Robust fractional charge localized at disclination defects has been recently found as a topological response in C6 symmetric 2D topological crystalline insulators (TCIs). In this article, we thoroughly investigate the fractional charge on disclinations in Cn symmetric TCIs, with or without time reversal symmetry, and including spinless and spin-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 e n for Cn symmetric TCIs; and for spin-1 2 TCIs, with additional time reversal symmetry, the disclination charge is fractionalized in units of 2e n . We extend our results to interacting phases and prove that the fractional disclination charge determined by our topological indices is robust against electronelectron interactions that preserve the Cn symmetry and many-body bulk gap. 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 Cn symmetric TCIs. Finally, we briefly discuss the connection between the Chern number dependence of our generalized indices and the Wen-Zee term.

Higher order topological insulators (HOTIs) are a new class of topological materials which host protected states at the corners or hinges of a crystal. HOTIs provide an intriguing alternative platform for helical and chiral edge states and Majorana modes, but there are very few known materials in this class. Recent studies have proposed Bi as a potential HOTI, however, its topological classification is not yet well accepted. In this work, we show that the (110) facets of Bi and BiSb alloys can be used to unequivocally establish the topology of these systems. Bi and Bi0.92Sb0.08 (110) films were grown on silicon substrates using molecular beam epitaxy and studied by scanning tunneling spectroscopy. The surfaces manifest rectangular islands which show localized hinge states on three out of the four edges, consistent with the theory for the HOTI phase. This establishes Bi and Bi0.92Sb0.08 as HOTIs, and raises questions about the topological classification of the full family of BixSb1−x alloys.

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We study the geometric response of three-dimensional non-Hermitian crystalline systems with non-trivial point gap topology. For systems with four-fold rotation symmetry, we show that in presence of disclination lines with a total Frank angle which is an integer multiple of 2π, there can be non-trivial, one-dimensional point gap topology along the direction of disclination lines. This results in disclination-induced non-Hermitian skin effects. We extend the recently proposed non-Hermitian field theory approach to describe this phenomenon as a Euclidean Wen-Zee term. Furthermore, by doubling a non-Hermitian Hamiltonian to a Hermitian 3D chiral topological insulator, we show that the disclination-induced skin modes are zero modes of the surface Dirac fermion(s) in the presence of a pseudo-magnetic flux induced by disclinations.

We study the entanglement spectrum (ES) of two-dimensional Cn-symmetric second-order topological insulators (TIs). We show that some characteristic higher order topological observables, e.g., the filling anomaly and its associated fractional corner charge, can be determined from the ES of atomic and fragile TIs. By constructing the relationship between the configuration of Wannier orbitals and the number of protected in-gap states in the ES for different symmetric cuts in real space, we express the fractional corner charge in terms of the number of protected in-gap states of the ES. We show that our formula is robust in the presence of electron-electron interactions as long as the interactions preserve Cn rotation symmetry and charge-conservation symmetry. Moreover, we discuss the possible signatures higher order topology in the many-body ES. Our methods allow the identification of some classes of higher order topology without requiring the usage of nested Wilson loops or nested entanglement spectra.