We consider the anisotropic one-dimensional $X-Y$ model of Lieb, Schultz, and Mattis and study the three zero-temperature correlation functions ${\rho }_{\mu \nu }\mathrm{}(R,t)=〈S_1^{\mu }\mathrm{}\left(0\right)\mathrm{}{S}_{R+1\mathrm{}}^{\nu }\mathrm{}\left(t\right)\mathrm{}〉$, where $\mu \nu =x,y,z$ but $\mu \ne \nu$. We obtain the first-order term of ${\rho }_{\mathrm{xy}}$ for $R\to \infty$ or $t\to \infty$ and exact expressions for ${\rho }_{\mathrm{xz}}$ and ${\rho }_{\mathrm{yz}}$.

• Received 20 July 1971

DOI:https://doi.org/10.1103/PhysRevA.4.2314