1. Topology induced emergent dynamic gauge theory in an extended Kane-Mele-Hubbard model Xi Luo (Fudan) @Taipei January 5, 2015 arXiv: 1408.5730

2. Collaborators Yue Yu (Chinese Academy of Science, Fudan, and Collaborative Innovation Center of Advanced Microstructures) Long Liang (Chinese Academy of Science)

3. Outline Introduction Emergence of the Proca theory Emergent QED 3 Conclusions and discussions

4. Introduction Confinement and deconfinement phase transition of gauge fields is crucial in particle physics and condensed matter physics.

5. Introduction Eg.1 QCD, still a mystery. – (Alford etal. Rev.Mod.Phys 2008)

6. Introduction Eg.2 spin-charge separation, superconductor, and etc. – (high field magnet laboratory, Radboud University)

7. Introduction Haldane model (Haldane PRL 88’) – NNN coupling with a phase – Breaking IS leads to a trivial insulator – Breaking TRS leads to a topological Chern insulator – Intrinsic property of band structure without external magnetic field, Hall conductance, QAHE

8. Introduction Experimental realization (Jotzu etal. Nature 2014) – Ultracold fermionic atoms in a periodically modulated optical honeycomb lattice

9. Introduction Thirring model (Fradkin & Schaposnik, PLB 94’) – Bosonization in 3 dimensions – Massive Thirring model – Maxwell-Chern-Simons theory – Hubbard-Stratonovich transformation

10. Outline Introduction Emergence of the Proca theory Emergent QED 3 Conclusions and discussions

11. Emergence of the Proca theory Kane-Mele model without Rashba interaction (Kane and Mele, PRL, 2005) Dirac fermion dispersion – Around Dirac points – Mass gap with – Chern number TI (SPT) in 2D

12. Emergence of the Proca theory With a current-current interaction – where is the physical current: – NN Hubbard interaction for j 0 - terms

13. Emergence of the Proca theory Effective field theory (doubled Thrring model) After a Hubbard-Stratonovich transformation and integrate out the Fermions (mutual CS)

14. Emergence of the Proca theory Effective field theory – As U becomes stronger, the excitation energy for the gauge field will be lower than the charge gap. Then we can integrate out the charge current. – Define

15. Emergence of the Proca theory Effective field theory – The spin gap is lower than the charge gap means that m A is smaller than min{Δ, t}. This requires, – In this case, a Proca theory emerges. The efforts to determine the limit of the photon mass are going along a long time. Our results give a playground to see what happens if there is a massive photon.

16. Emergence of the Proca theory Collective excitation lies in the charge gap, the GS is still TI.

17. Emergence of the Proca theory Effective field theory – Finite Proca mass is consistent with no gapless excitation in the bulk of a topological insulator. – Emergence of gauge field due to topology other than mean field theory.

18. Emergence of the Proca theory Correlation function and Bragg spectroscopy (Stamper-Kurn etal. PRL 99’) – M≠0

19. Emergence of the Proca theory In the zero mass limit (c->0, or large λ), a compact U(1) Maxwell theory emerges and the monopole condensation will induce charge confinement. Correlation function and Bragg spectroscopy – M=0, monopole condensation

20. Outline Introduction Emergence of the Proca theory Emergent QED 3 Conclusions and discussions

21. Emergent QED 3 Model – Emergence of a Chern-Simons term in the gauge theory requires to break the TRS. Distinguishing the hoppings and the couplings by spins will do so. – an extra fermion χ is put in, which may or may not have a non-trivial Chern number and serves as the matter field in the emergent QED 3.

22. Emergent QED 3 Interaction Effective theory “Charge” carried by χ fermion

23. Emergent QED 3 Experimental phenomena (PHE and QAHE) – Integrate out the χ fermion – The current response

24. Emergent QED 3 When there is a static spatial distribution of the densities of the spinful fermions in the bulk, the Proca equations are simplified – The PHE and QAHE responding to the "electric" field, i.e., the fluctuation of the gradient of the spin density, can be observed either individually or combinatorially.

25. Conclusions and Discussions 1) Emergent Proca and QED 3 from a weak interacting Kane-Mele model 2) Emergence of gauge field due to topology 3) Confinement 4) PHE and QAHE 5) Rashba, non abelian 6) Experiments for Proca theory 7) (4+1)d generalization, second Chern number?