Rational Q-systems at Root of Unity I. Closed Chains

Abstract

The solution of Bethe ansatz equations for XXZ spin chain with the parameter $q$ being a root of unity is infamously subtle. In this work, we develop the rational $Q$-system for this case, which offers a systematic way to find all physical solutions of the Bethe ansatz equations at root of unity. The construction contains two parts. In the first part, we impose additional constraints to the rational $Q$-system. These constraints eliminate the so-called Fabricius-McCoy (FM) string solutions, yielding all primitive solutions. In the second part, we give a simple procedure to construct the descendant tower of any given primitive state. The primitive solutions together with their descendant towers constitute the complete Hilbert space. We test our proposal by extensive numerical checks and apply it to compute the torus partition function of the 6-vertex model at root of unity.

41 pages, 6 figures

References in corpus (27)

• How Algebraic Bethe Ansatz works for integrable model
• Quasilocal charges in integrable lattice systems
• Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations
• Families of quasi-local conservation laws and quantum spin transport
• Quasilocal conservation laws in XXZ spin-1/2 chains: open, periodic and twisted boundary conditions
• Bethe's equation is incomplete for the XXZ model at roots of unity
• PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain
• Fast analytic solver of rational Bethe equations
• Quasilocal conservation laws from semicyclic irreducible representations of $U_q({\mathfrak{sl}}_2)$ in $XXZ$ spin-$1/2$ chains
• Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity
• Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains
• Rational $Q$-systems, Higgsing and Mirror Symmetry
• XXZ Bethe states as highest weight vectors of the $s{l}_2$ loop algebra at roots of unity
• Geometric Algebra and Algebraic Geometry of Loop and Potts Models
• On Generalized $Q$-systems
• On zero-remainder conditions in the Bethe ansatz
• Phantom Bethe roots in the integrable open spin $1/2$ $XXZ$ chain
• Q-systems with boundary parameters
• Bethe Ansatz and Classical Hirota Equation
• Conjectures on Hidden Onsager Algebra Symmetries in Interacting Quantum Lattice Models
• On the reality of spectra of ${\mathbf{U}}_{\mathbf{q}}(\mathbf{s}{\mathbf{l}}_2)$-invariant XXZ Hamiltonians
• On the computation of fusion over the affine Temperley-Lieb algebra
• The $s{l}_2$ loop algebra symmetry of the twisted transfer matrix of the six vertex model at roots of unity
• Spin-$s$ Rational $Q$-system
• The $A_m^{(1)}$ Q-system
• Extended T-systems, Q matrices and T-Q relations for $s\ell (2)$ models at roots of unity
• Conserved Currents in the Six-Vertex and Trigonometric Solid-On-Solid Models

Cited by in corpus (4)

• Exact Spin Correlators of Integrable Quantum Circuits from Algebraic Geometry
• Spectrum-preserving deformations of integrable spin chains with open boundaries
• Spin-$s$ $Q$-systems: Twist and Open Boundaries
• Periodic revivals from supersymmetry in a fermionic kinetically constrained model