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A quadratic band crossing (QBC) is a crossing of two bands with quadraticdispersion, which has been intensively investigated due to its appearance inBernal-stacked bilayer graphene. Here, we study an extension of QBCs, thetriply degenerate quadratic band crossing (TQBC), which is a three-bandcrossing node containing two quadratic dispersing bands and a flat band. Wefocus on two types of TQBCs. The first type contains a symmetry-protected QBCand a free-electron band, the prototype of which is the AA-stacked bilayersquareoctagon lattice. In a magnetic field, such a TQBC exhibits an anomalousLandau level structure, leading to a distinctive quantum Hall effect whichdisplays an infinite ladder of Hall plateaus when the chemical potentialapproaches zero. The other type of TQBC can be viewed as a pseudospin-1extension of the bilayer-graphene QBC. Under perturbations, this type of TQBCsmay split into linear pseudospin-1 Dirac-Weyl fermions. When tunneling througha potential barrier, the transmission probability of the first type decaysexponentially with the barrier width for any incident angle, similar to thefree-electron case, while the second type hosts an all-angle perfect reflectionwhen the energy of the incident particles is equal to half the barrier height.

Unconventional transport properties in systems with triply degenerate quadratic band crossings