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.
Hanging about a hypothetical connections between the ground state vector for some special spin systems and the alternating-sign matrices, we have found a numerical evidence for the fact that the numbers of the states of the fully packed loop model with fixed link-patterns coincide with the components of the ground state vector of the dense O(1) loop model considered by Batchelor, de Gier and Nienhuis. Our conjecture generalizes in a sense the conjecture of Bosley and Fidkowski, refined by Cohn and Propp, and proved by Wieland.
I would say that imagination is a form of memory.(V. Nabokov)In paper [1] we made some conjectures related to combinatorial properties of the ground state vector of the XXZ spin chain for the asymmetry parameter ∆ = −1/2 and an odd number of cites. In the subsequent paper [2] Batchelor, de Gier and Nienhuis considered two variations of this model along with the corresponding O(n) loop model at n = 1 and notably increased the number of models and related combinatorial objects (see also [3]).During last months we have accumulated a lot of data on relation between the spin systems and alternating sign matrices (ASMs). In this note we limit ourselves to one qualitative conjecture which relates some classes of the states of the fully packed loop (FPL) model with the ground state vector of the dense O(n) loop model (see [4] and references therein). The states the FPL model is in bijective correspondence with the ASMs. Therefore if our conjecture is true then one can investigate the FPL model, and the ASMs, using the methods elaborated in mathematical physics for integrable model (see, for example, [5]).The background information on the ASMs and their different combinatorial forms can be found in the recent review by Propp [6] and in references therein. For more details on enumerative problems related to the states of the FPL model see the article by Wieland [7].Following the review paper by Propp [6] we define the "generalized tic-tac-toe" graph as the graph formed by n horizontal lines and n vertical lines meeting n 2 intersections of degree 4, with 4n vertices of degree 1 at the boundary. Then we number the vertices of degree 1. We start with the left top vertex and number clockwise every other vertex. Now 1
We analyse the famous Baxter's T −Q equations for XXX (XXZ) spin chain and show that apart from its usual polynomial (trigonometric) solution, which provides the solution of Bethe-Ansatz equations, there exists also the second solution which should corresponds to Bethe-Ansatz beyond N/2. This second solution of Baxter's equation plays essential role and together with the first one gives rise to all fusion relations.
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