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To calculate excited states in quantum many-body systems, multi-configurationmixing has often been employed. However, it has been still unclear how tochoose important Slater determinants from a huge model space. We propose anovel efficient method as the Replica Exchange Monte-Carlo (RXMC) method tosample important Slater determinants and optimize and analyze the obtainedresults. 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 detailedstructure of the obtained states. The RXMC method enables us to efficientlysample Slater determinants following the Boltzmann distribution on themulti-dimensional potential energy surface (PES) under a given model space. Toanalyze the obtained excited states, we embed sampled basis functions onto thePES calculated with the $\beta$-$\gamma$ constraint method and discuss the maincomponent in the state. The RXMC method can efficiently perform the samplingswith a temperature parameter of $T_L=2.5$ MeV in $^{12}$C. We obtain thegas-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 andstretching vibrational modes in the third and fourth $0^+$ states,respectively. In the fifth $0^+$ state, the main component of the basisfunctions contains expanded equilateral triangle configurations. The second$0^+$ gas-like state emerges at the local minimum in the PES, which is thebeginning of the valley structure connected to the linear-chain breakupchannel. The third and fourth linear-chain-like states emerge in this valleystructure. We conclude that the RXMC method is a powerful method to calculatethe excited states of nuclei, which would be applied to much complicatednuclear fission dynamics in heavier nuclei.

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