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Crystalline symmetries can generate exotic band-crossing features, which canlead to unconventional fermionic excitations with interesting physicalproperties. We show how a cubic Dirac point---a four-fold-degenerateband-crossing point with cubic dispersion in a plane and a linear dispersion inthe third direction---can be stabilized through the presence of a nonsymmorphicglide mirror symmetry in the space group of the crystal. Notably, the cubicDirac point in our case appears on a threefold axis, even though it has beenbelieved previously that such a point can only appear on a sixfold axis. Weshow that a cubic Dirac point involving a threefold axis can be realized closeto the Fermi level in the non-ferroelectric phase of LiOsO$_3$. Upon loweringtemperature, LiOsO$_3$ has been shown experimentally to undergo a structuralphase transition from the non-ferroelectric phase to the ferroelectric phasewith spontaneously broken inversion symmetry. Remarkably, we find that thebroken symmetry transforms the cubic Dirac point into three mutually-crossednodal rings. There also exist several linear Dirac points in the low-energyband structure of LiOsO$_3$, each of which is transformed into a single nodalring across the phase transition.

Nonsymmorphic cubic Dirac point and crossed nodal rings across the ferroelectric phase transition in LiOsO$_3$