Quantized electromagnetic response of three-dimensional chiral topological insulators

S.-T. Wang, D.-L. Deng, Joel E. Moore, Kai Sun, and L.-M. Duan

Phys. Rev. B 91, 035108 – Published 7 January 2015

Abstract

Protected by the chiral symmetry, three-dimensional chiral topological insulators are characterized by an integer-valued topological invariant. How this invariant could emerge in physical observables is an important question. Here, we show that the magnetoelectric polarization can identify the integer-valued invariant if we gap the system without coating a quantum Hall layer on the surface. The quantized response is demonstrated to be robust against weak perturbations. We also study the topological properties by adiabatically coupling two nontrivial phases, and find that gapless states appear and are localized at the boundary region. Finally, an experimental scheme is proposed to realize the Hamiltonian and measure the quantized response with ultracold atoms in optical lattices.

Received 22 August 2014
Revised 17 November 2014

DOI:https://doi.org/10.1103/PhysRevB.91.035108

Authors & Affiliations

S.-T. Wang^1,2, D.-L. Deng^1,2, Joel E. Moore^3,4, Kai Sun¹, and L.-M. Duan^1,2

¹Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
²Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China
³Department of Physics, University of California, Berkeley, California 94720, USA
⁴Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

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Vol. 91, Iss. 3 — 15 January 2015

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Images

Figure 1
The winding number $Γ$ as a function of the parameter $h$ . The Hamiltonians are $H_{1} (k)$ in (a) and $H_{2} (k)$ in (b), respectively. $δ = 0.5$ for both panels.
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Figure 2
Coupling two topologically nontrivial phases by varying the parameter $h$ adiabatically from 0 to 2 along the $z$ direction. $x$ and $y$ directions are periodic. Sixty slabs are taken for the $z$ direction. (a) The energy dispersion for the lowest conduction band and highest valence band. (b) and (c) show the surface states near the respective Dirac points. The $Γ$ point is displaced from the center for a better display of the Dirac cones.
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Figure 3
Charge $Q$ accumulated on the surface in the $z$ direction due to a uniform magnetic field with total flux $Φ$ at $k_{x} = k_{y} = 0$ . $N = 199, L_{z} = 20$ for all three panels. (a) and (b) consider the perturbative effect of an intra-site nearest neighbor hopping. (c) adds a random on-site potential characterized by $Δ_{rp}$ . The insets show the charge density at $Φ = 3 Φ_{0}$ without perturbations, and a closeup for the slope. The parameters in each panel are (a) Hamiltonian $H_{1} (k)$ with $h = 2, δ = 0.5, Δ_{S} = 1, Γ = 1$ ; (b) Hamiltonian $H_{2} (k)$ with $h = 0, δ = 0, Δ_{S} = 1, Γ = - 4$ ; (c) Hamiltonian $H_{1} (k)$ with $h = 2, δ = 0, Δ_{S} = 1, Γ = 1$ . In all panels, the accumulated charge $Q$ is relative to the case when $Φ = 0$ and is summed over all particle species for the upper half of the sample at half-fillings.
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Figure 4
Schematics to realize the Hamiltonian $H_{1}$ with cold atoms. (a) Atomic level structure of $^{6} Li$ and the four internal states used to represent the spin and orbital degrees of freedom. (b) The optical lattice is tilted with a homogeneous energy gradient along each direction. (c) Laser configurations to create the first term in $H_{rx}$ . The superscript on each Rabi frequency denotes the polarization of the beam.
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Figure 5
Spectrum for the surface states showing the number of Dirac cones. The upper panels in (a) and (b) show the lowest conduction and highest valence band. The lower panels show the next two bands closest to the Fermi energy. (a) For $H_{1} (k), h = 2, δ = 0.5$ with winding number $Γ = 1$ and 1 Dirac cone. (b) For $H_{2} (k), h = 0, δ = 0$ with winding number $Γ = 4$ and 4 Dirac cones in total. The $Γ$ point is displaced from the center for better display of the Dirac cones.
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Figure 6
(a) Energy spectrum for the one-layer Hamiltonian $H_{1} (k)$ with a strong uniform magnetic field and unit cell flux as $\frac{1}{3} Φ_{0}$ . An additional weak flux tube is inserted through the center lattice cell of the layer, with flux up to $Φ / Φ_{0} = 0.1$ . (b) and (c) [(d) and (e)] correspond to the charge polarization with respect to the increasing flux tube at a chemical potential $μ_{1} [μ_{2}]$ . Charge is accumulated around the flux tube, and by changing the chemical potential and hence the Landau level occupancy, the charge accumulation rate can be modified by an integer.
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