Monday, June 17, 2013

1306.3268 (Gerardo Ortiz et al.)

Repulsive Interactions in Quantum Hall Systems as a Pairing Problem    [PDF]

Gerardo Ortiz, Zohar Nussinov, Jorge Dukelsky, Alexander Seidel
A subtle relation between Quantum Hall physics and the phenomenon of pairing is unveiled. By use of second quantization, we establish a connection between (i) a broad class of rotationally symmetric two-body interactions within the lowest Landau level and (ii) integrable hyperbolic Richardson-Gaudin type Hamiltonians that arise in (p_x+ip_y) superconductivity. Specifically, we show that general Haldane pseudopotentials (and their sums) can be expressed as a sum of repulsive non-commuting (p_x+ip_y)-type pairing Hamiltonians. The determination of the spectrum and individual null spaces of each of these non-commuting Richardson-Gaudin type Hamiltonians is non-trivial yet is Bethe Ansatz solvable. We prove separability of arbitrary Haldane pseudopotentials, providing explicit expressions for their second quantized forms, and further show by explicit construction how to exploit the topological equivalence between different geometries (disk, cylinder, and sphere) sharing the same topological genus number, in the second quantized formalism, through similarity transformations. As an application of the second quantized approach, we establish a "squeezing principle" that applies to the zero modes of a general class of Hamiltonians, which includes but is not limited to Haldane pseudopotentials. We also show how one may establish (bounds on) "incompressible filling factors" for those Hamiltonians. By invoking properties of symmetric polynomials, we provide explicit second quantized quasi-hole generators; the generators that we find directly relate to bosonic chiral edge modes and further make aspects of dimensional reduction in the Quantum Hall systems precise.
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