  Entropy accumulation near a quantum critical point was expected based on
general scaling arguments, and has recently been explicitly observed. We
explore this issue further in two canonical models for quantum criticality,
with particular attention paid to the potential effects beyond hyperscaling. In
the case of a one-dimensional transverse field Ising model, we derive the
specific scaling form of the free energy. It follows from this scaling form
that the singular temperature dependence at the critical field has a vanishing
prefactor but the singular field dependence at zero temperature is realized.
For the spin-density-wave model above its upper critical dimension, we show
that the dangerously irrelevant quartic coupling comes into the free energy in
a delicate way but in the end yields only subleading contributions beyond
hyperscaling. We conclude that entropy accumulation near quantum critical point
is a robust property of both models.
