  As a paradigmatic example of multi-scale quantum criticality, we consider the
Pomeranchuk instability of an isotropic Fermi liquid in two spatial dimensions,
d=2. The corresponding Ginzburg-Landau theory for the quadrupolar fluctuations
of the Fermi surface consists of two coupled modes, critical at the same point,
and characterized by different dynamical exponents: one being ballistic with
dynamical exponent z=2 and the other one is Landau-damped with z=3, thus giving
rise to multiple dynamical scales. We find that at temperature T=0, the
ballistic mode governs the low-energy structure of the theory as it possesses
the smaller effective dimension d+z. Its self-interaction leads to logarithmic
singularities, which we treat with the help of the renormalization group. At
finite temperature, the coexistence of two different dynamical scales gives
rise to a modified quantum-to-classical crossover. It extends over a
parametrically large regime with intricate interactions of quantum and
classical fluctuations leading to a universal T-dependence of the correlation
length independent of the interaction amplitude. The multiple scales are also
reflected in the phase diagram and in the critical thermodynamics. In
particular, we find that the latter cannot be interpreted in terms of only a
single dynamical exponent: whereas, e.g., the critical specific heat is
determined by the z=3 mode, the critical compressibility is found to be
dominated by the z=2 fluctuations.
