Yu. G. Stroganov

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.

The importance of being odd

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 + 1y + Δxa0σjzσj + 1z}. As it turns out, the ground state energy is proportional to the number of sites E = -3N/2 for a special value of the asymmetry parameter Δ = -1/2. The trigonometric polynomial Q(u), the zeros of which are parameters of the ground state Bethe eigenvector, is explicitly constructed. This polynomial of degree n = (N-1)/2 satisfies the Baxter T-Q equation. Using the second independent solution of this equation that corresponds to the same eigenvalue of the transfer matrix, it is possible to find a derivative of the ground state energy w.r.t. the asymmetry parameter. This derivative is closely connected with the correlation function σjzσj + 1z = -1/2 + 3/2N2. This correlation function is related to the average number of spin strings for the ground state Nstring = ¾(N-1/N). I would like to stress that all the above simple formulae are not applicable to the case of an even number of sites which is usually considered.

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.

Bethe equations `on the wrong side of the equator'

We analyse Baxters famous T-Q equations for the XXX (XXZ) spin chain and show that, apart from its usual polynomial (trigonometric) solution, which provides the solution of Bethe ansatz equations, there also exists a second solution which should correspond to the Bethe ansatz beyond . This second solution of Baxters equation plays an essential role and together with the first one gives rise to all fusion relations.

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.

Families of solutions of the nested Bethe ansatz for the A2 spin chain

The full set of polynomial solutions of the nested Bethe ansatz is constructed for the case of the A2 rational spin chain. The structure and properties of these associated solutions are more various than the case of the usual XXX (A1) spin chain but their role is similar.

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

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 Uq (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.

New conjecture for the SUq(N) Perk–Schultz models

We present a new conjecture for the SUq(N) Perk–Schultz models. This conjecture extends a conjecture presented in our article (Alcaraz F C and Stroganov Yu G J. Phys. A: Math. Gen. 35 6767–87).