Finite-size scaling behaviors of eigen microstates near the critical point

In the eigen microstate approach (EMA), phase emergence is signaled by the condensation of an eigen microstate with a nite eigenvalue. The associated eigenvalue acts as an order parameter, exhibiting nite-size scaling (FSS) behaviors within the critical regime. The eigen microstate derived from EMA precisely characterizes the spatial structure of the emergent phase, which remains less explored in systems lacking translational invariance. Here, we propose an FSS form for the eigen microstate itself and numerically validate it using the two-dimensional Ising model under various boundary conditions. This result enables the derivation of FSS relations for correlation functions, which we verify through numerical simulations. Our results complete the FSS formalism for the EMA.