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

## TL;DR

This paper reveals that the Abelian Jain composite fermion state effectively describes the low-energy physics of the non-Abelian Gaffnian state in the $
u=2/5$ fractional quantum Hall effect, highlighting their close relationship.

## Contribution

It demonstrates that the Jain phase can be viewed as an effective Abelian low-energy theory of the Gaffnian phase under short-range interactions.

## Key findings

- Both states share the same low-energy quasielectrons from the Laughlin phase.
- Differences arise due to interaction range affecting excitation energies.
- Jain phase emerges as an effective Abelian description of the Gaffnian phase.

## 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.