Hofstadter Butterflies in Phononic Structures: Commensurate Spectra, Wave Localisation and Metal-Insulator Transitions

Summary We present a new, simple, and easy-to-implement one-dimensional phononic system whose spectrum exactly corresponds to the Hofstadter butterfly. The system consists of masses that are coupled by linear springs and are mounted on flexural beams whose cross-sections (and, hence, stiffness) are modulated. We show that this system is the simplest version possible to achieve the Hofstadter butterfly exactly; in particular, the local resonances due to the beams are an essential component for this achievement. The butterfly emerges when a phase parameter that controls the spatial frequency of the cross-sectional modulation is varied. We examine the various approaches to producing spectral butterflies, including Bloch spectra for rational parameter choices, resonances of finite-sized systems and transmission coefficients of sections of finite length. For finite-size systems, we study the localisation of the modes by calculating the inverse participation ratio, and detect a phase transition characterised by a critical value of the stiffness modulation amplitude, where the state of the system changes from mainly extended to localised, corresponding to a metal-insulator phase transition. The obtained results offer a practical strategy to realise experimentally a system with similar dynamical properties. The transmission coefficient for sections of finite length is benchmarked through the comparison with Bloch spectra of the same finite-sized systems. The numerical results for the transmission spectra confirm the evidence of a phase transition in the dynamical state of the system. Our approach opens significant new perspectives in order to design mechanical systems able to support phase transitions in their vibrational properties.