1. Fermionic Symmetry Protected Topological Phase Induced by Interaction Shangqiang NING First year PHD student Institute For Advanced Study, Tsinghua University Email: nsq12382@126.com

2. Collaborators Hongchen Jiang Slac, Stanford Zhengxin Liu IASTU, Beijing

3. Outline 1. Background and motivation 2. Study of a specific model 3. Conclusion

4. Background and Motivation 2D topological insulator Key properties of TI: Band insulator Mass terms breaks T breaks U(1) X. Qi, S.C. Zhang RevModPhys.83.1057

5. Symmetry Protected Topological phase (SPT phase) gapped gapless

6. Classification of SPT For more symmetry group in

7. ∎ SPT phases also exist in Bosonic systems S=1 Haldane phase (Bosonic) Finite excitation gap No symmetry breaking SO(3) symmetry protected spin-1/2 edge states Degeneracy of entanglement spectrum Spin-1/2 Spin-1 Haldane 83; Affleck, Kennedy, Lieb, Tasaki 87 Gu, Wen 09; Pollmann et.al 09; Chen,Gu,Wen 11 bulk gap SO(3) symmetry

8. A B

9. Entanglement spectrum J 2 /J 1 =0.9 SO(3) symmetry Non trivial Trivial

10. Classification of Bosonic SPT SO(3) symmetry : 2 phases Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen,2011

11. Classification of Interacting Fermionic SPT 1D superconductor 2D superconductor Time reversal and mirror reflect ZZ8Z8 It seems that interactions reduce the classification. However, it maybe not true! Hong Yao and Shinsei Ryu, arXiv :1202.5805 Evelyn Tang and Xiao-Gang Wen arXiv:1204.0520v2

12. Model

13. Symmetry and classification free case, only the trivial phase. 2 projective representation  two SPT phases.

14. Non-interacting limit U=0, J=0 = 2-fold

15. Non-interacting limit U=0, J=0 SPT only focus on gapped phases. Open boundary, no zero mode, indicating the trivialness of phase

16. Add Hubbard interaction the band insulator and the Mott insulator are adiabatically connected

17. Interacting case

18. Competition

19. Interacting case

31. We find two phases in the interacting case ： trivial insulator and Haldane insulator

32. Conclusion

33. Thanks for your attention!