  We have used exact numerical diagonalization to study the excitation spectrum
and the dynamic spin correlations in the $s=1/2$ next-next-nearest neighbor
Heisenberg antiferromagnet on the square lattice, with additional 4-spin ring
exchange from higher order terms in the Hubbard expansion. We have varied the
ratio between Hubbard model parameters, $t/U$, to obtain different relative
strengths of the exchange parameters, while keeping electrons localized. The
Hubbard model parameters have been parametrized via an effective ring exchange
coupling, $J_r$, which have been varied between 0$J$ and 1.5$J$. We find that
ring exchange induces a quantum phase transition from the $(\pi, \pi)$ ordered
Ne\`el state to a $(\pi/2, \pi/2)$ ordered state. This quantum critical point
is reduced by quantum fluctuations from its mean field value of $J_r/J = 2$ to
a value of $\sim 1.1$. At the quantum critical point, the dynamical correlation
function shows a pseudo-continuum at $q$-values between the two competing
ordering vectors.
