Fractional charge and inter-Landau–level states at points of singular curvature

Abstract

The quest for universal properties of topological phases is fundamentally important because these signatures are robust to variations in system-specific details. Aspects of the response of quantum Hall states to smooth spatial curvature are well-studied, but challenging to observe experimentally. Here we go beyond this prevailing paradigm and obtain general results for the response of quantum Hall states to points of singular curvature in real space; such points may be readily experimentally actualized. We find, using continuum analytical methods, that the point of curvature binds an excess fractional charge and sequences of quantum states split away, energetically, from the degenerate bulk Landau levels. Importantly, these inter-Landau-level states are bound to the topological singularity and have energies that are universal functions of bulk parameters and the curvature. Our exact diagonalization of lattice tight-binding models on closed manifolds demonstrates that these results continue to hold even when lattice effects are significant. An important technological implication of these results is that these inter-Landau-level states, being both energetically and spatially isolated quantum states, are promising candidates for constructing qubits for quantum computation.quantum Hall | geometry | gravitational response | singularity | quantum computation Q uantum Hall (QH) states (1) were among the first known examples of topologically nontrivial quantum states, a rapidly expanding category that now includes other exotica, such as Chern and topological insulators (2, 3), topological superconductors (2, 3), and spin liquids (4). The distinguishing feature of QH states is that they possess quantized values of Hall conductance, which are rational multiples of the conductance quantum, e 2 =h, a fundamental physical constant. This characteristic property results from the interplay between the degeneracy of QH states and their topological response to phase twists across their boundaries (5, 6).An additional topological feature of QH states, less wellknown and appreciated several years after the first, is that they possess a universal coupling to the intrinsic geometry of the 2D real-space manifold where the electrons reside (7). This coupling causes a real-space curvature field, KðxÞ, to induce an excess carrier number density,KðxÞ.[1]Here, and subsequently, we set the magnetic length ℓ = ffiffiffiffiffiffiffiffiffiffi Z=eB p to unity; ν, the filling fraction, is equal to 1 for an isolated filled Landau level (LL). The value of the so-called "gravitational" coupling constant, κ, is a characteristic of the QH state, and is robust to small perturbations to the Hamiltonian (7). LLs with indices n = 0,1, . . ., are characterized by corresponding distinct values (7, 8) of the coupling constant, κ n = n + 1=2, allowing us to distinguish between continuum LLs on the basis of a local measurement. Nonintuitive and interesting new behaviors resulting from this gravitational coupling are being increasingly appreciated: It has rece...

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