Nonperturbative Complete Energy Eigenbasis for Strongly Coupled Systems

We develop a non-pertrubative approach to the strongly coupled $\phi^4$ theory by using an eigen-energy basis that solves the full equations of motion. By rewriting the action in terms of this basis we are able to implement a nonperturbative ``energy-shell'' renormalization procedure, which yields a critical exponent of $\nu=0.6308$. We then identify and characterize an additional fixed point at even stronger coupling. All flows are relevant at this additional fixed point and the correlation exponent $\nu=\frac{2}{3}$ in three dimensions. We then discuss the differences between the value obtained for the anomalous dimension $\eta=0.10$ and that found in the literature $\eta=0.03$. Finally we report precise mean field exponents and logarithmic corrections in four dimensions.