[Paper Review] Completeness of Wronskian Bethe equations for rational gl(m|n) spin chains

Dmitry Chernyak, Sébastien Leurent+1 more|arXiv (Cornell University)|Apr 6, 2020

Algebraic structures and combinatorial models4 citations

TL;DR

This paper proves the completeness of Wronskian Bethe equations for rational gl(m|n) supersymmetric spin chains by establishing an isomorphism between the Bethe algebra of conserved charges and a polynomial Wronskian algebra defined by functional relations among Baxter Q-functions. The key result is that the algebraic number of solutions to these equations matches the Hilbert space dimension, and all solutions are physical for generic inhomogeneities, resolving long-standing issues with standard nested Bethe Ansatz.

ABSTRACT

We consider rational integrable supersymmetric gl(m|n) spin chains in the defining representation and prove the isomorphism between a commutative algebra of conserved charges (the Bethe algebra) and a polynomial ring (the Wronskian algebra) defined by functional relations between Baxter Q-functions that we call Wronskian Bethe equations. These equations, in contrast to standard nested Bethe equations, admit only physical solutions for any value of inhomogeneities and furthermore we prove that the algebraic number of solutions to these equations is equal to the dimension of the spin chain Hilbert space (modulo relevant symmetries). Both twisted and twist-less periodic boundary conditions are considered, the isomorphism statement uses, as a sufficient condition, that the spin chain inhomogeneities $θ_\ell$, $\ell=1,\ldots,L$ satisfy $θ_\ell+\hbar eqθ_{\ell'}$ for $\ell

Motivation & Objective

• To establish a rigorous isomorphism between the Bethe algebra of conserved charges and a polynomial Wronskian algebra defined by functional relations among Baxter Q-functions.
• To prove that the Wronskian Bethe equations yield only physical solutions for any generic values of inhomogeneities θℓ.
• To show that the algebraic number of solutions to the Wronskian equations equals the dimension of the spin chain Hilbert space, modulo symmetries.
• To provide a systematic labelling of solutions using standard Young tableaux in the twist-less case.
• To demonstrate the faithfulness and maximality of the Bethe algebra via character computations and specialisation techniques.

Proposed method

• Define the Wronskian Bethe equations as functional relations between Baxter Q-functions, replacing standard nested Bethe equations.
• Use the Wronskian determinant condition to characterize the algebra of conserved charges as a polynomial ring over symmetric functions χℓ.
• Compute the Hilbert series (character) of the Wronskian algebra to count solutions algebraically.
• Perform explicit solution counting in scaling regimes (e.g., θℓ+1/θℓ ≫ 1) to verify the algebraic count matches the Hilbert space dimension.
• Apply the analytic implicit function theorem to continue limiting solutions (in large inhomogeneity limits) to finite inhomogeneities.
• Use rotational transformations and degeneracy analysis (via Lemma D.1) to identify divergent root scaling and corner boxes in Young diagrams.

Experimental results

Research questions

• RQ1 Does the Wronskian Bethe equation formulation yield only physical solutions for all generic inhomogeneities θℓ?
• RQ2 Is the number of solutions to the Wronskian Bethe equations equal to the dimension of the Hilbert space of the spin chain?
• RQ3 Can solutions to the Wronskian equations be systematically labelled by standard Young tableaux in the twist-less case?
• RQ4 Is the Wronskian algebra isomorphic to the Bethe algebra, and does this isomorphism remain faithful under specialisation to fixed inhomogeneity values?
• RQ5 Can limiting solutions in large-inhomogeneity regimes be analytically continued to finite inhomogeneities?

Key findings

• The Wronskian Bethe equations admit only physical solutions for any generic inhomogeneities θℓ satisfying θℓ+ℏ ≠ θℓ′ for ℓ < ℓ′.
• The algebraic number of solutions to the Wronskian Bethe equations is exactly equal to the dimension of the spin chain Hilbert space, confirming completeness.
• In the twist-less case, solutions are in one-to-one correspondence with standard Young tableaux of shape λ, with explicit labelling provided via scaling limits.
• The Wronskian algebra is a free module over the ring of symmetric functions C[χ], and its Hilbert series matches the character of the Bethe algebra.
• The isomorphism between the Wronskian and Bethe algebras is preserved under specialisation to fixed inhomogeneity values, ensuring faithfulness.
• Solutions obtained in the large-inhomogeneity limit (θL → ∞) can be analytically continued to finite inhomogeneities using the implicit function theorem, with non-vanishing Jacobian.