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# Topology on a new facet of bismuth

Edited by Philip Hofmann, Department of Physics and Astronomy, Aarhus University, and accepted by Editorial Board Member Angel Rubio May 20, 2019 (received for review January 10, 2019)

## Significance

We uncover the presence of a new topological crystalline insulator (TCI) state in bismuth, which is protected by a twofold rotational symmetry. In contrast to the recently discovered higher-order topological phase in bismuth, the present TCI phase hosts unpinned Dirac cone surface states that could be accessed directly through photoemission experiments. Our study provides a comprehensive understanding of the rich topological electronic structure of bismuth.

## Abstract

Bismuth-based materials have been instrumental in the development of topological physics, even though bulk bismuth itself has been long thought to be topologically trivial. A recent study has, however, shown that bismuth is in fact a higher-order topological insulator featuring one-dimensional (1D) topological hinge states protected by threefold rotational and inversion symmetries. In this paper, we uncover another hidden facet of the band topology of bismuth by showing that bismuth is also a first-order topological crystalline insulator protected by a twofold rotational symmetry. As a result, its

Bismuth is well known for its peculiar physical properties. It was long considered to be the stable element with the highest atomic mass, but relatively recent experiments have shown that bismuth is in fact weakly radioactive (1). It is a semimetal with a vanishingly small carrier density (^{−3}) but an exceptionally high electron mobility (^{−1}⋅V^{−1}) (2⇓⇓–5). As a result, the ultraquantum regime is reached in bismuth at a magnetic field as small as 9 T, beyond which a number of correlated electron states have been observed (4⇓⇓–7). Because of bismuth’s large spin–orbit coupling, bismuth-based materials have also played a fundamental role in topological physics (8, 9). A bismuth–antimony alloy (Bi_{1−x}Sb_{x}) was the first experimental realization of a 3D topological insulator (TI) (10). The

Theoretical advances on topological crystalline insulators (TCIs) (21) have greatly expanded the topological classification of band insulators beyond the *et al*. (47) and Khalaf *et al*. (48) found that, when certain additional symmetry Y is present, topological invariants of TCIs protected by symmetry X can be inferred from the Y-related symmetry eigenvalues of the energy bands. Such proposals of symmetry indicators and topological quantum chemistry have facilitated first-principles studies of new topological materials (52⇓⇓⇓⇓⇓–58).

Building upon these theoretical advances, a recent work (41) showed that pure bismuth hosts a second-order band topology that is protected by threefold rotational and inversion symmetries. As a result, it supports 1D topological hinge states in a crystal whose shape preserves the threefold rotational and inversion symmetries. Here we show that the band topology of bismuth is even richer in that it also hides a first-order TCI state, which is protected by its twofold rotational symmetries, resulting in a pair of unpinned topological Dirac surface states on its

## Results

First-principles calculations were carried out using the VASP package (59, 60). An ultrasoft pseudopotential and the generalized-gradient approximation were applied in the self-consistency process. The experimental crystal structure of Bi was used (61) in the calculations. A Wannier-basis–based tight-binding model, where the s and p orbitals of Bi were included, was obtained via the Wannier90 package (62).

Bismuth crystalizes in a rhombohedral structure (61). Its space group and point group are *R*-3*m* (*A*). In this construction, the out-of-plane *B*). The time-reversal invariant momentum (TRIM) points include one Γ, one T, three L, and three F symmetry points. Our band structure (Fig. 1*C*) confirms that bismuth is a compensated semimetal with a continuous band gap. To unfold the nature of band inversions in bismuth, we calculated the parity eigenvalues at the TRIM points, and with the exception of the Γ point, the lowest three valence bands at all of the TRIM points were found to have one state with positive parity and two states with negative parity. In contrast, at Γ, all three valence bands have positive parity eigenvalues, so that bismuth has two band inversions at the Γ point.

We turn next to discuss the symmetry-based indicators. As shown in refs. 47 and 48, crystals in the space group *SI Appendix*, section 1 for details. Our analysis thus reveals that bismuth is a purely rotational-symmetry–protected TCI with

To obtain a deeper understanding of *C*). The nontrivial topology protected by the twofold rotation **1**) as**2**). This rearrangement may lead to new topology beyond the **3**, we see that **4** shows that

With the preceding discussion in mind, we computed the surface band structure throughout the *B*. These are generic k points, which are related only by the twofold rotational symmetry *E* and *F*) directly show the presence of gapless Dirac surface states. Interestingly, these unpinned Dirac fermions are of type II (64) in that the velocities of the two surface bands that cross possess the same sign. By contrast, along the path *D*). To further confirm the nontrivial nature of these states, we studied the Wannier charge centers (WCCs) (65⇓–67) *G* and *H*, are seen to be disjointed in the whole surface Brillouin zone, except at *H*). These results unambiguously demonstrate the presence of unpinned Dirac surface states associated with the twofold rotational-symmetry–protected topology.

We now demonstrate the 1D topological hinge states protected by *A* and *B*), which are periodic along the *A*). In the second rod, the side surfaces are (001) and (111) and the cross-section is a parallelogram (Fig. 3*B*). Both rods are invariant under the twofold rotation *C* and *D*, from which we can clearly identify the existence of 1D helical edge states lying inside the bulk band gap. We also investigated the real-space distribution of the wave function of these 1D helical edge states. Fig. 3 *E* and *F* shows that these helical states are localized on the edges shared by adjacent side surfaces, further confirming that these are topological hinge states.

Finally, we show that the

In summary, we have investigated topological properties of the bulk and surface band structures of bismuth. We show that bismuth is a TCI with multiple nontrivial topological invariants. In the first order, bismuth features unpinned Dirac surface states on the

## Acknowledgments

T.-R.C. and X.Z. were supported by the Young Scholar Fellowship Program from the Ministry of Science and Technology (MOST) in Taiwan, under a MOST grant for the Columbus Program MOST108-2636-M-006-002, National Cheng Kung University, Taiwan, and National Center for Theoretical Sciences, Taiwan. This work was supported partially by the MOST, Taiwan, Grant MOST107-2627-E-006-001. This research was supported in part by Higher Education Sprout Project, Ministry of Education to the Headquarters of University Advancement at National Cheng Kung University (NCKU). H.L. acknowledges Academia Sinica, Taiwan, for the support under Innovative Materials and Analysis Technology Exploration (AS-iMATE-107-11). L.F. was supported by the US Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES), Division of Materials Sciences and Engineering under Award DE-SC0018945. The work at Northeastern University was supported by the US DOE, Office of Science, BES Grant DE-FG02-07ER46352, and benefited from Northeastern University’s Advanced Scientific Computation Center and the National Energy Research Scientific Computing Center (NERSC) supercomputing center through DOE Grant DE-AC02-05CH11231. N.G. and S.-Y.X. acknowledge support by the Science and Technology Center for Integrated Quantum Materials, NSF Grant DMR-1231319.

## Footnotes

↵

^{1}C.-H.H., X.Z., and T.-R.C. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: nilnish{at}gmail.com, suyangxu{at}mit.edu, or liangfu{at}mit.edu.

Author contributions: H.L. designed research; C.-H.H., X.Z., T.-R.C., Q.M., N.G., A.B., S.-Y.X., H.L., and L.F. performed research; C.-H.H., X.Z., T.-R.C., S.-Y.X., H.L., and L.F. analyzed data; and C.-H.H., A.B., S.-Y.X., H.L., and L.F. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. P.H. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1900527116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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