SO(5) Quantum Nonlinear sigma Model Theory of the High Tc Superconductivity

We show that the complex phase diagram of high $T_c$ superconductors can be deduced from a simple symmetry principle, a $SO(5)$ symmetry which unifies antiferromagnetism with $d$ wave superconductivity. We derive the approximate $SO(5)$ symmetry from the microscopic Hamiltonian and show furthermore that this symmetry becomes exact under the renormalization group flow towards a bicritical point. With the help of this symmetry, we construct a $SO(5)$ quantum nonlinear $\sigma$ model to describe the effective low energy degrees of freedom of the high $T_c$ superconductors, and use it to deduce the phase diagram and the nature of the low lying collective excitations of the system. We argue that this model naturally explains the basic phenomenology of the high $T_c$ superconductors from the insulating to the underdoped and the optimally doped region.