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Register a new userGaplessness of Landau Hamiltonians on hyperbolic half-planes via coarse geometry
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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.
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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.
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].
We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({mathbb R^2})$, whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues $Lambda_q$, $q in {mathbb Z}_+$. We perturb $H_0$ by a non-local potential written as a bounded pseudo-differential operator ${rm Op}^{rm w}({mathcal V})$ with real-valued Weyl symbol ${mathcal V}$, such that ${rm Op}^{rm w}({mathcal V}) H_0^{-1}$ is compact. We study the spectral properties of the perturbed operator $H_{{mathcal V}} = H_0 + {rm Op}^{rm w}({mathcal V})$. First, we construct symbols ${mathcal V}$, possessing a suitable symmetry, such that the operator $H_{mathcal V}$ admits an explicit eigenbasis in $L^2({mathbb R^2})$, and calculate the corresponding eigenvalues. Moreover, for ${mathcal V}$ which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of $H_{mathcal V}$ adjoining any given $Lambda_q$. We find that the effective Hamiltonian in this context is the Toeplitz operator ${mathcal T}_q({mathcal V}) = p_q {rm Op}^{rm w}({mathcal V}) p_q$, where $p_q$ is the orthogonal projection onto ${rm Ker}(H_0 - Lambda_q I)$, and investigate its spectral asymptotics.
We prove that a spectral gap-filling phenomenon occurs whenever a Hamiltonian operator encounters a coarse index obstruction upon compression to a domain with boundary. Furthermore, the gap-filling spectra contribute to quantised current channels, which follow and are localised at the possibly complicated boundary. This index obstruction is shown to be insensitive to deformations of the domain boundary, so the phenomenon is generic for magnetic Laplacians modelling quantum Hall systems and Chern topological insulators. A key construction is a quasi-equivariant version of Roes algebra of locally compact finite propagation operators.
We discuss the homological aspects of the connection between quantum string generating function and the formal power series associated to the dimensions of chains and homologies of suitable Lie algebras. Our analysis can be considered as a new straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of $SL(2,{mathbb Z})$) to the partition functions of Lagrangian branes, refined vertex and open string partition functions, represented by means of formal power series that encode Lie algebra properties. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras and in the role of Selberg-type spectral functions of an hyperbolic three-geometry associated with $q$-series in the computation of the string amplitudes.
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