Josephson-coupled Moore-Read states

Abstract

We study a quantum Hall bilayer system of bosons at total filling factor ν = 1, and study the phase that results from short-ranged pair tunneling combined with short-ranged interlayer interactions. We introduce two exactly solvable model Hamiltonians which both yield the coupled Moore-Read state [Phys. Rev. Lett. 108, 256809 (2012)] as a ground state, when projected onto fixed particle numbers in each layer. One of these Hamiltonians describes a gapped topological phase, while the other is gapless. However, on introduction of a pair-tunneling term, the second system becomes gapped and develops the same topological order as the gapped Hamiltonian. Supported by the exact solution of the full zero-energy quasihole spectrum and a conformal field-theory approach, we develop an intuitive picture of this system as two coupled composite fermion superconductors. In this language, pair tunneling provides a Josephson coupling of the superconducting phases of the two layers, and gaps out the Goldstone mode associated with particle transport between the layers. In particular, this implies that quasiparticles are confined between the layers. In the bulk, the resulting phase has the topological order of the Halperin 220 phase with U(1) 2 ×U(1) 2 topological order, but it is realized in the symmetric/antisymmetric basis of the layer index. Consequently, the edge spectrum at a fixed particle number reveals an unexpected U(1) 4 ×U(1) structure.

Phase diagram ofν=12+12bilayer bosons with interlayer couplings

“…We can actually make a stronger statement: as shown in Ref. 21, with the proper amount of inter-layer interaction (U 0 = 2.0) and Josephson coupling (t J = 1), and setting U 1 = 0, the resulting Hamiltonian is nothing but the model Hamiltonian for the π/2 rotated Halperin (220) state. Adding some small U 1 or changing a little bit either U 0 or t J does not change this picture.…”

“…lower) layer. This wave function is the exact densest zero energy state of the following model Hamiltonian 21 (once projected onto the lowest Landau level)…”

“…Recent works have considered such a setup either on different geometry and a slightly long-range interaction [20][21][22] or using a lattice analogue via two copies of fractional Chern insulators 23,24 . Other studies have also considered such a system at larger filling factors in the context of the nonAbelian spin-singlet state 25 or the integer quantum Hall effect for bosons 20,26,27 .…”

Anyon Condensation and Its Applications

“…To discuss critical points, we require some understanding of the underlying Hamiltonian that describes the transition itself. The more experimentally relevant Hamiltonian realizations of TSB involve bilayer quantum Hall systems, [78][79][80]82,113 78,130 and a bilayer of the Moore-Read state. 76,113 However, to study the critical point in a systematic way it is advantageous to use more tractable (but less physically realistic) lattice models, 42,60,76,77,84,86,129,[131][132][133][134][135][136][137][138] which will be our focus here.…”

Crater function approach to ion-induced nanoscale pattern formation: Craters for flat surfaces are insufficient

“…The second terms on the right-hand sides of Eqs. (29) and (53) therefore yield nonzero contributions to C 11 and C 22 . These contributions were computed explicitly in the preceding section.…”