Effect of strong disorder in a three-dimensional topological insulator: Phase diagram and maps of the ${\mathbb{Z}}_{2}$ invariant

Effect of strong disorder in a three-dimensional topological insulator: Phase diagram and maps of the $Z_{2}$ invariant

Bryan Leung and Emil Prodan

Phys. Rev. B 85, 205136 – Published 25 May 2012

Abstract

We study the effect of strong disorder in a three-dimensional topological insulator with time-reversal symmetry and broken-inversion symmetry. First, using level-statistics analysis, we demonstrate the persistence of delocalized bulk states even at large disorder. The delocalized spectrum is seen to display the levitation and pair annihilation effect, indicating that the delocalized states continue to carry the $Z_{2}$ invariant after the onset of disorder. Second, the $Z_{2}$ invariant is computed via twisted boundary conditions using an efficient numerical algorithm. We demonstrate that the $Z_{2}$ invariant remains quantized and nonfluctuating even after the spectral gap becomes filled with dense localized states. In fact, our results indicate that the $Z_{2}$ invariant remains quantized until the mobility gap closes or until the Fermi level touches the mobility edges. Based on such data, we compute the phase diagram of the Bi $_{2}$ Se $_{3}$ topological material as a function of disorder strength and position of the Fermi level.

Received 9 February 2012

DOI:https://doi.org/10.1103/PhysRevB.85.205136

Authors & Affiliations

Bryan Leung¹ and Emil Prodan²

¹Center for Materials Theory, Department of Physics & Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
²Department of Physics, Yeshiva University, New York, New York 10016, USA

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Vol. 85, Iss. 20 — 15 May 2012

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Figure 1
Example of a time-reversal-invariant path in the Brillouin torus and its discretization.Reuse & Permissions
Figure 2
(a)–(c) The band structure of the model for a slab configuration $0 < n_{3} < 30$ , plotted as function of $k_{1}$ with $k_{2}$ fixed at $k_{2} = 0$ . The hopping parameter $t$ takes the values $t = 14$ meV in panel (a), $t = 22.6$ meV in panel (b), and $t = 40$ meV in panel (c). Panel (d) reports a calculation of the strong $Z_{2}$ invariant as $t$ was varied from 14 to 40 meV.Reuse & Permissions
Figure 3
Level statistics analysis for (left column) the topological insulator $t = 40$ meV and (right column) the trivial insulator $t = 14$ meV. Each panel displays the variance of the level-spacings ensembles as a function of the energy where the level spacings were collected. The gray lines in each panel represent the integrated density of states (IDOS), which can be used to assess the evolution of the spectral gap, corresponding to the flat IDOS, and especially to determine when the gap is closing and becoming completely filled with localized states. The horizontal dash lines mark the value 0.104, the variance of the GSE ensemble. The vertical range in each panel goes from 0 to 1. The shaded regions represent the emerging phase diagram of the topological model.Reuse & Permissions
Figure 4
Illustration of the phase diagram of the model as derived in Fig. 3, and the two paths used in the mapping of the $Z_{2}$ invariant.Reuse & Permissions
Figure 5
The upper panels show the $Z_{2}$ invariant computed along the path (1) of Fig. 4 on an 8 $\times$ 8 $\times$ 8 unit-cell lattice via twisted boundary conditions. The dimension of the occupied space was slowly reduced from 1024 to 124, as indicated in the figure. For each $Z_{2}$ , the calculation was repeated for 10 random disorder configurations and the output is shown by the full dots, exactly how it occurs in the actual calculation. The percentages of the $Z_{2} = \pm 1$ occurrences are displayed in each panel. The lower panel shows the variance of the level spacings for $W = 300$ meV, and the averaged Fermi levels (see the vertical lines) corresponding to each $Z_{2}$ calculation.Reuse & Permissions
Figure 6
This figure reports the results of a computation of the strong $Z_{2}$ invariant along the path (2) of Fig. 4, completed on an 8 $\times$ 8 $\times$ 8 unit-cell lattice via twisted boundary conditions. The disorder strength was increased from $W = 100$ to 1200 meV, as indicated in the figure. For each $Z_{2}$ , the calculation was repeated for 10 random disorder configurations and the output is shown by the full dots, exactly how it occurs in the actual calculation. The percentages of the $Z_{2} = \pm 1$ occurrences are displayed in each panel. The accompanying panels show the variance of the level spacings at the corresponding $W$ 's, from where one can determine when the Fermi level is in a region of localized/delocalized spectrum. The Fermi level, represented by the dotted vertical line, was kept at 68 meV during these calculations.Reuse & Permissions

Physical Review B

covering condensed matter and materials physics

Effect of strong disorder in a three-dimensional topological insulator: Phase diagram and maps of the $Z_{2}$ invariant

Bryan Leung and Emil Prodan

Phys. Rev. B 85, 205136 – Published 25 May 2012

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Physical Review B

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Effect of strong disorder in a three-dimensional topological insulator: Phase diagram and maps of the Z2 invariant

Bryan Leung and Emil Prodan

Phys. Rev. B 85, 205136 – Published 25 May 2012

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Effect of strong disorder in a three-dimensional topological insulator: Phase diagram and maps of the $Z_{2}$ invariant