A. V. Razumov

Spin chains and combinatorics

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 = -3N/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.

Spin chains and combinatorics: twisted boundary conditions

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 the system possesses an eigenvalue -3N?/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.

O(1) loop model with different boundary conditions and symmetry classes of alternating-sign matrices

This work is a continuation of our recent paper where we discussed numerical evidence that the numbers of the states of the fully packed loop model with fixed pairing patterns coincide with the components of the ground state vector of the O(1) loop model with periodic boundary conditions and an even number of sites. We give two new conjectures related to different boundary conditions: we suggest and numerically verify that the numbers of the half-turn symmetric states of the fully packed loop model with fixed pairing patterns coincide with the components of the ground state vector of the O(1) loop model with periodic boundary conditions and an odd number of sites and that the corresponding numbers of the vertically symmetric states describe the case of open boundary conditions and an even number of sites.

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

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 anisotropy parameter Δ equal to −1/2 and an odd number of sites. The obtained integral representations for the components of this eigenvector allow us to prove some conjectures on its properties formulated earlier. A new statement relating the ground-state components of XXZ spin chains and Temperley–Lieb loop models is formulated and proved.

ENUMERATION OF QUARTER-TURN-SYMMETRIC ALTERNATING-SIGN MATRICES OF ODD ORDER

Kuperberg showed that the partition function of the square-ice model related to half-turn-symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijective correspondence with half-turn-symmetric alternating-sign matrices of odd order. The partition function of this model is expressed via the above factors. We find the contributions to the partition function that correspond to the alternating-sign matrices having 1 or −1 as the central entry and establish the related enumerations.

Refined enumerations of some symmetry classes of alternating-sign matrices

Using determinant representations for partition functions of the corresponding square ice models and the method proposed recently by one of the authors, we investigate refined enumerations of vertically symmetric alternating-sign matrices, off-diagonally symmetric alternating-sign matrices and alternating-sign matrices with U-turn boundary. For all these cases the explicit formulas for refined enumerations are found. It particular, Kutin–Yuen conjecture is proved.

A possible combinatorial point for the XYZ spin chain

We formulate and discuss several conjectures related to the ground state vectors of odd-length XYZ spin chains with periodic boundary conditions and a special choice of the Hamiltonian parameters. In particular, we argue for the validity of a sum rule for the vector components that in a sense describes the degree of antiferromagneticity of the chain.

Bethe roots and refined enumeration of alternating-sign 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 2M+1.

Three-coloring statistical model with domain wall boundary conditions: Functional equations

We consider the Baxter three-coloring model with boundary conditions of the domain wall type. In this case, it can be proved that the partition function satisfies some functional equations similar to the functional equations satisfied by the partition function of the six-vertex model for a special value of the crossing parameter.