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Partons as unique ground states of quantum Hall parent Hamiltonians: The case of Fibonacci anyons

by M. Tanhayi Ahari, S. Bandyopadhyay, Z. Nussinov, A. Seidel, and G. Ortiz

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Gerardo Ortiz
Submission information
Preprint Link: scipost_202207_00032v2  (pdf)
Date accepted: 2023-05-22
Date submitted: 2023-02-18 20:49
Submitted by: Ortiz, Gerardo
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We present microscopic, multiple Landau level, (frustration-free and positive semi-definite) parent Hamiltonians whose ground states, realizing different quantum Hall fluids, are parton-like and whose excitations display either Abelian or non-Abelian braiding statistics. We prove ground state energy monotonicity theorems for systems with different particle numbers in multiple Landau levels, demonstrate S-duality in the case of toroidal geometry, and establish complete sets of zero modes of special Hamiltonians stabilizing parton-like states, specifically at filling factor $\nu=2/3$. The emergent Entangled Pauli Principle (EPP), introduced in Phys. Rev. B {\bf 98}, 161118(R) (2018) and which defines the ``DNA'' of the quantum Hall fluid, is behind the exact determination of the topological characteristics of the fluid, including charge and braiding statistics of excitations, and effective edge theory descriptions. When the closed-shell condition is satisfied, the densest (i.e., the highest density and lowest total angular momentum) zero-energy mode is a unique parton state. We conjecture that parton-like states generally span the subspace of many-body wave functions with the two-body $M$-clustering property within any given number of Landau levels, that is, wave functions with $M$th-order coincidence plane zeroes and both holomorphic and anti-holomorphic dependence on variables. General arguments are supplemented by rigorous considerations for the $M=3$ case of fermions in four Landau levels. For this case, we establish that the zero mode counting can be done by enumerating certain patterns consistent with an underlying EPP. We apply the coherent state approach of Phys. Rev. X {\bf 1}, 021015 (2011) to show that the elementary (localized) bulk excitations are Fibonacci anyons. This demonstrates that the DNA associated with fractional quantum Hall states encodes all universal properties. Specifically, for parton-like states, we establish a link with tensor network structures of {\em finite} bond dimension that emerge via root level entanglement.

Author comments upon resubmission

Dear Editor,

We thank you for taking care of our paper, and the referees for such positive reviews. It is our
understanding that we are required to put the latest revised version of our manuscript in the arxiv. I have already done it. On the
other hand, we would like to highlight all the changes introduced in the revised manuscript
prompted by the referees comments. It is simpler for us to provide a (color) highlighted version
of our revised manuscript. Is that possible and if so, how do we send such a version to you ?

Response to Third referee:

We thank the referee for a positive review. As the referee emphasized this present work is
part of a program that our group started a few years back. However, this is not a review of
prior work. Our work centers on new original results, including a complete new theory of multivariate
non-holomorphic polynomials relevant for parton-like states, technical results (theorems)
on monotonicity of ground states energies in special k-body Hamiltonians and S-duality, and
derivation of a non-Abelian fluid with Fibonacci topological excitations (rigorously proved
within the coherent state approach), among others. Since we understand that people may
get confused about what is new and what is not, we have modified the Introduction to
make it more clear.

Published as SciPost Phys. 15, 043 (2023)

Reports on this Submission

Report 3 by Mikael Fremling on 2023-4-13 (Invited Report)


I have read the revised parts of the manuscript, And I think the authors have in a satisfactory way addressed all of my concerns.

I now recommend publications in SciPost.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 2 on 2023-3-23 (Invited Report)


The authors have convinced me of the novelty of their work. I now recommend publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Report 1 by Bo Yang on 2023-3-19 (Invited Report)

  • Cite as: Bo Yang, Report on arXiv:scipost_202207_00032v2, delivered 2023-03-19, doi: 10.21468/SciPost.Report.6923


The revised manuscript is better in terms of letting the readers understand the new contributions of the current manuscript.

The authors did not respond to one technical questions I asked in the previous report:

“The full two-fermion basis for NL = 4 LLs is of dimension 40”, with respect to the definition of the two-fermion basis. If they are just states containing two fermions, then of course the dimension grows exponentially with the system size.

Thanks to private discussions with Li Chen from WUSTL, I now understand here the authors are defining the two-fermion basis as states containing two fermions that are highest weight (so lifting the center of mass degeneracy) with relative angular momentum penalised by the TK Hamiltonian.

While this definition should be there when we look carefully at the mathematical expressions, it is not obvious and one has to go through those different notations. Thus for readers who are not keen in reproducing the technical machinery and just want to get the general idea, it is better for the "two-fermion basis" to be properly defined by physically intuitive terms.

Requested changes

Proper definition of "the two-fermion basis" as the basis of highest weight (center of mass angular momentum 0) states with relative angular momentum having finite energy cost with respect to the TK Hamiltonian.

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

Author:  Gerardo Ortiz  on 2023-04-08  [id 3565]

(in reply to Report 1 by Bo Yang on 2023-03-19)
answer to question

Dear Dr Yang,
Thank you very much for your positive and encouraging report. We have now added explanation about the way this 40 vectors basis is generated. Essentially, the basis is cutoff by the number of Landau levels one uses (in this case 4 Landau levels) and the maximum m (7 in this case) such that matrix elements of the TK Hamiltonian are non-vanishing. The new version of the manuscript has been uploaded in the arxiv. (Section II C and Appendix C have been modified accordingly).
Best regards,
Gerardo Ortiz

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Anonymous on 2023-02-20  [id 3383]

Dear Benjamin,
Please find attached a pdf file with highlighted (in blue) changes addressing reviewers comments. Let me know if you receive it in good shape. Best, Gerardo