Effective potential of latticeφ4theory

The structure of the effective potential for lattice ${\phi }^4$ theory is discussed. It is shown that the effective potential $V\left(\hat{\phi }\right)$ is undefined in an infinite lattice system for certain values of $\hat{\phi }$ if spontaneous symmetry breaking occurs. It is also shown that $\frac{d^{2}V}{d{\hat{\phi }}^2}\ge 0$ for all $\phi$, thus precluding the familiar "double-well" shape suggested by the classical potential. These results do not depend on the spacetime dimensionality of the lattice or upon the particulars of any loop expansion. A graphical approximation procedure for the effective potential is formulated and compares very favorably with Monte Carlo results. Comparisons with strong-coupling-expansion predictions are also made.