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Topological bulk states and their currents

Chris N. Self, Alvaro Rubio-García, Juan Jose García-Ripoll, and Jiannis K. Pachos
Phys. Rev. B 102, 045424 – Published 23 July 2020
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  • INTRODUCTION
  • EDGE AND BULK CURRENTS
  • CURRENTS IN LATTICE MODELS
  • BULK CURRENTS FROM PERTURBATION THEORY
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We provide evidence that, alongside topologically protected edge states, two-dimensional Chern insulators also support localized bulk states deep in their valence and conduction bands. These states manifest when local potential gradients are applied to the bulk, while all parts of the system remain adiabatically connected to the same phase. In turn, the bulk states produce bulk current transverse to the potential difference. This occurs even when the potential is always below the energy gap, where one expects only edge currents to appear. Bulk currents are topologically protected and behave as edge currents under an external influence, such as temperature or local disorder. Detecting topologically resilient bulk currents offers a direct means to probe the localized bulk states.

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    • Received 13 October 2019
    • Accepted 14 July 2020

    DOI:https://doi.org/10.1103/PhysRevB.102.045424

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Edge statesSurface statesTopological insulators
    1. Techniques
    Matrix product states
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Chris N. Self1,*, Alvaro Rubio-García2,*, Juan Jose García-Ripoll3, and Jiannis K. Pachos1

    • 1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom
    • 2Instituto de Estructura de la Materia IEM-CSIC, Calle Serrano 123, E-28006 Madrid, Spain
    • 3Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, E-28006 Madrid, Spain

    • *These authors contributed equally to this work.

    See Also

    Seeing topological edge and bulk currents in time-of-flight images

    Alvaro Rubio-García, Chris N. Self, Juan Jose García-Ripoll, and Jiannis K. Pachos
    Phys. Rev. B 102, 041123(R) (2020)

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    Vol. 102, Iss. 4 — 15 July 2020

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    Images

    • Figure 1
      Figure 1

      Edge currents of the Haldane model as a function of temperature T and chemical potential μ. (a) Hexagonal plaquette of the Haldane lattice model. Arrows denote hopping directions for which νij=+1 in Eq. (14). (b) The Haldane model on a cylinder showing the direction of the edge currents and the line l used to define Iedge. (c) Edge currents vs chemical potential for kBT=0.1t1 and 10t1. At low temperatures the current is independent of temperature and follows Eq. (2); at large temperatures it tends to zero. (d) Edge current vs temperature for μ=0.2t1 and 3.2t1. When μ is small the edge current is stable against temperature. These results are obtained for a cylinder with Lx×Ly=300×30.

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    • Figure 2
      Figure 2

      Spatial variations of the local potential give rise to bulk currents. The local potential is kept always below the energy gap, so the whole cylinder is always in the same topological phase. (a) Within a central band the potential linearly increases and decreases, with its maximum at the center of the cylinder. (b) The currents I as a function of height on the cylinder y. We observe bulk currents (blue bars) traverse to the y axis around the whole cylinder with local strength that depends on the magnitude and sign of the potential gradient ∇V, in agreement with Eq. (3). We also compute the total current along the top (orange diamond marker) and bottom (green circle) halves of the cylinder, finding that they are equal to the total change of potential along those regions. Data are presented for a cylinder with Lx×Ly=1000×30 at T=0.

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    • Figure 3
      Figure 3

      Bulk currents localized on potential steps at y=Ly/4 and y=3Ly/4. This is achieved by a stripe of potential around the cylinder. The currents are zero everywhere except at the potential steps, up to a regularization by the lattice spacing. Data are presented for a cylinder with Lx×Ly=1000×30 at T=0.

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    • Figure 4
      Figure 4

      Systematic investigation of the dependence of the bulk currents on the topological properties of the system. We study a cylinder with a central stripe of increased potential [illustrated in Fig. 3]. (a) Bulk currents scale linearly with the potential gradient when the system is in a topological phase M=0, ν=±1. (b) Outside the topological phase, M=0.9t1, ν=0, the bulk currents vanish in the gapless region, consistent with Eq. (3). (c) In a phase with nonzero Chern number, M=0, localized bulk currents appear even for exponentially small steps w/t1. Computations were carried out on a system of size Lx×Ly=1000×30 at T=0.

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    • Figure 5
      Figure 5

      Investigating the robustness of the bulk currents. We study a cylinder with a central stripe of increased potential, illustrated in Fig. 3. Bulk currents are robust against (a) temperature and (b) disorder (here, T=0). Data for (a) are presented for a cylinder with Lx×Ly=1000×30, while (b) is presented for a cylinder Lx×Ly=10000×30. Disorder is averaged over 100 samples and error bars on the mean 〈I〉 are only noticeable around the energy gap.

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    • Figure 6
      Figure 6

      The spectrum of the system showing localization measures of the states. The valence and conductance bands are displayed together with the edge states. The setup is a cylinder with a central stripe of increased potential, as shown in Fig. 3. (a) The IPR, Eq. (20) is shown for all states in the system, highlighting strongly localized edge states as well as localized bulk states. Inset: Dispersion relation of the full spectrum. (b) The asymmetrical IPR Fn,p, Eq. (21), plotted for states close to zero energy and momentum. (c) Zooming in on the states localized within the bulk, we can infer they give rise to bulk currents.

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    • Figure 7
      Figure 7

      Population distribution of a simple SSH chain inside the step potential of Eq. (23). (a) The t1 coupling connects pairs of alternated sites, or dimers. An example of such is formed by the pair of sites (b2,a3), shown inside the dashed circle. The two edge sites are disconnected from the rest of the chain. (b) The population density is evenly distributed inside each dimer, except for the dimer at the potential step. To illustrate the population imbalance at the step we show w=0.2,0.4,0.8. The edge modes are occupied or unoccupied depending on whether they have negative or positive potential, respectively, with an average population 0.5 between them. Data are show for a chain of Ly=6 at T=0.

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    • Figure 8
      Figure 8

      Currents in the Haldane cylinder with a potential step w, arising from the Fourier mode p=π. (a) Inside the dashed-dotted rectangle we show a sample of periodic unit cell, for which we also show the couplings inside of it (t1: dashed and continuous lines; t2: arrow lines). The other continuous and dashed lines are the t1 couplings of the whole Haldane model inside a cylinder. (b) Localized bulk currents and edge currents for a step potential w=0.1. Data presented for Ly=6, T=0.

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