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 systems 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 materials symmetry-protected topological order.
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