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      Almost commuting self-adjoint matrices --- the real and self-dual cases

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          Abstract

          We show that a pair of almost commuting self-adjoint, symmetric matrices is close to a pair of commuting self-adjoint, symmetric matrices (in a uniform way). Moreover we prove that the same holds with self-dual in place of symmetric. The notion of self-dual Hermitian matrices is important in physics when studying fermionic systems that have time reversal symmetry. Since a symmetric, self-adjoint matrix is real, we get a real version of Huaxin Lin's famous theorem on almost commuting matrices. Similarly the self-dual case gives a version for matrices over the quaternions. We prove analogous results for element of real C^*-algebras of "low rank." In particular, these stronger results apply to paths of almost commuting Hermitian matrices that are real or self-dual. Along the way we develop a theory of semiprojectivity for real C^*-algebras.

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          Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures

          Normal-conducting mesoscopic systems in contact with a superconductor are classified by the symmetry operations of time reversal and rotation of the electron's spin. Four symmetry classes are identified, which correspond to Cartan's symmetric spaces of type C, CI, D, and DIII. A detailed study is made of the systems where the phase shift due to Andreev reflection averages to zero along a typical semiclassical single-electron trajectory. Such systems are particularly interesting because they do not have a genuine excitation gap but support quasiparticle states close to the chemical potential. Disorder or dynamically generated chaos mixes the states and produces novel forms of universal level statistics. For two of the four universality classes, the n-level correlation functions are calculated by the mapping on a free 1D Fermi gas with a boundary. The remaining two classes are related to the Laguerre orthogonal and symplectic random-matrix ensembles. For a quantum dot with an NS-geometry, the weak localization correction to the conductance is calculated as a function of sticking probability and two perturbations breaking time-reversal symmetry and spin-rotation invariance. The universal conductance fluctuations are computed from a maximum-entropy S-matrix ensemble. They are larger by a factor of two than what is naively expected from the analogy with normal-conducting systems. This enhancement is explained by the doubling of the number of slow modes: owing to the coupling of particles and holes by the proximity to the superconductor, every cooperon and diffuson mode in the advanced-retarded channel entails a corresponding mode in the advanced-advanced (or retarded-retarded) channel.
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            Blind signal separation: statistical principles

            J. Cardoso (1998)
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              The quantum spin Hall effect and topological insulators

              , (2010)
              In topological insulators, spin-orbit coupling and time-reversal symmetry combine to form a novel state of matter predicted to have exotic physical properties.
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                Author and article information

                Journal
                2010-12-15
                2013-07-12
                Article
                1012.3494
                eeff4153-6657-455e-a9a0-23def99c1212

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                CPH-SYM-00
                Expanded references. 33 pages
                math.OA

                Algebra
                Algebra

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