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Exact generalized Bethe eigenstates of the non-integrable alternating Heisenberg chain
by Ronald Melendrez, Bhaskar Mukherjee, Marcin Szyniszewski, Christopher J. Turner, Arijeet Pal, Hitesh J. Changlani
Submission summary
| Authors (as registered SciPost users): | Hitesh Changlani |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202503_00022v1 (pdf) |
| Date submitted: | 2025-03-12 20:54 |
| Submitted by: | Changlani, Hitesh |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Exact solutions of quantum lattice models serve as useful guides for interpreting physical phenomena in condensed matter systems. Prominent examples of integrability appear in one dimension, including the Heisenberg chain, where the Bethe ansatz method has been widely successful. Recent work has noted that certain non-integrable models harbor quantum many-body scar states, which form a superspin of regular states hidden in an otherwise chaotic spectrum. Here we consider one of the simplest examples of a non-integrable model, the alternating ferromagnetic-antiferromagnetic (bond-staggered) Heisenberg chain, a close cousin of the spin-1 Haldane chain and a spin analog of the Su-Schrieffer-Heeger model, and show the presence of exponentially many zero-energy states. We highlight features of the alternating chain that allow treatment with the Bethe ansatz (with important modifications) and surprisingly for a non-integrable system, we find simple compact expressions for zero-energy eigenfunctions for a few magnons including solutions with fractionalized particle momentum. We discuss a general numerical recipe to diagnose the existence of such generalized Bethe ansatz (GBA) states and also provide exact analytic expressions for the entanglement of such states. We conclude by conjecturing a picture of magnon pairing which may generalize to multiple magnons. Our work opens the avenue to describe certain eigenstates of partially integrable systems using the GBA.
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Referee Report on the Manuscript:
"Exact generalized Bethe eigenstates of the non-integrable alternating Heisenberg chain"
I have read the manuscript and have several substantive concerns regarding its
claims and conclusions.
The authors assert the existence of "partial integrability" in the alternating
Heisenberg chain (AHC) for a larger number of magnons. Upon closer
examination, what is actually demonstrated is that in spin models with
conserved particle number (i.e., magnon number), one- and two-particle states
can be constructed in accordance with standard quantum mechanical
principles. The two-magnon states presented are simply conventional
two-particle scattering states. This behavior is generic and does not
constitute a signature of integrability.
In the case of three magnons, the authors construct specific states as
superpositions of plane waves, with additional matching conditions. Similar
constructions are attempted for higher numbers of magnons, but all are
restricted to the special case of zero-energy states. Although these states
formally resemble Bethe ansatz-type wavefunctions, the results apply only to
narrow and atypical subsets of the full spectrum.
The authors do succeed in analytically constructing a large number of
zero-energy states, consistent with prior numerical diagonalizations. However,
these zero-energy states lie in the "middle" of the many-body spectrum. The
manuscript does not convincingly articulate why these states are of particular
physical interest or relevance.
Furthermore, the manuscript contains a significant misconception. The authors
state:
“The key reason for the success of the Bethe ansatz is the tractability of the
momenta of the particles in an integrable model – when two particles collide,
their momenta after the collision are simply exchanged.”
This characterization is misleading. The simple exchange of momenta upon
two-particle scattering is a common feature of many quantum systems, not
exclusive to integrable ones. The true hallmark of integrability is the
absence of intrinsic many-particle scattering — that is, the factorization of
the full scattering matrix into two-particle S-matrices. Importantly, the
Yang-Baxter equation provides a necessary (but not sufficient) condition for
such factorization.
Given these points, I do not consider the manuscript suitable for publication
in SciPost in its current form.
Recommendation
Reject
Report
In the present manuscript, the authors explore the spin 1/2 alternating ferromagnetic-antiferromagnetic Heisenberg chain and identify exactly solvable states. They introduce a generalized Bethe Ansatz (GBA) calculation capable of generating many-particle zero-energy scar states, which they systematically analyze both theoretically and numerically. While the manuscript presents comprehensive discussions, several key points require further clarification for publication:
The manuscript rigorously presents wavefunctions and GBA equations for one-, two-, and three-magnon sectors (sections II-IV), and solves the GBA numerically up to the four-magnon sector (section V). However, it remains unclear whether these scar states persist in arbitrarily large N-magnon sectors—an assertion that warrants explicit proof or argumentation.
The authors assert area-law entanglement properties for the studied scar states observed in few-magnon states. Given the thermodynamic limit, where all zero particle density states exhibit area-law entanglement trivially, it is crucial to investigate how the entanglement scales in scar states with finite magnon densities. Specifically, understanding the scaling behavior towards the thermodynamic limit, including potential deviations from area law such as critical or volume-law entanglement, would enhance the manuscript's completeness.
As scar states are distinguished by their violation of the Eigenstate Thermalization Hypothesis (ETH), it would be insightful to explore observable quantities—like magnetization—that differentiate scar states from thermal equilibrium states. This analysis would further elucidate the physical implications and experimental relevance of the identified scar states.
Overall, with these modifications and clarifications, the manuscript presents a valuable contribution to the field and deserves publication.
Recommendation
Ask for minor revision