Magnetic Hofstadter cascade in a twisted semiconductor homobilayer

TL;DR

This work probes spin-resolved Hofstadter physics in twisted WSe$_2$ homobilayers using a scanning SET to map the chemical potential and extract $M_z$ via a Maxwell relation. It reveals a cascade of magnetic transitions corresponding to filling spin-minority Hofstadter subbands, enabling spin-resolved spectroscopy of Hofstadter's butterfly in a moiré platform. The onset of spin polarization is largely independent of twist angle, implying that exchange interactions from the constituent WSe$_2$ dominate, while moiré potentials primarily shape insulating gaps and correlate with nearby phase transitions. Collectively, the study demonstrates a quantitative, spin-resolved view of Hofstadter physics in a strongly interacting moiré system and elucidates how twist, density, and electric fields tune the resulting ground states for potential topological and correlated phenomena.

Abstract

Transition metal dichalcogenide moiré homobilayers have emerged as a platform in which magnetism, strong correlations, and topology are intertwined. In a large magnetic field, the energetic alignment of states with different spin in these systems is dictated by both strong Zeeman splitting and the structure of the Hofstadter's butterfly spectrum, yet the latter has been difficult to probe experimentally. Here we conduct local thermodynamic measurements of twisted WSe$_2$ homobilayers that reveal a cascade of magnetic phase transitions. We understand these transitions as the filling of individual Hofstadter subbands, allowing us to extract the structure and connectivity of the Hofstadter spectrum of a single spin. The onset of magnetic transitions is independent of twist angle, indicating that the exchange interactions of the component layers are only weakly modified by the moiré potential. In contrast, the magnetic transitions are associated with stark changes in the insulating states at commensurate filling. Our work achieves a spin-resolved measurement of Hofstadter's butterfly despite overlapping states, and it disentangles the role of material and moiré effects on the nature of the correlated ground states.

Figures (8)

• Figure 1: Cascade of electronic phase transitions in twisted WSe$_2$ (tWSe$_2$).a, Schematic of the scanning single-electron transistor (sSET) and device. b, Momentum space structure of tWSe$_2$: spin-valley locked moiré bands are localized in mini Brillouin zones (MBZs) near the $K$ and $K'$ points. c, Schematic of the electronic density of states (DOS) of tWSe$_2$ valence bands for each spin at both zero magnetic flux quanta per unit cell $\Phi / \Phi_0$ and at a large, rational value $\Phi/\Phi_0 = p/q$, where $\Phi_0$ is the flux quantum, and $p,q$ are integers. At nonzero flux, the Zeeman energy $E_Z$ shifts the valence band edges and each moiré band splits into $q$ Hofstadter subbands. d, Inverse electronic compressibility d$\mu$/d$n$ at twist angle $\theta = 1.44^\circ$ as a function of moiré filling factor $\nu$ and magnetic field $B$ (left axis) or equivalently $\Phi/\Phi_0$ (right axis). e, Wannier style plot schematically showing incompressible gaps, the phase transitions from d, and the region of full spin polarization.
• Figure 1: Measurement of magnetization across phase transitions, a-b, Linecuts of d$\mu$/d$n$ (a) and d$M_z$/d$n$ (b) at selected magnetic fields from the datasets presented in Fig. \ref{['fig:Fig1']}d and Fig. \ref{['fig:Fig2']}a. Purple arrows indicate positions of phase transitions. c, Magnetization $M_z$ relative to the magnetization at $\nu = -1$ as a function of $\nu$ at fixed $B$. This quantity is a numerical integration of b. The total change in magnetization in the two limits is similar because total number of carriers reversing spin is the same whether the underlying moiré bands have split into Hofstadter subbands or not.
• Figure 2: Magnetization of phase transitions and Hofstadter subband crossings.a, Derivative of the magnetization in the $z$ direction, d$M_z$/d$n$, as a function of $\nu$ and $B$. b, Schematic of Hofstadter subband crossings as carrier density $n$ is changed at fixed field. As $|n|$ is increased, individual spin-minority ($K',\downarrow$) Hofstadter subbands cross the Fermi level, denoted by $\mu$. c, Calculated d$M_z$/d$n$ from a Stoner model using the Hofstadter spectra shown in d-e (Methods). d-e, Calculated spin-resolved Hofstadter spectra based on the continuum model tWSe$_2$ bands (Methods).
• Figure 2: Magnetic phase transitions at additional twist angles.a-c, d$\mu$/d$n$ as a function of $n$ and $B$ at $\theta = 1.60^\circ$ (a), $\theta = 1.42^\circ$ (b) and $\theta = 1.12^\circ$ (c). d, Schematic showing the most prominent transitions from a-c alongside the monolayer WSe$_2$ transition line highlighted in Fig. \ref{['fig:Fig4']}.
• Figure 3: Determining the Hofstadter spectrum structure.a, d$\mu$/d$n$ as a function of $\nu$ for a set of rational magnetic flux quanta per unit cell $\Phi/\Phi_0$, showing $q$ dips when $\Phi/\Phi_0 = 1/q$. The data are taken from Fig. \ref{['fig:Fig1']}d. b, d$\mu$/d$n$ as a function of $B$ and $\nu - \nu^*$, the difference in filling factor relative to the onset of magnetic transitions. The span of the colorbar is from $-1 \times 10^{-11}$ meV cm$^{2}$ to 0. Dashed boxes highlight how subbands merge going from $\Phi/\Phi_0 = \frac{1}{6} \rightarrow \frac{1}{5}$ (brown), $\Phi/\Phi_0 = \frac{1}{5} \rightarrow \frac{1}{4}$ (blue) and $\Phi/\Phi_0 = \frac{1}{4} \rightarrow \frac{1}{3}$ (green). c, Calculated spin-minority Hofstadter spectrum, plotted in energy relative to the valence band edge $E - E_{\rm{bandedge}}$ and magnetic field. Dashed boxes highlight regions that can be identified in b.