Diamagnetism of nodal fermions in semimetals: graphene and significant others

Nodal fermionic excitations are interesting examples of the simplest fermionic quantum criticality in which the dynamic critical exponent $z=1$, and the quasiparticles are well defined. They arise in a number of physical contexts. We derive the scaling form of the diamagnetic susceptibility, $\chi$, at finite temperatures, $T$, and finite chemical potential, $\mu$. From measurements in graphene, or in $Bi_{1-x}Sb_{x}$ ($x=0.4$), one may be able to infer the striking quantum critical Landau diamagnetic susceptibility of the system at $T=0$ and $\mu=0$, $\chi\propto - H^{-1/2 }, \; H\to 0$, where $H$ is the magnetic field. Although the quasiparticles in the mean field description of the proposed $d$-density wave (DDW) condensate in a high temperature superconductors is another example of nodal quasiparticles, the crossover from the high temperature behavior, $\chi\propto - T^ {-1} $, and the quantum critical behavior takes place at a far lower temperature due to the reduction of the velocity scale from the fermi velocity $v_{F}$ in graphene to $\sqrt{v_{F}v_{DDW}}$, where $v_{DDW}$ is the velocity in the direction orthogonal to the nodal direction at the Fermi point of the spectra of the DDW condensate.