Effective Abelian theory from a non-Abelian topological order in $\nu =2/5$ fractional quantum Hall effect

Abstract

Topological phases of matter are distinguished by topological invariants, such as Chern numbers and topological spins, that quantize their response to electromagnetic currents and changes of ambient geometry. Intriguingly, in the $\nu =2/5$ fractional quantum Hall effect, prominent theoretical approaches -- the composite fermion theory and conformal field theory -- have constructed two distinct states, the Jain composite fermion (CF) state and the Gaffnian state, for which many of the topological indices coincide and even the microscopic ground state wave functions have high overlap with each other in system sizes accessible to numerics. At the same time, some aspects of these states are expected to be very different, e.g., their elementary excitations should have either Abelian (CF) or non-Abelian (Gaffnian) statistics. In this paper we investigate the close relationship between these two states by considering not only their ground states, but also the low-energy charged excitations. We show that the low-energy physics of both phases is spanned by the same type of quasielectrons of the neighbouring Laughlin phase. The main difference between the two states arises due to an implicit assumption of short-range interaction in the CF approach, which causes a large splitting of the variational energies of the Gaffnian excitations. We thus propose that the Jain phase emerges as an effective Abelian low-energy description of the Gaffnian phase when the Hamiltonian is dominated by two-body interactions of sufficiently short range.

11 pages, 3 figures, comments very welcome

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• Transition from two-component 332 Halperin state to one-component Jain state at filling factor ν=2/5

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• Geometric Fluctuation of Conformal Hilbert Spaces and Multiple Graviton Modes in Fractional Quantum Hall Effect
• Analytic exposition of the graviton modes in fractional quantum Hall effects and its physical implications
• The Gaffnian and Haffnian: physical relevance of non-unitary CFT for incompressible fractional quantum Hall effect
• Fractionalisation and dynamics of anyons at $\nu =n+1/3$ in fractional quantum Hall effect and their experimental signatures
• Statistical interactions and boson-anyon duality in fractional quantum Hall fluids
• Fractional quantum Hall effect from frustration-free Hamiltonians
• Non-Abelian Fermionization and the Landscape of Quantum Hall Phases
• Self-similarity of spectral response functions for fractional quantum Hall states
• The composite fermion theory revisited: a microscopic derivation without Landau level projection
• From the Gaffnian critical point to the incompressible 2/5 quantum Hall state
• Crossover from Integer to Fractional Quantum Hall Effect