  In contrast to strongly frustrated classical systems, their quantum
counterparts typically have a non-degenerate ground state. A counterexample is
the celebrated Heisenberg sawtooth spin chain with ferromagnetic zigzag bonds
$J_1$ and competing antiferromagnetic basal bonds $J_2$. At a quantum phase
transition point $|J_2/J_1|=1/2$, this model exhibits a flat one-magnon
excitation band leading to a massively degenerate ground-state manifold which
results in a large residual entropy. Thus, for the spin-half model, the
residual entropy amounts to exactly one half of its maximum value
$\lim_{T\to\infty} S(T)/N = \ln2$. In the present paper we study in detail the
role of the spin quantum number $s$ and the magnetic field $H$ in the parameter
region around the transition (flat-band) point. For that we use full exact
diagonalization up to $N=20$ lattice sites and the finite-temperature Lanczos
method up to $N=36$ sites to calculate the density of states as well as the
temperature dependence of the specific heat, the entropy and the
susceptibility. The study of chain lengths up to $N=36$ allows a careful
finite-size analysis. At the flat-band point we find extremely small
finite-size effects for spin $s=1/2$, i.e., the numerical data virtually
correspond to the thermodynamic limit. In all other cases the finite-size
effects are still small and become visible at very low temperatures. In a
sizeable parameter region around the flat-band point the former massively
degenerate ground-state manifold acts as a large manifold of low-lying
excitations leading to extraordinary thermodynamic properties at the transition
point as well as in its vicinity such as an additional low-temperature maximum
in the specific heat. Moreover, there is a very strong influence of the
magnetic field on the low-temperature thermodynamics including an enhanced
magnetocaloric effect.
