Spin chains and combinatorics

Abstract

In this letter we continue the investigation of finite XXZ spin chains with periodic boundary conditions and odd number of sites, initiated in paper [1]. As it turned out, for a special value of the asymmetry parameter ∆ = −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 of paper [1] 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 wave function components are noticed. It is turned out, for example, that the ratio of the largest component to the least one is equal to the number of the alternating sing matrices.

Calculation of multi-fractal dimensions in spin chains

“…[5] For this model, there exists a special point, = −1/2, called the combinatorial point, where more information about the GS wave function is available. From the Razumov-Stroganov conjecture [33] proved in [34], it follows that at this value of and odd N = 2R + 1 the following statements are valid:…”

On Polynomials Interpolating Between the Stationary State of a O(n) Model and a Q.H.E. Ground State

“…, 1) can be normalized to be positive integers. These integers are conjectured to count ASM [1]. At this value, the sum of the components of Ψ(z i ) is a symmetric polynomial in the z i which can be evaluated explicitly [5].…”

“…It is convenient to introduce the operators e ij having the expression (109) and acting in C 2 i ⊗ C 2 j . T ij = t 1 2 + e ij , and the operators x ij are defined as in (C.1), x ij = T ij P ij . In the same basis as above, x ij is realized as a triangular matrices as: …”

The Raise and Peel Model of a Fluctuating Interface

“…Another bijection relates the ice model with special boundary conditions to a fully packed loop model (FPLM) [17,18,19]. A second conjecture (conjecture II) states that the weight of an RSOS path in the PDF of the stationary state of our model is given by the number of FPLM configurations with the same topology [7,20]. (In Section 3 we will review these facts).…”