Quantum phase transitions with eigen microstate approach in one-dimensional transverse-field Ising model

We propose an eigen microstate approach (EMA) for analyzing quantum phase transitions in quantum many-body systems, introducing a novel framework that does not require prior knowledge of an order parameter. Using the transverse-field Ising model (TFIM) as a case study, we demonstrate the effectiveness of EMA by identifying key features of the phase transition through the scaling behavior of eigenvalues and the structure of associated eigen microstates. Our results reveal substantial changes in the ground state of the TFIM as it undergoes a phase transition, as reflected in the behavior of specific components ${\xi }_i^{(k)}$ within the eigen microstates. This method is expected to be applicable to other quantum systems where predefining an order parameter is challenging.