59 research outputs found
Derivation of the probability distribution function for the local density of states of a disordered quantum wire via the replica trick and supersymmetry
We consider the statistical properties of the local density of states of a
one-dimensional Dirac equation in the presence of various types of disorder
with Gaussian white-noise distribution. It is shown how either the replica
trick or supersymmetry can be used to calculate exactly all the moments of the
local density of states. Careful attention is paid to how the results change if
the local density of states is averaged over atomic length scales. For both the
replica trick and supersymmetry the problem is reduced to finding the ground
state of a zero-dimensional Hamiltonian which is written solely in terms of a
pair of coupled ``spins'' which are elements of u(1,1). This ground state is
explicitly found for the particular case of the Dirac equation corresponding to
an infinite metallic quantum wire with a single conduction channel. The
calculated moments of the local density of states agree with those found
previously by Al'tshuler and Prigodin [Sov. Phys. JETP 68 (1989) 198] using a
technique based on recursion relations for Feynman diagrams.Comment: 39 pages, 1 figur
Phase diagrams of the metallic zigzag carbon nanotube
We investigate a metallic zigzag carbon nanotube by means of a Hubbard model
which includes both on-site and nearest neighbour interactions. Assuming weak
interactions, a renormalization group analysis of the equivalent two-leg ladder
followed by bosonization and refermionization results in a Gross-Neveu model
with an enlarged symmetry relative to the original Hamiltonian. For the undoped
case the symmetry of the Gross-Neveu model is SO(8), but for the doped case the
particle-hole symmetry is broken and the symmetry reduces to SO(6). Four ground
state phases are found in the undoped carbon nanotube with repulsive
interactions, a d-wave Mott insulator, an s-wave Mott insulator, a p-density
wave and a charge density wave. The doped case has two ground state phases, a
d-wave superconductor and a phase where a p-density wave and a charge density
wave co-exist. We also explore the global phase diagram with a general
interaction profile and find several additional states, including a chiral
current phase where current flows around the nanotube along the zigzag bonds.Comment: 16 pages, 9 figure
Ruderman-Kittel-Kasuya-Yosida interactions on a bipartite lattice
Carrier-mediated exchange coupling, known as Ruderman-Kittel-Kasuya-Yosida
(RKKY) interaction, plays a fundamental role in itinerant ferromagnetism and
has great application potentials in spintronics. A recent theorem based on the
imaginary-time method shows that the oscillatory RKKY interaction becomes
commensurate on bipartite lattice and predicts that the effective exchange
coupling is always ferromagnetic for the same sublattice but antiferromagnetic
for opposite sublattices. We revisit this important problem by real- and
imaginary-time methods and find the theorem misses important contributions from
zero modes. To illustrate the importance of zero modes, we study the spin
susceptibility in graphene nanoribbons numerically. The effective exchange
coupling is largest on the edges but does not follow the predictions from the
theorem
Accurate macroscale modelling of spatial dynamics in multiple dimensions
Developments in dynamical systems theory provides new support for the
macroscale modelling of pdes and other microscale systems such as Lattice
Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically
resolving subgrid microscale dynamics the dynamical systems approach constructs
accurate closures of macroscale discretisations of the microscale system. Here
we specifically explore reaction-diffusion problems in two spatial dimensions
as a prototype of generic systems in multiple dimensions. Our approach unifies
into one the modelling of systems by a type of finite elements, and the
`equation free' macroscale modelling of microscale simulators efficiently
executing only on small patches of the spatial domain. Centre manifold theory
ensures that a closed model exist on the macroscale grid, is emergent, and is
systematically approximated. Dividing space either into overlapping finite
elements or into spatially separated small patches, the specially crafted
inter-element/patch coupling also ensures that the constructed discretisations
are consistent with the microscale system/PDE to as high an order as desired.
Computer algebra handles the considerable algebraic details as seen in the
specific application to the Ginzburg--Landau PDE. However, higher order models
in multiple dimensions require a mixed numerical and algebraic approach that is
also developed. The modelling here may be straightforwardly adapted to a wide
class of reaction-diffusion PDEs and lattice equations in multiple space
dimensions. When applied to patches of microscopic simulations our coupling
conditions promise efficient macroscale simulation.Comment: some figures with 3D interaction when viewed in Acrobat Reader. arXiv
admin note: substantial text overlap with arXiv:0904.085
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