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

Journal of Physics Conference Series

On the completeness of the set of Bethe-Hulthén solutions of the linear Heisenberg system

W.J. Caspers, M. Łabuz, Andrzej Wal

Published Feb 28, 2006• 0 citations
Publisher

Abstract

Abstract. In this work we formulate the standard form of the solutions of the Heisenberg chain with periodic boundary conditions and show that these solutions can be transformed into the well-known Bethe-Hulthén solutions. The standard form is found by solving the secular problem, separated according to the irreducible representations of the translation group. The relevant parameters exp(ik j) of the Bethe-Hulthén solutions are found from a set of linear equations with coefficients derived from the standard solutions. This correspondence between standard and Bethe-Hulthén solutions realizes the completeness of the Bethe-Hulthén method. 1.

DOI
10.1088/1742-6596/30/1/011

Related papers

TOWARDS A GENERAL SOLUTION OF THE LINEAR HEISENBERG PROBLEM

W.J. Caspers, Tadeusz Lulek, Barbara Lulek +2 more

The construction of a complete set of stationary states of the linear Heisenberg system with periodic boundary conditions (rings) has been a topic of intensive research for many decades. Many eminent theoreticians have made their contribution to this joint eort, but we should mention in the rst place the epoch-making work of Bethe 1 and Hulthen. 5 Their work is the basis of many papers on this topic that have appeared during the past 70 years. We do no have the intention to give a complete survey of all this work but we will try to indicate that especially an asymptotic approach starting from the Hypothesis of Strings gives the prospect of a general and complete solution of the Heisenberg chain.

International School on Symmetry and Structural Properties of Condensed Matter (SSPCM)•Jul 1, 2003•1 citation

Completeness of solutions of Bethe's equations

Wenrui Hao, Rafael I. Nepomechie, Andrew John Sommese

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.

arXiv:1308.4645•Physical Review E•Nov 11, 2013•84 citations

Singular solutions, repeated roots and completeness for higher-spin chains

Wenrui Hao, Rafael I. Nepomechie, Andrew John Sommese

We investigate the completeness of the solutions of the Bethe equations for the integrable spin- s isotropic (XXX) spin chain with periodic boundary conditions. Solutions containing the exact string i s , i( s − 1), ..., −i( s − 1), −i s are singular. For s > 1/2, there exist also 'strange' solutions with repeated roots, which nevertheless are physical (i.e., correspond to eigenstates of the Hamiltonian). We derive conditions for the singular solutions and the solutions with repeated roots to be physical. We formulate a conjecture for the number of solutions with pairwise distinct roots in terms of the numbers of singular and strange solutions. Using homotopy continuation, we solve the Bethe equations numerically for s = 1 and s = 3/2 up to eight sites, and find some support for the conjecture. We also present several examples of strange solutions.

arXiv:1312.2982•Journal of Statistical Mechanics Theory and Experiment•Mar 26, 2014•32 citations

On the Bethe Hypothesis for the Anisotropic Heisenberg Chain

Michael Flicker, Harvey S. Leff

The two-reversed-spin problem for the anisotropic Heisenberg linear chain is solved in a straightforward and rigorous way. The method involves the solution of a second-order partial difference equation and does not necessitate the use of the Bethe hypothesis. It is shown that, although most of the eigenfunctions are of the Bethe form, there exist special states which, a priori, cannot be so written. In addition, a simple method is presented for enumerating the bound states. Detailed connections between the present solution and that given by Bethe, and later by Orbach, are discussed.

Physical Review•Apr 10, 1968•2 citations

Completeness and Bethe root distribution of the spin-1/2 Heisenberg chain with arbitrary boundary fields

Yuzhu Jiang, Shuai Cui, Cao, Junpeng +2 more

Recently, the XXX spin chain with arbitrary boundary fields was successfully solved [1] via the off-diagonal Bethe ansatz method [2]. The correctness and the completeness of this solution were numerically verified by Nepomechie for one choice of the parameterizations [3]. In this paper, we discuss further the completeness of another parameterization of the Bethe ansatz equations and its reduction to the parallel boundary field case. The numerical results show that when the relative angle between the boundary fields turns to zero, both the T-Q relations and the Bethe ansatz equations are reduced to the ones obtained by the conventional Bethe ansatz methods. This allows us to establish a one-to-one correspondence between the Bethe roots of the unparallel boundary field case and those of the parallel boundary field case. In the thermodynamic limit, those two sets of Bethe roots tend to the same and the contribution of the relative angle to the energy is in the order of 1/N.

arXiv:1309.6456•Sep 25, 2013•7 citations

Spin-Wave Spectrum of the Antiferromagnetic Linear Chain

J. des Cloizeaux, James J. Pearson

The methods of Bethe and Hulth\'en are used to build spin-wave states for the antiferromagnetic linear chain. These states, of spin 1 and translational quantum number $k$, are eigenstates of the Hamiltonian $H=\ensuremath{\Sigma}{j}^{}{\mathrm{S}}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{j+1}$ with periodic boundary conditions. For an infinite chain, their spectrum is ${\ensuremath{\epsilon}}_{k}=(\frac{\ensuremath{\pi}}{2})|sink|$, whereas Anderson's spin-wave theory gives ${\ensuremath{\epsilon}}_{k}=|sink|$. For finite chains it has been verified by numerical computation that these states are the lowest states of given $k$, but no rigorous proof has been given for an infinite chain.

Physical Review•Dec 1, 1962•931 citations
Abstract. In this work we formulate the standard form of the solutions of the Heisenberg chain with periodic boundary conditions and show that these solutions can be transformed into the well-known Bethe-Hulthén solutions. The standard form is found by solving the secular problem, separated according to the irreducible representations of the translation group. The relevant parameters exp(ik j) of the Bethe-Hulthén solutions are found from a set of linear equations with coefficients derived from the standard solutions. This correspondence between standard and Bethe-Hulthén solutions realizes the completeness of the Bethe-Hulthén method. 1.
The construction of a complete set of stationary states of the linear Heisenberg system with periodic boundary conditions (rings) has been a topic of intensive research for many decades. Many eminent theoreticians have made their contribution to this joint eort, but we should mention in the rst place the epoch-making work of Bethe 1 and Hulthen. 5 Their work is the basis of many papers on this topic that have appeared during the past 70 years. We do no have the intention to give a complete survey of all this work but we will try to indicate that especially an asymptotic approach starting from the Hypothesis of Strings gives the prospect of a general and complete solution of the Heisenberg chain.
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.
We investigate the completeness of the solutions of the Bethe equations for the integrable spin- s isotropic (XXX) spin chain with periodic boundary conditions. Solutions containing the exact string i s , i( s − 1), ..., −i( s − 1), −i s are singular. For s > 1/2, there exist also 'strange' solutions with repeated roots, which nevertheless are physical (i.e., correspond to eigenstates of the Hamiltonian). We derive conditions for the singular solutions and the solutions with repeated roots to be physical. We formulate a conjecture for the number of solutions with pairwise distinct roots in terms of the numbers of singular and strange solutions. Using homotopy continuation, we solve the Bethe equations numerically for s = 1 and s = 3/2 up to eight sites, and find some support for the conjecture. We also present several examples of strange solutions.
The two-reversed-spin problem for the anisotropic Heisenberg linear chain is solved in a straightforward and rigorous way. The method involves the solution of a second-order partial difference equation and does not necessitate the use of the Bethe hypothesis. It is shown that, although most of the eigenfunctions are of the Bethe form, there exist special states which, a priori, cannot be so written. In addition, a simple method is presented for enumerating the bound states. Detailed connections between the present solution and that given by Bethe, and later by Orbach, are discussed.
Recently, the XXX spin chain with arbitrary boundary fields was successfully solved [1] via the off-diagonal Bethe ansatz method [2]. The correctness and the completeness of this solution were numerically verified by Nepomechie for one choice of the parameterizations [3]. In this paper, we discuss further the completeness of another parameterization of the Bethe ansatz equations and its reduction to the parallel boundary field case. The numerical results show that when the relative angle between the boundary fields turns to zero, both the T-Q relations and the Bethe ansatz equations are reduced to the ones obtained by the conventional Bethe ansatz methods. This allows us to establish a one-to-one correspondence between the Bethe roots of the unparallel boundary field case and those of the parallel boundary field case. In the thermodynamic limit, those two sets of Bethe roots tend to the same and the contribution of the relative angle to the energy is in the order of 1/N.
The methods of Bethe and Hulth\'en are used to build spin-wave states for the antiferromagnetic linear chain. These states, of spin 1 and translational quantum number kkk, are eigenstates of the Hamiltonian $H=\ensuremath{\Sigma}{j}^{}{\mathrm{S}}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{j+1}$ with periodic boundary conditions. For an infinite chain, their spectrum is ${\ensuremath{\epsilon}}_{k}=(\frac{\ensuremath{\pi}}{2})|sink|$, whereas Anderson's spin-wave theory gives ${\ensuremath{\epsilon}}_{k}=|sink|$. For finite chains it has been verified by numerical computation that these states are the lowest states of given kkk, but no rigorous proof has been given for an infinite chain.