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We study quantum phase transitions (QPTs) associated with splitting nodalFermi points, motivated by topological phase transitions between Dirac and Weylsemi-metals. A Dirac point in Dirac semi-metals may be split into two Weylpoints by breaking a lattice symmetry or time reversal symmetry, and theLifshitz transition is commonly used to describe the phase transitions. Here,we show that the Lifshitz description is fundamentally incorrect in QPTs withsplitting nodal Fermi points. We argue that correlations between fermions,order parameter, and the long range Coulomb interaction { must} be incorporatedfrom the beginning. One of the most striking correlation effects we find is{\it infinite anisotropy} of physical quantities, which cannot appear in aLifshitz transition. By using the standard renormalization group (RG) method,two types of infinitely anisotropic quantum criticalities are found in threespatial 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 parameterexcitations becomes universal ($\sqrt{2}-1$) along the Dirac point splittingdirection . For $N_f >1$, we find that fermions are parametrically faster thanorder parameter excitations in all directions. Our RG analysis is fullycontrolled by the fact that order parameter and fermion fluctuations are at theupper critical dimension, and thus our stable fixed points demonstrate thepresence of weakly coupled quantum criticalities with infinite anisotropy.

Quantum Criticalities with Infinite Anisotropy in Topological Phase Transitions between Dirac and Weyl Semi-metals