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In condensed matter physics, Dirac cones can be found in graphene, topological insulators, cuprates, and iron-pnictides. This means that electrons behave as massless particles near the Dirac points.

What is the significance of this in condensed matter physics? Why is this surprising? Are there any potential applications?

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I think the most striking feature which got people like Edward Witten involved was the manifestation of Dirac-like dynamics, which are usually associated with highly relativistic situations, in a rather tame low-energy setting of solid state physics. So you could think of graphene as a nice testing ground of many notions of relativistic physics.

Beyond this, in and of itself, it seems that Dirac points present themselves in systems that are on the boundary between two topological phases, i.e., they are a manifestation of a system that is precisely on the topological phase boundary (in the space of all possible systems). But it is not clear to me whether all phase boundaries look like Dirac cones.

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