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Non-linear integral equations for the XXX spin-1/2 quantum chain with non-diagonal boundary fields
by Holger Frahm, Andreas Klümper, Dennis Wagner, Xin Zhang
Submission summary
| Authors (as registered SciPost users): | Holger Frahm · Andreas Klümper |
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| Preprint Link: | https://arxiv.org/abs/2502.07229v1 (pdf) |
| Date submitted: | April 3, 2025, 4:17 p.m. |
| Submitted by: | Klümper, Andreas |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
The XXX spin-$\frac{1}{2}$ Heisenberg chain with non-diagonal boundary fields represents a cornerstone model in the study of integrable systems with open boundaries. Despite its significance, solving this model exactly has remained a formidable challenge due to the breaking of $U(1)$ symmetry. Building on the off-diagonal Bethe Ansatz (ODBA), we derive a set of nonlinear integral equations (NLIEs) that encapsulate the exact spectrum of the model. For $U(1)$ symmetric spin-$\frac{1}{2}$ chains such NLIEs involve two functions $a(x)$ and $\bar{a}(x)$ coupled by an integration kernel with short-ranged elements. The solution functions show characteristic features for arguments at some length scale which grows logarithmically with system size $N$. For the non $U(1)$ symmetric case, the equations involve a novel third function $c(x)$, which captures the inhomogeneous contributions of the $T$-$Q$ relation. The kernel elements coupling this function to the standard ones are long-ranged and lead for the ground-state to a winding phenomenon. In $\log(1+a(x))$ and $\log(1+\bar a(x))$ we observe a sudden change by $2\pi$i at a characteristic scale $x_1$ of the argument. Other features appear at a value $x_0$ which is of order $\log N$. These two length scales, $x_1$ and $x_0$, are independent: their ratio $x_1/x_0$ is large for small $N$ and small for large $N$. Explicit solutions to the NLIEs are obtained numerically for these limiting cases, though intermediate cases ($x_1/x_0 \sim 1$) present computational challenges. This work lays the foundation for studying finite-size corrections and conformal properties of other integrable spin chains with non-diagonal boundaries, opening new avenues for exploring boundary effects in quantum integrable systems.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report
The manuscript is devoted to the derivation and analysis of the nonlinear integral equations, which solution describes the behavior of the isotropic (antiferromagnetic) spin-1/2 Heisenberg chain with fixed boundary conditions. The latter is achieved by the application of local boundary magnetic fields, one of which is coupled to two projections of the edge spin. The nonlinear equations are derived using so called inhomogeneous T-Q relations.
On the one hand, the topic of the manuscript is interesting, and the derivation of the nonlinear integral equations and their analysis is interesting and deserves publication.
On the other hand, the presentation of the results, to my mind, needs some changes, several corrections and additions are necessary:
(i) It is not clear from the text that the treatment addresses only features of the ground state. Also, I assume (though it is not written directly) that the number of sites of the chain is even. Those issues can be added to the manuscript.
(ii) The main achievement of the manuscript is the derivation of the nonlinear integral equations for the considered case. The authors state that unlike the set for periodic boundary conditions, the derived set contains three (not two) functions, which depend on spectral parameter and the values of boundary fields. In fact, I do not understand that comparison. The known solution of two nonlinear integral equations for periodic boundary conditions (for functions a and \bar a), describes the case of NONZERO temperature, see below, and it is namely the advantage of the so-called quantum transfer matrix method. Here the authors consider the T=0 case only. To my mind, the comparison is unclear. It seems more reasonable to compare obtained description with the ones obtained by other methods, see below.
(iii) The abstract contains the invalid statement. It is written "for the non U(1) symmetric case, the equations involve a novel third function c(x)". First, for the non U(1) symmetric case one can describe thermodynamics of the XYZ chain (clearly non U(1) symmetric case) using only two functions, a and \bar a, published many years ago by the second author, A. Kluemper, Z. Phys. B 91, 507 (1993). Perhaps the authors meant non U(1) symmetry caused by boundary fields. Second, one can see from the derived set of nonlinear equations that for the absence of the "non-diagonal" term in the Hamiltonian, \xi =0, one still has three nonlinear equations, including the one for c, at least some procedure to get two equations in that case has to be applied and explained.
(iv) In fact, the title implies the special role of the "non-diagonal boundary fields". However, it is not clear from the analysis that one needs namely those terms. Let us look at the main results of the manuscript, Eqs. (35)-(38), the nonlinear equations, and Eqs.(44)-(46), the ground state energy. If one puts \xi =0 to those equations, no qualitative changes appear, see also the remark (iii). Where is then the mentioned special role of the non-diagonal boundary terms?
(v) Unfortunately, there is not enough comparison with known results for boundary fields for spin-1/2 chains. For example, the case with \xi =0 has to coincide with the well-known results of P.A. de Sa and A.M. Tsvelik, Phys. Rev. B 52, 3067 (1995), A. Kapustin and S. Skorik, J. Phys. A 29, 1629 (1996). Non-diagonal boundary fields were considered, e.g, for the general XYZ chain W.L. Yang and Y.Z. Zhang, Nucl. Phys. B 744, 312 (2006), and reviewed in Ref. [14], for the XY chain A.A. Zvyagin, Phys. Rev. Lett. 110, 217207 (2013), and reviewed for the XX chain in A.A. Zvyagin, Finite Size Effects in Correlated Electron Systems: Exact Results, Imperial College Press, (2005). It is not clear from the text of the considered manuscript what are the special features of the obtained in the manuscript results comparing with the mentioned cases.
I support publication of the manuscript provided necessary corrections and additions made.
Recommendation
Ask for minor revision
Report
It is seen that a key feature of the NLIE approach developed for the model with non-diagonal boundary fields is the necessity to introduce an additional third function c(x) . In their preliminary analysis of the NLIEs the authors reproduce the known results for the bulk and boundary contributions for the ground state energy.
In summary this article represents a significant step forward in the study of the spin-1/2 XXX Heisenberg chain with non-diagonal boundary fields, by extending the NLIE approach to calculate bulk and boundary properties and lay the groundwork for future calculations of finite-size corrections and related properties. This progress via the NLIE approach is also made with an eye towards applyication to other models with U(1) symmetry breaking boundary terms, the most immediate being the XXZ chain. I have no hesitation in recommending publication of this article.
Requested changes
There are no suggested changes for improvement.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)