Thermal Hofstadter Butterflies

Abstract

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.

Figures (8)

• Figure 1: Top panels: (a) square, (b) honeycomb, and (c) triangular geometrical lattices. The light green rectangle in (a) and parallelograms in (b) and (c) show the magnetic unit cells, the cyan square in (a) and parallelograms in (b) and (c) represent the geometrical unit cells, $a_{\text{AB}}$ is the distance between A-B atoms in b. Ordered pairs label the nearest neighbor sites to the $(m,n)$ site in each lattice. Bottom panels show the corresponding elemental plaquettes with magnetic flux $\Phi=\alpha \Phi_0$, with rational $\alpha=p/q$.
• Figure 2: Chemical potential curves $\mu(\alpha,T)$ (solid lines) as a function of $\alpha=p/q$ ($q=1123$) and different values of temperature $T$ under the condition of half-filling in the triangular lattice; all energies in units of the hopping parameter $t$. The energy spectrum $\varepsilon$ is shown behind the curves, where the sharp drops in $\mu$ are seen to occur within the gaps in the spectrum.
• Figure 3: Square lattice. Contour plots for (a) electronic specific heat $C_e(\alpha, T)$, and (b) entropy $S(\alpha, T)$, as function of the magnetic flux parameter $\alpha=p/q$ ($q=1123$), and temperature $T/t$. Solid light-yellow lines represent selected constant-$C_e$ or constant-$S_e$ contours. Color bars in units of $k_B$. Notice rich oscillatory structure associated with spectrum.
• Figure 4: Honeycomb lattice. Contour plots for (a) electronic specific heat $C_e(\alpha,T)$, and (b) entropy $S_e(\alpha, T)$ as function of the magnetic flux parameter $\alpha=p/q$ ($q=1123$), and temperature $T/t$. Solid light-yellow lines represent selected constant-$C_e$ or constant-$S_e$ contours. Color bars in units of $k_B$.
• Figure 5: Triangular lattice. Contour plots for (a) electronic specific heat $C_e(\alpha, T)$, and (b) entropy $S_e(\alpha, T)$, as function of the magnetic flux parameter $\alpha=p/q$ ($q=1123$), and temperature $T/t$. Solid yellow lines represent selected contours in both panels. Color bars in units of $k_B$.