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arXiv:2010.08965 · SciPost Physics

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

Ajit Coimbatore Balram

Published Apr 15, 2021• 32 citations
arXivHTMLPDF

Abstract

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.

DOI
10.21468/scipostphys.10.4.083

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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.
We consider the fractional quantum Hall effect at the filling factor $\ensuremath{\nu}=4/11$, where two independent experiments have observed a well-developed and quantized Hall plateau. We examine the Abelian state described by the ``42‾134\overline{2}{1}^{3}4213'' parton wave function and numerically demonstrate it to be a plausible candidate for the ground state at $\ensuremath{\nu}=4/11$. We work out the low-energy effective theory of the 42‾134\overline{2}{1}^{3}4213 edge and make predictions for experimentally measurable properties of the state.
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.
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 ``32‾133\overline{2}{1}^{3}3213'' 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 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.
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.
The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer. The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE. An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions. One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality. Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus. Having explicit torus wave functions allows us to use the methods developed by Read and Read \\& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information. We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.