Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Origin of the singular Bethe ansatz solutions for the Heisenberg XXZ spin chain

Jae Dong Noh, Deok‐Sun Lee, Doochul Kim, Deok-sun Leeb B

Published Jan 1, 2000• 14 citations
OpenAlex

Abstract

We investigate symmetry properties of the Bethe ansatz wave functions for the Heisenberg XXZ spin chain. The XXZ Hamiltonian commutes simultaneously with the shift operator T and the lattice inversion operator V in the space of Ω=±1 with Ω the eigenvalue of T. We show that the Bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of T and V with quantum number (Ω, ϒ)=(±1, ∓1) where ϒ is the eigenvalue of V. Therefore, the Bethe ansatz wave functions should be singular for nondegenerate eigenstates of the Hamiltonian with quantum number (Ω, ϒ)=(±1, ∓1). It is also shown that such states exist in any nontrivial down-spin number sector and that the number of them diverges exponentially with the chain length.

Related papers

Singularities in the Bethe solution of the XXX and XXZ Heisenberg spin chains

Rahul Siddharthan

We examine the question of whether Bethe's ansatz reproduces all states in the periodic Heisenberg XXZ and XXX spin chains. As was known to Bethe himself, there are states for which the Bethe momenta $k_n$ diverge: these are in fact the simplest examples of ``string'' solutions. The coefficients of the Bethe wavefunction, too, diverge. When there are only two down spins in the system (the case considered by Bethe), we can renormalize these coefficients to get a sensible (and correct) wavefunction. We show that this is not always possible when there are more than two down spins. The Bethe equations have several such divergent solutions, and some of these correspond to genuine eigenfunctions of the Hamiltonian, but several do not. Nor do they reproduce the correct energy eigenvalues. Moreover, we point out that the algebraic Bethe ansatz, an alternative way to construct the wavefunctions proposed by Faddeev, Takhtajan et al., leads to vanishing wavefunctions for all these solutions. Thus, the Bethe ansatz solution of the Heisenberg model must be regarded as either incomplete, or inaccurate.

arXiv:cond-mat/9804210•Apr 20, 1998•18 citations

Singular eigenstates in the even(odd) length Heisenberg spin chain

Pulak Ranjan Giri, Tetsuo Deguchi

We study the implications of the regularization for the singular solutions on the even(odd) length spin- 1 / 2 XXX chains in some specific down-spin sectors. In particular, the analytic expressions of the Bethe eigenstates for three down-spin sector have been obtained along with their numerical forms in some fixed length chains. For an even-length chain if the singular solutions { λ α } are invariant under the sign changes of their rapidities { λ α } = { − λ α } , then the Bethe ansatz equations are reduced to a system of ( M − 2 ) / 2 ( ( M − 3 ) / 2 ) equations in an even (odd) down-spin sector. For an odd N length chain in the three down-spin sector, it has been analytically shown that there exist singular solutions in any finite length of the spin chain of the form N = 3 ( 2 k + 1 ) with k = 1 , 2 , 3 , ⋯ . It is also shown that there exist no singular solutions in the four down-spin sector for some odd-length spin- 1 / 2 XXX chains.

arXiv:1411.5839•Journal of Physics A Mathematical and Theoretical•Apr 8, 2015•7 citations

Anisotropic spin chain: Heisenberg’s xxz model for 4 inverted spins

А.М. Калиев

we study the numerical solutions of the Bethe ansatz equations for the inhomogeneous anisotropic XXZ Heisenberg model with a finite number of spins, with four inverted spins.Having determined the phases of the wave functions and using N-periodicity, we obtained numerical solutions for the system of Bethe equations of the anisotropic antiferromagnetic XXZ Heisenberg model.Using algebraic Bethe ansatz, various pulse and energy configurations for these states were given.

International scientific review•Jan 1, 2019•0 citations

Bethe ansatz solution of a closed spin 1 XXZ Heisenberg chain with quantum algebra symmetry

Jon Links, Angela Foerster, Michael Karowski

A quantum algebra invariant integrable closed spin 1 chain is introduced and analyzed in detail. The Bethe ansatz equations as well as the energy eigenvalues of the model are obtained. The highest weight property of the Bethe vectors with respect to Uq(sl(2)) is proved.

arXiv:solv-int/9809001•Journal of Mathematical Physics•Feb 1, 1999•11 citations

Bethe Ansatz for the Anisotropic Heisenberg Quantum Spin Chain

Hans-Peter Eckle

Abstract This chapter verifies the conjecture for the wave function, the Bethe ansatz wave function, of the anisotropic Heisenberg quantum spin chain by examining first the cases for one, two, and three spin deviations. The equations determining the quasi- momenta are the Bethe ansatz equations, now obtained from the coordinate Bethe ansatz. The Bethe ansatz equations derive from the eigenvalue equation in combination with boundary conditions, here periodic boundary conditions. These quasi-momenta also determine the energy eigenvalue. However, solving the Bethe ansatz equations to obtain a particular state requires more considerations. New variables, called rapidities, are useful. The consideration of the thermodynamic limit then allows to extract information about the ground state and low-lying excitations of the anisotropic quantum spin chain from the Bethe ansatz equations. Furthermore, complex solutions of the Bethe ansatz equations, called strings, are considered.

•Jul 25, 2019•0 citations

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

Tetsuo Deguchi, Pulak Ranjan Giri

Abstract 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 numbers for the physical solutions of BAE. For the spin-1/2 XXX chain we rigorously derive all the quantum numbers for the complete set of the Bethe-ansatz eigenvectors in the two down-spin sector with any chain length N . Here we obtain them both for real and complex solutions. We also show that all the solutions associated with them are distinct. Consequently, we prove the completeness of the Bethe ansatz and give an exact expression for the number of real solutions which correspond to collapsed bound-state solutions (i.e., two-string solutions) in the sector: in terms of Gauss’ symbol. Moreover, we prove in the sector the scheme conjectured by Takahashi for solving BAE systematically. We also suggest that by applying the present method we can derive the quantum numbers for the spin-1/2 XXZ chain.

arXiv:1509.00108•Journal of Physics A Mathematical and Theoretical•Mar 17, 2016•8 citations
We investigate symmetry properties of the Bethe ansatz wave functions for the Heisenberg XXZ spin chain. The XXZ Hamiltonian commutes simultaneously with the shift operator T and the lattice inversion operator V in the space of Ω=±1 with Ω the eigenvalue of T. We show that the Bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of T and V with quantum number (Ω, ϒ)=(±1, ∓1) where ϒ is the eigenvalue of V. Therefore, the Bethe ansatz wave functions should be singular for nondegenerate eigenstates of the Hamiltonian with quantum number (Ω, ϒ)=(±1, ∓1). It is also shown that such states exist in any nontrivial down-spin number sector and that the number of them diverges exponentially with the chain length.
We examine the question of whether Bethe's ansatz reproduces all states in the periodic Heisenberg XXZ and XXX spin chains. As was known to Bethe himself, there are states for which the Bethe momenta knk_nkn​ diverge: these are in fact the simplest examples of ``string'' solutions. The coefficients of the Bethe wavefunction, too, diverge. When there are only two down spins in the system (the case considered by Bethe), we can renormalize these coefficients to get a sensible (and correct) wavefunction. We show that this is not always possible when there are more than two down spins. The Bethe equations have several such divergent solutions, and some of these correspond to genuine eigenfunctions of the Hamiltonian, but several do not. Nor do they reproduce the correct energy eigenvalues. Moreover, we point out that the algebraic Bethe ansatz, an alternative way to construct the wavefunctions proposed by Faddeev, Takhtajan et al., leads to vanishing wavefunctions for all these solutions. Thus, the Bethe ansatz solution of the Heisenberg model must be regarded as either incomplete, or inaccurate.
We study the implications of the regularization for the singular solutions on the even(odd) length spin- 1 / 2 XXX chains in some specific down-spin sectors. In particular, the analytic expressions of the Bethe eigenstates for three down-spin sector have been obtained along with their numerical forms in some fixed length chains. For an even-length chain if the singular solutions { λ α } are invariant under the sign changes of their rapidities { λ α } = { − λ α } , then the Bethe ansatz equations are reduced to a system of ( M − 2 ) / 2 ( ( M − 3 ) / 2 ) equations in an even (odd) down-spin sector. For an odd N length chain in the three down-spin sector, it has been analytically shown that there exist singular solutions in any finite length of the spin chain of the form N = 3 ( 2 k + 1 ) with k = 1 , 2 , 3 , ⋯ . It is also shown that there exist no singular solutions in the four down-spin sector for some odd-length spin- 1 / 2 XXX chains.
Abstract This chapter verifies the conjecture for the wave function, the Bethe ansatz wave function, of the anisotropic Heisenberg quantum spin chain by examining first the cases for one, two, and three spin deviations. The equations determining the quasi- momenta are the Bethe ansatz equations, now obtained from the coordinate Bethe ansatz. The Bethe ansatz equations derive from the eigenvalue equation in combination with boundary conditions, here periodic boundary conditions. These quasi-momenta also determine the energy eigenvalue. However, solving the Bethe ansatz equations to obtain a particular state requires more considerations. New variables, called rapidities, are useful. The consideration of the thermodynamic limit then allows to extract information about the ground state and low-lying excitations of the anisotropic quantum spin chain from the Bethe ansatz equations. Furthermore, complex solutions of the Bethe ansatz equations, called strings, are considered.
Abstract 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 numbers for the physical solutions of BAE. For the spin-1/2 XXX chain we rigorously derive all the quantum numbers for the complete set of the Bethe-ansatz eigenvectors in the two down-spin sector with any chain length N . Here we obtain them both for real and complex solutions. We also show that all the solutions associated with them are distinct. Consequently, we prove the completeness of the Bethe ansatz and give an exact expression for the number of real solutions which correspond to collapsed bound-state solutions (i.e., two-string solutions) in the sector: in terms of Gauss’ symbol. Moreover, we prove in the sector the scheme conjectured by Takahashi for solving BAE systematically. We also suggest that by applying the present method we can derive the quantum numbers for the spin-1/2 XXZ chain.