  The equilibrium phase behavior of microphase-forming substances and models is
notoriously difficult to obtain because of the extended metastability of the
modulated phases. We develop a simulation method based on thermodynamic
integration that avoids this problem and with which we obtain the phase diagram
of the canonical three-dimensional axial next-nearest-neighbor Ising model. The
equilibrium devil's staircase, magnetization, and susceptibility are obtained.
The critical exponents confirm the XY nature of the disorder-modulated phase
transition beyond the Lifshitz point. The results identify the limitations of
various approximation schemes used to analyze this basic microphase-forming
model.
