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Free fermions with no Jordan-Wigner transformation
by Paul Fendley, Balazs Pozsgay
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Submission summary
| Authors (as registered SciPost users): | Paul Fendley · Balázs Pozsgay |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2310.19897v2 (pdf) |
| Date submitted: | 2023-12-18 12:26 |
| Submitted by: | Fendley, Paul |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
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.
Current status:
Reports on this Submission
Report #2 by Bernard Nienhuis (Referee 2) on 2024-1-22 (Invited Report)
- Cite as: Bernard Nienhuis, Report on arXiv:2310.19897v2, delivered 2024-01-22, doi: 10.21468/SciPost.Report.8442
Strengths
1 A very detailed and complete analysis of a new quantum chain.
2 The place in the brief history of the subject is well documented in the text.
Weaknesses
The paper is highly technical. It takes a lot of stamina to read it through, and diagonal reading is virtually impossible.
Report
This interesting paper discusses a new quantum chain, with a Hamiltonian trilinear in Pauli operators. It has the property that its spectrum is that of free fermions, in which the fermion energies are the roots of a polynomial. Remarkably it combines this property with supersymmetry and has two supersymmetry generators. The model shares these properties with several others in the literature, and in fact interpolates between two of them.
The model is analysed extensively, resulting in a list of local and non-local conserved quantities, as well as the spectrum. The analysis is highly technical, but to a large extent self contained.
In spite of the technicality in most places it is clear where the text is heading. I consider it very positive that the connection with related models already in the literature is msde very explicit.
In my opinion this paper brings together several research lines in mathematical statistical physics: integrability, supersymmetry, free fermions, and possibly Cooper pairs.
It is written clearly, and the derivations can be reproduced, with only small number of necessary references.
I recommend publication in SciPost.
Requested changes
1)
In the title: Aside from the ugly "with no" I find the title misleading. The Jordan-Wigner transformation is quite explicit in eq. (2.5). What is different from the well known applications of this transformation is that the Hamiltonian is not bilinear in the resulting (Majorana) fermion operators. What I find the most important properties of the paper is that it deals with (i) free fermions, (ii) supersymmetry, (iii) explicit spectrum and (iv) explicit conserved quantities, local and non-local.
2) Between eq. (2.2) and (2.3) it says: "If the Hamiltonian is derived via (2.1) then ${\cal H}_2$ is always the next local charge..."
What is the meaning here of always? Always under what variation? And what is kept fixed; not only (2.1) I am convinced, but that is what is implied.
Something similar plays in page 4 line 6: in "for all known solutions of (2.3) they do exist", the side conditions are not clear, presumably (2.2) and the embedded expression for $\cal H$.
3) Sec 2.4 is very casual about the Hamiltonian being equal to the square of two supersymmetry generators up to a different additive constant, stating that the larger constant is less a useful bound (I would say more useful). It does not spend a word on the fact that the lower bound of the two bounds can not be reached, which implies that there are no supersymmetry singlets with respect to the corresponding generator. And if the effective groundstates are singlets with respect to one of the generators (the one with the larger constant), they must form doublets with respect to the other generator.
Report #1 by Anonymous (Referee 1) on 2024-1-12 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2310.19897v2, delivered 2024-01-12, doi: 10.21468/SciPost.Report.8401
Strengths
1-Relevant new materials and methods to the area of exact integrable models
2- Possible extensions and applications of the ideas presented in the paper
2-Clear presentation
Report
Most of the known quantum chains with a free fermionic spectrum are the ones
that after a Jordan-Wigner transformation brings the quantum chain in a sum
of bilinear fermion operators. Some exceptions however appear in the
literature. These are models that although not bilinear after the
Jordan-Wigner transformation, also show a free-fermion eigenspectra.
Two examples of models with three-spin multispin interactions are given by the
Hamiltonian DFNR (ref. [9]) and FFD (ref. [10]). The connection between these
quantum chains is not known. In the present paper the authors introduce an
extended model that interpolate both Hamiltonians continuously, clarifying the
free-fermion nature of these exceptional models.
The proof of their results were obtained by exploiting some general results
(refs.[11-12], for free-fermion defined in graphs. They also show that the
free-fermionic excitations are obtained from the roots of a polynomial.
The paper is well written and the results are significant for the area of
exact integrable models. We recommend the publication.
Suggestion: The title seems misleading, perhaps "Free fermions models not solved by Jordan-Wigner transformation", or something on this line ...
Question: Is it possible to write in the paper the general recursion relations
for the polynomial p(u)?, from (4.2) and (4.3)? In the positive case it will
be interesting to show this relation.