Quantum criticality with infinite anisotropy in topological phase transitions between Dirac and Weyl semimetals

We study quantum phase transitions associated with splitting nodal Fermi points, motivated by topological phase transitions between Dirac and Weyl semimetals. A Dirac point in Dirac semimetals may be split into two Weyl points by breaking a lattice symmetry or time-reversal symmetry, and the Lifshitz transition is commonly used to describe the phase transitions. Here, we show that the Lifshitz description is fundamentally incorrect in quantum phase transitions with splitting nodal Fermi points. We argue that correlations between fermions, order parameter, and the long-range Coulomb interaction must be incorporated from the beginning. One of the most striking correlation effects we find is infinite anisotropy of physical quantities, which cannot appear in a Lifshitz transition. By using the standard renormalization group method, two types of infinitely anisotropic quantum criticalities are found in three spatial dimensions, varying with the number of the Dirac points $({N}_f)$. For ${N}_f=1$, the ratio of the fermion velocity to the velocity of order-parameter excitations becomes universal $(1+\sqrt{2})$ along the Dirac point splitting direction. For ${N}_f>1$, we find that fermions are parametrically faster than order-parameter excitations in all directions. Our renormalization group analysis is fully controlled by the fact that order parameter and fermion fluctuations are at the upper critical dimension, and thus our stable fixed points demonstrate the presence of weakly coupled quantum criticalities with infinite anisotropy.