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Register a new userThe Geometry of (non-Abelian) Landau levels
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The purpose of this paper is threefold: First of all the topological aspects of the Landau Hamiltonian are reviewed in the light (and with the jargon) of theory of topological insulators. In particular it is shown that the Landau Hamiltonian has a generalized even time-reversal symmetry (TRS). Secondly, a new tool for the computation of the topological numbers associated with each Landau level is introduced. The latter is obtained by combining the Dixmier trace and the (resolvent of the) harmonic oscillator. Finally, these results are extended to models with non-Abelian magnetic fields. Two models are investigated in details: the Jaynes-Cummings model and the Quaternionic model.
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We develop the differential aspects of a noncommutative geometry for the Quantum Hall Effect in the continuous, with the ambition of proving Kubos formula. Taking inspiration from the ideas developed by Bellissard during the 80s we build a Fredholm module for the $C^*$-algebra of continuous magnetic operators, based on a Dirac operator closely related to the quantum harmonic oscillator. An important piece of Bellissards theory (the so-called second Connes formula) is proved. This work provides the continuation of the recent article [DS].
It is an important open problem to understand the landscape of non-Abelian fractional quantum Hall phases which can be obtained starting from physically motivated theories of Abelian composite particles. We show that progress on this problem can be made using recently proposed non-Abelian bosonization dualities in 2+1 dimensions, which morally relate $U(N)_k$ and $SU(k)_{-N}$ Chern-Simons-matter theories. The advantage of these dualities is that regions of the phase diagram which may be obscure on one side of the duality can be accessed by condensing local operators on the other side. Starting from parent Abelian states, we use this approach to construct Landau-Ginzburg theories of non-Abelian states through a pairing mechanism. In particular, we obtain the bosonic Read-Rezayi sequence at fillings $ u=k/(kM+2)$ by starting from $k$ layers of bosons at $ u=1/2$ with $M$ Abelian fluxes attached. The Read-Rezayi states arise when $k$-clusters of the dual non-Abelian bosons condense. We extend this construction by showing that $N_f$-component generalizations of the Halperin $(2,2,1)$ bosonic states have dual descriptions in terms of $SU(N_f+1)_1$ Chern-Simons-matter theories, revealing an emergent global symmetry in the process. Clustering $k$ layers of these theories yields a non-Abelian $SU(N_f)$-singlet state at filling $ u = kN_f / (N_f + 1 + kMN_f)$.
We use coarse index methods to prove that the Landau Hamiltonian on the hyperbolic half-plane, and even on much more general imperfect half-spaces, has no spectral gaps. Thus the edge states of hyperbolic quantum Hall Hamiltonians completely fill up the gaps between Landau levels, just like those of the Euclidean counterpart.
The Landau levels of cold atomic gases in non-Abelian gauge fields are analyzed. In particular we identify effects on the energy spectrum and density distribution which are purely due to the non-Abelian character of the fields. We investigate in detail non-Abelian generalizations of both the Landau and the symmetric gauge. Finally, we discuss how these non-Abelian Landau and symmetric gauges may be generated by means of realistically feasible lasers in a tripod scheme.
An almost non-abelian extension of the Rieffel deformation is presented in this work. The non-abelicity comes into play by the introduction of unitary groups which are dependent of the infinitesimal generators of $SU(n)$. This extension is applied to quantum mechanics and quantum field theory.
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