Thermal Hofstadter Butterflies

Fractal electronic spectra arising from the competition between lattice periodicity and magnetic flux are a fundamental hallmark of two-dimensional quantum systems. While the spectral properties of Hofstadter butterflies are well documented, their thermodynamic response has remained remarkably unexplored. We present an original characterization of the electronic entropy $S_{e}$, and specific heat $C_{e}$, at half-filling, for square, honeycomb, and triangular lattices under a magnetic field. We demonstrate that these observables exhibit fast and slow magneto-thermo oscillations and pronounced magnetocaloric effects. We identify striking self-similarity in $S_e$ and $C_e$, tracing heart-shaped specific heat and tunnel-like entropy contours that repeat at specific lattice-dependent magnetic fluxes. Entropy minima at low temperatures play a remarkable role, acting as fingerprints for the butterfly spines, resolving the underlying fractal spectra. These findings may establish thermal measurements as high-resolution spectroscopic probes, providing a robust framework for recognizing fractal signatures through thermodynamics in diverse nanostructures.