Higher-order topological insulators in amorphous solids

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Higher-order topological insulators in amorphous solids

Adhip Agarwala, Vladimir Juričić, and Bitan Roy

Phys. Rev. Research 2, 012067(R) – Published 17 March 2020

Abstract

We identify the possibility of realizing higher order topological (HOT) phases in noncrystalline or amorphous materials. Starting from two- and three-dimensional crystalline HOT insulators, accommodating topological corner states, we gradually enhance structural randomness in the system. Within a parameter regime, as long as amorphousness is confined by an outer crystalline boundary, the system continues to host corner states, yielding amorphous HOT insulators. However, as structural disorder percolates to the edges, corner states start to dissolve into amorphous bulk, and ultimately the system becomes a trivial insulator when amorphousness plagues the entire system. These outcomes are further substantiated by computing the quadrupolar (octupolar) moment in two (three) dimensions. Therefore, HOT phases can be realized in amorphous solids, when wrapped by a thin (lithographically grown, for example) crystalline layer. Our findings suggest that crystalline topological phases can be realized even in the absence of local crystalline symmetry.

Received 10 April 2019
Revised 8 September 2019
Accepted 21 February 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.012067

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Structural properties Topological materials Topological phases of matter

Amorphous materials Amorphous semiconductors Topological insulators

Lattice models in condensed matter Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Adhip Agarwala^1,2,*, Vladimir Juričić^3,†, and Bitan Roy^2,4,‡

¹International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
²Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany
³Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden
⁴Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015, USA

^*adhip.agarwala@icts.res.in
^†vladimir.juricic@nordita.org
^‡bitan.roy@lehigh.edu

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Vol. 2, Iss. 1 — March - May 2020

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Images

Figure 1
Setup for two-dimensional amorphous HOT phases (see Fig. 2). Here, $L$ is the linear dimension of the system. Structural disorder is confined within a radius $R_{s}$ (scrambling radius), and litters the edges when $R_{s} \geq L / 2$ . In this region, the sites are randomly distributed. This construction can be generalized to higher dimensions (see Fig. 5).
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Figure 2
Energy spectra (top) and the local density of states (LDOS) near (due to finite system size) zero-energy modes (bottom) with increasing scrambling radius $R_{s}$ (from left to right) in a system with linear dimension $L = 20$ . Here we set $R = 4, M = 0, r_{o} = 1, t_{1} = t_{2} = g = 1$ , and $n$ is the index for energy eigenvalues $(E_{n})$ . A crystalline HOTI supports four in-gap corner modes near zero energy, well separated from the bulk states (first column). With increasing $R_{s}$ the bulk gap shrinks, but the system continues to support “sharp” corner states, as long as $R_{s} ≲ L / 2$ , yielding an amorphous HOTI. When $R_{s} > L / 2$ (entire system is scrambled), the bulk gap closes and the corner states melt into the glassy bulk (fourth column). The system then represents a trivial insulator. The scaling of the corresponding quadrupolar moment $(Q_{x y})$ with $R_{s}$ is shown in Fig. 3.
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Figure 3
Scaling of (a) quadrupolar $(Q_{x y})$ and (b) octupolar $(Q_{x y z})$ moments with $R_{s}$ for (a) $L = 20$ and (b) $L = 10$ . In both crystalline and amorphous systems $Q_{x y} = Q_{x y z} = 0.5$ , when they host corner modes, see Figs. 2 and 5. For large $R_{s}$ , the corner states disappear and $Q_{x y}$ and $Q_{x y z}$ deviate from 0.5, yielding a trivial insulator. Origin $(x_{0}, y_{0})$ dependence of $Q_{x y}$ in (c) crystalline $(M = - 1, g = 1)$ and (e) amorphous $(M = 0, g = 1, R_{s} = 6, R = 4, r_{0} = 1)$ HOTIs in periodic systems with $L = 20$ . For these sets of parameters, both open crystalline and amorphous systems support four corner-localized zero-energy modes. Scaling of $β$ , fraction of the area yielding $Q_{x y} = 0.5$ , with $1 / L$ in periodic (d) crystalline and (f) amorphous (with $R_{s} = \frac{L}{2} - 4)$ HOTIs [46]. Therefore, minor origin dependence of quantized $Q_{x y} = 0.5$ disappears in the thermodynamic limit $(L \to \infty$ , with fixed $R_{s} / L)$ , promoting $Q_{x y}$ as a bulk topological invariant. A similar behavior is expected for the octupolar moment $Q_{x y z}$ in three-dimensional crystalline and amorphous HOTI.
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Figure 4
Global phase diagram of a two-dimensional amorphous insulator in the $(M, g)$ plane [see Eq. (1)], obtained by computing the quadrupolar moment $(Q_{x y})$ in a system with $L = 20$ and $R_{s} = 6$ with periodic (left) and open (right) boundaries. We set $R = 4$ and $r_{o} = 1$ . Deep inside the amorphous HOTI (trivial phase) $Q_{x y} = 0.5 (0)$ . Here we average over 100 random and independent realizations of structural disorder. In a crystalline system, the HOTI phase occupies the white dashed region. HOTI can only be found for $g \neq 0$ . Along the entire $g = 0$ axis, $Q_{x y} = 0$ in both quantum spin Hall (for small $| M |)$ and trivial (for large $| M |)$ insulators, which is independent of the choice of origin for any $L$ [46]. Note that $Q_{x y}$ in periodic and open crystalline systems are identical (the white shaded region) [51]. In an amorphous system with PBC and OBC, $Q_{x y}$ agree (within the numerical accuracy) when the scale of amorphousness or structural disorder $(| g |)$ is comparable to the bandwidth $(t_{1})$ , i.e., $| g | ≲ 2 t_{1}$ or much larger than the bandwidth $(| g | ≫ t_{1})$ .
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Figure 5
LDOS for near-zero-energy states in a three-dimensional cubic system with $L = 10$ for (a) $R_{s} = 0$ (crystalline HOTI), (b) $R_{s} = 4$ (amorphous HOTI), and (c) completely scrambled (trivial insulator). We set $R = 2$ and $r_{0} = 0.5$ .
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