Dynamical T=0 correlations of the S=1/2 1D Heisenberg Anti-Ferromagnet with 1/r^2 exchange in a magnetic field

Abstract

We present a new selection rule for matrix elements of local spin operators in the $S=1/2$ ``Haldane-Shastry'' model. Based on this rule we extend a recent exact calculation \cite{H93} of the ground-state dynamical spin correlation function $S^{ab}(n,t)$ = $\langle 0 | S^a(n,t)S^b(0,0)| 0 \rangle$ and its Fourier-transform $S^{ab}(Q,E)$ of this model to a finite magnetic field. In zero field, only {\it two-spinon} excitations contribute to the spectral function; in the (positively) partially-spin-polarized case, there are two types of elementary excitations: {\it spinons} ($ΔS^z = \pm 1/2$) and {\it magnons} ($ΔS^z = -1 $). The magnons are divided into left- or right-moving branches. The only classes of excited states contributing to the spectral functions are: (I) two spinons, (II) two spinons + one magnon, (IIIa) two spinons + two magnons (moving in opposite directions), (IIIb) one magnon. The contributions to the various correlations are: $S^{-+}$: (I); $S^{zz}$: (I)+(II); $S^{+-}$: (I)+(II)+(III). In the zero-field limit there are no magnons, while in the fully-polarized case, there are no spinons. We discuss the relation of the spectral functions to correlations of the Calogero-Sutherland model at coupling $λ= 2$.