arXiv:cond-mat/0102247 · Journal of Physics A Mathematical and General

Spin chains and combinatorics: twisted boundary conditions

A. V. Razumov, Yu. G. Stroganov

Published Jun 22, 2001• 80 citations

Abstract

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.

DOI: 10.1088/0305-4470/34/26/304

Related papers

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/0012141Journal of Physics A Mathematical and GeneralMar 30, 2001217 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/0605004Journal of Statistical Mechanics Theory and ExperimentJul 13, 200618 citations

2002 Spectrum of a duality–twisted Ising quantum chain

Uwe Grimm

Abstract. The Ising quantum chain with a peculiar twisted boundary condition is considered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, which becomes a symmetry of the model at the critical point. Thus, at the critical point, the Ising quantum chain with the duality-twisted boundary is translationally invariant, similar as in the case of the usual periodic or antiperiodic boundary conditions. The complete energy spectrum of the Ising quantum chain is calculated analytically for finite systems, and the conformal properties of the scaling limit are investigated. This provides an explicit example of a conformal twisted boundary condition and a corresponding generalised twisted partition function. PACS numbers: 05.50.+q, 75.10.Jm, 71.10.Fd, 11.25.Hf Recently, there has been a lot of interest in generalised twisted partition functions of conformal field theory [1, 2, 3, 4] and the corresponding conformal twisted boundary conditions in solvable lattice models of statistical mechanics [5]. Examples of such boundary conditions leading to the appearance of “exotic ” spinor operators in the

Nov 19, 200114 citations

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:2008.03220Journal of Statistical Mechanics Theory and ExperimentJan 1, 20211 citation

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

Murray T. Batchelor, Jan de Gier, Bernard Nienhuis

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.

arXiv:cond-mat/0101385Journal of Physics A Mathematical and GeneralMay 2, 2001145 citations

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

A. V. Razumov, Yu. G. Stroganov, Paul Zinn-Justin

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.

Aug 4, 201330 citations