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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. As a consequence, Dirac matter can be distinguished in fermionic, bosonic or anyonic Dirac matter. Prominent examples of Dirac matter^{[1][2][3][4][5]} are graphene and other Dirac semimetals, topological insulators, Weyl semimetals, various high-temperature superconductors with ${\displaystyle d}$-wave pairing and liquid helium-3. The effective theory of such systems is classified by a specific choice of the Dirac mass, the Dirac velocity, the gamma matrices and the space-time curvature. The universal treatment of the class of Dirac matter in terms of an effective theory leads to a common features with respect to the density of states, the heat capacity and impurity scattering.

Definition

Members of the class of Dirac matter differ significantly in nature. However, all examples of Dirac matter are unified by similarities within the algebraic structure of an effective theory describing them.

General

The general definition of Dirac matter is a condensed matter system where the quasi-particle excitations can be described in curved spacetime by the generalised Dirac equation:

${\displaystyle \left\Psi =0.}$

In the above definition ${\displaystyle d_{\mu }}$ denotes a covariant vector depending on the ${\displaystyle (d+1)}$-dimensional momentum ${\displaystyle p}$ (${\displaystyle d}$ space ${\displaystyle +1}$ time dimension), ${\displaystyle e_{a}^{\mu }}$ is the vierbein describing the curvature of the space, ${\displaystyle m}$ the quasi-particle mass and ${\displaystyle v_{\rm {D}}}$ the Dirac velocity. Note that since in Dirac matter the Dirac equation gives the effective theory of the quasiparticles, the energy from the mass term is ${\displaystyle mv_{\rm {D}}^{2}}$, not the rest mass ${\displaystyle mc^{2}}$ of a massive particle. ${\displaystyle \gamma ^{\mu }}$ refers to a set of Dirac matrices, where the defining for the construction is given by the anticommutation relation,

${\displaystyle \left\{\gamma ^{\mu },\gamma ^{\nu }\right\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=\eta ^{\mu \nu }I_{d}.}$

${\displaystyle \eta ^{\mu \nu }}$ is the Minkowski metric with signature (+ - - -) and ${\displaystyle I_{d}}$ is the ${\displaystyle d\times d}$-dimensional unit matrix. In all equations, implicit summation over ${\displaystyle a}$ and ${\displaystyle \mu }$ is used (Einstein convention). Furthermore, ${\displaystyle \Psi }$ is the wavefunction. The unifying feature of all Dirac matter is the matrix structure of the equation describing the quasi-particle excitations.

In the limit where ${\displaystyle d_{\mu }(p)=D_{\mu }}$, i.e. the covariant derivative, conventional Dirac matter is obtained. However, this general definition allows the description of matter with higher order dispersion relations and in curved spacetime as long as the effective Hamiltonian exhibits the matrix structure specific to the Dirac equation.

Common (conventional)

The majority of experimental realisations of Dirac matter to date are in the limit of ${\displaystyle d_{\mu }(p)=D_{\mu }}$ which therefore defines conventional Dirac matter in which the quasiparticles are described by the Dirac equation in curved space-time,

${\displaystyle \left\Psi =0.}$

Here, ${\displaystyle D_{\mu }}$ denotes the covariant derivative. As an example, for the flat metric, the energy of a free Dirac particle differs significantly from the classical kinetic energy where energy is proportional to momentum squared:

Zdroj:https://en.wikipedia.org?pojem=Dirac_matter
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