Fermionic quantum criticality and the fractal nodal surface

The complete lack of theoretical understanding of the quantum critical states found in the heavy-fermion metals and the normal states of the high-T-c superconductors is rooted in a deep fundamental problem of condensed-matter physics: the infamous minus signs associated with Fermi-Dirac statistics render the path integral nonprobabilistic and do not allow the establishment of a connection to critical phenomena in classical systems. Using Ceperley's constrained path-integral formalism, we demonstrate that the workings of scale invariance and Fermi-Dirac statistics can be reconciled. The latter is self-consistently translated into a geometrical constraint structure. We show that this "nodal hypersurface" encodes the scales of the Fermi liquid, and we conjecture that it turns fractal when the system becomes quantum critical. To substantiate this, we analyze the nodal structures of fermionic Feynman backflow wave functions to find that the nodal surface indeed turns into a scale-invariant fractal when the backflow becomes hydrodynamical. Moreover, by following the evaluation of the quasiparticle momentum distribution, we demonstrate that the emergence of scale invariance in the nodal structure is accompanied by a divergence in the effective quasiparticle mass. Such a collapse of a Fermi liquid at a critical point has been observed in the heavy-fermion intermetallics in a spectacular fashion.