Advertisement

The European Physical Journal Special Topics

, Volume 226, Issue 12, pp 2781–2791 | Cite as

Fractional quantization of charge and spin in topological quantum pumps

  • Pasquale MarraEmail author
  • Roberta Citro
Regular Article
Part of the following topical collections:
  1. Quantum Gases and Quantum Coherence

Abstract

Topological quantum pumps are topologically equivalent to the quantum Hall state: In these systems, the charge pumped during each pumping cycle is quantized and coincides with the Chern invariant. However, differently from quantum Hall insulators, quantum pumps can exhibit novel phenomena such as the fractional quantization of the charge transport, as a consequence of their distinctive symmetries in parameter space. Here, we report the analogous fractional quantization of the spin transport in a topological spin pump realized in a one-dimensional lattice via a periodically modulated Zeeman field. In the proposed model, which is a spinfull generalization of the Harper-Hofstadter model, the amount of spin current pumped during well-defined fractions of the pumping cycle is quantized as fractions of the spin Chern number. This fractional quantization of spin is topological, and is a direct consequence of the additional symmetries ensuing from the commensuration of the periodic field with the underlying lattice.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M.Z. Hasan, C.L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010)ADSCrossRefGoogle Scholar
  2. 2.
    X.-L. Qi, S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011)ADSCrossRefGoogle Scholar
  3. 3.
    K.v. Klitzing, G. Dorda, M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45, 494 (1980)ADSCrossRefGoogle Scholar
  4. 4.
    D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982)ADSCrossRefGoogle Scholar
  5. 5.
    C.L. Kane, E.J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005)ADSCrossRefGoogle Scholar
  6. 6.
    C.L. Kane, E.J. Mele, Z2 topological order and the quantum spin Hall effect, Phys. Rev. Lett. 95, 146802 (2005)ADSCrossRefGoogle Scholar
  7. 7.
    B.A. Bernevig, S.-C. Zhang, Quantum spin Hall effect, Phys. Rev. Lett. 96, 106802 (2006)ADSCrossRefGoogle Scholar
  8. 8.
    D.N. Sheng, Z.Y. Weng, L. Sheng, F.D.M. Haldane, Quantum spin-Hall effect and topologically invariant Chern numbers, Phys. Rev. Lett. 97, 036808 (2006)ADSCrossRefGoogle Scholar
  9. 9.
    D.J. Thouless, Quantization of particle transport, Phys. Rev. B 27, 6083 (1983)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Q. Niu, D.J. Thouless, Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction, J. Phys. A: Math. Gen. 17, 2453 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    L. Fu, C.L. Kane, Time reversal polarization and a Z2 adiabatic spin pump, Phys. Rev. B 74, 195312 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    R. Shindou, Quantum spin pump in S=1/2 antiferromagnetic chains – holonomy of phase operators in sine-Gordon theory, J. Phys. Soc. Jpn. 74, 1214 (2005)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    R. Citro, F. Romeo, N. Andrei, Electrically controlled pumping of spin currents in topological insulators, Phys. Rev. B 84, 161301 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    F. Mei, S.-L. Zhu, Z.-M. Zhang, C.H. Oh, N. Goldman, Simulating Z2 topological insulators with cold atoms in a one-dimensional optical lattice, Phys. Rev. A 85, 013638 (2012)ADSCrossRefGoogle Scholar
  15. 15.
    D. Ferraro, G. Dolcetto, R. Citro, F. Romeo, M. Sassetti, Spin current pumping in helical Luttinger liquids, Phys. Rev. B 87, 245419 (2013)ADSCrossRefGoogle Scholar
  16. 16.
    C.Q. Zhou, Y.F. Zhang, L. Sheng, R. Shen, D.N. Sheng, D.Y. Xing, Proposal for a topological spin Chern pump, Phys. Rev. B 90, 085133 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    W.Y. Deng, W. Luo, H. Geng, M.N. Chen, L. Sheng, D.Y. Xing, Non-adiabatic topological spin pumping, New J. Phys. 17, 103018 (2015)ADSCrossRefGoogle Scholar
  18. 18.
    M.N. Chen, L. Sheng, R. Shen, D.N. Sheng, D.Y. Xing, Spin Chern pumping from the bulk of two-dimensional topological insulators, Phys. Rev. B 91, 125117 (2015)ADSCrossRefGoogle Scholar
  19. 19.
    M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(De), U. Sen, Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond, Adv. Phys. 56, 243 (2007)ADSCrossRefGoogle Scholar
  20. 20.
    I. Bloch, J. Dalibard, W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008)ADSCrossRefGoogle Scholar
  21. 21.
    J. Dalibard, F. Gerbier, G. Juzeliūnas, P. Öhberg, Colloquium: Artificial gauge potentials for neutral atoms, Rev. Mod. Phys. 83, 1523 (2011)ADSCrossRefGoogle Scholar
  22. 22.
    M. Aidelsburger, M. Atala, S. Nascimbène, S. Trotzky, Y.-A. Chen, I. Bloch, Experimental realization of strong effective magnetic fields in an optical lattice, Phys. Rev. Lett. 107, 255301 (2011)ADSCrossRefGoogle Scholar
  23. 23.
    K. Jiménez-García, L.J. LeBlanc, R.A. Williams, M.C. Beeler, A.R. Perry, I.B. Spielman, Peierls substitution in an engineered lattice potential, Phys. Rev. Lett. 108, 225303 (2012)ADSCrossRefGoogle Scholar
  24. 24.
    J. Struck, C. Ölschläger, M. Weinberg, P. Hauke, J. Simonet, A. Eckardt, M. Lewenstein, K. Sengstock, P. Windpassinger, Tunable gauge potential for neutral and spinless particles in driven optical lattices, Phys. Rev. Lett. 108, 225304 (2012)ADSCrossRefGoogle Scholar
  25. 25.
    Y.V. Kartashov, V.V. Konotop, F.K. Abdullaev, Gap solitons in a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. Lett. 111, 060402 (2013)ADSCrossRefGoogle Scholar
  26. 26.
    N. Goldman, G. Juzeliūnas, P. Ohberg, I.B. Spielman, Light-induced gauge fields for ultracold atoms, Rep. Prog. Phys. 77, 126401 (2014)ADSCrossRefGoogle Scholar
  27. 27.
    L.-J. Lang, X. Cai, S. Chen, Edge states and topological phases in one-dimensional optical superlattices, Phys. Rev. Lett. 108, 220401 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    M. Aidelsburger, M. Atala, M. Lohse, J.T. Barreiro, B. Paredes, I. Bloch, Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices, Phys. Rev. Lett. 111, 185301 (2013)ADSCrossRefGoogle Scholar
  29. 29.
    H. Miyake, G.A. Siviloglou, C.J. Kennedy, W.C. Burton, W. Ketterle, Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices, Phys. Rev. Lett. 111, 185302 (2013)ADSCrossRefGoogle Scholar
  30. 30.
    M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J.T. Barreiro, S. Nascimbene, N.R. Cooper, I. Bloch, N. Goldman, Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms, Nat. Phys. 11, 162 (2015)CrossRefGoogle Scholar
  31. 31.
    M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, I. Bloch, A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice, Nat. Phys. 12, 350 (2016)CrossRefGoogle Scholar
  32. 32.
    S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, Y. Takahashi, Topological Thouless pumping of ultracold fermions, Nat. Phys. 12, 296 (2016)CrossRefGoogle Scholar
  33. 33.
    R. Citro, Ultracold atoms: A topological charge pump, Nat. Phys. 12, 288 (2016)CrossRefGoogle Scholar
  34. 34.
    C. Schweizer, M. Lohse, R. Citro, I. Bloch, Spin pumping and measurement of spin currents in optical superlattices, Phys. Rev. Lett. 117, 170405 (2016)ADSCrossRefGoogle Scholar
  35. 35.
    P.L. e.S. Lopes, P. Ghaemi, S. Ryu, T.L. Hughes, Competing adiabatic Thouless pumps in enlarged parameter spaces, Phys. Rev. B 94, 235160 (2016)ADSCrossRefGoogle Scholar
  36. 36.
    T.-S. Zeng, W. Zhu, D.N. Sheng, Fractional charge pumping of interacting bosons in one-dimensional superlattice, Phys. Rev. B 94, 235139 (2016)ADSCrossRefGoogle Scholar
  37. 37.
    F. Ronetti, M. Carrega, D. Ferraro, J. Rech, T. Jonckheere, T. Martin, M. Sassetti, Polarized heat current generated by quantum pumping in two-dimensional topological insulators, Phys. Rev. B 95, 115412 (2017)ADSCrossRefGoogle Scholar
  38. 38.
    S. Gangadharaiah, L. Trifunovic, D. Loss, Localized end states in density modulated quantum wires and rings, Phys. Rev. Lett. 108, 136803 (2012)ADSCrossRefGoogle Scholar
  39. 39.
    J.-H. Park, G. Yang, J. Klinovaja, P. Stano, D. Loss, Fractional boundary charges in quantum dot arrays with density modulation, Phys. Rev. B 94, 075416 (2016)ADSCrossRefGoogle Scholar
  40. 40.
    P. Marra, R. Citro, C. Ortix, Fractional quantization of the topological charge pumping in a one-dimensional superlattice, Phys. Rev. B 91, 125411 (2015)ADSCrossRefGoogle Scholar
  41. 41.
    D.-W. Zhang, Z.-D. Wang, S.-L. Zhu, Relativistic quantum effects of Dirac particles simulated by ultracold atoms, Front. Phys. 7, 31 (2012)CrossRefGoogle Scholar
  42. 42.
    D.-W. Zhang, L.-B. Shao, Z.-Y. Xue, H. Yan, Z.D. Wang, S.-L. Zhu, Particle-number fractionalization of a one-dimensional atomic Fermi gas with synthetic spin-orbit coupling, Phys. Rev. A 86, 063616 (2012)ADSCrossRefGoogle Scholar
  43. 43.
    D.-W. Zhang, S.-L. Zhu, Z.D. Wang, Simulating and exploring Weyl semimetal physics with cold atoms in a two-dimensional optical lattice, Phys. Rev. A 92, 013632 (2015)ADSCrossRefGoogle Scholar
  44. 44.
    D.-W. Zhang, Y.X. Zhao, R.-B. Liu, Z.-Y. Xue, S.-L. Zhu, Z.D. Wang, Quantum simulation of exotic PT-invariant topological nodal loop bands with ultracold atoms in an optical lattice, Phys. Rev. A 93, 043617 (2016)ADSCrossRefGoogle Scholar
  45. 45.
    D. Xiao, M.-C. Chang, Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Y.E. Kraus, Z. Ringel, O. Zilberberg, Four/dimensional quantum Hall effect in a two-dimensional quasicrystal, Phys. Rev. Lett. 111, 226401 (2013)ADSCrossRefGoogle Scholar
  47. 47.
    A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I.B. Spielman, G. Juzeliūnas, M. Lewenstein, Synthetic gauge fields in synthetic dimensions, Phys. Rev. Lett. 112, 043001 (2014)ADSCrossRefGoogle Scholar
  48. 48.
    P. Marra, M. Cuoco, Pinning Majorana states to domain walls in amplitude-modulated magnetic textures, arXiv:1606.08450 (2016)
  49. 49.
    D.R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239 (1976)ADSCrossRefGoogle Scholar
  50. 50.
    P.G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. A 68, 874 (1955)ADSCrossRefzbMATHGoogle Scholar
  51. 51.
    D. Osadchy, J.E. Avron, Hofstadter butterfly as quantum phase diagram, J. Math. Phys. 42, 5665 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    D.-W. Zhang, S. Cao, Measuring the spin Chern number in time-reversal-invariant Hofstadter optical lattices, Phys. Lett. A 380, 3541 (2016)ADSCrossRefGoogle Scholar
  53. 53.
    D. Prezzi, D. Eom, K.T. Rim, H. Zhou, S. Xiao, C. Nuckolls, T.F. Heinz, G.W. Flynn, M.S. Hybertsen, Edge structures for nanoscale graphene islands on Co(0001) surfaces, ACS Nano 8, 5765 (2014)CrossRefGoogle Scholar
  54. 54.
    L.L. Patera, F. Bianchini, G. Troiano, C. Dri, C. Cepek, M. Peressi, C. Africh, G. Comelli, Temperature-driven changes of the graphene edge structure on Ni(111): Substrate vs hydrogen passivation, Nano Letters 15, 56 (2015)ADSCrossRefGoogle Scholar
  55. 55.
    J. Klinovaja, P. Stano, D. Loss, Transition from fractional to Majorana fermions in Rashba nanowires, Phys. Rev. Lett. 109, 236801 (2012)ADSCrossRefGoogle Scholar
  56. 56.
    S. Nadj-Perge, I.K. Drozdov, B.A. Bernevig, A. Yazdani, Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor, Phys. Rev. B 88, 020407 (2013)ADSCrossRefGoogle Scholar
  57. 57.
    B. Braunecker, P. Simon, Interplay between classical magnetic moments and superconductivity in quantum one/dimensional conductors: toward a self/sustained topological Majorana phase, Phys. Rev. Lett. 111, 147202 (2013)ADSCrossRefGoogle Scholar
  58. 58.
    F. Pientka, L.I. Glazman, F. von Oppen, Topological superconducting phase in helical Shiba chains, Phys. Rev. B 88, 155420 (2013)ADSCrossRefGoogle Scholar
  59. 59.
    J. Klinovaja, P. Stano, A. Yazdani, D. Loss, Topological superconductivity and majorana fermions in RKKY systems, Phys. Rev. Lett. 111, 186805 (2013)ADSCrossRefGoogle Scholar
  60. 60.
    M.M. Vazifeh, M. Franz, Self-organized topological state with Majorana fermions, Phys. Rev. Lett. 111, 206802 (2013)ADSCrossRefGoogle Scholar
  61. 61.
    Y. Kim, M. Cheng, B. Bauer, R.M. Lutchyn, S. Das Sarma, Helical order in one-dimensional magnetic atom chains and possible emergence of Majorana bound states, Phys. Rev. B 90, 060401 (2014)ADSCrossRefGoogle Scholar
  62. 62.
    A. Saha, D. Rainis, R.P. Tiwari, D. Loss, Quantum charge pumping through fractional fermions in charge density modulated quantum wires and Rashba nanowires, Phys. Rev. B 90, 035422 (2014)ADSCrossRefGoogle Scholar
  63. 63.
    A. Tadjine, G. Allan, C. Delerue, From lattice Hamiltonians to tunable band structures by lithographic design, Phys. Rev. B 94, 075441 (2016)ADSCrossRefGoogle Scholar
  64. 64.
    M. Polini, F. Guinea, M. Lewenstein, H.C. Manoharan, V. Pellegrini, Artificial honeycomb lattices for electrons, atoms and photons, Nat. Nano. 8, 625 (2013)CrossRefGoogle Scholar
  65. 65.
    N.R. Cooper, A.M. Rey, Adiabatic control of atomic dressed states for transport and sensing, Phys. Rev. A 92, 021401 (2015)ADSCrossRefGoogle Scholar
  66. 66.
    L. Taddia, E. Cornfeld, D. Rossini, L. Mazza, E. Sela, R. Fazio, Topological fractional pumping with alkaline-earth(-like) ultracold atoms, arXiv:1607.07842 (2016)

Copyright information

© EDP Sciences and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.CNR-SPINFisciano (Salerno)Italy
  2. 2.Department of Physics “E. R. Caianiello”University of SalernoFisciano (Salerno)Italy

Personalised recommendations