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

Submission summary

 As Contributors: Ajit Coimbatore Balram Arxiv Link: https://arxiv.org/abs/2010.08965v1 (pdf) Date submitted: 2020-10-20 04:31 Submitted by: Coimbatore Balram, Ajit Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational Approaches: Theoretical, Computational

Abstract

Fascinating structures have arisen from the study of the fractional quantum Hall effect (FQHE) at the even denominator fraction of $5/2$. We consider the FQHE at another even denominator fraction, namely $\nu=2+3/8$, where a well-developed and quantized Hall plateau has been observed in experiments. We examine the non-Abelian state described by the "$\bar{3}\bar{2}^{2}1^{4}$" parton wave function and numerically demonstrate it to be a feasible candidate for the ground state at $\nu=2+3/8$. We make predictions for experimentally measurable properties of the $\bar{3}\bar{2}^{2}1^{4}$ state that can reveal its underlying topological structure.

Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission 2010.08965v2 on 4 March 2021

Submission 2010.08965v1 on 20 October 2020

Reports on this Submission

Strengths

1- comprehensive analysis of the CF parton wave function for the nu=2+3/8 quantum Hall effect, including quantum Monte-Carlo and exact diagonalization calculations
2- discussion of experimental signatures distinguishing the proposed trial state from alternative wave functions
3- nice review of the relevant background material
4- additional results on related wave functions presented in the appendix

Weaknesses

1- Some features of the proposed candidate wave function show unsatisfactory agreement with the exact diagonalization results: in particular:
a) the pair wave function has a pronounced shoulder at short distances, which is absent in the exact results
b) Its trial energy is higher than that of the competing Bonderson-Slingerland state

Report

The paper provides a comprehensive analysis of the different trial wave functions for the FQHE at nu=2+3/8, as well as discussing their topological properties. The results are a valuable addition to the relevant literature, and the paper is suitable for publication in SciPost.

While I agree with the author's argument that the trial wave functions should not be judged uniquely on the basis of accuracy of the energy, I wonder if the proposed state corresponds to a sufficiently robust quantum Hall state. Given the numerics on the sphere at N=12, and flux 2l=35, the correlation function of the ground state shown in Fig. 3 does reveal a large degree of oscillations even at large particle separation. This may well be a signature of instability. In order to reassure me in this respect, the author should add corresponding data for the aternative topological sectors, also, displaying the correlation functions for the shift of the competing Bonderson-Slingerland wave function.
The level of agreement between the correlation functions of the proposed parton state and the exact eigenstate should also be put in relation to that obtained for other trial wave functions in the second Landau level. For example consider the Moore-Read state where excellent agreement is reached at short distances, especially when the pair wave function is optimized (see Moller, G. & Simon, S. H. Paired composite-fermion wave functions. Phys. Rev. B 77, 075319 (2008)).

Some additional consideration should be given to these points:

- The author states that the overlap with the exact state or the entanglement spectrum could not be given, as they do not possess a Fock space representation. However, both these quantities can be efficiently evaluated using Monte-Carlo methods, also. The overlap calculation is standard, and for the ES, see e.g.: Rodriguez, I. D., Simon, S. H. & Slingerland, J. K. Evaluation of Ranks of Real Space and Particle Entanglement Spectra for Large Systems. Phys. Rev. Lett. 108, 256806 (2012).

-In the background section, it felt odd that no reference was made to the relevant trial states for the nu=3/8 state in the lowest Landau-level. Indeed, it appears to me that the 3/2 filled composite fermion LL was initially proposed as a candidate in the LLL.

Requested changes

1- Add a comparison on the correlation functions for the exact ground state at the shift of the Bonderson-Slingerland wave function.
2- Quantify arguments about geometries for N=12(18) being (in)accessible by quoting the relevant Hilbert space dimensions of the largest $L_z$ subspace.
3- Add a calculation of the overlap for the parton state with the exact ground state, and possibly its entanglement spectrum
4- Amend Fig. 2 so that figure captions do not overlap with axis ticks to increase clarity.

• validity: high
• significance: good
• originality: good
• clarity: top
• formatting: excellent
• grammar: perfect

Author Ajit Coimbatore Balram on 2021-01-23
(in reply to Report 1 by Gunnar Moller on 2021-01-22)
Category:
pointer to related literature

Thanks for carefully reading the manuscript, suggesting changes, and recommending publication. Here is a detailed response to the comments:

In the background section, it felt odd that no reference was made to the relevant trial states for the nu=3/8 state in the lowest Landau-level. Indeed, it appears to me that the 3/2 filled composite fermion LL was initially proposed as a candidate in the LLL. The reason for leaving out the 3/8 state in the LLL (which as the referee points out can be described as the 3/2 filled composite fermion LL state) is that the most recent experiments (see Samkharadze et. al. https://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.081109 and Pan et al. https://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.041301) suggest that there is no FQHE in the LLL at 3/8. Incompressibility between 1/3 and 2/5 in the LLL has only been established at filling factors 4/11 and 5/13.

Requested changes: 1- Add a comparison on the correlation functions for the exact ground state at the shift of the Bonderson-Slingerland wave function. This data is already available in the literature (see Fig. 1 of https://journals.aps.org/prb/abstract/10.1103/PhysRevB.95.125302) and has now been referred to in the text. The pair wave function of the exact SLL Coulomb ground state at the BS shift also shows oscillations at large distances (though less pronounced than that at the proposed parton shift [in terms of Hilbert space dimensions, the system at the BS shift is smaller than that at the parton shift]). Note that the BS pair wave function also has a pronounced shoulder at short distances, which is again absent in the exact results.

2 - Quantify arguments about geometries for N=12(18) being (in)accessible by quoting the relevant Hilbert space dimensions of the largest Lz subspace. We have now stated the relevant Hilbert space dimensions for the N=12 and 18 systems in the text.

3- Add a calculation of the overlap for the parton state with the exact ground state, and possibly its entanglement spectrum The Monte Carlo estimate of the overlap of the trial state with the exact SLL Coulomb ground state is 0.63(4). This estimate has now been added to the text. The reason for not including the real-space entanglement spectrum (ES) is that for states carrying modes in both the forward and backward directions that is not described as the ground-state of a model Hamiltonian (our parton state is of this kind), it is hard to glean much information from the ES (see, for example, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.95.125302).

4- Amend Fig. 2 so that figure captions do not overlap with axis ticks to increase clarity. The figure captions do not overlap with the axis ticks. Does the referee mean axis labels instead of figure captions? We have provided a vector graphics version of Fig. 2 which can be zoomed-in for clarity.

Intended changes: 1) Include a Monte Carlo estimate of the overlap of the proposed parton state with the exact SLL Coulomb ground state. 2) Include Hilbert space dimensions of the N=12 and 18 systems. 3) Add the Monte Carlo estimate of the overlap of the parton state with the exact SLL Coulomb ground state. 4) Add a reference to G. Moller, S. H. Simon, Paired composite-fermion wave functions. Phys. Rev. B 77, 075319 (2008).