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

arXiv:hep-th/9912252 · Journal of Physics A Mathematical and General

Ground state of the quantum symmetric finite-size XXZ spin chain with anisotropy parameter Delta = ½

V Fridkin, Yu Stroganov, D Zagier

Published Mar 23, 2000• 39 citations
arXivHTMLPDF

Abstract

We find an analytic solution of the Bethe Ansatz equations for the special case of a finite XXZ spin chain with free boundary conditions and with a complex surface field which provides for U q ( sl (2)) symmetry of the Hamiltonian. More precisely, we find one nontrivial solution, corresponding to the ground state of the system with anisotropy parameter = ½ corresponding to q 3 = -1.

DOI
10.1088/0305-4470/33/13/101

Related papers

The open XXZ chain at Δ = −1/2 and the boundary quantum Knizhnik–Zamolodchikov equations

Christian Hagendorf, Jean Liénardy

Abstract The open XXZ spin chain with the anisotropy parameter Δ = − 1 2 and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R -matrix and diagonal K -matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.

arXiv:2008.03220•Journal of Statistical Mechanics Theory and Experiment•Jan 1, 2021•1 citation

Excited States in the Twisted XXZ Spin Chain

Tarek Nassar, Olav Tirkkonen

We compute the finite size spectrum for the spin 1/2 XXZ chain with twisted boundary conditions, for anisotropy in the regime 0 < γ < π/2, and arbitrary twist θ. The string hypothesis is employed for treating complex excitations. The Bethe Ansatz equtions are solved within a coupled non-linear integral equation approach, with one equation for each type of string. The root-of-unity quantum group invariant periodic chain reduces to the XXZ1/2 chain with a set of twist boundary conditions (π/γ ∈ ZZ, θ an integer multiple of γ). For this model, the restricted Hilbert space corresponds to an unitary conformal field theory, and we recover all primary states in the Kač table in terms of states with specific twist and strings.

•Jan 1, 1997•4 citations

Analytical results on the Heisenberg spin chain in a magnetic field

Etienne Granet, Jesper Lykke Jacobsen, Hubert Saleur

Abstract We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field h as a series in , where h c is the smallest field for which the ground state is fully polarized. All the coefficients of the series can be computed in closed form through a recurrence formula that involves only algebraic manipulations. For some values of the anisotropy parameter the expansion is numerically observed to be convergent in the full range . To that end we express the free energy at mean magnetization per site as a series in whose coefficients can be similarly recursively computed in closed form. This series converges for all . The recurrence is nothing but the Bethe equations when their roots are written as a double series in their corresponding Bethe number and in . It can also be used to derive the corrections in finite size, that correspond to the spectrum of a free compactified boson whose Luttinger parameter can be expanded as a similar series. The method presumably applies to a large class of models: it also successfully applies to a case where the Bethe roots lie on a curve in the complex plane.

arXiv:1901.05878•Journal of Physics A Mathematical and Theoretical•May 7, 2019•15 citations

Exact solution of the deformed biquadratic spin-1 chain

R. Köberle, A. Lima-Santos

We solve for the spectrum of the deformed biquadratic spin-1 chain by a generalization of the coordinate Bethe ansatz. The model is invariant under the quantum group U q sl(2) and, depending on the deformation parameter q, it's Hamiltonian may not be Hermitian. We consider several boundary conditions and find in particular that for periodic ones, the complete spectrum of the spin- 1/2 XXZ model is always contained in the spectrum of the biquadratic model. For free boundary conditions the spectrum is identical to the XXZ model spectrum, only the multiplicities are, of course, different. Bethe states are highest-weight states of the quantum group, except possibly for zero-energy states.

arXiv:hep-th/9302140•Journal of Physics A Mathematical and General•Aug 21, 1994•14 citations

Bethe Ansatz from functional relations of open XXZ chain for new special cases

Rajan Murgan, Rafael I. Nepomechie

The transfer matrix of the general integrable open XXZ quantum spin chain obeys certain functional relations at roots of unity. By exploiting these functional relations, we determine the Bethe Ansatz solution for the transfer matrix eigenvalues for the special cases that all but one of the boundary parameters are zero, and the bulk anisotropy parameter is i\\pi/3, i\\pi/5 ,...

•Apr 13, 2005•0 citations

The solution of an open XXZ chain with arbitrary spin revisited

Rajan Murgan, Chris Silverthorn

The Bethe ansatz solutions for an open XXZ spin chain with arbitrary spin with N sites and nondiagonal boundary terms are revisited. The anisotropy parameter, for cases considered here, has values , where r and q are positive integers with q restricted to odd integers. Numerical results are presented to support the solutions.

arXiv:1403.4313•Journal of Statistical Mechanics Theory and Experiment•Feb 3, 2015•3 citations
We find an analytic solution of the Bethe Ansatz equations for the special case of a finite XXZ spin chain with free boundary conditions and with a complex surface field which provides for U q ( sl (2)) symmetry of the Hamiltonian. More precisely, we find one nontrivial solution, corresponding to the ground state of the system with anisotropy parameter = ½ corresponding to q 3 = -1.
Abstract The open XXZ spin chain with the anisotropy parameter Δ = − 1 2 and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R -matrix and diagonal K -matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.
We compute the finite size spectrum for the spin 1/2 XXZ chain with twisted boundary conditions, for anisotropy in the regime 0 < γ < π/2, and arbitrary twist θ. The string hypothesis is employed for treating complex excitations. The Bethe Ansatz equtions are solved within a coupled non-linear integral equation approach, with one equation for each type of string. The root-of-unity quantum group invariant periodic chain reduces to the XXZ1/2 chain with a set of twist boundary conditions (π/γ ∈ ZZ, θ an integer multiple of γ). For this model, the restricted Hilbert space corresponds to an unitary conformal field theory, and we recover all primary states in the Kač table in terms of states with specific twist and strings.
Abstract We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field h as a series in , where h c is the smallest field for which the ground state is fully polarized. All the coefficients of the series can be computed in closed form through a recurrence formula that involves only algebraic manipulations. For some values of the anisotropy parameter the expansion is numerically observed to be convergent in the full range . To that end we express the free energy at mean magnetization per site as a series in whose coefficients can be similarly recursively computed in closed form. This series converges for all . The recurrence is nothing but the Bethe equations when their roots are written as a double series in their corresponding Bethe number and in . It can also be used to derive the corrections in finite size, that correspond to the spectrum of a free compactified boson whose Luttinger parameter can be expanded as a similar series. The method presumably applies to a large class of models: it also successfully applies to a case where the Bethe roots lie on a curve in the complex plane.
We solve for the spectrum of the deformed biquadratic spin-1 chain by a generalization of the coordinate Bethe ansatz. The model is invariant under the quantum group U q sl(2) and, depending on the deformation parameter q, it's Hamiltonian may not be Hermitian. We consider several boundary conditions and find in particular that for periodic ones, the complete spectrum of the spin- 1/2 XXZ model is always contained in the spectrum of the biquadratic model. For free boundary conditions the spectrum is identical to the XXZ model spectrum, only the multiplicities are, of course, different. Bethe states are highest-weight states of the quantum group, except possibly for zero-energy states.