$XXZ$ Spin Chain with the Asymmetry Parameter $\Delta=-1/2$: Evaluation of the Simplest Correlators

We consider a finite $XXZ$ spin chain with periodic boundary conditions and an odd number of sites. It appears that for the special value of the asymmetry parameter $\Delta=-1/2$, the ground state of this system described by the Hamiltonian $H_{xxz}=-\sum_{j=1}^{N}\bigl\{\sigma_j^{x}\sigma_{j+1}^{x}+ \sigma_j^{y}\sigma_{j+1}^{y}-\frac12sigma_j^z\sigma_{j+1}^z\bigr\}$ has the energy $E_0=-3N/2$. Although the ground state is antiferromagnetic, we can find the corresponding solution of the Bethe equations. Specifically, we can explicitly construct a trigonometric polynomial $Q(u)$ of degree $n=(N-1)/2$, whose zeros are the parameters of the Bethe wave function for the ground state of the system. As is known, this polynomial satisfies the Baxter $T$–$Q$ equation. This equation also has a second independent solution corresponding to the same eigenvalue of the transfer matrix T. We use this solution to find the derivative of the ground-state energy of the $XXZ$ chain with respect to the crossing parameter $\eta$. This derivative is directly related to one of the spin-spin correlators, which appears to be $\langle\sigma_j^z\sigma_{j+1}^z\rangle=-1/2+3/2N^2$. In turn, this correlator gives the average number of spin strings for the ground state of the chain $\langle N_{\text{string}}\rangle={(3/8)(N-1)/N}$. All these simple formulas fail if the number $N$ of chain sites is even.