Universal Spin Response in Copper Oxide Materials

The spin response in the copper oxide materials at finite temperatures in the underdoped and optimal doped regimes is studied within the framework of the fermion-spin theory. The integrated dynamical spin structure factor is almost temperature independent, the integrated susceptibility shows the particularly universal behavior as $I(\omega, T) \propto arctan [ a_{1}\omega/T + a_{3}(\omega/T)^{3}]$, and the spin-lattice relaxation time is weakly temperature dependent, which are consistent with experiments and numerical simulations.