  We present a comprehensive theoretical study of the static spin response in
HgTe quantum wells, revealing distinctive behavior for the topologically
nontrivial inverted structure. Most strikingly, the q=0 (long-wave-length) spin
susceptibility of the undoped topological-insulator system is constant and
equal to the value found for the gapless Dirac-like structure, whereas the same
quantity shows the typical decrease with increasing band gap in the
normal-insulator regime. We discuss ramifications for the ordering of localized
magnetic moments present in the quantum well, both in the insulating and
electron-doped situations. The spin response of edge states is also considered,
and we extract effective Lande g-factors for the bulk and edge electrons. The
variety of counter-intuitive spin-response properties revealed in our study
arises from the system's versatility in accessing situations where the
charge-carrier dynamics can be governed by ordinary Schrodinger-type physics,
mimics the behavior of chiral Dirac fermions, or reflects the material's
symmetry-protected topological order.
