  A universal phenomenon in phase transitions is critical slowing down (CSD) -
systems, after an initial perturbation, take an exceptionally long time to
return to equilibrium. It is universally observed in the dynamics of bosonic
excitations, like order-parameter collective modes, but it is not generally
expected to occur for fermionic excitations because of the half-integer nature
of the fermionic spin. Direct observation of CSD in fermionic excitations or
quasiparticles would therefore be of fundamental significance. Here, we observe
fermionic CSD in the heavy-fermion (HF) compound YbRh$_2$Si$_2$ by terahertz
time-domain spectroscopy. HFs are compound objects with a strongly enhanced
effective mass, composed of itinerant and localized electronic states. We see
that near the quantum phase transition in YbRh$_2$Si$_2$ the build-up of
spectral weight of the HFs towards the Kondo temperature $T_K\approx 25$ K is
followed by a logarithmic rise of the quasiparticle excitation rate on the
heavy-Fermi-liquid side of the quantum phase transition below $10$ K. A
critical two-band HF liquid theory shows that this is indicative of fermionic
CSD. This CSD is a clear indication that the HF quasiparticles experience a
breakdown near the quantum phase transition, and the critical exponent of this
breakdown introduces a classification of fermionic quantum phase transitions
analogous to thermodynamic phase transitions - solution to a long-standing
problem.
