The Importance of being Odd

Abstract

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 $H_{xxz}=-{\sum }_{j=1}^N\{{\sigma }_j^x{\sigma }_{j+1}^x+{\sigma }_j^y{\sigma }_{j+1}^y+\Delta {\sigma }_j^z{\sigma }_{j+1}^z\}$. As it turned out, its ground state energy is exactly proportional to the number of sites $E=-3N/2$ for a special value of the asymmetry parameter $\Delta =-1/2$. The trigonometric polynomial $q(u)$, zeroes of which being the parameters of the ground state Bethe eigenvector is explicitly constructed. This polynomial of degree $n=(N-1)/2$ 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 $<{\sigma }_j^z{\sigma }_{j+1}^z>=-1/2+3/2N^2$. In its turn this correlation function is related to an average number of spin strings for the ground state of the system under consideration: $<N_{string}>=3/8(N-1/N)$. 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.

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

References in corpus (2)

• Spin chains and combinatorics
• Is the purely biquadratic spin 1 chain always massive?

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