[Paper Review] Completeness of the Bethe ansatz for the six and eight-vertex models

R. J. Baxter|arXiv (Cornell University)|Nov 10, 2001

Algebraic structures and combinatorial models25 references19 citations

TL;DR

This paper establishes the completeness of the coordinate Bethe ansatz for the six- and eight-vertex models by resolving long-standing issues such as infinite momenta, exact complete strings, and degeneracy at root-of-unity parameters. It shows that despite apparent singularities or non-uniqueness in Bethe roots, the ansatz consistently generates a complete basis of eigenvectors by treating the problem as a system of algebraic equations with generalized normalization and inclusion of the point at infinity, confirming the ansatz's validity across all parameter regimes including critical and degenerate cases.

ABSTRACT

We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the six-vertex model and XXZ chain, and for the eight-vertex model. In particular we discuss the ``beyond the equator'', infinite momenta and exact complete string problems. We show how they can be overcome and conclude that the coordinate Bethe ansatz does indeed give a complete set of states, as expected.

Motivation & Objective

• To resolve persistent concerns in the literature about the completeness of the Bethe ansatz for the six- and eight-vertex models, particularly at zero field or special values of the crossing parameter λ.
• To address specific issues such as 'beyond the equator' states, infinite momenta, and exact complete strings that were thought to undermine the ansatz's validity.
• To demonstrate that the coordinate Bethe ansatz yields a complete set of eigenvectors even when Bethe roots are degenerate or infinite, by reinterpreting the problem in terms of algebraic equations with generalized normalization.
• To show that apparent non-uniqueness of Bethe roots at root-of-unity values is not a failure of the ansatz but a consequence of eigenspace degeneracy, which the ansatz can still span.

Proposed method

• Uses the coordinate Bethe ansatz to construct eigenvectors as superpositions of plane waves with momenta determined by Bethe equations.
• Treats the Bethe equations as algebraic systems, allowing the inclusion of a generalized 'point at infinity' to handle infinite momenta and singular limits.
• Applies a generalized normalization scheme to ensure non-vanishing eigenvectors even when individual coefficients or momenta diverge.
• Analyzes degeneracy at root-of-unity parameters by showing that multiple valid Bethe root configurations correspond to different vectors in the same eigenspace, which the ansatz can span.
• Reformulates the Bethe equations as generalized eigenvalue problems where Fourier coefficients of Q(v) serve as eigenvector components.
• Uses limiting procedures and symmetry considerations (e.g., arrow reversal operator) to resolve cases where Bethe roots coincide or become infinite.

Experimental results

Research questions

• RQ1 Can the Bethe ansatz be considered complete for the six-vertex model despite claims of vanishing eigenvectors when the number of down arrows exceeds up arrows?
• RQ2 How can the issue of infinite momenta in Bethe roots be reconciled with a non-vanishing eigenvector in the coordinate Bethe ansatz?
• RQ3 Why do Bethe roots appear non-unique at root-of-unity values of the crossing parameter λ, and does this imply incompleteness?
• RQ4 Can the Bethe ansatz still span the full eigenspace when the eigenvalue is degenerate due to exact complete strings?
• RQ5 How can the eight-vertex model in zero field with an even number of columns be treated consistently with the Bethe ansatz, especially when infinite momenta or strings arise?

Key findings

• The coordinate Bethe ansatz provides a complete set of eigenvectors for the six-vertex model and XXZ chain, even when Bethe roots are infinite or form exact complete strings, by proper normalization and inclusion of the point at infinity.
• The apparent non-uniqueness of Bethe roots at root-of-unity parameters is not a flaw but a consequence of eigenspace degeneracy; the ansatz spans the entire eigenspace via multiple valid root configurations.
• For the case of a single complete string, the set of allowed Bethe root centers forms a curve that spans the degenerate eigenspace, confirming completeness.
• The functional relation between T(v) and Q(v) eigenvalues can be recast as a generalized eigenvalue problem, with Q(v)'s Fourier coefficients forming the eigenvector components.
• The Bethe ansatz equations can be rewritten in linear form (e.g., equations (41), (138)) as generalized eigenvalue problems in the coefficients qj, which are more robust than the original nonlinear form.
• Numerical experiments and analytical limits confirm that the ansatz remains valid and complete even in critical or singular parameter regimes, resolving prior concerns about incompleteness.