Phase transitions on strange sets: the Ising quasicrystal

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Abstract
A quantum Ising spin chain with nearest-neighbour couplings arranged in a quasiperiodic sequence is considered. The Cantor set structure of the energy spectrum is analysed in terms of the thermodynamic description of multifractals. Evidence is given that the spectrum of scales develops a singular behaviour: this is associated with a first-order phase transition of a new type. It is argued that this effect involves, not only quantum spins, but the whole class of phonon-like propagation problems on quasiperiodic chains.