Spin dynamics of the doped t-J model

We present a new method for studying the low energy excitations in finite cluster systems and apply it to clarify the nature of the low energy spin excitations in the moderately doped t-J model. We find that the character of the low energy spin excitations changes within the Brillouin zone, with the dominant excitation for momentum transfer (pi,pi) being a spin wave-like collective mode. Spin excitations with momentum transfer different from (pi,pi) have particle-hole character. We give an explicit construction of a low energy collective spin mode in a system with `almost broken symmetry' and demonstrate its validity by explicit numerical check.