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Register a new userQuantization of fractional corner charge in $C_n$-symmetric higher-order topological crystalline insulators
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Added by
Wladimir A. Benalcazar
Publication date
2018
fields
Physics
and research's language is
English
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In the presence of crystalline symmetries, certain topological insulators present a filling anomaly: a mismatch between the number of electrons in an energy band and the number of electrons required for charge neutrality. In this paper, we show that a filling anomaly can arise when corners are introduced in $C_n$-symmetric crystalline insulators with vanishing polarization, having as consequence the existence of corner-localized charges quantized in multiples of $frac{e}{n}$. We characterize the existence of this charge systematically and build topological indices that relate the symmetry representations of the occupied energy bands of a crystal to the quanta of fractional charge robustly localized at its corners. When an additional chiral symmetry is present, $frac{e}{2}$ corner charges are accompanied by zero-energy corner-localized states. We show the application of our indices in a number of atomic and fragile topological insulators and discuss the role of fractional charges bound to disclinations as bulk probes for these crystalline phases.
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In this paper, we derive a general formula for the quantized fractional corner charge in two-dimensional C_n-symmetric higher-order topological insulators. We assume that the electronic states can be described by the Wannier functions and that the edges are charge neutral, but we do not assume vanishing bulk electric polarization. We expand the scope of the corner charge formula obtained in previous works by considering more general surface conditions, such as surfaces with higher Miller index and surfaces with surface reconstruction. Our theory is applicable even when the electronic states are largely modulated near system boundaries. It also applies to insulators with non-vanishing bulk polarization, and we find that in such cases the value of the corner charge depends on the surface termination even for the same bulk crystal with C_3 or C_4 symmetry, via a difference in the Wyckoff position of the center of the C_n-symmetric crystal.
Robust fractional charge localized at disclination defects has been recently found as a topological response in $C_{6}$ symmetric 2D topological crystalline insulators (TCIs). In this article, we thoroughly investigate the fractional charge on disclinations in $C_n$ symmetric TCIs, with or without time reversal symmetry, and including spinless and spin-$frac{1}{2}$ cases. We compute the fractional disclination charges from the Wannier representations in real space and use band representation theory to construct topological indices of the fractional disclination charge for all $2D$ TCIs that admit a (generalized) Wannier representation. We find the disclination charge is fractionalized in units of $frac{e}{n}$ for $C_n$ symmetric TCIs; and for spin-$frac{1}{2}$ TCIs, with additional time reversal symmetry, the disclination charge is fractionalized in units of $frac{2e}{n}$. We furthermore prove that with electron-electron interactions that preserve the $C_n$ symmetry and many-body bulk gap, though we can deform a TCI into another which is topologically distinct in the free fermion case, the fractional disclination charge determined by our topological indices will not change in this process. Moreover, we use an algebraic technique to generalize the indices for TCIs with non-zero Chern numbers, where a Wannier representation is not applicable. With the inclusion of the Chern number, our generalized fractional disclination indices apply for all $C_n$ symmetric TCIs. Finally, we briefly discuss the connection between the Chern number dependence of our generalized indices and the Wen-Zee term.
Based on first-principles calculations and symmetry analysis, we predict atomically thin ($1-N$ layers) 2H group-VIB TMDs $MX_2$ ($M$ = Mo, W; $X$ = S, Se, Te) are large-gap higher-order topological crystalline insulators protected by $C_3$ rotation symmetry. We explicitly demonstrate the nontrivial topological indices and existence of the hallmark corner states with quantized fractional charge for these familiar TMDs with large bulk optical band gaps ($1.64-1.95$ eV for the monolayers), which would facilitate the experimental detection by STM. We find that the well-defined corner states exist in the triangular finite-size flakes with armchair edges of the atomically thin ($1-N$ layers) 2H group-VIB TMDs, and the corresponding quantized fractional charge is the number of layers $N$ divided by 3 modulo integers, which will simply double including spin degree of freedom.
Two-dimensional higher-order topological insulators can display a number of exotic phenomena such as half-integer charges localized at corners or disclination defects. In this paper, we analyze these phenomena, focusing on the paradigmatic example of the quadrupole insulator with $C_4$ rotation symmetry, and present a topological field theory description of the mixed geometry-charge responses. Our theory provides a unified description of the corner and disclination charges in terms of a physical geometry (which encodes disclinations), and an effective geometry (which encodes corners). We extend this analysis to interacting systems, and predict the response of fractional quadrupole insulators, which exhibit charge $e/2(2k+1)$ bound to corners and disclinations.
We construct the symmetric-gapped surface states of a fractional topological insulator with electromagnetic $theta$-angle $theta_{em} = frac{pi}{3}$ and a discrete $mathbb{Z}_3$ gauge field. They are the proper generalizations of the T-pfaffian state and pfaffian/anti-semion state and feature an extended periodicity compared with their of integer topological band insulators counterparts. We demonstrate that the surface states have the correct anomalies associated with time-reversal symmetry and charge conservation.
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