Fluctuations in Phase Transition of Quantum Materials and ML-Assisted Spectral Characterization

New states of matter are often identified by the emergence of long-range order through a thermodynamic phase transition. Traditionally, such transitions are probed by electron or optical spectroscopy, with the opening of an energy gap serving as a proxy for an order parameter under the mean-field approximation. However, in quantum materials, especially many low-dimensional and strongly correlated materials, quantum fluctuations are significant, causing a decoupling between gap opening and symmetry breaking. In this talk, I will present several examples of such fluctuating states, identified through a range of spectroscopic techniques, with a focus on unconventional superconductivity and structural transitions. These cases illustrate that simple spectroscopic analyses are insufficient for reliably identifying quantum phases. To overcome this limitation, I will introduce a machine-learning-based approach capable of distinguishing between fluctuations and long-range order using subtle spectral features. By combining angle-resolved photoemission spectroscopy with a domain-adversarial neural network, trained through mixed experimental and simulation spectral data, our model can detect thermodynamic phase transitions directly from single-snapshot electronic spectra. More broadly, this framework represents a solution to the limited availability of experimental data in material spectroscopy by effectively and properly incorporating high-fidelity simulated data.