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arXiv:1608.06504 · Journal of Physics A Mathematical and Theoretical

Fast analytic solver of rational Bethe equations

C Marboe, Dmytro Volin

Published Apr 5, 2017• 81 citations
arXivHTMLPDF

Abstract

Abstract In this note we propose an approach for a fast analytic determination of all possible sets of Bethe roots corresponding to eigenstates of rational G L ( N | M ) integrable spin chains of given not too large length, in terms of Baxter Q-functions. We observe that all exceptional solutions, if any, are automatically correctly accounted. The key intuition behind the approach is that the equations on the Q-functions are determined solely by the Young diagram, and not by the choice of the rank of the G L symmetry. Hence we can choose arbitrary N and M that accommodate the desired representation. Then we consider all distinguished Q-functions at once, not only those following a certain Kac–Dynkin path.

DOI
10.1088/1751-8121/aa6b88

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Abstract In this note we propose an approach for a fast analytic determination of all possible sets of Bethe roots corresponding to eigenstates of rational G L ( N | M ) integrable spin chains of given not too large length, in terms of Baxter Q-functions. We observe that all exceptional solutions, if any, are automatically correctly accounted. The key intuition behind the approach is that the equations on the Q-functions are determined solely by the Young diagram, and not by the choice of the rank of the G L symmetry. Hence we can choose arbitrary N and M that accommodate the desired representation. Then we consider all distinguished Q-functions at once, not only those following a certain Kac–Dynkin path.
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.
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.
We propose a basis for rational gl(N) spin chains in an arbitrary rectangular representation (SA) that factorises the Bethe vectors into products of Slater determinants in Baxter Q-functions. This basis is constructed by repeated action of fused transfer matrices on a suitable reference state. We prove that it diagonalises the so-called B-operator; hence, the operatorial roots of the latter are the separated variables. The spectrum of the separated variables is also explicitly computed, and it turns out to be labeled by Gelfand-Tsetlin patterns. Our approach utilises a special choice of the spin chain twist which substantially simplifies derivations.
Abstract Bethe equations, whose solutions determine exact eigenvalues and eigenstates of corresponding integrable Hamiltonians, are generally hard to solve. We implement a Variational Quantum Eigensolver approach to estimating Bethe roots of the spin-1/2 XXZ quantum spin chain, by using Bethe states as trial states, and treating Bethe roots as variational parameters. In numerical simulations of systems of size up to 6, we obtain estimates for Bethe roots corresponding to both ground states and excited states with up to 5 down-spins, for both the closed and open XXZ chains. This approach is not limited to real Bethe roots.
We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum spin chain with periodic boundary conditions. We formulate a conjecture for the number of solutions with pairwise distinct roots of these equations, in terms of numbers of so-called singular (or exceptional) solutions. Using homotopy continuation methods, we find all such solutions of the Bethe equations for chains of length up to 14. The numbers of these solutions are in perfect agreement with the conjecture. We also discuss an indirect method of finding solutions of the Bethe equations by solving the Baxter T-Q equation. We briefly comment on implications for thermodynamical computations based on the string hypothesis.