On zero-remainder conditions in the Bethe ansatz

Abstract

We prove that physical solutions to the Heisenberg spin chain Bethe ansatz equations are exactly obtained by imposing two zero-remainder conditions. This bridges the gap between different criteria, yielding an alternative proof of a recently devised algorithm based on $QQ$ relations, and solving its minimality issue.

arXiv comment14 pages. New version: corrected typos, modified abstract and added references

References in corpus (22)

• A Shortcut to the Q-Operator
• Observing complex bound states in the spin-1/2 Heisenberg XXZ chain using local quantum quenches
• Bethe's equation is incomplete for the XXZ model at roots of unity
• Bethe Equations "on the Wrong Side of Equator"
• On the completeness of solutions of Bethe's equations
• New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains
• Fast analytic solver of rational Bethe equations
• Deformed strings in the Heisenberg model
• Algebraic Bethe ansatz for singular solutions
• Completeness of Bethe's states for generalized $XXZ$ model
• Low-Temperature Dynamical Structure Factor of the Two-Leg Spin-1/2 Heisenberg Ladder
• String hypothesis for gl(n|m) spin chains: a particle/hole democracy
• On Generalized $Q$-systems
• Completeness of the Bethe ansatz for the six and eight-vertex models
• Completeness of ``Good'' Bethe Ansatz Solutions of a Quantum Group Invariant Heisenberg Model
• Exceptional Operators in N=4 super Yang-Mills
• Twisting singular solutions of Bethe's equations
• Origin of the singular Bethe ansatz solutions for the Heisenberg XXZ spin chain
• Singularities in the Bethe solution of the XXX and XXZ Heisenberg spin chains
• Bases of Bethe Vectors and Difference Equations with Regular Singular Points
• Algebraic Bethe ansatz
• Spontaneous symmetry breaking in 2D supersphere sigma models and applications to intersecting loop soups

Cited by in corpus (16)

• Rational $Q$-systems, Higgsing and Mirror Symmetry
• Geometric Algebra and Algebraic Geometry of Loop and Potts Models
• On Generalized $Q$-systems
• A review of the AdS/CFT Quantum Spectral Curve
• Completeness of Wronskian Bethe equations for rational gl(m|n) spin chains
• Lattice regularisation of a non-compact boundary conformal field theory
• The action of the Virasoro algebra in quantum spin chains. I. The non-rational case
• Cylinder partition function of the 6-vertex model from algebraic geometry
• Spin-$s$ Rational $Q$-system
• Wronskian-type formula for inhomogeneous TQ-equations
• Rational Q-systems at Root of Unity I. Closed Chains
• Analytic continuation of Bethe energies and application to the thermodynamic limit of the $SL(2,\mathbb{C})$ non-compact spin chains
• Exact Spin Correlators of Integrable Quantum Circuits from Algebraic Geometry
• On the Q operator and the spectrum of the XXZ model at root of unity
• Spectrum-preserving deformations of integrable spin chains with open boundaries
• Spin-$s$ $Q$-systems: Twist and Open Boundaries