In twisted van der Waals materials, local atomic relaxation can alter the underlying electronic structure. Characterizing lattice reconstruction and its susceptibility to strain is essential for understanding emergent electronic states, especially in multilayers in which interference between moiré lattices yields larger supermoiré patterns whose energy is highly sensitive to local stacking. Here we image spatial modulations in the electronic character of helical trilayer graphene, which indicate relaxation into a superstructure of large domains with uniform moiré periodicity. We show that the supermoiré domain size is increased by strain and can be altered in the same device while preserving the local properties within each domain. Finally, we observe a higher conductance at the domain boundaries, consistent with predictions that they host counterpropagating edge modes. Our work provides a real-space visualization of moiré-periodic domains, reveals two independently tunable length scales and demonstrates strain engineering as a route towards designing correlated topological networks at the supermoiré scale. A scanning single-electron transistor is used to probe the strain dependence of moiré and supermoiré domains. It is observed that these can be considered nearly independent of each other.

Twisted graphene multilayers exhibit strong electronic correlations leading to a range of experimental signatures. Yet how these signatures relate to each other and to the microscopic ground states-and how twist angle and band structure reshape them-remains poorly understood. Here we study this interplay by correlating local thermodynamic and transport measurements in twisted trilayer graphene with unequal angles and flat electronic bands. We use a scanning single-electron transistor to map the impact of electron-electron interactions in the sample by selecting a region with smooth local twist angle evolution. We observe gapped correlated insulator states and asymmetric oscillations in the inverse electronic compressibility, both exhibiting pronounced electron-hole asymmetry with distinct 'magic' angles for conduction and valence bands. Subsequent transport measurements in the same region reveal robust superconductivity with a similar electron-hole asymmetry. Our measurements indicate that superconductivity is not directly tied to the correlated insulator states. Instead, its critical temperature correlates closely with the strength of the compressibility oscillations, suggesting a common origin or link between the two. By combining a local probe with transport measurements, we uncover connections between superconductivity and thermodynamic correlation signatures that are not apparent from either technique in isolation. These findings highlight the power of our dual approach.

Excitons in atomically thin transition metal dichalcogenides (TMDs) possess intriguing optical properties, drawing interest for both technology and fundamental research. However, as the demands for nanodevice applications and the exploration of fundamental physics push toward smaller, subwavelength scales, studying them locally is challenging. In this work, we introduce a cryogenic scanning probe photoelectrical sensing technique, termed exciton-resonant microwave impedance microscopy (ER-MIM), to measure the excitonic responses in a monolayer MoSe2 device at 1.5K. From the microwave signal changes, we identify exciton polarons and their Rydberg states. Building on these observations, we systemically reveal the local and nonlocal effects of carrier density, inhomogeneous electric fields, as well as dielectric screening on excitons, beyond the reach of conventional probes. By further integrating deep learning techniques, we precisely extracted the electrical parameters surrounding excitons, demonstrating a quantified, exciton-assisted nanoscale electrometry. Our results provide new insight into exciton-environment interactions, establish ER-MIM as a powerful optoelectronic sensing platform, and open avenues for exciton-based quantum control and device technologies.

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.

Determining the symmetry breaking order of correlated quantum phases is essential for understanding the microscopic interactions in their host systems. The flat bands in magic angle twisted bilayer graphene (MATBG) provide an especially rich arena to investigate such interaction-driven ground states, and while progress has been made in identifying the correlated insulators and their excitations at commensurate moire filling factors, the spin-valley polarizations of the topological states that emerge at high magnetic field remain unknown. Here we introduce a new technique based on twist-decoupled van der Waals layers that enables measurements of their electronic band structure and, by studying the backscattering between counter-propagating edge states, determination of relative spin polarization of the their edge modes. Applying this method to twist-decoupled MATBG and monolayer graphene, we find that the broken-symmetry quantum Hall states that extend from the charge neutrality point in MATBG are spin-unpolarized at even integer filling factors. The measurements also indicate that the correlated Chern insulator emerging from half filling of the flat valence band is spin-unpolarized, but suggest that its conduction band counterpart may be spin-polarized. Our results constrain models of spin-valley ordering in MATBG and establish a versatile approach to study the electronic properties of van der Waals systems.

Semiconductor moiré superlattices have been shown to host a wide array of interaction-driven ground states. However, twisted homobilayers have been difficult to study in the limit of large moiré wavelengths, where interactions are most dominant. In this study, we conducted local electronic compressibility measurements of twisted bilayer WSe2 (tWSe2) at small twist angles. We demonstrated multiple topological bands that host a series of Chern insulators at zero magnetic field near a "magic angle" around 1.23°. Using a locally applied electric field, we induced a topological quantum-phase transition at one hole per moiré unit cell. Our work establishes the topological phase diagram of a generalized Kane-Mele-Hubbard model in tWSe2, demonstrating a tunable platform for strongly correlated topological phases.

Low-temperature scanning probe microscopes (SPMs) are critical for the study of quantum materials and quantum information science. Due to the rising costs of helium, cryogen-free cryostats have become increasingly desirable. However, they typically suffer from comparatively worse vibrations than cryogen-based systems, necessitating the understanding and mitigation of vibrations for SPM applications. Here we demonstrate the construction of two cryogen-free dilution refrigerator SPMs with minimal modifications to the factory default and we systematically characterize their vibrational performance. We measure the absolute vibrations at the microscope stage with geophones, and use both microwave impedance microscopy and a scanning single electron transistor to independently measure tip-sample vibrations. Additionally, we implement customized filtering and thermal anchoring schemes, and characterize the cooling power at the scanning stage and the tip electron temperature. This work serves as a reference to researchers interested in cryogen-free SPMs, as such characterization is not standardized in the literature or available from manufacturers.

In a lattice model subject to a perpendicular magnetic field, when the lattice constant is comparable to the magnetic length, one enters the "Hofstadter regime," where continuum Landau levels become fractal magnetic Bloch bands. Strong mixing between bands alters the nature of the resulting quantum phases compared to the continuum limit; lattice potential, magnetic field, and Coulomb interaction must be treated on equal footing. Using determinant quantum Monte Carlo (DQMC) and density matrix renormalization group (DMRG) techniques, we study this regime numerically in the context of the Hubbard-Hofstadter model on a triangular lattice. In the field-filling phase diagram, we find a broad wedge-shaped region of ferromagnetic ground states for filling factor $\nu \leq 1$, bounded below by filling factor $\nu = 1$ and bounded above by half-filling the lowest Hofstadter subband. We observe signatures of SU(2) quantum Hall ferromagnetism at filling factors $\nu=1$ and $\nu=3$. The phases near $\nu=1$ are particle-hole asymmetric, and we observe a rapid decrease in ground state spin polarization consistent with the formation of skyrmions only on the electron doped side. At large fields, above the ferromagnetic wedge, we observe a low-spin metallic region with spin correlations peaked at small momenta. We argue that the phenomenology of this region likely results from exchange interaction mixing fractal Hofstadter subbands. The phase diagram derived beyond the continuum limit points to a rich landscape to explore interaction effects in magnetic Bloch bands.

First-order phase transitions produce abrupt changes to the character of both ground and excited electronic states. Here we conduct electronic compressibility measurements to map the spin phase diagram and Landau level (LL) energies of monolayer WSe2 in a magnetic field. We resolve a sequence of firstorder phase transitions between completely spin-polarized LLs and states with LLs of both spins. Unexpectedly, the LL gaps are roughly constant over a wide range of magnetic fields below the transitions, which we show reflects spin-polarized ground states with opposite spin excitations. These transitions also extend into compressible regimes, with a sawtooth boundary between full and partial spin polarization. We link these observations to the important influence of LL filling on the exchange energy beyond a smooth density-dependent contribution. Our results show that WSe2 realizes a unique hierarchy of energy scales where such effects induce reentrant magnetic phase transitions tuned by density and magnetic field.

While the recent advances in topology have led to a classification scheme for electronic bands described by the standard theory of metals, a similar scheme has not emerged for strongly correlated systems such as Mott insulators in which a partially filled band carries no current. By including interactions in the topologically nontrivial Haldane model, we show that a quarter-filled state emerges with a nonzero Chern number provided the interactions are sufficiently large. We first motivate this result on physical grounds and then by two methods: Analytically by solving exactly a model in which interactions are local in momentum space and then numerically through the corresponding Hubbard model. All methods yield the same result: For sufficiently large interaction strengths, the quarter-filled Haldane model is a ferromagnetic topological Mott insulator with a Chern number of unity. Possible experimental realizations in cold-atom and solid-state systems are discussed.

Motivated by recent experimental work on moiré systems in a strong magnetic field, we compute the compressibility as well as the spin correlations and Hofstadter spectrum of spinful electrons on a honeycomb lattice with Hubbard interactions using the determinantal quantum Monte Carlo method. While the interactions in general preserve quantum and anomalous Hall states, emergent features arise corresponding to an antiferromagnetic insulator at half-filling and other incompressible states following the Chern sequence ± (2 N + 1). These odd integer Chern states exhibit strong ferromagnetic correlations and arise spontaneously without any external mechanism for breaking the spin-rotation symmetry. Analogs of these magnetic states should be observable in general interacting quantum Hall systems. In addition, the interacting Hofstadter spectrum is qualitatively similar to the experimental data at intermediate values of the on-site interaction.

Moiré superlattices formed from transition metal dichalcogenides (TMDs) have been shown to support a variety of quantum electronic phases that are highly tunable using applied electromagnetic fields. While the valley character of the low-energy states dramatically affects optoelectronic properties in the constituent TMDs, this degree of freedom has yet to be fully explored in moiré systems. Here, we establish twisted double bilayer WSe_2 as an experimental platform to study electronic correlations within Γ-valley moiré bands. Through a combination of local and global electronic compressibility measurements, we identify charge-ordered phases at multiple integer and fractional moiré band fillings ν. By measuring the magnetic field dependence of their energy gaps and the chemical potential upon doping, we reveal spin-polarized ground states with novel spin polaron quasiparticle excitations. In addition, an applied displacement field allows us to realize a new mechanism of metal-insulator transition at ν = -1 driven by tuning between Γ- and K-valley moiré bands. Together, our results demonstrate control over both the spin and valley character of the correlated ground and excited states in this system.

The flat electronic bands in magic-angle twisted bilayer graphene (MATBG) host a variety of correlated insulating ground states, many of which are predicted to support charged excitations with topologically non-trivial spin and/or valley skyrmion textures. However, it has remained challenging to experimentally address their ground state order and excitations, both because some of the proposed states do not couple directly to experimental probes, and because they are highly sensitive to spatial inhomogeneities in real samples. Here, using a scanning single-electron transistor, we observe thermodynamic gaps at even integer moiré filling factors at low magnetic fields. We find evidence of a field-tuned crossover from charged spin skyrmions to bare particle-like excitations, suggesting that the underlying ground state belongs to the manifold of strong-coupling insulators. From the spatial dependence of these states and the chemical potential variation within the flat bands, we infer a link between the stability of the correlated ground states and local twist angle and strain. Our work advances the microscopic understanding of the correlated insulators in MATBG and their unconventional excitations.

Semiconductor moir\'e superlattices have been shown to host a wide array of interaction-driven ground states. However, twisted homobilayers have been difficult to study in the limit of large moir\'e wavelength, where interactions are most dominant. Here, we conduct local electronic compressibility measurements of twisted bilayer WSe$_2$ (tWSe$_2$) at small twist angles. We demonstrate multiple topological bands which host a series of Chern insulators at zero magnetic field near a 'magic angle' around $1.23^\circ$. Using a locally applied electric field, we induce a topological quantum phase transition at one hole per moir\'e unit cell. Our work establishes the topological phase diagram of a generalized Kane-Mele-Hubbard model in tWSe$_2$, demonstrating a tunable platform for strongly correlated topological phases.

Excitons -- quasiparticles formed by the binding of an electron and a hole through electrostatic attraction -- hold promise in the fields of quantum light confinement and optoelectronic sensing. Atomically thin transition metal dichalcogenides (TMDs) provide a versatile platform for hosting and manipulating excitons, given their robust Coulomb interactions and exceptional sensitivity to dielectric environments. In this study, we introduce a cryogenic scanning probe photoelectrical sensing platform, termed exciton-resonant microwave impedance microscopy (ER-MIM). ER-MIM enables ultra-sensitive probing of exciton polarons and their Rydberg states at the nanoscale. Utilizing this technique, we explore the interplay between excitons and material properties, including carrier density, in-plane electric field, and dielectric screening. Furthermore, we employ deep learning for automated data analysis and quantitative extraction of electrical information, unveiling the potential of exciton-assisted nano-electrometry. Our findings establish an invaluable sensing platform and readout mechanism, advancing our understanding of exciton excitations and their applications in the quantum realm.

Local thermodynamic measurements of a twisted transition metal dichalcogenide heterostructure reveal competition between unconventional charge order and Hofstadter states. This results from the presence of both flat and dispersive electronic bands, whose energetic ordering can be experimentally tuned.

In non-interacting systems, bands from non-trivial topology emerge strictly at half-filling and exhibit either the quantum anomalous Hall or spin Hall effects. Here we show using determinantal quantum Monte Carlo and an exactly solvable strongly interacting model that these topological states now shift to quarter filling. A topological Mott insulator is the underlying cause. The peak in the spin susceptibility is consistent with a possible ferromagnetic state at $T=0$. The onset of such magnetism would convert the quantum spin Hall to a quantum anomalous Hall effect. While such a symmetry-broken phase typically is accompanied by a gap, we find that the interaction strength must exceed a critical value for this to occur. Hence, we predict that topology can obtain in a gapless phase but only in the presence of interactions in dispersive bands. These results explain the recent quarter-filled quantum anomalous Hall effects seen in moire systems.

The emergence of moiré materials with flat bands provides a platform to systematically investigate and precisely control the correlated electronic phases. Here we report on a rich phase diagram of interpenetrating Hofstadter states—also called Chern insulators—and electron solids in a twisted WSe 2 /MoSe 2 heterobilayer using local electronic compressibility measurements. We show that this reflects the presence of both flat and dispersive moiré bands whose relative energies, and therefore occupations, are tuned by the density and magnetic field. At low density, the competition between moiré bands leads to a transition from the commensurate arrangements of singlets at doubly occupied sites to triplet configurations at high fields. Hofstadter states are generally favoured at high density as dispersive bands are populated, but are suppressed by an intervening region of re-entrant charge-ordered states in which holes originating from multiple bands cooperatively crystallize. Our results reveal the key microscopic ingredients that favour distinct correlated ground states in semiconductor moiré systems, and they demonstrate an emergent lattice model system in which both interactions and band dispersion can be experimentally controlled.

In magic angle twisted bilayer graphene (MATBG), the moir\'e superlattice potential gives rise to narrow electronic bands1 which support a multitude of many-body quantum phases. Further richness arises in the presence of a perpendicular magnetic field, where the interplay between moir\'e and magnetic length scales leads to fractal Hofstadter subbands. In this strongly correlated Hofstadter platform, multiple experiments have identified gapped topological and correlated states, but little is known about the phase transitions between them in the intervening compressible regimes. Here, using a scanning single-electron transistor microscope to measure local electronic compressibility, we simultaneously unveil novel sequences of broken-symmetry Chern insulators (CIs) and resolve sharp phase transitions between competing states with different topological quantum numbers and spin/valley flavor occupations. Our measurements provide a complete experimental mapping of the energy spectrum and thermodynamic phase diagram of interacting Hofstadter subbands in MATBG. In addition, we observe full lifting of the degeneracy of the zeroth Landau levels (zLLs) together with level crossings, indicating moir\'e valley splitting. We propose a unified flavor polarization mechanism to understand the intricate interplay of topology, interactions, and symmetry breaking as a function of density and applied magnetic field in this system.

Observations of an electronic state where rotational symmetry is broken show that this could be a generic feature of moiré materials.