Free nodal fermionic excitations are simple but interesting examples of 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 and for finite chemical potential. From measurements in graphene, or in ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ $(x=0.04)$, one may be able to infer the striking Landau diamagnetic susceptibility of the system at the quantum critical point. Although the quasiparticles in the mean field description of the proposed $d$-density wave (DDW) condensate in high-temperature superconductors are another example of nodal quasiparticles, the crossover from the high-temperature behavior to 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 $\sqrt{v_F{v}_{\mathrm{DDW}}}$, where $v_{\mathrm{DDW}}$ is the velocity in the direction orthogonal to the nodal direction at the Fermi point of the spectra of the DDW condensate.

• Received 29 November 2006

DOI:https://doi.org/10.1103/PhysRevB.75.115123