  An important yet largely unsolved problem in the statistical mechanics of
disordered quantum systems is to understand how quenched disorder affects
quantum phase transitions in systems of itinerant fermions. In the clean limit,
continuous quantum phase transitions of the symmetry-breaking type in Dirac
materials such as graphene and the surfaces of topological insulators are
described by relativistic (2+1)-dimensional quantum field theories of the
Gross-Neveu-Yukawa (GNY) type. We study the universal critical properties of
the chiral Ising, XY, and Heisenberg GNY models perturbed by quenched
random-mass disorder, both uncorrelated or with long-range power-law
correlations. Using the replica method combined with a controlled triple
epsilon expansion below four dimensions, we find a variety of new
finite-randomness critical and multicritical points with nonzero Yukawa
coupling between low-energy Dirac fields and bosonic order parameter
fluctuations, and compute their universal critical exponents. Analyzing
bifurcations of the renormalization-group flow, we find instances of the
fixed-point annihilation scenario---continuously tuned by the power-law
exponent of long-range disorder correlations and associated with an
exponentially large crossover length---as well as the transcritical bifurcation
and the supercritical Hopf bifurcation. The latter is accompanied by the birth
of a stable limit cycle on the critical hypersurface, which represents the
first instance of fermionic quantum criticality with emergent discrete scale
invariance.
