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arXiv:2105.07462 · Physical Review Research

Parton wave function for the fractional quantum Hall effect at ν=6/17

Ajit Coimbatore Balram, A. Wójs

Published Jul 23, 2021• 17 citations
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Abstract

We consider the fractional quantum Hall effect at the filling $\ensuremath{\nu}=6/17$, where experiments have observed features of incompressibility in the form of a minimum in the longitudinal resistance. We propose a parton state denoted as ``$3\overline{2}{1}^{3}$'' and show it to be a feasible candidate to capture the ground state at $\ensuremath{\nu}=6/17$. We work out the low-energy effective theory of the $3\overline{2}{1}^{3}$ 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 $3\overline{2}{1}^{3}$ state likely lies in the same universality class as the state obtained from composite-fermionizing the $1+1/5$ Laughlin state.

DOI
10.1103/physrevresearch.3.033087

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We consider the fractional quantum Hall effect at the filling \ensuremathν=6/17\ensuremath{\nu}=6/17\ensuremathν=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 ``32‾133\overline{2}{1}^{3}3213'' and show it to be a feasible candidate to capture the ground state at \ensuremathν=6/17\ensuremath{\nu}=6/17\ensuremathν=6/17. We work out the low-energy effective theory of the 32‾133\overline{2}{1}^{3}3213 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 32‾133\overline{2}{1}^{3}3213 state likely lies in the same universality class as the state obtained from composite-fermionizing the 1+1/51+1/51+1/5 Laughlin state.
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A non-Abelian parton state for the ν=2+3/8ν=2+3/8ν=2+3/8 fractional quantum Hall effect
Fascinating structures have arisen from the study of the fractional quantum Hall effect (FQHE) at the even denominator fraction of 5/2 5 / 2 . We consider the FQHE at another even denominator fraction, namely \nu=2+3/8 ν = 2 + 3 / 8 , where a well-developed and quantized Hall plateau has been observed in experiments. We examine the non-Abelian state described by the `` \bar{3}\bar{2}^{2}1^{4} 3 ‾ 2 ‾ 2 1 4 " parton wave function and numerically demonstrate it to be a feasible candidate for the ground state at \nu=2+3/8 ν = 2 + 3 / 8 . We make predictions for experimentally measurable properties of the \bar{3}\bar{2}^{2}1^{4} 3 ‾ 2 ‾ 2 1 4 state that can reveal its underlying topological structure.
The fractional quantum Hall effect at $\ensuremath{\nu}=2+3/8$, which has been definitively observed, is one of the last fractions for which no viable explanation has so far been demonstrated. Our detailed study suggests that it belongs to a new class of exotic states described by the Bonderson-Slingerland wave function. Its excitations are non-Abelian anyons similar to those of the well studied Pfaffian state at 5/2, but its wave function has a more complex structure. Using the effective edge theory, we make predictions for various measurable quantities that should enable a confirmation of the underlying topological order of this state.
We investigate the feasibility of many candidate quantum Hall states for two-component bosons in the lowest Landau level. We identify interactions for which spin-singlet incompressible states occur at filling factors $\ensuremath{\nu}=2/3$, 4/54/54/5, and 4/34/34/3, and partially spin-polarized states at filling factors 3/43/43/4 and 3/23/23/2, where ``spin'' serves as a generic label for the two components. We study ground states, excitations, edge states, and entanglement spectrum for systems with up to 16 bosons and construct explicit trial wave functions to clarify the underlying physics. The composite fermion theory very accurately describes the ground states as well as excitations at $\ensuremath{\nu}=2/3$, 4/54/54/5, and 3/43/43/4, although it is less satisfactory for the $\ensuremath{\nu}=3/2$ state. For $\ensuremath{\nu}=4/3$ a ``non-Abelian spin-singlet'' state, which is the exact ground state of a three-body contact interaction, has been proposed to occur even for a two-body contact interaction; our trial wave functions are very accurate for the excitations of the three-body interaction, but they do not describe the excitations of the two-body interaction very well. Instead, we find that the $\ensuremath{\nu}=4/3$ state is more likely to be a spin-singlet state of reverse-flux-attached composite fermions at filling ${\ensuremath{\nu}}^{*}=4$. We also consider incompressible states at integral filling factors $\ensuremath{\nu}=1$ and 2. The incompressible state at $\ensuremath{\nu}=1$ is shown to be well described by the parton-based Jain spin-singlet wave function, and the incompressible state at $\ensuremath{\nu}=2$ as the spin-singlet state of reverse-flux-attached composite fermions at ${\ensuremath{\nu}}^{*}=2$, which provides an example of the bosonic integer topological phase.
The fractional quantum Hall effect (FQHE) in the second Landau level (SLL) likely stabilizes non-Abelian topological orders. Recently, a parton sequence has been proposed to capture many of the fractions observed in the SLL [A. C. Balram, SciPost Phys. 10, 083 (2021)]. We consider the first member of this sequence which has not yet been studied, which is a non-Abelian state that occurs at 4/114/114/11. As yet FQHE in the SLL at this fraction has not been observed in experiments. Nevertheless, by studying its competition with other candidate FQHE states in the SLL we show that this parton state might be viable. We also make predictions for experimentally measurable properties of the parton state which can distinguish it from other topological orders.
Trial wave functions which, in the thermodynamic limit, describe states relevant to the fractional quantum Hall effect are explicitly constructed for up to eight electrons. Estimates are given for the ground-state energies and pair-correlation functions and for the low-lying excitation energies for states associated with the filling factors \ensuremath{\nu}=(1/3), (2/7), and (2/5). The results help explain why the filling factors \ensuremath{\nu}=${\ensuremath{\nu}}_{m}$=m/(2m+1), m=1,2,..., and \ensuremath{\nu}=1-${\ensuremath{\nu}}_{m}$ show strong quantum Hall effects.