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arXiv:cond-mat/0012035 · Journal of Physics A Mathematical and General

The importance of being odd

Yu. G. Stroganov

Published Mar 26, 2001• 108 citations

Abstract

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 H xxz = -∑ j = 1 N {σ j x σ j + 1 x + σ j y σ j + 1 y + Δ σ j z σ j + 1 z }. As it turns out, the ground state energy is proportional to the number of sites E = -3 N /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 ⟨σ j z σ j + 1 z ⟩ = -1/2 + 3/2 N 2 . This correlation function is related to the average number of spin strings for the ground state ⟨ N string ⟩ = 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.

DOI: 10.1088/0305-4470/34/13/104

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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 H xxz = -∑ j = 1 N {σ j x σ j + 1 x + σ j y σ j + 1 y + Δ σ j z σ j + 1 z }. As it turns out, the ground state energy is proportional to the number of sites E = -3 N /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 ⟨σ j z σ j + 1 z ⟩ = -1/2 + 3/2 N 2 . This correlation function is related to the average number of spin strings for the ground state ⟨ N string ⟩ = 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.

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 scalar product of XXZ spin chain revisited. Application to the ground state at

Δ=-1/2

For the scalar product

S_n

of the XXZ

s=1/2

spin chain we derive a new determinant expression which is symmetric in the Bethe roots. We consider an application of this formula to the inhomogeneous groundstate of the model with

Δ=-1/2

with twisted periodic boundary conditions. At this point the ground state eigenvalue

τ_n

of the transfer matrix is known and has a simple form that does not contain the Bethe roots. We use the knowledge of

τ_n(μ)

to obtain a closed expression for the scalar product. The result is written in terms of Schur functions. The computations of the normalization of the ground state and the expectation value of

σ^z

are also presented.

Abstract. 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 e ±2πi/3 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 ∆ equal to −1/2 and odd number of sites. The obtained integral representations for the components of this eigenvector allow to prove some conjectures on its properties formulated earlier. A new conjecture relating the ground state components of XXZ spin chains and Temperley–Lieb loop models is formulated and proved. 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.