Topological insulators are a newly discovered class of materials which are bulk insulators with metallic surface states. The existence of gapless surface states is a direct consequence of the non-trivial topology of the bulk band structure of these materials. As a result theses states are robust against local disorder and currents propagate without back-scattering from non-magnetic defect. These exceptional electronic properties open the possibility to realize new phenomena in condensed matter physics ranging from fundamental physics (Majorana fermions) to technological applications (quantum computing, spintronics). In Z2 topological insulators, the topological phase originates from time reversal symmetry and strong spin-orbit interaction. As a consequence, bismuth, which is the stable element with the highest spin-orbit coupling strength, is a key ingredient of many topologically non-trivial compounds. Two well known examples of such compounds are Bi2Se3 and Bi2Te3 which are strong topological insulators with Dirac surface states.
Here I will present a density functional theory (DFT) study of other bismuth compounds with interesting topological phases. I will rst discuss the case of BiTeI, a layered semiconductor with giant Rashba splitting  which is expected to undergo a topological phase transition under pressure. It was predicted that pressure lead to gap closure with further reopening resulting in a strong Z2 topological insulator type band structure . Using GW approximation calculations we show that the predicted topological phase transition is hindered by the structural phase transition taking place at a lower pressure . I will then discuss the discovery of a new topological phase in the quasi one-dimensional compound Bi4I4. While DFT predicts a weak topological phase, the correct estimation of band ordering using the GW approximation reveals this material is a strong topological insulator characterized by Z2 invariants (1,110). The existence of this strong topological phase is con rmed experimentally using ARPES.
 A. Crepaldi et al., Phys. Rev. Lett. 109, 096803 (2012).
 M.S. Bahramy, B.-J. Yang, R. Arita, and N. Nagaosa, Nature Commun. 3, 679 (2012).
 M. K. Tran et al., Phys. Rev. Lett. 112, 047402 (2014).