Fractional quantum Hall effect at ν=2+4/9

Abstract

We consider the fractional quantum Hall effect at the filling ν = 6/17, where experiments have observed features of incompressibility in the form of a minimum in the longitudinal resistance. We propose a parton state denoted as "321 3 " and show it to be a feasible candidate to capture the ground state at ν = 6/17. We work out the low-energy effective theory of the 321 3 edge and make several predictions for experimentally measurable properties of the state which can help detect its underlying topological order. Intriguingly, we find that the 321 3 state likely lies in the same universality class as the state obtained from composite-fermionizing the 1+1/5 Laughlin state.

A non-Abelian parton state for the $ν=2+3/8$ fractional quantum Hall effect

“…In comparison, the32 2 1 4 state has an energy of −0.3993(2) for the same interaction, where the number in the parenthesis is the statistical uncertainty in the Monte Carlo estimate of the energy of thē 32 2 1 4 state. Although not definitive, this level of agreement is comparable with that of other candidate states in the SLL [18,23]. Next, we turn to charge and neutral gaps.…”

“…Then21 3 states (this notation is elucidated in detail in Sec. 1.2) for n = 1, 2, 3 capture the FQHE seen at 8/3, 5/2 and 2+6/13 respectively [1,22,23] (see also Appendix B where we show new results to further demonstrate the viability of the321 3 state for 2 + 6/13). Furthermore, then2 2 1 4 states for n = 1, 2 capture the experimentally observed plateaus at 5/2 and 12/5 [20].…”

“…The g(r) of both states show oscillations that decay at long distances, which is a characteristic feature of an incompressible state [56][57][58]. The agreement between the g(r) of the exact ground state and the parton state is on par with trial states at other filling factors in the SLL [23,59]. We note that the32 2 1 4 state shows a "shoulder"-like feature in the g(r) at short to intermediate lengths, which is considered a typical fingerprint of clustering in non-Abelian states [27].…”

“…The neutral and charge gaps for N = 12, evaluated using exact diagonalization with both the spherical and disk pseudopotentials, are about 0.015 e 2 /(ε ) and 0.003 e 2 /(ε ) respectively. For comparison, these gaps are smaller than the corresponding gaps at 7/3, which indicates that the 2 + 3/8 state is more fragile compared to the 7/3 state [23]. The gap calculations indicate strong finite-size effects in the second LL as evidenced by the fact that the neutral gap is larger than the charge gap.…”

“…Here E(2l) is the exact ground state energy of N electrons at 2l, C(2l) is the average charging energy at 2l which accounts for the background contribution [23], and n q is the number of fundamental quasiholes (quasiparticles) produced upon the insertion (removal) of a single flux quantum in the ground state. As mentioned above, for the 1/r Coulomb interaction, C(2l) = 1/ l − n e 2 /(ε ) = 1/ Q e 2 /(ε ).…”