Exponential complex-scalar potential

In rescaled units, $V(\varphi )=-|\varphi {|}^2+e^{\lambda |\varphi {|}^2}$ is bounded below only for $\lambda >0$. With $s=|\varphi {|}^2$, its derivative is $-1+\lambda e^{\lambda s}$. For $\lambda \ge 1$ the unique minimum is $s=0$; for $0<\lambda <1$ the minima form a circle $s=-\ln \lambda /\lambda$. The latter phase has a massless angular Goldstone boson and canonical radial squared mass $-2\ln \lambda$. At $\lambda =1$ the quadratic curvature vanishes at the single symmetric minimum; that accidental critical masslessness is not a Goldstone mode.