Identifying Cn -symmetric higher-order topology and fractional corner charge using entanglement spectra

Abstract

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.

Fermionic tensor networks for higher-order topological insulators from charge pumping

“…The convenience of this bulk characterization makes the ES an important tool for the numerical analysis of topological phases. Recently it was also observed, as expected, that the ES of higher order TIs displays characteristic (d − n)-dimensional boundary states as long as the entanglement cut preserves the crystal symmetries of the phase [24][25][26].…”

“…Any small perturbation of φ triv by a non-zero β increases the number of entangled modes to Eq. (25). Moreover, these additional decoupled modes disappear when blocking the two columns A col and B col , i.e.…”

“…II A, the number of entangled modes is also given by Eq. (25). For φ triv , the number of entangled modes is given by 2L when L ≤ N y /2.…”

“…This shows that Eq. (25) gives the maximal number of entangled modes compatible with the U(1) symmetry of the SSH model MPS. For the PEPS defined by the trivial cycle φ triv from Eq.…”

Higher-order topological insulator in cubic semiconductor quantum wells

“…3A represents a 2D unit cell at the corner of an effective 2D system in the QW plane. The special arrangement of atoms in the vicinity of the corner preserving C n symmetry of the bulk may lead to the fractional charge of the corner states, known as a fractional corner anomaly [22,23,[61][62][63]. We note that since fractional corner anomaly requires a theoretical description on the unit cell scale, it cannot be treated on the basis of multi-band k•p Hamiltonian.…”