Evidence for Three-$\alpha$ Breathing Modes Uncovered by Control Neural Network

This work introduces a new Control Neural Network (Ctrl.NN) method to uncover evidence of exotic quantum state, \textit{i.e.}, the breathing modes in 3-$\alpha$ resonant states of $^{12}$C nucleus. We provide the most precise microscopic description to date for the $^{12}$C energy spectrum, identify two new exotic breathing states, and uncover strong evidence that directly connects the recent experimental observations to the breathing modes. The Ctrl.NN method significantly simplifies numerical calculations of quantum systems under multiple constraints and offers a new perspective for solving the nuclear many-body problem.