Optimization of basis functions for the multi-configuration mixing using the Replica Exchange Monte-Carlo method and its application to $^{12}$C

To calculate excited states in quantum many-body systems, multi-configuration mixing has often been employed. However, it has been still unclear how to choose important Slater determinants from a huge model space. We propose a novel efficient method as the Replica Exchange Monte-Carlo (RXMC) method to sample important Slater determinants and optimize and analyze the obtained results. As an application, we apply it to the ground and excited states of $^{12}$C based on the Bloch-Brink $\alpha$ cluster model and show the detailed structure of the obtained states. The RXMC method enables us to efficiently sample Slater determinants following the Boltzmann distribution on the multi-dimensional potential energy surface (PES) under a given model space. To analyze the obtained excited states, we embed sampled basis functions onto the PES calculated with the $\beta$-$\gamma$ constraint method and discuss the main component in the state. The RXMC method can efficiently perform the samplings with a temperature parameter of $T_L=2.5$ MeV in $^{12}$C. We obtain the gas-like state with a wide density distribution in the tail part in the second $0^+$ state. We also obtain the linear-chain-like states with the bending and stretching vibrational modes in the third and fourth $0^+$ states, respectively. In the fifth $0^+$ state, the main component of the basis functions contains expanded equilateral triangle configurations. The second $0^+$ gas-like state emerges at the local minimum in the PES, which is the beginning of the valley structure connected to the linear-chain breakup channel. The third and fourth linear-chain-like states emerge in this valley structure. We conclude that the RXMC method is a powerful method to calculate the excited states of nuclei, which would be applied to much complicated nuclear fission dynamics in heavier nuclei.