. (To appear in the Proceedings of the Conference: Recent Developments in
Statistical Mechanics and Quantum Field Theory (Trieste, 1995))Comment: 16 pages, Late
On the distribution of transmission eigenvalues in disordered wires
We solve the Dorokhov-Mello-Pereyra-Kumar equation which describes the
evolution of an ensamble of disordered wires of increasing length in the three
cases . The solution is obtained by mapping the problem in that of
a suitable Calogero-Sutherland model. In the case our solution is in
complete agreement with that recently found by Beenakker and Rejaei.Comment: 4 pages, Revtex, few comments added at the end of the pape
Disordered quantum wires: microscopic origins of the DMPK theory and Ohm's law
We study the electronic transport properties of the Anderson model on a
strip, modeling a quasi one-dimensional disordered quantum wire. In the
literature, the standard description of such wires is via random matrix theory
(RMT). Our objective is to firmly relate this theory to a microscopic model. We
correct and extend previous work (arXiv:0912.1574) on the same topic. In
particular, we obtain through a physically motivated scaling limit an ensemble
of random matrices that is close to, but not identical to the standard transfer
matrix ensembles (sometimes called TOE, TUE), corresponding to the Dyson
symmetry classes \beta=1,2. In the \beta=2 class, the resulting conductance is
the same as the one from the ideal ensemble, i.e.\ from TUE. In the \beta=1
class, we find a deviation from TOE. It remains to be seen whether or not this
deviation vanishes in a thick-wire limit, which is the experimentally relevant
regime. For the ideal ensembles, we also prove Ohm's law for all symmetry
classes, making mathematically precise a moment expansion by Mello and Stone.
This proof bypasses the explicit but intricate solution methods that underlie
most previous results.Comment: Corrects and extends arXiv:0912.157
Equivalence of Fokker-Planck approach and non-linear -model for disordered wires in the unitary symmetry class
The exact solution of the Dorokhov-Mello-Pereyra-Kumar-equation for quasi
one-dimensional disordered conductors in the unitary symmetry class is employed
to calculate all -point correlation functions by a generalization of the
method of orthogonal polynomials. We obtain closed expressions for the first
two conductance moments which are valid for the whole range of length scales
from the metallic regime () to the insulating regime () and
for arbitrary channel number. In the limit (with )
our expressions agree exactly with those of the non-linear -model
derived from microscopic Hamiltonians.Comment: 9 pages, Revtex, one postscript figur
On Quantum Lie Algebras and Quantum Root Systems
As a natural generalization of ordinary Lie algebras we introduce the concept
of quantum Lie algebras . We define these in terms of certain
adjoint submodules of quantized enveloping algebras endowed with a
quantum Lie bracket given by the quantum adjoint action. The structure
constants of these algebras depend on the quantum deformation parameter and
they go over into the usual Lie algebras when .
The notions of q-conjugation and q-linearity are introduced. q-linear
analogues of the classical antipode and Cartan involution are defined and a
generalised Killing form, q-linear in the first entry and linear in the second,
is obtained. These structures allow the derivation of symmetries between the
structure constants of quantum Lie algebras.
The explicitly worked out examples of and illustrate the
results.Comment: 22 pages, latex, version to appear in J. Phys. A. see
http://www.mth.kcl.ac.uk/~delius/q-lie.html for calculations and further
informatio
Localization and delocalization in dirty superconducting wires
We present Fokker-Planck equations that describe transport of heat and spin
in dirty unconventional superconducting quantum wires. Four symmetry classes
are distinguished, depending on the presence or absence of time-reversal and
spin rotation invariance. In the absence of spin-rotation symmetry, heat
transport is anomalous in that the mean conductance decays like
instead of exponentially fast for large enough length of the wire. The
Fokker-Planck equations in the presence of time-reversal symmetry are solved
exactly and the mean conductance for quasiparticle transport is calculated for
the crossover from the diffusive to the localized regime.Comment: 4 pages, RevTe
DIFFUSION IN ONE DIMENSIONAL RANDOM MEDIUM AND HYPERBOLIC BROWNIAN MOTION
Classical diffusion in a random medium involves an exponential functional of
Brownian motion. This functional also appears in the study of Brownian
diffusion on a Riemann surface of constant negative curvature. We analyse in
detail this relationship and study various distributions using stochastic
calculus and functional integration.Comment: 18 page