37,460 research outputs found
Congruences and Canonical Forms for a Positive Matrix: Application to the Schweinler-Wigner Extremum Principle
It is shown that a real symmetric [complex hermitian] positive
definite matrix is congruent to a diagonal matrix modulo a
pseudo-orthogonal [pseudo-unitary] matrix in [ ], for any
choice of partition . It is further shown that the method of proof in
this context can easily be adapted to obtain a rather simple proof of
Williamson's theorem which states that if is even then is congruent
also to a diagonal matrix modulo a symplectic matrix in
[]. Applications of these results considered include a
generalization of the Schweinler-Wigner method of `orthogonalization based on
an extremum principle' to construct pseudo-orthogonal and symplectic bases from
a given set of linearly independent vectors.Comment: 7 pages, latex, no figure
Determining the spin-orbit coupling via spin-polarized spectroscopy of magnetic impurities
We study the spin-resolved spectral properties of the impurity states
associated to the presence of magnetic impurities in two-dimensional, as well
as one-dimensional systems with Rashba spin-orbit coupling. We focus on Shiba
bound states in superconducting materials, as well as on impurity states in
metallic systems. Using a combination of a numerical T-matrix approximation and
a direct analytical calculation of the bound state wave function, we compute
the local density of states (LDOS) together with its Fourier transform (FT). We
find that the FT of the spin-polarized LDOS, a quantity accessible via
spin-polarized STM, allows to accurately extract the strength of the spin-orbit
coupling. Also we confirm that the presence of magnetic impurities is strictly
necessary for such measurement, and that non-spin-polarized experiments cannot
have access to the value of the spin-orbit coupling.Comment: 26 pages, 6 figure
Hamilton's Turns for the Lorentz Group
Hamilton in the course of his studies on quaternions came up with an elegant
geometric picture for the group SU(2). In this picture the group elements are
represented by ``turns'', which are equivalence classes of directed great
circle arcs on the unit sphere , in such a manner that the rule for
composition of group elements takes the form of the familiar parallelogram law
for the Euclidean translation group. It is only recently that this construction
has been generalized to the simplest noncompact group , the double cover of SO(2,1). The present work develops a theory of
turns for , the double and universal cover of SO(3,1) and ,
rendering a geometric representation in the spirit of Hamilton available for
all low dimensional semisimple Lie groups of interest in physics. The geometric
construction is illustrated through application to polar decomposition, and to
the composition of Lorentz boosts and the resulting Wigner or Thomas rotation.Comment: 13 pages, Late
RKKY interaction on the surface of three-dimensional Dirac semimetals
We study the RKKY interaction between two magnetic impurities located on the
surface of a three-dimensional Dirac semimetal with two Dirac nodes in the band
structure. By taking into account both bulk and surface contributions to the
exchange interaction between the localized spins, we demonstrate that the
surface contribution in general dominates the bulk one at distances larger than
the inverse node separation due to a weaker power-law decay. We find a strong
anisotropy of the surface term with respect to the spins being aligned along
the node separation axis or perpendicular to it. In the many impurity dilute
regime, this implies formation of quasi-one-dimensional magnetic stripes
orthogonal to the node axis. We also discuss the effects of a surface
spin-mixing term coupling electrons from spin-degenerate Fermi arcs.Comment: 7,5 pages, 3 figures (+4 pages of Appendixes
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