We extend a renormalisation operator for the transition by breaking of analyticity of incommensurate minimum energy states of Frenkel-Kontorova chains to one for the quantum statistical mechanics. The known fixed point for the former operator induces two for the latter, both at temperature zero, with β~ = 0 and ∞, respectively. Scaling exponents are deduced from the those for the former case. This is joint work with N.Catarino.