  We propose a new approach to constructing a phase diagram using the effective
Hamiltonian derived only from a single real-space image produced by scanning
tunneling microscopy (STM). Currently, there have been two main methods to
construct phase diagrams in material science: ab initio calculations and
CALPHAD with thermodynamic information obtained by experiments and/or
theoretical calculations. Although the two methods have successfully revealed a
number of unsettled phase diagrams, their results sometimes contradicted when
it is difficult to construct an appropriate Hamiltonian that captures the
characteristics of materials, e.g., for a system consisting of multiple-scale
objects whose interactions are essential to the system's characteristics.
Meanwhile, the advantage of our approach over existing methods is that it can
directly and uniquely determine the effective Hamiltonian without any
thermodynamic information. The validity of our approach is demonstrated through
an Mg--Zn--Y long-period stacking-ordered structure, which is a challenging
system for existing methods, leading to contradictory results. Our result
successfully reproduces the ordering tendency seen in STM images that previous
theoretical study failed to reproduce and clarifies its previously unknown
phase diagram. Thus, our approach can be used to clear up contradictions shown
by existing methods.
