  The ferrimagnetic phase of the sawtooth chain with mixed ferromagnetic
nearest-neighbour interactions $J$ and antiferromagnetic next-nearest-neighbour
interactions $J'$ (within the isotropic Heisenberg model) was previously
characterized as a phase with commensurate order. In this paper, we demonstrate
that the system in fact exhibits an incommensurate quantum spin spiral. Even
though the ground state is translationally invariant in terms of the local spin
expectations $\avg{\vec{S}_i}$, the spiral can be detected via the connected
spin-spin correlations
$\avg{\vec{S}_i\cdot\vec{S}_j}-\avg{\vec{S}_i}\cdot\avg{\vec{S}_j}$ between the
apical spins. It has a long wavelength that grows with $J'$ and that soon
exceeds finite-system sizes typically employed in numerical simulations. A
faithful treatment thus requires the use of state-of-the-art simulations for
large, periodic systems.
  In this work, we are able to accurately treat up to $L=400$ sites (200 unit
cells) with periodic boundary conditions using the density-matrix
renormaliztion group (DMRG). Exploiting the SU(2) symmetry allows us to
directly compute the lowest-energy state for a given total spin. Our results
are corroborated by variational uniform matrix product state (VUMPS)
calculations, which work directly in the thermodynamic limit at the cost of a
lower accuracy.
