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Exploring 4D quantum Hall physics with a 2D topological charge pump

Abstract

The discovery of topological states of matter has greatly improved our understanding of phase transitions in physical systems. Instead of being described by local order parameters, topological phases are described by global topological invariants and are therefore robust against perturbations. A prominent example is the two-dimensional (2D) integer quantum Hall effect1: it is characterized by the first Chern number, which manifests in the quantized Hall response that is induced by an external electric field2. Generalizing the quantum Hall effect to four-dimensional (4D) systems leads to the appearance of an additional quantized Hall response, but one that is nonlinear and described by a 4D topological invariant—the second Chern number3,4. Here we report the observation of a bulk response with intrinsic 4D topology and demonstrate its quantization by measuring the associated second Chern number. By implementing a 2D topological charge pump using ultracold bosonic atoms in an angled optical superlattice, we realize a dynamical version of the 4D integer quantum Hall effect5,6. Using a small cloud of atoms as a local probe, we fully characterize the nonlinear response of the system via in situ imaging and site-resolved band mapping. Our findings pave the way to experimentally probing higher-dimensional quantum Hall systems, in which additional strongly correlated topological phases, exotic collective excitations and boundary phenomena such as isolated Weyl fermions are predicted4.

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Figure 1: 4D quantum Hall system and the corresponding 2D topological charge pump.
Figure 2: 4D-like nonlinear centre-of-mass (COM) response.
Figure 3: Local probing of the quantized nonlinear bulk response for θ = 0.54(3) mrad.
Figure 4: Scaling of the 4D-like response with the tilt angle θ.

References

  1. Klitzing, K. v., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980)

    Article  ADS  Google Scholar 

  2. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982)

    Article  ADS  CAS  Google Scholar 

  3. Fröhlich, J. & Pedrini, B. in Mathematical Physics 2000 (eds Fokas, A. et al.) 9–47 (Imperial College Press, 2000)

  4. Zhang, S.-C. & Hu, J. A four-dimensional generalization of the quantum Hall effect. Science 294, 823–828 (2001)

    Article  ADS  CAS  PubMed  Google Scholar 

  5. Thouless, D. J. Quantization of particle transport. Phys. Rev. B 27, 6083–6087 (1983)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  6. Kraus, Y. E., Ringel, Z. & Zilberberg, O. Four-dimensional quantum Hall effect in a two-dimensional quasicrystal. Phys. Rev. Lett. 111, 226401 (2013)

    Article  ADS  PubMed  Google Scholar 

  7. Yang, C. N. & Mills, R. L. Conservation of isotopic spin and isotopic gauge invariance. Phys. Rev. 96, 191–195 (1954)

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  9. 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)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  10. Laughlin, R. B. Quantized Hall conductivity in two dimensions. Phys. Rev. B 23, 5632–5633 (1981)

    Article  ADS  Google Scholar 

  11. Lu, L. et al. Experimental observation of Weyl points. Science 349, 622–624 (2015)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  12. Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015)

    Article  ADS  CAS  PubMed  Google Scholar 

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

    Article  ADS  CAS  PubMed  Google Scholar 

  14. Qi, X.-L., Hughes, T. L. & Zhang, S.-C. Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008)

    Article  ADS  Google Scholar 

  15. Goldman, N., Budich, J. C. & Zoller, P. Topological quantum matter with ultracold gases in optical lattices. Nat. Phys. 12, 639–645 (2016)

    Article  CAS  Google Scholar 

  16. Lu, L., Joannopoulos, J. D. & Soljacic, M. Topological photonics. Nat. Photon. 8, 821–829 (2014)

    Article  ADS  CAS  Google Scholar 

  17. Sugawa, S., Salces-Carcoba, F., Perry, A. R., Yue, Y. & Spielman, I. B. Observation of a non-abelian Yang monopole: from new Chern numbers to a topological transition. Preprint at http://arxiv.org/abs/1610.06228 (2016)

  18. Mancini, M. et al. Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. Science 349, 1510–1513 (2015)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  19. Stuhl, B. K., Lu, H.-I., Aycock, L. M., Genkina, D. & Spielman, I. B. Visualizing edge states with an atomic Bose gas in the quantum Hall regime. Science 349, 1514–1518 (2015)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  20. Price, H. M., Zilberberg, O., Ozawa, T., Carusotto, I. & Goldman, N. Four-dimensional quantum Hall effect with ultracold atoms. Phys. Rev. Lett. 115, 195303 (2015)

    Article  ADS  CAS  PubMed  Google Scholar 

  21. Price, H. M., Zilberberg, O., Ozawa, T., Carusotto, I. & Goldman, N. Measurement of Chern numbers through center-of-mass responses. Phys. Rev. B 93, 245113 (2016)

    Article  ADS  Google Scholar 

  22. Kraus, Y. E. & Zilberberg, O. Topological equivalence between the Fibonacci quasicrystal and the Harper model. Phys. Rev. Lett. 109, 116404 (2012)

    Article  ADS  PubMed  Google Scholar 

  23. Switkes, M., Marcus, C. M., Campman, K. & Gossard, A. C. An adiabatic quantum electron pump. Science 283, 1905–1908 (1999)

    Article  ADS  CAS  PubMed  Google Scholar 

  24. Kraus, Y. E., Lahini, Y., Ringel, Z., Verbin, M. & Zilberberg, O. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012)

    Article  ADS  PubMed  Google Scholar 

  25. Lohse, M., Schweizer, C., Zilberberg, O., Aidelsburger, M. & Bloch, I. A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nat. Phys. 12, 350–354 (2016)

    Article  CAS  Google Scholar 

  26. Nakajima, S. et al. Topological Thouless pumping of ultracold fermions. Nat. Phys. 12, 296–300 (2016)

    Article  CAS  Google Scholar 

  27. Rice, M. J. & Mele, E. J. Elementary excitations of a linearly conjugated diatomic polymer. Phys. Rev. Lett. 49, 1455–1459 (1982)

    Article  ADS  CAS  Google Scholar 

  28. Zhu, S.-L., Wang, Z.-D., Chan, Y.-H. & Duan, L.-M. Topological Bose–Mott insulators in a one-dimensional optical superlattice. Phys. Rev. Lett. 110, 075303 (2013)

    Article  ADS  PubMed  Google Scholar 

  29. Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  30. Hosur, P. & Qi, X. Recent developments in transport phenomena in Weyl semimetals. C. R. Phys. 14, 857–870 (2013)

    Article  ADS  CAS  Google Scholar 

  31. Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, https://doi.org/10.1038/nature25011 (2018)

    Article  ADS  CAS  PubMed  Google Scholar 

  32. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  33. Harper, P. G. The general motion of conduction electrons in a uniform magnetic field, with application to the diamagnetism of metals. Proc. Phys. Soc. A 68, 879–892 (1955)

    Article  ADS  Google Scholar 

  34. Azbel, M. Ya . Energy spectrum of a conduction electron in a magnetic field. Zh. Eksp. Teor. Fiz. 46, 929–946 (1964); Sov. Phys. JETP 19, 634–645 (1964) [transl.]

    CAS  Google Scholar 

  35. Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976)

    Article  ADS  CAS  Google Scholar 

  36. Hatsugai, Y. & Kohmoto, M. Energy spectrum and the quantum Hall effect on the square lattice with next-nearest-neighbor hopping. Phys. Rev. B 42, 8282–8294 (1990)

    Article  ADS  CAS  Google Scholar 

  37. Roux, G. et al. Quasiperiodic Bose–Hubbard model and localization in one-dimensional cold atomic gases. Phys. Rev. A 78, 023628 (2008)

    Article  ADS  Google Scholar 

  38. Marra, P., Citro, R. & Ortix, C. Fractional quantization of the topological charge pumping in a one-dimensional superlattice. Phys. Rev. B 91, 125411 (2015)

    Article  ADS  Google Scholar 

  39. Widera, A. et al. Coherent collisional spin dynamics in optical lattices. Phys. Rev. Lett. 95, 190405 (2005)

    Article  ADS  PubMed  Google Scholar 

  40. Gerbier, F., Widera, A., Fölling, S., Mandel, O. & Bloch, I. Resonant control of spin dynamics in ultracold quantum gases by microwave dressing. Phys. Rev. A 73, 041602 (2006)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We acknowledge discussions with M. Aidelsburger and I. Carusotto. This work was funded by the European Commission (UQUAM, SIQS), the Deutsche Forschungsgemeinschaft (DIP, FOR2414) and the Nanosystems Initiative Munich. M.L. was additionally supported by the Elitenetzwerk Bayern (ExQM), H.M.P. by the European Commission (FET Proactive, grant no. 640800 ‘AQuS’, and Marie Skłodowska–Curie Action, grant no. 656093 ‘SynOptic’) and the Autonomous Province of Trento (SiQuro), and O.Z. by the Swiss National Science Foundation.

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Authors and Affiliations

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Contributions

M.L. and C.S. performed the experiment and data analysis. O.Z. proposed the experiment. All authors contributed to the theoretical analysis and to writing the paper. I.B. supervised the project.

Corresponding author

Correspondence to Immanuel Bloch.

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Extended data figures and tables

Extended Data Figure 1 Nonlinear response versus depth of the short lattice along y.

Slope of the nonlinear response at and θ = 0.54(3) mrad as a function of Vs,y, with all other lattice parameters as in Figs 3 and 4. with is the maximum intra-double-well tunnelling rate along y, which is calculated from the corresponding lattice depth. The solid line indicates the theoretically expected slope and the error bars show the fit error for the slope. The dashed line at Vs,y = 6.25Er,s marks the point at which a topological transition occurs in the first excited subband along y, indicating the transition between the Landau regime for Vs,y < 6.25Er,s and the Hofstadter regime for Vs,y > 6.25Er,s.

Extended Data Figure 2 Pump cycle of the 2D topological charge pump.

The 4D tight-binding parameter space (δJx, Δx, δJy, Δy) is visualized using the transformation in equation (4). a, Changing the pump parameter φx leads to a periodic modulation of δJx and Δx along a closed trajectory, as shown in the inset for a full pump cycle φx = 0 → 2π. This pump path (green) encircles the degeneracy point at the origin (grey), at which the gap between the two lowest subbands of the Rice–Mele model closes. The surface in the main plot shows the same trace transformed according to equation (4) and with φy [0.46π, 0.54π]. The spacing of the mesh grid illustrating φx is π/10. b, For a given φx, a large system simultaneously samples all values of φy. This corresponds to a closed path in δJyΔy parameter space, in which a singularity also occurs at the origin (inset). The main plot shows the transformed path for φx [0.46π, 0.54π]. c, In a full pump cycle, such a system therefore covers a closed surface in the 4D parameter space by translating the path shown in b along the trajectory from a. d, In the transformed parameter space, the singularities at (δJx = 0, Δx = 0) and (δJy = 0, Δy = 0) correspond to two planes that touch at the origin. e, Cut around r3 = 0 showing both the pump path from c (red/blue) and the singularities from d (grey). Whereas they intersect in the 3D space (r1, r2, r3), the value of r4 is different on both surfaces and the 4D pump path thus fully encloses the degeneracy planes.

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Lohse, M., Schweizer, C., Price, H. et al. Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature 553, 55–58 (2018). https://doi.org/10.1038/nature25000

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