Quantized heat flow in the Hofstadter butterfly

Abstract

When subjected to a strong magnetic field, electrons on a two-dimensional lattice acquire a fractal energy spectrum called Hofstadter's butterfly. In addition to its unique recursive structure, the Hofstadter butterfly is intimately linked to non-trivial topological orders, hosting a cascade of ground states characterized by non-zero topological invariants. These states, called Chern insulators, are usually understood as replicas of the ground states of the quantum Hall effect, with electrical and thermal conductances that should be quantized, reflecting their topological order. The Hofstadter butterfly is now commonly observed in van-der-Waals heterostructures-based moiré superlattices. However, its thermal properties, particularly the quantized heat flow expected in the Chern insulators, have not been investigated, potentially questioning their similarity with standard quantum Hall states. Here we probe the heat transport properties of the Hofstadter butterfly, obtained in a graphene~/~hexagonal boron nitride moiré superlattice. We observe a quantized heat flow, uniquely set by the topological invariant, for all investigated states of the Hofstadter butterfly: quantum Hall states, Chern insulators, and even symmetry-broken Chern insulators emerging from strong electronic interactions. Our work firmly establishes the universality of the quantization of heat transport and its intimate link with topology.

Figures (4)

• Figure 1: $\vert$ Hofstadter spectrum in a graphene/hBN moiré heterostructure. a, Sample schematic and experimental wiring. b, Optical micrograph of the sample. c, 2-point differential conductance $(dV_R/dI_R)^{-1}$ (in log scale) versus moiré unit cell filling $n/n_0$ and magnetic field $\lvert B\rvert$ (left Y-axis) / reduced magnetic flux $\phi/\phi_0$, measured at $T\approx20~$mK. Dashed lines indicate the probed QH and Chern insulators states, along with their corresponding $(t,s)$. The line color correspond to the value of $s$. Symbols indicate where we performed the heat transport measurements shown here: $\diamond$: (2,-4) at $B=+4~$T; $\circ$: (-4,-2) at $B=-7.2~$T; $\circ$: (-2,-2) at $B=-10.5~$T; $\circ$: (-3,-1.5) at $B=-10.5~$T; $\varhexagon$: (1,0) at $B=+4~$T; $\varhexagon$: (2,0) at $B=+4~$T; $\varhexagon$: (3,0) at $B=+4~$T; $\medtriangleup$: (2,0) at $B=+10.5~$T; $\medtriangleup$: (3,0) at $B=+10.5~$T. d, e, and f, zoom of the data shown in c in the vicinity of the satellite peak (d) and of the CNP clones at $\phi/\phi_0=1/3$ (e) and $\phi/\phi_0=1/2$ (f).
• Figure 2: $\vert$ Chiral electronic current splitting in Chern and QH insulators. Measured differential conductances versus $n/n_0$ at $B=-10.5~$T (a), $B=-7.2~$T (b), and $B=+4~$T (c), for $T=20~$mK. Full lines: 2-point conductances $(dV_R/dI_R)^{-1}$ (blue) and $(dV_T/dI_T)^{-1}$ (red). Dashed lines: reflected (dark blue) and transmitted (orange) transconductances $(dV_{R,T}/dI_0)^{-1}$. Shaded areas indicate the QH and Chern insulators states probed in our experiment, with colors corresponding to the symbols of Figs. 1, 3, and 4.
• Figure 3: $\vert$ Electron thermometry in Hofstadter states. Measured $\Delta T_{\mathrm{c}}$ versus dc heating current $I_0$, at $B=+10.5~$T (a), $-10.5~$T (b), $-7.2~$T (c), and $+4~$T (d). Symbols: experimental data (shape and color corresponding to the symbols of Fig. 1), taken at $T=20~$mK. Lines: HCB predictions (see text).
• Figure 4: $\vert$ Universal quantized heat flow. Heat flow $J_{\mathrm{Q}}/(0.5 \kappa_0)$ versus $T_{\mathrm{c}}^2-T_0^2$. Symbols correspond to the data shown in Fig. 3. Dashed lines: quantized heat flow with full suppression of a single heat channel $(2N-1)J_{\mathrm{Q}}^\mathrm{e}$, with $N=1\rightarrow4$ (from bottom to top) the number of ballistic heat channels on each side of the metallic island. Full lines: HCB predictions for the corresponding $N$ at $T_0=20~$mK.