Seeing Hofstadter's butterfly in atomic Fermi gases

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 89, Issue 1, 2014, Pages

  Wang, Lei ^a   Troyer, Matthias ^a  

^a ETH ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84894466735     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.89.011603     Document Type: Article

References (51)

1. D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976). 0556-2805 10.1103/PhysRevB.14.2239
2. C. Albrecht, J. H. Smet, K. von Klitzing, D. Weiss, V. Umansky, and H. Schweizer, Phys. Rev. Lett. 86, 147 (2001). PRLTAO 0031-9007 10.1103/PhysRevLett.86.147 (Pubitemid 32131754)
3. C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, Nature (London) 497, 598 (2013). NATUAS 0028-0836 10.1038/nature12186
4. B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori, Science 340, 1427 (2013). SCIEAS 0036-8075 10.1126/science.1237240
5. L. A. Ponomarenko, R. V. Gorbachev, G. L. Yu, D. C. Elias, R. Jalil, A. A. Patel, A. Mishchenko, A. S. Mayorov, C. R. Woods, J. R. Wallbank, M. Mucha-Kruczynski, B. A. Piot, M. Potemski, I. V. Grigorieva, K. S. Novoselov, F. Guinea, V. I. Fal'ko, and A. K. Geim, Nature (London) 497, 594 (2013). NATUAS 0028-0836 10.1038/nature12187
6. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982). PRLTAO 0031-9007 10.1103/PhysRevLett.49.405
7. D. Osadchy and J. E. Avron, J. Math. Phys. 42, 5665 (2001). JMAPAQ 0022-2488 10.1063/1.1412464
8. E. J. Mueller, Phys. Rev. A 70, 041603 (R) (2004). PLRAAN 1050-2947 10.1103/PhysRevA.70.041603
9. R. O. UmucalIlar, H. Zhai, and M. Ö. Oktel, Phys. Rev. Lett. 100, 070402 (2008). PRLTAO 0031-9007 10.1103/PhysRevLett.100.070402 (Pubitemid 351393606)
10. F. Gerbier and J. Dalibard, New J. Phys. 12, 033007 (2010). NJOPFM 1367-2630 10.1088/1367-2630/12/3/033007
11. A. R. Kolovsky, Europhys. Lett. 93, 20003 (2011). EULEEJ 0295-5075 10.1209/0295-5075/93/20003
12. C. E. Creffield and F. Sols, EPL 101, 40001 (2013). EULEEJ 0295-5075 10.1209/0295-5075/101/40001
13. M. Aidelsburger, M. Atala, S. Nascimbène, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011). PRLTAO 0031-9007 10.1103/PhysRevLett.107.255301
14. M. Aidelsburger, M. Atala, S. Nascimbène, S. Trotzky, Y.-A. Chen, and I. Bloch, Appl. Phys. B 113, 1 (2013). 10.1007/s00340-013-5418-1
15. D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003). NJOPFM 1367-2630 10.1088/1367-2630/5/1/356
16. M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Phys. Rev. Lett. 111, 185301 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.111.185301
17. H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, Phys. Rev. Lett. 111, 185302 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.111.185302
18. E. Alba, X. Fernandez-Gonzalvo, J. Mur-Petit, J. K. Pachos, and J. J. Garcia-Ripoll, Phys. Rev. Lett. 107, 235301 (2011). PRLTAO 0031-9007 10.1103/PhysRevLett.107.235301
19. N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Proc. Nat. Acad. Sci. USA 110, 6736 (2013). PNASA6 0027-8424 10.1073/pnas.1300170110
20. D. A. Abanin, T. Kitagawa, I. Bloch, and E. Demler, Phys. Rev. Lett. 110, 165304 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.110.165304
21. L. Wang, A. A. Soluyanov, and M. Troyer, Phys. Rev. Lett. 110, 166802 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.110.166802
22. X.-J. Liu, K. T. Law, T. K. Ng, and P. A. Lee, Phys. Rev. Lett. 111, 120402 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.111.120402
23. M. Jarrell and J. E. Gubernatis, Phys. Rep. 269, 133 (1996). PRPLCM 0370-1573 10.1016/0370-1573(95)00074-7 (Pubitemid 126175303)
24. Y. Hasegawa, P. Lederer, T. M. Rice, and P. B. Wiegmann, Phys. Rev. Lett. 63, 907 (1989). PRLTAO 0031-9007 10.1103/PhysRevLett.63.907
25. W. H. Xu, L. P. Yang, M. P. Qin, and T. Xiang, Phys. Rev. B 78, 241102 (R) (2008). PRBMDO 1098-0121 10.1103/PhysRevB.78.241102
26. L.-P. Yang, W.-H. Xu, M.-P. Qin, and T. Xiang, J. Phys. Soc. Jpn. 81, 044605 (2012). JUPSAU 0031-9015 10.1143/JPSJ.81.044605
27. Other thermodynamic quantities like entropy and specific heat also reveal the DOS in the zero-temperature limit; however, unlike the compressibility, they are not directly accessible in the cold atom toolbox.
28. We need the local chemical potential in the trap to scan through the energy band, i.e., the density in the trap center to reach one. However, one can utilize the particle-hole symmetry of the Hofstadter lattice and only probe ρ < 0.5 to restore the whole spectrum.
29. N. Gemelke, X. Zhang, C.-L. Hung, and C. Chin, Nature (London) 460, 995 (2009). NATUAS 0028-0836 10.1038/nature08244
30. K. Van Houcke, F. Werner, E. Kozik, N. Prokof'ev, B. Svistunov, M. J. H. Ku, A. T. Sommer, L. W. Cheuk, A. Schirotzek, and M. W. Zwierlein, Nat. Phys. 8, 366 (2012). 1745-2473 10.1038/nphys2273
31. M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Science 335, 563 (2012). SCIEAS 0036-8075 10.1126/science.1214987
32. Y.-R. Lee, M.-S. Heo, J.-H. Choi, T. T. Wang, C. A. Christensen, T. M. Rvachov, and W. Ketterle, Phys. Rev. A 85, 063615 (2012). PLRAAN 1050-2947 10.1103/PhysRevA.85.063615
33. μ 0 and α will vary in different experimental runs. One can nevertheless utilize the particle-hole symmetry of the Hofstadter model and chose the origin of chemical potential at ρ = 0.5 for each measurement.
34. D. C. McKay and B. DeMarco, Rep. Prog. Phys. 74, 054401 (2011). RPPHAG 0034-4885 10.1088/0034-4885/74/5/054401
35. S. Fuchs, E. Gull, L. Pollet, E. Burovski, E. Kozik, T. Pruschke, and M. Troyer, Phys. Rev. Lett. 106, 030401 (2011). PRLTAO 0031-9007 10.1103/PhysRevLett.106.030401
36. A. W. Sandvik, Phys. Rev. B 57, 10287 (1998). PRBMDO 0163-1829 10.1103/PhysRevB.57.10287
37. A. S. Mishchenko, N. V. Prokof'ev, A. Sakamoto, and B. V. Svistunov, Phys. Rev. B 62, 6317 (2000). PRBMDO 0163-1829 10.1103/PhysRevB.62.6317 (Pubitemid 32336014)
38. K. S. D. Beach, arXiv:cond-mat/0403055.
39. S. Fuchs, T. Pruschke, and M. Jarrell, Phys. Rev. E 81, 056701 (2010). PLEEE8 1539-3755 10.1103/PhysRevE.81.056701
40. N. Prokof'ev and B. Svistunov, arXiv:1304.5198.
41. R. N. Silver, D. S. Sivia, and J. E. Gubernatis, Phys. Rev. B 41, 2380 (1990). PRBMDO 0163-1829 10.1103/PhysRevB.41.2380
42. J. Shi, S.-J. Tang, B. Wu, P. T. Sprunger, W. L. Yang, V. Brouet, X. J. Zhou, Z. Hussain, Z.-X. Shen, Z. Zhang, and E. W. Plummer, Phys. Rev. Lett. 92, 186401 (2004). PRLTAO 0031-9007 10.1103/PhysRevLett.92.186401
43. See Supplemental Material http://link.aps.org/supplemental/10.1103/ PhysRevA.89.011603 for details of the maximum entropy method and the stochastic inference approach.
44. Q. Zhou and T.-L. Ho, Phys. Rev. Lett. 106, 225301 (2011). PRLTAO 0031-9007 10.1103/PhysRevLett.106.225301
45. C. Sanner, E. J. Su, A. Keshet, R. Gommers, Y.-i. Shin, W. Huang, and W. Ketterle, Phys. Rev. Lett. 105, 040402 (2010). PRLTAO 0031-9007 10.1103/PhysRevLett.105.040402
46. T. Müller, B. Zimmermann, J. Meineke, J.-P. Brantut, T. Esslinger, and H. Moritz, Phys. Rev. Lett. 105, 040401 (2010). PRLTAO 0031-9007 10.1103/PhysRevLett.105.040401
47. T.-L. Ho and Q. Zhou, Nat. Phys. 6, 131 (2009). 1745-2473 10.1038/nphys1477
48. P. N. Ma, L. Pollet, and M. Troyer, Phys. Rev. A 82, 033627 (2010). PLRAAN 1050-2947 10.1103/PhysRevA.82.033627
49. The LDA correction to the density profile is less than 0.1 % [44] at the typical experimental temperatures (T / J ∼ 1). We have also performed the deconvolution using the LDA density profiles. The images are only slightly different from the one using the exact density data, indicating the the LDA corrections to the final results are indeed small. These corrections will further decrease if one uses a weaker trapping potential experimentally.
50. L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T. Esslinger, Nature (London) 483, 302 (2012). NATUAS 0028-0836 10.1038/nature10871
51. G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vishwanath, and D. M. Stamper-Kurn, Phys. Rev. Lett. 108, 045305 (2012). PRLTAO 0031-9007 10.1103/PhysRevLett.108.045305