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:cond-mat/0101385 · Journal of Physics A Mathematical and General

The quantum symmetricXXZchain at Δ = -½, alternating-sign matrices and plane partitions

Murray T. Batchelor, Jan de Gier, Bernard Nienhuis

Published May 2, 2001• 145 citations
arXivHTMLPDF

Abstract

We consider the ground-state wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at Δ = -½, along with the ground-state wavefunction of the corresponding O( n ) loop model at n = 1. Based on exact results for finite-size systems, sums involving the wavefunction components, and in some cases the largest component itself, are conjectured to be directly related to the total number of alternating-sign matrices and plane partitions in certain symmetry classes.

DOI
10.1088/0305-4470/34/19/101

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

The XXZ spin chain at Δ=−1/2: Bethe roots, symmetric functions, and determinants

Jan de Gier, Murray T. Batchelor, Bernard Nienhuis +1 more

A number of conjectures have been given recently concerning the connection between the antiferromagnetic XXZ spin chain at Δ=−1/2 and various symmetry classes of alternating sign matrices. Here we use the integrability of the XXZ chain to gain further insight into these developments. In doing so we obtain a number of new results using Baxter’s Q function for the XXZ chain for periodic, twisted and open boundary conditions. These include expressions for the elementary symmetric functions evaluated at the ground state solution of the Bethe roots. In this approach Schur functions play a central role and enable us to derive determinant expressions which appear in certain natural double products over the Bethe roots. When evaluated these give rise to the numbers counting different symmetry classes of alternating sign matrices.

arXiv:math-ph/0110011•Journal of Mathematical Physics•Aug 1, 2002•37 citations

Spin chains and combinatorics

A. V. Razumov, Yu. G. Stroganov

In this paper we continue the investigation of finite XXZ spin chains with periodic boundary conditions and odd number of sites, initiated in our previous article (Stroganov Yu G 2001 J. Phys. A: Math. Gen. 34 L179-85 ). As it turned out, for a special value of the asymmetry parameter Δ = -1/2 the Hamiltonian of the system has an eigenvalue, which is exactly proportional to the number of sites E = -3 N /2. Using Mathematica we have found explicitly the corresponding eigenvectors for N ⩽17. The obtained results support the conjecture our paper that this special eigenvalue corresponds to the ground state vector. We make a lot of conjectures concerning the correlations of the model. Many remarkable relations between the wavefunction components are noted. It turns out, for example, that the ratio of the largest component to the least one is equal to the number of the alternating sign matrices.

arXiv:cond-mat/0012141•Journal of Physics A Mathematical and General•Mar 30, 2001•217 citations

Spin chains and combinatorics: twisted boundary conditions

A. V. Razumov, Yu. G. Stroganov

The finite XXZ Heisenberg spin chain with twisted boundary conditions is considered. For the case of an even number of sites N , anisotropy parameter -1/2 and twisting angle 2π/3 the Hamiltonian of the system possesses an eigenvalue -3 N /2. The explicit form of the corresponding eigenvector was found for N ⩽12. Conjecturing that this vector is the ground state of the system we made and verified several conjectures related to the norm of the ground state vector, its component with maximal absolute value and some correlation functions, which have combinatorial nature. In particular, we conjecture that the squared norm of the ground state vector coincides with the number of half-turn symmetric alternating sign N × N matrices.

arXiv:cond-mat/0102247•Journal of Physics A Mathematical and General•Jun 22, 2001•80 citations

Bethe roots and refined enumeration of alternating-sign matrices

A. V. Razumov, Yu. G. Stroganov

The properties of the most probable ground state candidate for the XXZ spin chain with the anisotropy parameter equal to −1/2 and an odd number of sites is considered. Some linear combinations of the components of the state considered, divided by the maximal component, coincide with the elementary symmetric polynomials in the corresponding Bethe roots. It is proved that these polynomials are equal to the numbers providing the refined enumeration of the alternating-sign matrices of order M +1 divided by the total number of the alternating-sign matrices of order M , for the chain of length 2 M +1.

arXiv:math-ph/0605004•Journal of Statistical Mechanics Theory and Experiment•Jul 13, 2006•18 citations

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

V Fridkin, Yu Stroganov, D Zagier

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.

arXiv:hep-th/9912252•Journal of Physics A Mathematical and General•Mar 23, 2000•39 citations
We consider the ground-state wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at Δ = -½, along with the ground-state wavefunction of the corresponding O( n ) loop model at n = 1. Based on exact results for finite-size systems, sums involving the wavefunction components, and in some cases the largest component itself, are conjectured to be directly related to the total number of alternating-sign matrices and plane partitions in certain symmetry classes.
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.
A number of conjectures have been given recently concerning the connection between the antiferromagnetic XXZ spin chain at Δ=−1/2 and various symmetry classes of alternating sign matrices. Here we use the integrability of the XXZ chain to gain further insight into these developments. In doing so we obtain a number of new results using Baxter’s Q function for the XXZ chain for periodic, twisted and open boundary conditions. These include expressions for the elementary symmetric functions evaluated at the ground state solution of the Bethe roots. In this approach Schur functions play a central role and enable us to derive determinant expressions which appear in certain natural double products over the Bethe roots. When evaluated these give rise to the numbers counting different symmetry classes of alternating sign matrices.
In this paper we continue the investigation of finite XXZ spin chains with periodic boundary conditions and odd number of sites, initiated in our previous article (Stroganov Yu G 2001 J. Phys. A: Math. Gen. 34 L179-85 ). As it turned out, for a special value of the asymmetry parameter Δ = -1/2 the Hamiltonian of the system has an eigenvalue, which is exactly proportional to the number of sites E = -3 N /2. Using Mathematica we have found explicitly the corresponding eigenvectors for N ⩽17. The obtained results support the conjecture our paper that this special eigenvalue corresponds to the ground state vector. We make a lot of conjectures concerning the correlations of the model. Many remarkable relations between the wavefunction components are noted. It turns out, for example, that the ratio of the largest component to the least one is equal to the number of the alternating sign matrices.
The finite XXZ Heisenberg spin chain with twisted boundary conditions is considered. For the case of an even number of sites N , anisotropy parameter -1/2 and twisting angle 2π/3 the Hamiltonian of the system possesses an eigenvalue -3 N /2. The explicit form of the corresponding eigenvector was found for N ⩽12. Conjecturing that this vector is the ground state of the system we made and verified several conjectures related to the norm of the ground state vector, its component with maximal absolute value and some correlation functions, which have combinatorial nature. In particular, we conjecture that the squared norm of the ground state vector coincides with the number of half-turn symmetric alternating sign N × N matrices.
The properties of the most probable ground state candidate for the XXZ spin chain with the anisotropy parameter equal to −1/2 and an odd number of sites is considered. Some linear combinations of the components of the state considered, divided by the maximal component, coincide with the elementary symmetric polynomials in the corresponding Bethe roots. It is proved that these polynomials are equal to the numbers providing the refined enumeration of the alternating-sign matrices of order M +1 divided by the total number of the alternating-sign matrices of order M , for the chain of length 2 M +1.