Quantum metric and localization in a quasicrystal

We use the quantum metric to understand the properties of quasicrystals, represented by the one-dimensional (1D) Fibonacci chain. We show that the quantum metric can relate the localization properties of the eigenstates to the scale-invariance of both the chain and its energy spectrum. In particular, the quantum metric incorporates information about distances between the local symmetry centers of each eigenstate, making it much more sensitive to the localization properties of quasicrystals than other measures of localization, such as the inverse participation ratio. We further find that a full description of localization requires us to introduce a new phasonic component to the quantum metric, and a mixed phason-position Chern number. Finally, we show that the sum of both position and phasonic components of the quantum metric is lower-bounded by the gap labels associated with each energy gap of the Fibonacci chain. This establishes a direct link between the spatial localization and fractal energy spectrum of quasicrystals. Taken together, the quantum metric provides a unifying, yet accessible, understanding of quasicrystals, rooted in their scale-invariance and with intriguing consequences also for many-body physics.