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The two bands touch at the zone corners (the K point in the Brillouin zone), where there is a zero density of states but no band gap. The graphene sheet thus displays a semimetallic (or zero-gap semiconductor) character. Two of the six Dirac points are independent, while the rest are equivalent by symmetry.
In k-space, this shows up as a hypercone, which have doubly degenerate bands which also meet at Dirac points. [11] Dirac semimetals contain both time reversal and spatial inversion symmetry; when one of these is broken, the Dirac points are split into two constituent Weyl points, and the material becomes a Weyl semimetal.
The Dirac points are six locations in momentum space on the edge of the Brillouin zone, divided into two non-equivalent sets of three points. These sets are labeled K and K'. These sets give graphene a valley degeneracy of =. In contrast, for traditional semiconductors, the primary point of interest is generally Γ, where momentum is zero. [60]
The term Dirac matter refers to a class of condensed matter systems which can be effectively described by the Dirac equation.Even though the Dirac equation itself was formulated for fermions, the quasi-particles present within Dirac matter can be of any statistics.
Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case.
In mathematics a Dirac structure is a geometric structure generalizing both symplectic structures and Poisson structures, and having several applications to mechanics. It is based on the notion of the Dirac bracket constraint introduced by Paul Dirac and was first introduced by Ted Courant and Alan Weinstein .
In Dirac's theory the fields are quantized for the first time and it is also the first time that the Planck constant enters the expressions. In his original work, Dirac took the phases of the different electromagnetic modes ( Fourier components of the field) and the mode energies as dynamic variables to quantize (i.e., he reinterpreted them as ...
This approach is called the 'differential' and 'field' approach by Schwinger, as opposed to the 'integral' and 'particle' approach of the Feynman diagrams. [ 12 ] [ 13 ] The core idea is that if the interaction has a small coupling constant (i.e. in the case of electromagnetism of the order of the fine structure constant) successive ...