# POLYNOMIAL SOLUTIONS OF qKZ EQUATION AND GROUND STATE OF XXZ SPIN CHAIN AT = −1/2

@article{Razumov2007POLYNOMIALSO, title={POLYNOMIAL SOLUTIONS OF qKZ EQUATION AND GROUND STATE OF XXZ SPIN CHAIN AT = −1/2}, author={A. V. Razumov and Yu. G. Stroganov and Paul Zinn-Justin}, journal={Journal of Physics A}, year={2007}, volume={40}, pages={11827-11847} }

Integral formulae for polynomial solutions of the quantum Knizhnik–Zamolodchikov equations associated with the R-matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit, it is a ground-state eigenvector of the antiferromagnetic XXZ spin chain with the… Expand

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#### References

SHOWING 1-10 OF 66 REFERENCES

The Quantum Symmetric Xxz Chain at ∆ = − , Alternating Sign Matrices and Plane Partitions

- 2008

We consider the groundstate wavefunction of the quantum symmetric antifer-romagnetic XXZ chain with open and twisted boundary conditions at ∆ = − 1 2 , along with the groundstate wavefunction of the… Expand

LETTER TO THE EDITOR: The quantum symmetric XXZ chain at Delta = - 1/2 , alternating-sign matrices and plane partitions

- Physics, Mathematics
- 2001

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… Expand

Spin chains and combinatorics: twisted boundary conditions

- Physics, Mathematics
- 2001

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 2?/3 the Hamiltonian of… Expand

LETTER TO THE EDITOR: The quantum Knizhnik Zamolodchikov equation, generalized Razumov Stroganov sum rules and extended Joseph polynomials

- Mathematics, Physics
- 2005

We prove higher rank analogues of the Razumov–Stroganov sum rule for the ground state of the O(1) loop model on a semi-infinite cylinder: we show that a weighted sum of components of the ground state… Expand

The importance of being odd

- Mathematics, Physics
- 2000

In this Letter I mainly consider a finite XXZ spin chain with periodic boundary conditions and an odd number of sites. This system is described by the Hamiltonian Hxxz = -∑j = 1N{σjxσj + 1x + σjyσj +… Expand

Spin chains and combinatorics

- Mathematics, Physics
- 2000

In this letter we continue the investigation of finite XXZ spin chains with periodic boundary conditions and odd number of sites, initiated in paper \cite{S}. As it turned out, for a special value of… Expand

Combinatorial Nature of the Ground-State Vector of the O(1) Loop Model

- Mathematics, Physics
- 2001

Studying a possible connection between the ground-state vector for some special spin systems and the so-called alternating-sign matrices, we find numerical evidence that the components of the… Expand

Eight-vertex Model and Non-stationary Lame Equation

- Physics, Mathematics
- 2005

We study the ground state eigenvalues of Baxter's Q-operator for the eight-vertex model in a special case when it describes the off-critical deformation of the six-vertex model. We show that these… Expand

Bethe roots and refined enumeration of alternating-sign matrices

- Mathematics, Physics
- 2006

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… Expand

Quantum Knizhnik-Zamolodchikov equation, totally symmetric self-complementary plane partitions, and alternating sign matrices

- Mathematics, Physics
- 2007

AbstractWe present multiple-residue integral formulas for partial sums in the basis of link patterns of the polynomial solution of the level-1
$$U_q (\widehat{\mathfrak{s}\mathfrak{l}_2 })$$
… Expand