We have performed first principles study for bismuth crystal structure at pressure from 0~14 GPa based on structure search and quasi-harmonic approximation. A new stable phase with Pbcm symmetry is predicted by structure search method. We find that the most stable structures of bismuth at 0 K are R-3m(0 ~ 3.29 GPa), Pbcm (3.29 GPa ~ 4.91 GPa), Cmce (4.91 GPa ~ 10.57 GPa) and Im-3m ( > 10.57 GPa), respectively. By quasi-harmonic approximation, we predicted the phase diagram of bismuth from first principles calculations. We found that the phase transition pressure reduces with increasing temperature. Our calculation agrees with the trend of experimental phase diagram. The P4/ncc model structure for the incommensurate Bi-III phase is not a stable phase in our calculation. A better model for the Bi-III phase is still needed. We also note that the spin-orbital interaction is very important for phase-diagram simulation of bismuth. By using first principle based structure search method, we successfully determine the low temperature high pressure phase diagram of bismuth, showing that the structure search method can effectively find the most stable structure of given material at high pressure even with high Z elements.

We investigate the valence transition in three-dimensional topological Kondo insulator through slave-boson analysis of periodic Anderson model. By including the effect of intra-atomic Coulomb correlation $U_{fc}$ between conduction and local electrons, we find a first-order valence transition from Kondo region to mixed valence upon ascending of local level above a critical $U_{fc}$, and this valence transition usually occurs very close to or simultaneously with a topological transition. Near the parameter region of zero-temperature valence transition, rise of temperature can generate a thermal valence transition from mixed valence to Kondo region, accompanied by a first-order topological transition. Remarkably, above a critical $U_{fc}$ which is considerable smaller than that generating paramagnetic valence transition, the original continuous antiferromagnetic transition is shifted to first order one, at which a discontinuous valence shift takes place. Upon increased $U_{fc}$, the paramagnetic valence transition approaches then converges with the first-order antiferromagnetic transition, leaving an significant valence shift on the magnetic boundary. The continuous antiferromagnetic transition, first-order antiferromagnetic transition, paramagnetic valence transition and topological transitions are all summarized in a global phase diagram. Our proposed exotic transition processes can help to understand the thermal valence variation as well as the valence shift around the pressure-induced magnetic transition in topological Kondo insulator candidates and in other heavy-fermion systems.