Unification of Laughlin and Moore–Read states in SUSY quantum Hall effect

Abstract

Based on the recently proposed SUSY quantum Hall effect, we show that Laughlin and MooreRead states are related by a hidden SUSY transformation. Regarding the SUSY Laughlin wavefunction as a master wavefunction, Laughlin and Moore-Read states appear as two extreme limits of component wavefunctions. Realizations of topological excitations on Laughlin and Moore-Read states are also discussed in the SUSY formalism. We develop a streographically projected formulation of the SUSY quantum Hall effect. With appropriate interpretation of Grassmann odd coordinates, we illustrate striking analogies between SUSY quantum Hall effect and superfluidity.Quantum Hall effects (QHE) have been experimentally confirmed at odd-denominator fillings and evendenominator fillings in monolayer and bilayer systems [1,2,3,4] In this paper, we discuss a hidden SUSY relation between Laughlin and Moore-Read states. We use a set-up of the SUSY QHE [16,17], which was recently proposed as a SUSY extension of the Haldane's spherical QHE [18]. The SUSY QHE contains non-anticommutative geometry as its mathematical background [19] and its low energy sector is described by a particular SUSY Chern-Simons field theory [20]. Though the SUSY QHE may not have apparent relevance to real QHE, the SUSY QHE provides an interesting unified formulation of Laughlin and Moore-Read states. Further, with appropriate interpretation of the Grassmann odd coordinates, like other QH liquids, we show the SUSY QH liquid may be understood as an exotic superfluid state.SUSY Quantum Hall Effect-The spherical SUSY QHE is constructed on a supersphere in a supermonopole background [16]. The supersphere S 2|2 = OSp(1|2)/U (1) is a supermanifold whose coordinates satisfy the constraint x 2 a + C αβ θ α θ β = R 2 , where x a (a = 1, 2, 3) and θ α (α = 1, 2) are Grassmann even and odd coordinates, respectively, and C αβ is a charge conjugation antisymmetric matrix with C 12 = 1. The one-particle Hamiltonian is given by

Supersymmetric Chern-Simons theory in the presence of a boundary

“…This model describes the quantization of the Hall conductance and the existence of vortex and antivortex excitations. Lately supersymmetric generalization of the fractional quantum Hall effect has also been investigated [54][55][56][57]. In particular physical properties of the topological excitations in the supersymmetric quantum Hall liquid have been discussed in a dual supersymmetric Chern-Simons theory [58].…”

SUSY Quantum Hall Effect on Non-Anti-Commutative Geometry

“…5 The planar SUSY quantum Hall ef fect 5.1 The generators on the superplane and stereographic projection [40,41] Using the Inönü-Wigner contraction, we derive the symmetry generators on the superplane from the OSp(1|2) generators. We apply a symmetric scaling to the OSp(1|2) generators as…”

Supersymmetric quantum hall liquid with a deformed supersymmetry

“…Many-body problem on supermanifolds under magnetic fields, i.e. SUSY QHE, have also been explored [22][23][24][25][26][27]. In Refs.…”