# Singular solutions to the Bethe ansatz equations and rigged configurations

@article{Kirillov2014SingularST, title={Singular solutions to the Bethe ansatz equations and rigged configurations}, author={Anatol N. Kirillov and Reiho Sakamoto}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2014}, volume={47} }

We provide a conjecture for the following two quantities related to the spin-12?> isotropic Heisenberg model defined over rings of even lengths: (i) the number of solutions to the Bethe ansatz equations which correspond to non-zero Bethe vectors; (ii) the number of physical singular solutions of the Bethe ansatz equations in the sense of Nepomechie and Wang 2013 J. Phys. A: Math. Theor. 46 325002. The conjecture is based on a natural relationship between the solutions to the Bethe ansatz…

## 22 Citations

### Observations on the Bethe ansatz solutions of the spin-1/2 isotropic anti-ferromagnetic Heisenberg chain

- PhysicsJournal of Physics Communications
- 2020

Evidence is presented that the solutions of the Bethe ansatz equations for spin-½ isotropic Heisenberg chains in fixed total spin and momentum sectors are the roots of single variable polynomials…

### Heisenberg model and rigged configurations

- Physics
- 2015

We show a correspondence of all the solutions of the Bethe ansatz equations of the one-dimensional spin-1/2 isotropic Heisenberg model with periodic boundary conditions for N = 12 to the rigged…

### Twisting singular solutions of Betheʼs equations

- Mathematics
- 2014

The Bethe equations for the periodic XXX and XXZ spin chains admit singular solutions, for which the corresponding eigenvalues and eigenvectors are ill-defined. We use a twist regularization to…

### Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE

- Mathematics
- 2018

A bstractIn this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry…

### Rigged configurations approach for the spin-1/2 isotropic Heisenberg model

- Physics, Mathematics
- 2015

We continue the rigged configurations analysis of the solutions to the Bethe ansatz equations for the spin-1/2 isotropic Heisenberg model. We analyze the non self-conjugate strings of Deguchi–Giri…

### Completeness of the Bethe Ansatz for the Periodic Isotropic Heisenberg Model

- Mathematics
- 2018

For the periodic isotropic Heisenberg model with arbitrary spins and inhomogeneities, we describe the system of algebraic equations whose solutions are in bijection with eigenvalues of the…

### Completeness of the Bethe Ansatz solution for the rational, spin-1/2 Richardson--Gaudin system

- Mathematics
- 2016

Establishing the completeness of a Bethe Ansatz solution for an exactly solved model is a perennial challenge, which is typically approached on a case by case basis. For the rational, spin-1/2…

### Topics in Bethe ansatz

- Mathematics
- 2017

of a dissertation at the University of Miami. Dissertation supervised by Professor Rafael I. Nepomechie. No. of pages in text. (116) Integrable quantum spin chains have close connections to…

### Exact quantum numbers of collapsed and non-collapsed two-string solutions in the spin-1/2 Heisenberg spin chain

- Mathematics
- 2016

Every solution of the Bethe-ansatz equations (BAEs) is characterized by a set of quantum numbers, by which we can evaluate it numerically. However, no general rule is known how to give quantum…

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