32 research outputs found
Random Dirac operators with time-reversal symmetry
Quasi-one-dimensional stochastic Dirac operators with an odd number of
channels, time reversal symmetry but otherwise efficiently coupled randomness
are shown to have one conducting channel and absolutely continuous spectrum of
multiplicity two. This follows by adapting the criteria of Guivarch-Raugi and
Goldsheid-Margulis to the analysis of random products of matrices in the group
SO, and then a version of Kotani theory for these operators. Absence of
singular spectrum can be shown by adapting an argument of Jaksic-Last if the
potential contains random Dirac peaks with absolutely continuous distribution.Comment: parts of introduction made more precise, corrections as follow-up on
referee report
Winding Numbers, Complex Currents, and Non-Hermitian Localization
The nature of extended states in disordered tight binding models with a
constant imaginary vector potential is explored. Such models, relevant to
vortex physics in superconductors and to population biology, exhibit a
delocalization transition and a band of extended states even for a one
dimensional ring. Using an analysis of eigenvalue trajectories in the complex
plane, we demonstrate that each delocalized state is characterized by an
(integer) winding number, and evaluate the associated complex current. Winding
numbers in higher dimensions are also discussed.Comment: 4 pages, 2 figure
Enumeration of simple random walks and tridiagonal matrices
We present some old and new results in the enumeration of random walks in one
dimension, mostly developed in works of enumerative combinatorics. The relation
between the trace of the -th power of a tridiagonal matrix and the
enumeration of weighted paths of steps allows an easier combinatorial
enumeration of the paths. It also seems promising for the theory of tridiagonal
random matrices .Comment: several ref.and comments added, misprints correcte
Population Dynamics and Non-Hermitian Localization
We review localization with non-Hermitian time evolution as applied to simple
models of population biology with spatially varying growth profiles and
convection. Convection leads to a constant imaginary vector potential in the
Schroedinger-like operator which appears in linearized growth models. We
illustrate the basic ideas by reviewing how convection affects the evolution of
a population influenced by a simple square well growth profile. Results from
discrete lattice growth models in both one and two dimensions are presented. A
set of similarity transformations which lead to exact results for the spectrum
and winding numbers of eigenfunctions for random growth rates in one dimension
is described in detail. We discuss the influence of boundary conditions, and
argue that periodic boundary conditions lead to results which are in fact
typical of a broad class of growth problems with convection.Comment: 19 pages, 11 figure
Eigenvector statistics in non-Hermitian random matrix ensembles
We study statistical properties of the eigenvectors of non-Hermitian random
matrices, concentrating on Ginibre's complex Gaussian ensemble, in which the
real and imaginary parts of each element of an N x N matrix, J, are independent
random variables. Calculating ensemble averages based on the quantity , where and are left and right eigenvectors of J, we show for large N that
eigenvectors associated with a pair of eigenvalues are highly correlated if the
two eigenvalues lie close in the complex plane. We examine consequences of
these correlations that are likely to be important in physical applications.Comment: 4 pages, no figure
Vortices in a cylinder: Localization after depinning
Edge effects in the depinned phase of flux lines in hollow superconducting
cylinder with columnar defects and electric current along the cylinder are
investigated. Far from the ends of the cylinder vortices are distributed almost
uniformly (delocalized). Nevertheless, near the edges these free vortices come
closer together and form well resolved dense bunches. A semiclassical picture
of this localization after depinning is described. For a large number of
vortices their density has square root singularity at the border of
the bunch ( is semicircle in the simplest case). However, by tuning
the strength of current, the various singular regimes for may be
reached. Remarkably, this singular behaviour reproduces the phase transitions
discussed during the past decade within the random matrix regularization of
2d-Gravity.Comment: 4 pages, REVTEX, 2 eps figure
Vortex Pinning and the Non-Hermitian Mott Transition
The boson Hubbard model has been extensively studied as a model of the zero
temperature superfluid/insulator transition in Helium-4 on periodic substrates.
It can also serve as a model for vortex lines in superconductors with a
magnetic field parallel to a periodic array of columnar pins, due to a formal
analogy between the vortex lines and the statistical mechanics of quantum
bosons. When the magnetic field has a component perpendicular to the pins, this
analogy yields a non-Hermitian boson Hubbard model. At integer filling, we find
that for small transverse fields, the insulating phase is preserved, and the
transverse field is exponentially screened away from the boundaries of the
superconductor. At larger transverse fields, a ``superfluid'' phase of tilted,
entangled vortices appears. The universality class of the transition is found
to be that of vortex lines entering the Meissner phase at H_{c1}, with the
additional feature that the direction of the tilted vortices at the transition
bears a non-trivial relationship to the direction of the applied magnetic
field. The properties of the Mott Insulator and flux liquid phases with tilt
are also discussed.Comment: 20 pages, 12 figures included in text; to appear in Physical Review
Delocalization in an open one-dimensional chain in an imaginary vector potential
We present first results for the transmittance, T, through a 1D disordered
system with an imaginary vector potential, ih, which provide a new analytical
criterion for a delocalization transition in the model. It turns out that the
position of the critical curve on the complex energy plane (i.e. the curve
where an exponential decay of is changed by a power-law one) is different
from that obtained previously from the complex energy spectra. Corresponding
curves for or are also different. This happens because of
different scales of the exponential decay of one-particle Green's functions
(GF) defining the spectra and many-particle GF governing transport
characteristics, and reflects higher-order correlations in localized
eigenstates of the non-Hermitian model.Comment: 4 pages in RevTex, 1 eps figure include
Ghost spins and novel quantum critical behavior in a spin chain with local bond-deformation
We study the boundary impurity-induced critical behavior in an integrable
SU(2)-invariant model consisting of an open Heisenberg chain of arbitrary
spin- (Takhatajian-Babujian model) interacting with an impurity of spin
located at one of the boundaries. For or , the
impurity interaction has a very simple form which
describes the deformed boundary bond between the impurity and the
first bulk spin with an arbitrary strength . With a weak
coupling , the impurity is completely compensated,
undercompensated, and overcompensated for , and as in the
usual Kondo problem. While for strong coupling , the
impurity spin is split into two ghost spins. Their cooperative effect leads to
a variety of new critical behaviors with different values of .Comment: 16 pages revtex, no figur