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Free fermions beyond Jordan and Wigner

by Paul Fendley, Balazs Pozsgay

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Submission summary

Authors (as registered SciPost users): Paul Fendley · Bal√°zs Pozsgay
Submission information
Preprint Link:  (pdf)
Date accepted: 2024-03-25
Date submitted: 2024-03-13 18:34
Submitted by: Fendley, Paul
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
Approach: Theoretical


The Jordan-Wigner transformation is frequently utilised to rewrite quantum spin chains in terms of fermionic operators. When the resulting Hamiltonian is bilinear in these fermions, i.e. the fermions are free, the exact spectrum follows from the eigenvalues of a matrix whose size grows only linearly with the volume of the system. However, several Hamiltonians that do not admit a Jordan-Wigner transformation to fermion bilinears still have the same type of free-fermion spectra. The spectra of such ``free fermions in disguise" models can be found exactly by an intricate but explicit construction of the raising and lowering operators. We generalise the methods further to find a family of such spin chains. We compute the exact spectrum, and generalise an elegant graph-theory construction. We also explain how this family admits an N=2 lattice supersymmetry.

Author comments upon resubmission

We thank the referees for the comments. We have made several minor changes, and hope that it is sufficient.

We have a comment regarding the "weakness" that Referee 2 stated ("The paper is highly technical. It takes a lot of stamina to read it through, and diagonal reading is virtually impossible.")

We recognise that we wrote a paper that does not allow for diagonal reading, but do not believe it is a weakness. We believe our results are interesting enough to make them worthy of a careful proof, and that we would not be serving readers well by omitting them. Our introduction and conclusion state the results in a clear and concise fashion, and we hope that this suffices for casual readers. In general we do not consider this style of theoretical physics to be weak, and hope that the SciPost editors agree.

List of changes

Regarding the requested changes from Referee 2:

1 (also from Referee 1). We note that while there are Jordan-Wigner fermions in the paper, they are not the free-fermion operators (as the original title indicated). We modified the title to avoid this confusion.

2. We clarified this sentence, without giving all details of the construction. This point is not central to the rest of the paper, and so we did not specify all the mathematical details, for example the regularity condition in question. We refer to the work [14] for details.

3. We note that the larger constant is less useful because Q^2 = const + H. Since eigenvalues of Q are non-negative, - const is a lower bound on the spectrum of H. We clarified the text, and also added a sentence on forming doublets.

Regarding the question from Referee 1:

There is indeed an analogous recursion relation for the polynomials, and we have added the relation to the paper in
equation (4.6).

Published as SciPost Phys. 16, 102 (2024)

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