Topological semimetal in a fermionic optical lattice


Optical lattices have an important role in advancing our understandingof correlated quantum matter. The recent implementation of orbital degrees of freedom in chequerboard1,2 and hexagonal3 optical lattices opens up a new avenue towards discovering novel quantum states of matter that have no prior analogues in solid-state electronic materials. Here, we predict that an exotic topological semimetal emerges as a parity-protected gapless state in the orbital bands of a two-dimensional fermionic optical lattice. This new quantum state is characterized by a parabolic band-degeneracy point with Berry flux 2π, in sharp contrast to the π flux of Dirac points as in graphene. We show that the appearance of this topological liquid is universal for all lattices with D4 point-group symmetry, as long as orbitals with opposite parities hybridize strongly with each other and the band degeneracy is protected by odd parity. Turning on inter-particle repulsive interactions, the system undergoes a phase transition to a topological insulator whose experimental signature includes chiral gapless domain-wall modes, reminiscent of quantum Hall edge states.

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Figure 1: The optical lattice potential and its experimental implementation.
Figure 2: The single-particle energy spectrum (measured in units of ER) for the lowest four bands and the topological structure near band-degeneracy points.
Figure 3: Topologically protected edge states and domain-wall modes.


  1. 1

    Wirth, G., Ölschläger, M. & Hemmerich, A. Evidence for orbital superfluidity in the p-band of a bipartite optical square lattice. Nature Phys. 7, 147–153 (2011).

  2. 2

    Ölschläger, M., Wirth, G. & Hemmerich, A. Unconventional superfluid order in the f band of a bipartite optical square lattice. Phys. Rev. Lett. 106, 015302 (2011).

  3. 3

    Soltan-Panahi, P., Lühmann, D-S., Struck, J., Windpassinger, P. & Sengstock, K. Quantum phase transition to unconventional multi-orbital superfluidity in optical lattices. Preprint at (2010).

  4. 4

    Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

  5. 5

    Kane, C. & Mele, E. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

  6. 6

    Bernevig, B. A., Hughes, T. L. & Zhang, S-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

  7. 7

    Konig, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).

  8. 8

    Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).

  9. 9

    Moore, J. E. & Balents, L. Topological invariants of time-reversal-invariant band structures. Phys. Rev. B 75, 121306 (2007).

  10. 10

    Roy, R. Z2 classification of quantum spin Hall systems: An approach using time-reversal invariance. Phys. Rev. B 79, 195321 (2009).

  11. 11

    Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970–974 (2008).

  12. 12

    Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

  13. 13

    Qi, X-L. & Zhang, S-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

  14. 14

    Kitaev, A. Periodic table for topological insulators and superconductors. Preprint at (2009).

  15. 15

    Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).

  16. 16

    Isacsson, A. & Girvin, S. M. Multi-flavor bosonic Hubbard models in the first excited Bloch band of an optical lattice. Phys. Rev. A 72, 053604 (2005).

  17. 17

    Liu, W. V. & Wu, C. Atomic matter of non-zero momentum Bose–Einstein condensation and orbital current order. Phys. Rev. A 74, 013607 (2006).

  18. 18

    Kuklov, A. B. Unconventional strongly interacting Bose–Einstein condensates in optical lattices. Phys. Rev. Lett. 97, 110405 (2006).

  19. 19

    Köhl, M., Moritz, H., Stöferle, T., Günter, K. & Esslinger, T. Fermionic atoms in a three dimensional optical lattice: Observing Fermi surfaces, dynamics, and interactions. Phys. Rev. Lett. 94, 080403 (2005).

  20. 20

    Browaeys, A. et al. Transport of atoms in a quantum conveyor belt. Phys. Rev. A 72, 053605 (2005).

  21. 21

    Lee, P. J. et al. Sublattice addressing and spin-dependent motion of atoms in a double-well lattice. Phys. Rev. Lett. 99, 020402 (2007).

  22. 22

    Müller, T., Fölling, S., Widera, A. & Bloch, I. State preparation and dynamics of ultracold atoms in higher lattice orbitals. Phys. Rev. Lett. 99, 200405 (2007).

  23. 23

    Sun, K., Yao, H., Fradkin, E. & Kivelson, S. A. Topological insulators and nematic phases from spontaneous symmetry breaking in 2d Fermi systems with a quadratic band crossing. Phys. Rev. Lett. 103, 046811 (2009).

  24. 24

    Sun, K. & Fradkin, E. Time-reversal symmetry breaking and spontaneous anomalous Hall effect in Fermi fluids. Phys. Rev. B 78, 245122 (2008).

  25. 25

    Das Sarma, S., Adam, S., Hwang, E. H. & Rossi, E. Electronic transport in two-dimensional graphene. Rev. Mod. Phys. 83, 407–470 (2011).

  26. 26

    Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the parity anomaly. Phys. Rev. Lett. 61, 2015–2018 (1988).

  27. 27

    Wu, C. Orbital analogue of the quantum anomalous Hall effect in p-band systems. Phys. Rev. Lett. 101, 186807 (2008).

  28. 28

    Stanescu, T. D., Galitski, V. & Das Sarma, S. Topological states in two-dimensional optical lattices. Phys. Rev. A 82, 013608 (2010).

  29. 29

    Ernst, P. T. et al. Probing superfluids in optical lattices by momentum-resolved Bragg spectroscopy. Nature Phys. 6, 56–61 (2010).

  30. 30

    Varney, C. N., Sun, K., Rigol, M. & Galitski, V. Interaction effects and quantum phase transitions in topological insulators. Phys. Rev. B 82, 115125 (2010).

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We appreciate the very helpful discussions with L. Fu, C. L. Kane and X-L. Qi. The work of K.S. and S.D.S. is supported by JQI-NSF-PFC, AFOSR-MURI, ARO-DARPA-OLE, and ARO-MURI. W.V.L. is supported by ARO (W911NF-07-1-0293 and 11-1-0230), ARO-DARPA-OLE (W911NF-07-1-0464) and the National Basic Research Program of China (Grant No 2012CB922101). A.H. acknlowledges support by DFG-SFB925. We thank the Kavli Institute for Theoretical Physics at UCSB for its hospitality where this research is supported in part by National Science Foundation Grant No. PHY05-51164.

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W.V.L., K.S., and S.D.S. planned the work. K.S. and W.V.L. carried out most of the calculations with input from S.D.S. A.H. provided the experimental protocol. All authors contributed to the writing of the manuscript.

Correspondence to W. Vincent Liu.

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Sun, K., Liu, W., Hemmerich, A. et al. Topological semimetal in a fermionic optical lattice. Nature Phys 8, 67–70 (2012).

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