49 research outputs found
Edge Currents for Quantum Hall Systems, I. One-Edge, Unbounded Geometries
Devices exhibiting the integer quantum Hall effect can be modeled by
one-electron Schroedinger operators describing the planar motion of an electron
in a perpendicular, constant magnetic field, and under the influence of an
electrostatic potential. The electron motion is confined to unbounded subsets
of the plane by confining potential barriers. The edges of the confining
potential barrier create edge currents. In this, the first of two papers, we
prove explicit lower bounds on the edge currents associated with one-edge,
unbounded geometries formed by various confining potentials. This work extends
some known results that we review. The edge currents are carried by states with
energy localized between any two Landau levels. These one-edge geometries
describe the electron confined to certain unbounded regions in the plane
obtained by deforming half-plane regions. We prove that the currents are stable
under various potential perturbations, provided the perturbations are suitably
small relative to the magnetic field strength, including perturbations by
random potentials. For these cases of one-edge geometries, the existence of,
and the estimates on, the edge currents imply that the corresponding
Hamiltonian has intervals of absolutely continuous spectrum. In the second
paper of this series, we consider the edge currents associated with two-edge
geometries describing bounded, cylinder-like regions, and unbounded,
strip-like, regions.Comment: 68 page
Intermixture of extended edge and localized bulk energy levels in macroscopic Hall systems
We study the spectrum of a random Schroedinger operator for an electron
submitted to a magnetic field in a finite but macroscopic two dimensional
system of linear dimensions equal to L. The y direction is periodic and in the
x direction the electron is confined by two smooth increasing boundary
potentials. The eigenvalues of the Hamiltonian are classified according to
their associated quantum mechanical current in the y direction. Here we look at
an interval of energies inside the first Landau band of the random operator for
the infinite plane. In this energy interval, with large probability, there
exist O(L) eigenvalues with positive or negative currents of O(1). Between each
of these there exist O(L^2) eigenvalues with infinitesimal current
O(exp(-cB(log L)^2)). We explain what is the relevance of this analysis to the
integer quantum Hall effect.Comment: 29 pages, no figure
Diagonalization of an Integrable Discretization of the Repulsive Delta Bose Gas on the Circle
We introduce an integrable lattice discretization of the quantum system of n
bosonic particles on a ring interacting pairwise via repulsive delta
potentials. The corresponding (finite-dimensional) spectral problem of the
integrable lattice model is solved by means of the Bethe Ansatz method. The
resulting eigenfunctions turn out to be given by specializations of the
Hall-Littlewood polynomials. In the continuum limit the solution of the
repulsive delta Bose gas due to Lieb and Liniger is recovered, including the
orthogonality of the Bethe wave functions first proved by Dorlas (extending
previous work of C.N. Yang and C.P. Yang).Comment: 25 pages, LaTe
Large deviations for ideal quantum systems
We consider a general d-dimensional quantum system of non-interacting
particles, with suitable statistics, in a very large (formally infinite)
container. We prove that, in equilibrium, the fluctuations in the density of
particles in a subdomain of the container are described by a large deviation
function related to the pressure of the system. That is, untypical densities
occur with a probability exponentially small in the volume of the subdomain,
with the coefficient in the exponent given by the appropriate thermodynamic
potential. Furthermore, small fluctuations satisfy the central limit theorem.Comment: 28 pages, LaTeX 2
Mutual Exclusion Statistics in Exactly Solvable Models in One and Higher Dimensions at Low Temperatures
We study statistical characterization of the many-body states in exactly
solvable models with internal degrees of freedom. The models under
consideration include the isotropic and anisotropic Heisenberg spin chain, the
Hubbard chain, and a model in higher dimensions which exhibits the Mott
metal-insulator transition. It is shown that the ground state of these systems
is all described by that of a generalized ideal gas of particles (called
exclusons) which have mutual exclusion statistics, either between different
rapidities or between different species. For the Bethe ansatz solvable models,
the low temperature properties are well described by the excluson description
if the degeneracies due to string solutions with complex rapidities are taken
into account correctly. {For} the Hubbard chain with strong but finite
coupling, charge-spin separation is shown for thermodynamics at low
temperatures. Moreover, we present an exactly solvable model in arbitrary
dimensions which, in addition to giving a perspective view of spin-charge
separation, constitutes an explicit example of mutual exclusion statistics in
more than two dimensions
Two ways to solve ASEP
The purpose of this article is to describe the two approaches to compute
exact formulas (which are amenable to asymptotic analysis) for the probability
distribution of the current of particles past a given site in the asymmetric
simple exclusion process (ASEP) with step initial data. The first approach is
via a variant of the coordinate Bethe ansatz and was developed in work of Tracy
and Widom in 2008-2009, while the second approach is via a rigorous version of
the replica trick and was developed in work of Borodin, Sasamoto and the author
in 2012.Comment: 10 pages, Chapter in "Topics in percolative and disordered systems
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Beer-Lambert law along non-linear mean light pathways for the rational analysis of Photoplethysmography
Photoplethysmography (PPG) is a technique that uses light to noninvasively obtain a volumetric measurement of an organ with each cardiac cycle. A PPG-based system emits monochromatic light through the skin and measures the fraction of the light power which is transmitted through a vascular tissue and detected by a photodetector. Part of thereby transmitted light power is modulated by the vascular tissue volume changes due to the blood circulation induced by the heart beating. This modulated light power plotted against time is called the PPG signal. Pulse Oximetry is an empirical technique which allows the arterial blood oxygen saturation (SpO2 – molar fraction) evaluation from the PPG signals. There have been many reports in the literature suggesting that other arterial blood chemical components molar fractions and concentrations can be evaluated from the PPG signals. Most attempts to perform such evaluation on empirical bases have failed, especially for components concentrations. This paper introduces a non-empirical physical model which can be used to analytically investigate the phenomena of PPG signal. Such investigation would result in simplified engineering models, which can be used to design validating experiments and new types of spectroscopic devices with the potential to assess venous and arterial blood chemical composition in both molar fractions and concentrations non-invasively
Widths of the Hall Conductance Plateaus
We study the charge transport of the noninteracting electron gas in a
two-dimensional quantum Hall system with Anderson-type impurities at zero
temperature. We prove that there exist localized states of the bulk order in
the disordered-broadened Landau bands whose energies are smaller than a certain
value determined by the strength of the uniform magnetic field. We also prove
that, when the Fermi level lies in the localization regime, the Hall
conductance is quantized to the desired integer and shows the plateau of the
bulk order for varying the filling factor of the electrons rather than the
Fermi level.Comment: 94 pages, v2: a revision of Sec. 5; v3: an error in Sec. 7 is
corrected, major revisions of Sec. 7 and Appendix E, Sec. 7 is enlarged to
Secs. 7-12, minor corrections; v4: major revisions, accepted for publication
in Journal of Statistical Physics; v5: minor corrections, accepted versio
Localization on quantum graphs with random vertex couplings
We consider Schr\"odinger operators on a class of periodic quantum graphs
with randomly distributed Kirchhoff coupling constants at all vertices. Using
the technique of self-adjoint extensions we obtain conditions for localization
on quantum graphs in terms of finite volume criteria for some energy-dependent
discrete Hamiltonians. These conditions hold in the strong disorder limit and
at the spectral edges
Current Fluctuations of the One Dimensional Symmetric Simple Exclusion Process with Step Initial Condition
For the symmetric simple exclusion process on an infinite line, we calculate
exactly the fluctuations of the integrated current during time
through the origin when, in the initial condition, the sites are occupied with
density on the negative axis and with density on the positive
axis. All the cumulants of grow like . In the range where , the decay of the distribution of is
non-Gaussian. Our results are obtained using the Bethe ansatz and several
identities recently derived by Tracy and Widom for exclusion processes on the
infinite line.Comment: 2 figure