Tensor-Network Analysis of Root Patterns in the XXX Model with Open Boundaries

The manuscript concerns the integrable spin-1/2 XXX chain with arbitrary boundary magnetic fields. In the case of generic boundary fields, not even a U(1) symmetry survives, which severely challenges the use of traditional methods. Instead, more advanced techniques must be employed. One such approach is the off-diagonal Bethe ansatz. However, the resulting Bethe ansatz equations (BAE) become inhomogeneous in the sense that one site of the BAE involves a sum of two terms. This seemingly simple modification has drastic consequences: the BAEs can no longer be cast into a convenient logarithmic form, making their solution considerably more challenging.

The manuscript circumvents this problem by applying the previously developed zero root method, which focuses on the zeros of the transfer matrix eigenvalue rather than on the roots of the Q-function.

The method proposed by the authors can be summarized as follows. The starting point is that the transfer matrix is inherently a matrix product operator (MPO), and its action on a matrix product state (MPS) can therefore be computed efficiently. The authors exploit this by approximating the unique ground state of the open XXX Hamiltonian using a DMRG algorithm. Since the ground state is unique, the corresponding transfer matrix eigenvalue can be extracted simply by applying the transfer matrix to this state. By doing this for a variety of fixed values of the spectral parameter, the authors can reconstruct the full dependence on the spectral parameter of the transfer matrix eigenvalue of the ground state. The zeros of this polynomial can then be obtained straightforwardly using standard root-finding algorithms.

The authors subsequently use the scalability of this approach to extract the zero roots for intermediate system sizes up to N=100. In this way, they confirm the Bethe hypothesis for the zero roots proposed in [1] across the complete phase diagram of boundary parameters. From the transfer matrix eigenvalue, the Bethe roots can be reconstructed via the TQ relation. The Bethe roots of the ground state are classified into regular strings, (un-)paired line roots, and arc roots. The manuscript concludes with a detailed discussion of how the Bethe root configurations transform when moving between different boundary-parameter regimes.

I am, however, critical of the overall scope and significance of the work. The authors claim that they “present a breakthrough on a previously identified and long-standing research stumbling block.” In my view, the main achievement is limited to a numerical confirmation of the Bethe hypothesis for the zero roots that was already established in [1] and subsequently used therein to compute physical quantities such as the surface energy. In contrast to [1], the present work does not provide any new physical insights into the XXX model with generic open BCs.

Furthermore, no new Bethe hypothesis is obtained from which a clear potential benefit could be inferred. The authors claim that their work will “open a new pathway in an existing or new research direction, with clear potential for multi-pronged follow-up work.” However, the proposed follow-up work, as stated in the conclusion, amounts essentially to establishing a Bethe hypothesis for excited states or testing the method in more complex models. Both should have been done in the first place, as it would have constituted a genuinely new contribution beyond [1] and would have demonstrated a clear advantage of the present method. In detail: for the ground state, the same (indeed correct!) conclusions were already obtainable from small system sizes alone in [1]. Ground state root patterns are typically already visible at small sizes if no ground state crossings occur. Demonstrating that indeed no crossing occurs in the present case, appears to be the main outcome of the work. This is, in my opinion, not sufficient to justify publication. In regard to excited states, the root pattern of those indeed sometimes needs larger system sizes to be analysed. However, the work actively ignores the advantage of being able to go to higher system sizes by not classifying the root pattern of excited states.

On top of this, the obtained Bethe root configuration of the ground state is not analytically classified. For the arc roots, the authors do not provide any closed-form description that would allow for a starting point of non-standard analytic calculation.

Another concern arises from the presentation itself, which at times even calls the usefulness of the method into question. For example, the authors state: “Among the available data, the only quantity that can be regarded as fully reliable is
Λ(u) calculated by the DMRG method.” This naturally raises the question of why one should then compute zero roots and Bethe roots at all. If the precision of the method is called into question, I do not see the benefit in contrast to DMRG alone.

Also, the statement that the authors have chosen the nodes on page 12 in a very particular way in order to achieve sufficient precision for the zero roots needs to be quantified. If obtaining adequate precision is only possible when a pattern is already known, then it is unclear what advantage this approach has, compared to previous methods.

I also question the application of the methods of the modulus and the argument principle. While the framework is obvious from complex analysis, I am asking why directly solving the Bethe ansatz equations or the condition
Λ(u)=0 is deemed insufficient. In particular, it is unclear why the authors do not use their numerically obtained data as a very elaborated initial guess to solve the off-diagonal Bethe equations themselves by a root finding algorithm with high precision, e.g., using the open source Wolframscript language. This either should clearly give a no-doubt argument with very high precision, or if not possible, the authors should at least include a discussion on why this straigthforward approach fails. Instead, the authors focus on the benchmark of the relative error to the transfer matrix eigenvalue of DMRG, which intrinsically has errors, and also, as the authors point out correctly, it has not yet efficiently implemented for an arbitrary amount of precision.

In addition to the above, the authors focus exclusively on the simplest case: the XXX model, with the smallest local Hilbert space and a rational R-matrix. This choice is not very convincing. To demonstrate the real usefulness of the method, one should at least address the XXZ model with anisotropy, where the Bethe-root structure is much richer and one can challenge the DMRG algorithm by tuning q to criticality. This would provide a genuine stress test/benchmark of the method, for example, by obtaining Bethe-root configurations of the first few low-energy excitation directly for intermediate sizes (e.g. N∼100) without relying on the usual iterative procedure starting from small system sizes.

Finally, the manuscript does not discuss whether the method can be applied in situations where the transfer matrix eigenvalue is degenerate.

In the context of presenting the material, I also see room for improvement. In modern times, one does not rely on presenting the root data in 12 pages of tables in the appendix. A simple .txt file uploaded to an online repository like git would be a much more transparent way. No reader will actually type in/use the data presented in the current state, nor look at each row in great detail. Further, if the only goal of the paper is really just to present a method of getting the Bethe roots in a systematic way, then an example code of doing so should be provided, which is also not the case for the present manuscript.

In summary, while the idea of combining DMRG with the analysis of transfer matrix eigenvalues is interesting, the current manuscript fails to convincingly demonstrate the applicability and advantages of the proposed method. This is primarily due to the lack of results for different physical scenarios. I believe the idea itself deserves serious consideration, but this would require a substantially more extensive study, which would likely take significant additional effort before the manuscript could be reconsidered for publication.

[1] G.-L. Li, Y. Qiao, J. Cao, W.-L. Yang, K. Shi, Y. Wang, Exact surface energy and elementary excitations of the XXX spin-1/2 chain with arbitrary non-diagonal boundary fields.

1- Refined Research Proposal. ---> Show that the DMRG method can be applied to obtain a full Bethe Hypothesis e.g., the low energy excitations. ---> Show that the method remains useful for XXZ or higher-dimensional or higher rank models. ---> Use the obtained Bethe Hypothesis to actually calculate something physical i.e., demonstrate a physical application ---> Make a stress test: what is happening if DMRG is challenged e.g. in critical lattice models, for example, the critical regime of XXZ. Also, incorporate the case of degenerated transfer matrix eigenvalues.

Reject