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Traditional topological materials belong to different Altland-Zirnbauer symmetry classes (AZSCs) depending on their non-spatial symmetries. Here we introduce the notion of hybrid symmetry class topological insulators (HSCTIs): A fusion of two different AZSC topological insulators (TIs) such that they occupy orthogonal Cartesian hyperplanes and their universal massive Dirac Hamiltonian mutually anticommute, a mathematical procedure we name hybridization. The boundaries of HSCTIs can also harbor TIs, typically affiliated with an AZSC that is different from the ones for the parent two TIs. As such, a fusion or hybridization between planar class AII quantum spin Hall and vertical class BDI Su-Schrieffer-Heeger insulators gives birth to a three-dimensional class A HSCTI, accommodating quantum anomalous Hall insulators (class A) of opposite Chern numbers and quantized Hall conductivity of opposite signs on the top and bottom surfaces. Such a response is shown to be stable against weak disorder. We extend this construction to encompass crystalline HSCTI and topological superconductors (featuring half-quantized thermal Hall conductivity of opposite sings on the top and bottom surfaces), and beyond three spatial dimensions. Non-trivial responses of three-dimensional HSCTIs to crystal defects (namely edge dislocations) in terms of mid-gap bound states at zero energy around its core only on the top and bottom surfaces are presented. Possible (meta)material platforms to harness and engineer HSCTIs are discussed. 

Immersed in external magnetic fields (B), buckled graphene constitutes an ideal tabletop setup, manifesting a confluence of time-reversal symmetry (T) breaking Abelian (B) and T-preserving strain-induced internal axial (b) magnetic fields. In such a system, here we numerically compute two-terminal conductance (G), and four- as well as six-terminal Hall conductivity (σxy) for spinless fermions. On a flat graphene (b=0), the B field produces quantized plateaus at G=±|σxy|=(2n+1)e2/h, where n=0,1,2,⋯. The strain-induced b field lifts the twofold valley degeneracy of higher Landau levels and leads to the formation of additional even-integer plateaus at G=±|σxy|=(2,4,⋯)e2/h, when B>b. While the same sequence of plateaus is observed for G when b>B, the numerical computation of σxy in Hall bar geometries in this regime becomes unstable. A plateau at G=σxy=0 always appears with the onset of a charge-density-wave order, causing a staggered pattern of fermionic density between two sublattices of the honeycomb lattice. 

Topological classification of quantum solids often (if not always) groups all trivial atomic or normal insulators (NIs) into the same featureless family. As we argue here, this is not necessarily the case always. In particular, when the global phase diagram of electronic crystals harbors topological insulators with the band inversion at various time-reversal invariant momenta KTIinv in the Brillouin zone, their proximal NIs display noninverted band-gap minima at KNImin=KTIinv. In such systems, once topological superconductors nucleate from NIs, the inversion of the Bogoliubov de Gennes bands takes place at KBdGinv=KNImin, inheriting from the parent state. We showcase this (possibly general) proposal for two-dimensional time-reversal symmetry-breaking insulators. Then, distinct quantized thermal Hall conductivity and responses to dislocation lattice defects inside the paired states (tied with KBdGinv or KNImin), in turn, unambiguously identify different parent atomic NIs.