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Completeness and Bethe root distribution of the spin-1/2 Heisenberg chain with arbitrary boundary fields
Statistical Mechanics · Submitted 25 Sep 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In the thermodynamic limit, those two sets of Bethe roots tend to the same and the contribution of the relative angle to the energy is in the order of 1/N.
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The abstract, sentence by sentence
Recently, the XXX spin chain with arbitrary boundary fields was successfully solved [1] via the off-diagonal Bethe ansatz method [2]. The correctness and the completeness of this solution were numerically verified by Nepomechie for one choice of the parameterizations [3]. In this paper, we discuss further the completeness of another parameterization of the Bethe ansatz equations and its reduction to the parallel boundary field case. The numerical results show that when the relative angle between the boundary fields turns to zero, both the T-Q relations and the Bethe ansatz equations are reduced to the ones obtained by the conventional Bethe ansatz methods. This allows us to establish a one-to-one correspondence between the Bethe roots of the unparallel boundary field case and those of the parallel boundary field case. In the thermodynamic limit, those two sets of Bethe roots tend to the same and the contribution of the relative angle to the energy is in the order of 1/N.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Recently, the XXX spin chain with arbitrary boundary fields was successfully solved [1] via the off-diagonal Bethe ansatz method [2]. | background 1.00 | 0.03 | 0.03 | asserted 1.00 |
| s1 | The correctness and the completeness of this solution were numerically verified by Nepomechie for one choice of the parameterizations [3]. | background 0.92 | 0.03 | 0.02 | asserted 1.00 |
| s2 | In this paper, we discuss further the completeness of another parameterization of the Bethe ansatz equations and its reduction to the parallel boundary field case. | contribution 1.00 | 0.69 | 0.11 | asserted 0.78 |
| s3 | The numerical results show that when the relative angle between the boundary fields turns to zero, both the T-Q relations and the Bethe ansatz equations are reduced to the ones obtained by the conventional Bethe ansatz methods. | result 1.00 | 0.86 | 0.21 | asserted 1.00 |
| s4 | This allows us to establish a one-to-one correspondence between the Bethe roots of the unparallel boundary field case and those of the parallel boundary field case. | result 0.59 | 0.79 | 0.33 | asserted 0.99 |
| s5 | In the thermodynamic limit, those two sets of Bethe roots tend to the same and the contribution of the relative angle to the energy is in the order of 1/N. | result 0.93 | 0.84 | 0.36 | asserted 1.00 |
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Original abstract
Recently, the XXX spin chain with arbitrary boundary fields was successfully solved [1] via the off-diagonal Bethe ansatz method [2]. The correctness and the completeness of this solution were numerically verified by Nepomechie for one choice of the parameterizations [3]. In this paper, we discuss further the completeness of another parameterization of the Bethe ansatz equations and its reduction to the parallel boundary field case. The numerical results show that when the relative angle between the boundary fields turns to zero, both the T-Q relations and the Bethe ansatz equations are reduced to the ones obtained by the conventional Bethe ansatz methods. This allows us to establish a one-to-one correspondence between the Bethe roots of the unparallel boundary field case and those of the parallel boundary field case. In the thermodynamic limit, those two sets of Bethe roots tend to the same and the contribution of the relative angle to the energy is in the order of 1/N.
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9 arXiv papers cite this in 15 sentences
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Cites as background 3 papers
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This problem has attracted a lot of attentions [ref] not only for its mathematical structure but also for its relevance to the stochastic process and the spin current physics.
Complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms · Introduction · 0.83
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This problem has attracted a lot of attentions [ref] not only for its mathematical structure but also for its relevance to the stochastic process and the spin current physics.
SOV approach for integrable quantum models associated to general representations on spin-1/2 chains of the 8-vertex reflection algebra · Introduction · 0.83
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This problem has attracted a lot of attentions [ref] not only for its mathematical structure but also for its relevance to the stochastic process and the spin current physics.
Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz · Introduction · 0.83
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