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Exceptional Points in the Baxter-Fendley Free Parafermion Model
by Robert A. Henry, Murray T. Batchelor
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Submission summary
| Authors (as registered SciPost users): | Robert Henry |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202301_00039v1 (pdf) |
| Date submitted: | 2023-01-30 09:57 |
| Submitted by: | Henry, Robert |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Certain spin chains, such as the quantum Ising chain, have free fermion spectra which can be expressed as the sum of decoupled two-level fermionic systems. \textit{Free parafermions} are a simple generalisation of this idea to $\Zn$-symmetric models. In 1989 Baxter discovered a non-Hermitian but $\PT$-symmetric model directly generalising the Ising chain which was much later recognised by Fendley to be a free parafermion spectrum. By extending the model's magnetic field parameter to the complex plane, we show that a series of exceptional points emerges, where the quasienergies defining the free spectrum become degenerate. An analytic expression for the locations of these points is derived, and various numerical investigations are performed. These exceptional points also exist in the Ising chain with a complex transverse field. Although the model is not in general $\PT$-symmetric at these exceptional points, their proximity can have a profound impact on the model on the $\PT$-symmetric real line. Furthermore, in certain cases an exceptional point may appear on the real line (with negative field).
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-3-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202301_00039v1, delivered 2023-03-14, doi: 10.21468/SciPost.Report.6893
Strengths
- draws attention to a model still little explored in the literature
- EPs are easily determined
Weaknesses
see report
Report
The authors consider a simple solvable Z(N) Hamiltonian introduced by Baxter in the late 1980s [1]. This is one of the simplest possible generalizations of the Ising chain in a transverse field. Recently, the model was solved in terms of free parafermions by Fendley [3] and since then there has been renewed interest in its solution, although the model is non-Hermitian for N>2.
In the present manuscript, the authors' main result is the determination of the so-called exceptional points (EP). These points occur when eigenvalues degenerate and the associated eigenvectors become parallel (coalesce). They are relevant for non-Hermitian Physics. For Baxter's model the EPs are given by equations (13) and (14). The transverse field is general complex at an EP. The coalescence of the eigenvectors is verified numerically.
I think the results in the paper are nice and can be published in SciPost, provided that the following below are addressed.
Requested changes
1- The determination of EPs is quite simple, as an analytical form of the quasienergies - eqs.(4,5) - is known. Therefore, I think the authors could make an effort to understand the coalescence of the eigenvectors analytically/algebraically. This should be feasible at least for for the case $N=2$.
2- The authors should be precise about which correlation functions are divergent for the Baxter-Fendley model, and how this divergence is explained by the exceptional points. Although this seems to be the main motivation of the paper, little is said or done about it.
Other minor comments:
3- Verify Fig. 2. For example, the top right panel ($\phi=0$) seems inconsistent with Eq. (4).
4- Verify Fig. 3. The pictures seems to be associated with L=5 not L=4.
5- The quantity $\Delta\epsilon_{01}$ in Fig. 3 should be clearly defined, maybe in the main text. Since the quasienergies are complex, it is not clear how the authors defined smallest and second-smallest quasienergies. Same for $\Delta E_{01}$ in Fig. 6.
6- I think Fig. 4 should be improved. It is not very clear what is plotted. Since this picture is suposed to confirm an important result of the paper, I think readers would benefit from more clear and well defined labels.
7- It seems that there is a misprint in the definition of $\lambda$ in Fig. 7 , as it does not give $\lambda=1$ when $\gamma=1/2$.
8- In the Conclusion, the authors mention that the number of EPs is $NL$. Clarify if this number includes the trivial EPs associated $k=0$ and $k=\pi$ . Maybe this number should be mentioned in Sec. 3 (see also remark 4-).
Report #1 by Anonymous (Referee 1) on 2023-3-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202301_00039v1, delivered 2023-03-09, doi: 10.21468/SciPost.Report.6874
Strengths
1- Study a set of interesting many-body model with two ingredients, parafermion degrees of fredonn and non hermitian .
2- Give a possible explanation of unusual effects reported on these quantum
chains.
3- It can motivated further studies in the area of non hermitean quantum mechanics
Report
The authors study a set of Z(N) symmetric spin chains with a free-energy
eigenspectrum. It is
expressed as the composition of independent quasi energies.
These hamiltonians introduced by Baxter and solved by Fendley have
exclusion effects in the spectral combination of the quasi-energies, that
depend on N, and they are called free parafermion models. For N>2 they are
non hermitean, and may also show unusual physics as compared with usual
hermitian quantum mechanics.
In a previous publication the authors observe an usual behavior of
the correlation function, for small chains of the N=3 model. In this paper
the authors give a possible explanation for this behavior. For that, they
extend the usual real value coupling constant defining the quantum
chain to complex values, as is usual for the study of Yang-Lee zeros of
partition function, on this way the quasi-energies that are all disting for
real coupling values can become equal at special values of complex coupling
constant parameters.
At these points where the eigenspectrum becomes globally degenerated they
verify numerically for small lattice sizes, but they also verified that
these degenerated states are parallel, and they call this point as an
Exceptional Point. The authors mention that from general studies in non
hermitian quantum mechanics unusual effects happens in these cases.
They give numerical evidence that in the thermodynamic limit , for the N=3
case, exceptional points that happens for imaginary couplings can reach
a real value for the coupling constant, possibly the critical point.
In my opinion the paper should be published since it exemplifies an
exactly many-body integrable model with the unusual effects predicted in
non-hermitian quantum mechanics, and my be useful as a toy model in this
area of research. I have however some remarks.
1) At these EP, the coalescence of the eigenvector is not just the appearance
of a standard Jordan cell. If this would be the case, I think the authors
should mention.
2) Equation (25) has an important misprint