  The notion of Dirac cones, wherein two or more bands become degenerate at a
certain momentum, is the starting point for the study of topological phases.
Dirac cones have been thoroughly explored in fermionic systems such as
graphene, Weyl semimetals, etc. The underlying mathematical structure in these
systems is a Clifford algebra -- a rule for identifying sets of matrices that
span the Hamiltonian. This structure allows for the identification of suitable
`mass' terms to open band gaps. In this article, we extend these ideas to
bosonic systems. Due to the pseudo-orthogonal nature of eigenvectors, the
algebra of matrices takes a very different form. Taking the honeycomb XY
ferromagnet as a prototype, we show that a Dirac cone emerges in the magnon
spectrum. A gap can be opened by a suitable mass term involving next-nearest
neighbour interactions. We next construct a one-dimensional ladder model with
triplon excitations. Using the new Clifford algebra, we define winding number
as a topological invariant. In analogy with the Su-Schrieffer-Heeger model,
topological transitions occur when the band gap closes, leading to the
appearance (or disappearance) of protected edge states. Our results suggest a
new route to studying band touching and band topology in bosonic systems.
