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Completeness of Wronskian Bethe equations for rational gl(m|n) spin chain

Dmitry Chernyak, Sébastien Leurent, Dmytro Volin

Published Apr 6, 2020• 4 citations
OpenAlex

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 $\theta_\ell$, $\ell=1,\ldots,L$ satisfy $\theta_\ell+\hbar\neq\theta_{\ell'}$ for $\ell<\ell'$. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of $\theta_\ell$. In particular, we consider the regime $\theta_{\ell+1}/\theta_{\ell}\gg 1$ for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.

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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 θℓ\theta_\ellθℓ​, ℓ=1,…,L\ell=1,\ldots,Lℓ=1,…,L satisfy θℓ+ℏ≠θℓ′\theta_\ell+\hbar\neq\theta_{\ell'}θℓ​+ℏ=θℓ′​ for ℓ<ℓ′\ell<\ell'ℓ<ℓ′. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of θℓ\theta_\ellθℓ​. In particular, we consider the regime θℓ+1/θℓ≫1\theta_{\ell+1}/\theta_{\ell}\gg 1θℓ+1​/θℓ​≫1 for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.
Completeness of Wronskian Bethe equations for rational glm∣n\mathfrak{gl}_{m|n}glm∣n​ spin chain
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 θℓ\theta_\ellθℓ​, ℓ=1,…,L\ell=1,\ldots,Lℓ=1,…,L satisfy θℓ+ℏ≠θℓ′\theta_\ell+\hbar\neq\theta_{\ell'}θℓ​+ℏ=θℓ′​ for ℓ<ℓ′\ell<\ell'ℓ<ℓ′. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of θℓ\theta_\ellθℓ​. In particular, we consider the regime θℓ+1/θℓ≫1\theta_{\ell+1}/\theta_{\ell}\gg 1θℓ+1​/θℓ​≫1 for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.
Spin-sss QQQ-systems: Twist and open boundaries
In integrable spin chains, the spectral problem can be solved by the method of Bethe Ansatz, which transforms the problem of diagonalization of the Hamiltonian into the problem of solving a set of algebraic equations named Bethe equations. In this work, we systematically investigate the spin- s s XXX chain with twisted and open boundary conditions using the rational Q Q -system, which is a powerful tool to solve Bethe equations. We establish basic frameworks of the rational Q Q -system and confirm its completeness numerically in both cases. For twisted boundaries, we investigate the polynomiality conditions of the rational Q Q -system and derive physical conditions for singular solutions of Bethe equations. For open boundaries, we uncover novel phenomena such as hidden symmetries and boundary induced strings under specific boundary parameters. Hidden symmetries lead to the appearance of extra degeneracies in the Hilbert space, while the boundary induced string is a novel type of exact string configuration, whose length depends on the boundary magnetic fields. These findings, supported by both analytical and numerical evidences, offer new insights into the interplay between symmetries and boundary conditions.
A bstract We develop a non-nested Bethe ansatz description of rational 𝔤𝔩 ℓ spin chains in the vector representation. Starting from the quantum spectral curve and the separation-of-variables framework, we derive closed systems of Bethe equations involving only the momentum-carrying Bethe roots. The construction is worked out explicitly for the 𝔤𝔩 3 and 𝔤𝔩 4 spin chains and then generalized to arbitrary rank. A central result of this work is the identification of a recursive hierarchy associated with the fundamental transfer matrices. The hierarchy is generated by regularity conditions of the lower transfer matrices and closes through a universal rank- ℓ equation ℛ ℓ = 0. This equation replaces the final level of the conventional nested Bethe ansatz and eliminates all auxiliary Bethe roots. Consequently, the complete spectral data of an eigenstate are encoded solely in the first Baxter polynomial Q 1 ( u ). We further obtain explicit expressions for the eigenvalues of all fundamental transfer matrices in terms of the momentum-carrying roots alone. The resulting formulation provides a compact characterization of the spectrum of rational 𝔤𝔩 ℓ spin chains and reveals a direct connection between the quantum spectral curve, transfer-matrix fusion relations, and a truncated Q -system underlying the non-nested description. Finally, we investigate the quasi-classical (Gaudin) limit of the non-nested Bethe equations. For the 𝔤𝔩 3 spin chain, we show that the leading non-trivial contribution gives rise to Gaudin equations whose pole-free form naturally defines a scalar third-order 𝔤𝔩 3 oper.