  In this paper, by analyzing the underlying Lefschetz thimble structure, we
study quantum phases in zero-dimensional scalar field theories with complex
actions. Using first principles, we derive the Lefschetz thimble equations of
these models, and discuss the issues when we apply the same calculations for
more complicated systems. We also derive the conditional expressions involving
relations among the parameters of the model, that would help us predict quantum
phase transitions in these systems.
