  The antiferromagnetic sawtooth chain is a prototypical example of a
frustrated spin system with vertex-sharing triangles, giving rise to complex
quantum states. Depending on the interaction parameters, this system has three
phases, of which the gapless non-collinear phase (for strongly coupled basal
spins and loosely attached apical spins) has received little theoretical
attention so far. In this work, we comprehensively investigate the properties
of the non-collinear phase using large-scale tensor network computations which
exploit the full SU(2) symmetry of the underlying Heisenberg model. We study
the ground state both for finite systems using the density-matrix
renormalization group (DMRG) as well as for infinite chains via the variational
uniform matrix-product state (VUMPS) formalism. Finite temperatures and
correlation functions are tackled via imaginary- or real time evolutions, which
we implement using the time-dependent variational principle (TDVP). We find
that the non-collinear phase is characterized by a double-Q structure for the
apex-apex correlations. Deep into the phase, two peaks merge into a single one
indicating a 90-degree spiral. The apical spins are soft and highly susceptible
to external perturbations; they form a large number of gapless magnetic states
that are polarized by weak fields and cause a long low-temperature tail in the
specific heat. The dynamic spin-structure factor exhibits additive
contributions from a two-spinon continuum (excitations of the basal chain) and
a gapless peak at $k=\pi/2$ (excitations of the apical spins). Small
temperatures excite the gapless states and smear the spectral weight of the
$k=\pi/2$ peak out into a homogeneous flat-band structure. Our results are
relevant, e.g., for the material atacamite Cu$_2$Cl(OH)$_3$ in high magnetic
fields.
