$\beta-YbAlB_4$: a critical vortex metal

$\beta-YbAlB_4$ is the first $Yb$ based heavy fermion superconductor and has a non Fermi Liquid behavior in the normal state that develops without external tuning by pressure or doping, making it intrinsically quantum critical [1]. Application of a magnetic field is found to drive the development of a Fermi Liquid in which the Fermi temperature is determined by the Zeeman energy [2]. Here we present a theory for the intrinsic quantum criticality in which the main ingredient is an anisotropic hybridization matrix with line nodes in momentum space that carry a vorticity and resemble topological defects. Our theory predicts that the application of a field induces a novel Lifshitz transition, in which a quasi-two dimensional Fermi Liquid with density of states $N^*(B) \propto 1/\sqrt{B}$ nucleates around the line node in momentum space. We also discuss how the vortex metal picture can account for the ESR anomalies observed in this system [3]. \vskip 0.3truein \noindent[1] S. Nakatsuji \textit{et al.}, Nature Phys. 4, 603 (2008). \noindent[2] Y. Matsumoto \textit{et al.}, Science 21, 316 (2011). \noindent[3] L. M. Holanda \textit{et al.}, Phys Rev. Lett. 107, 026402 (2011).