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2000200620122018202402468
Abstract: (arXiv)
In this letter I consider mainly a finite XXZ spin chain with periodic boundary conditions and \bf{odd} \rm number of sites. This system is described by the Hamiltonian Hxxz=−∑j=1N{σjxσj+1x+σjyσj+1y+Δσjzσj+1z}H_{xxz}=-\sum_{j=1}^{N}\{\sigma_j^{x}\sigma_{j+1}^{x} +\sigma_j^{y}\sigma_{j+1}^{y} +\Delta \sigma_j^z\sigma_{j+1}^z\}. As it turned out, its ground state energy is exactly proportional to the number of sites E=−3N/2E=-3N/2 for a special value of the asymmetry parameter Δ=−1/2\Delta=-1/2. The trigonometric polynomial q(u)q(u), zeroes of which being the parameters of the ground state Bethe eigenvector is explicitly constructed. This polynomial of degree n=(N−1)/2n=(N-1)/2 satisfy the Baxter T-Q equation. Using the second independent solution of this equation corresponding 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<\sigma_j^z\sigma_{j+1}^z> =-1/2+3/2N^2. In its turn this correlation function is related to an average number of spin strings for the ground state of the system under consideration: <Nstring>=3/8(N−1/N)<N_{string}> = {3/8}(N-1/N). I would like to stress once more that all these simple formulas are \bf wrong \rm in the case of even number of sites. Exactly this case is usually considered.
Note:
  • 9 pages, based on the talk given at NATO Advanced Research Workshop "Dynamical Symmetries in Integrable Two-dimensional Quantum Field Theories and Lattice Models", 25-30 September 2000, Kyiv, Ukraine. New references are added plus some minor corrections Subj-class: Statistical Mechanics; Exactly Solvable and Integrable Systems
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