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Register a new userPoor mans scaling and Lie algebras
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E. Kogan
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We consider a general model, describing a quantum impurity with degenerate energy levels, interacting with a gas of itinerant electrons, derive general scaling equation for the model, and analyse the connection between its particular forms and the symmetry of interaction. On the basis of this analysis we write down scaling equations for the Hamiltonians which are the direct products of $su(3)$ Lie algebras and have either $SU(2)times U(1)$ or $SU(2)$ symmetry. We also put into a new context anisotropic Coqblin -- Schrieffer models proposed by us earlier.
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We discuss Kondo effect for a general model, describing a quantum impurity with degenerate energy levels, interacting with a gas of itinerant electrons, and derive scaling equation to the second order for such a model. We show how the scaling equation for the spin-anisotropic Kondo model with the power law density of states (DOS) for itinerant electrons follows from the general scaling equation. We introduce the anisotropic Coqblin--Schrieffer model, apply the general method to derive scaling equation for that model for the power law DOS, and integrate the derived equation analytically.
We consider homological finiteness properties $FP_n$ of certain $mathbb{N}$-graded Lie algebras. After proving some general results, see Theorem A, Corollary B and Corollary C, we concentrate on a family that can be considered as the Lie algebra version of the generalized Bestvina-Brady groups associated to a graph $Gamma$. We prove that the homological finiteness properties of these Lie algebras can be determined in terms of the graph in the same way as in the group case.
Topological properties of quantum systems could provide protection of information against environmental noise, and thereby drastically advance their potential in quantum information processing. Most proposals for topologically protected quantum gates are based on many-body systems, e.g., fractional quantum Hall states, exotic superconductors, or ensembles of interacting spins, bearing an inherent conceptual complexity. Here, we propose and study a topologically protected quantum gate, based on a one-dimensional single-particle tight-binding model, known as the Su-Schrieffer-Heeger chain. The proposed $Y$ gate acts in the two-dimensional zero-energy subspace of a Y junction assembled from three chains, and is based on the spatial exchange of the defects supporting the zero-energy modes. With numerical simulations, we demonstrate that the gate is robust against hopping disorder but is corrupted by disorder in the on-site energy. Then we show that this robustness is topological protection, and that it arises as a joint consequence of chiral symmetry, time-reversal symmetry and the spatial separation of the zero-energy modes bound to the defects. This setup will most likely not lead to a practical quantum computer, nevertheless it does provide valuable insight to aspects of topological quantum computing as an elementary minimal model. Since this model is non-interacting and non-superconducting, its dynamics can be studied experimentally, e.g., using coupled optical waveguides.
We discuss a version of the Chevalley--Eilenberg cohomology in characteristic $2$, where the alternating cochains are replaced by symmetric ones.
In this paper we give the relationship between the connected components of pure generalized Dynkin graphs and Nichols braided Lie algebras.
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