Topological crystalline phases in electronic structures can be generally classified using the spatial symmetry characters of the valence bands and mapping them onto appropriate symmetry indicators. These mappings have been recently applied to identify thousands of topological electronic materials. There can exist, however, topological crystalline non-trivial phases that go beyond this paradigm: they cannot be identified using spatial symmetry labels and consequently lack any classification. In this work, we achieve the first of such classifications showcasing the paradigmatic example of two-dimensional crystals with twofold rotation symmetry. We classify the gapped phases in timereversal invariant systems with strong spin-orbit coupling identifying a set of three Z2 topological invariants, which correspond to nested quantized partial Berry phases. By further isolating the set of atomic insulators representable in terms of exponentially localized symmetric Wannier functions, we infer the existence of topological crystalline phases of the fragile type that would be diagnosed as topologically trivial using symmetry indicators, and construct a number of microscopic models exhibiting this phase. Our work is expected to have important consequences given the central role fragile topological phases are expected to play in novel two-dimensional materials such as twisted bilayer graphene. arXiv:1906.08695v1 [cond-mat.mes-hall]

We prove the existence of higher-order topological insulators with protected chiral hinge modes in quasi-two-dimensional systems made out of coupled layers stacked in an inversion-symmetric manner. In particular, we show that an external magnetic field drives a stack of alternating pand n-doped buckled honeycomb layers into a higher-order topological phase, characterized by a non-trivial three-dimensional Z2 invariant. We identify silicene multilayers as a potential material platform for the experimental detection of this novel topological insulating phase.

We investigate theoretically the spectrum of a graphene-like sample (honeycomb lattice) subjected to a perpendicular magnetic field and irradiated by circularly polarized light. This system is studied using the Floquet formalism, and the resulting Hofstadter spectrum is analyzed for different regimes of the driving frequency. For lower frequencies, resonances of various copies of the spectrum lead to intricate formations of topological gaps. In the Landau-level regime, new wing-like gaps emerge upon reducing the driving frequency, thus revealing the possibility of dynamically tuning the formation of the Hofstadter butterfly. In this regime, an effective model may be analytically derived, which allows us to retrace the energy levels that exhibit avoided crossings and ultimately lead to gap structures with a wing-like shape. At high frequencies, we find that gaps open for various fluxes at E = 0, and upon increasing the amplitude of the driving, gaps also close and reopen at other energies. The topological invariants of these gaps are calculated and the resulting spectrum is elucidated. We suggest opportunities for experimental realization and discuss similarities with Landau-level structures in non-driven systems.arXiv:1803.04791v2 [cond-mat.mes-hall]

We consider weak topological insulators with a twofold rotation symmetry around their "dark" direction and show that these systems can be endowed with the topological crystalline structure of a higher-order topological insulator protected by rotation symmetry. These hybrid-order weak topological insulators display surface Dirac cones on all surfaces. Translational symmetry breaking perturbations gap the Dirac cones on the side surfaces leaving anomalous helical hinge modes behind. We also prove that the existence of this topological phase comes about due to a novel crystalline topological invariant of quantum spin-Hall insulators that can neither be revealed by symmetry indicators nor using Wilson loop invariants. Considering the minimal symmetry requirements, we anticipate that our findings could apply to a large number of weak topological insulators.

The topology of insulators is usually revealed through the presence of gapless boundary modes: this is the so-called bulk-boundary correspondence. However, the many-body wavefunction of a crystalline insulator is endowed with additional topological properties that do not yield surface spectral features, but manifest themselves as (fractional) quantized electronic charges localized at the crystal boundaries. Here, we formulate such bulk-corner correspondence for the physical relevant case of materials with time-reversal symmetry and spin-orbit coupling. To do so we develop partial real-space invariants that can be neither expressed in terms of Berry phases nor using symmetry-based indicators. These previously unknown crystalline invariants govern the (fractional) quantized corner charges both of isolated material structures and of heterostructures without gapless interface modes. We also show that the partial real-space invariants are able to detect all time-reversal symmetric topological phases of the recently discovered fragile type.