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Magic-angle twisted trilayer graphene (MATTG) hosts flat electronic bands, and exhibits correlated quantum phases with electrical tunability. In this work, we demonstrate a spectroscopy technique that allows for dissociation of intertwined bands and quantification of the energy gaps and Chern numbers C of the correlated states in MATTG by driving band crossings between Dirac cone Landau levels and energy gaps in the flat bands. We uncover hard correlated gaps with C = 0 at integer moiré unit cell fillings of ν = 2 and 3 and reveal charge density wave states originating from van Hove singularities at fractional fillings ν = 5/3 and 11/3. In addition, we demonstrate displacement-field-driven first-order phase transitions at charge neutrality and ν = 2, which are consistent with a theoretical strong-coupling analysis, implying C2T symmetry breaking. Overall, these properties establish a diverse electrically tunable phase diagram of MATTG and provide an avenue for investigating other related systems hosting both steep and flat bands.The authors demonstrate a spectroscopic method, based on magnetotransport measurements, to quantitatively measure the size of the correlated gaps in twisted trilayer graphene and infer their topology.

The robustness of the macroscopic quantum nature of a superconductor can be characterized by the superfluid stiffness, ρ s , a quantity that describes the energy required to vary the phase of the macroscopic quantum wavefunction. In unconventional superconductors, such as cuprates, the low-temperature behaviour of ρ s markedly differs from that of conventional superconductors owing to quasiparticle excitations from gapless points (nodes) in momentum space. Intensive research on the recently discovered magic-angle twisted graphene family has revealed, in addition to superconducting states, strongly correlated electronic states associated with spontaneously broken symmetries, inviting the study of ρ s to uncover the potentially unconventional nature of its superconductivity. Here we report the measurement of ρ s in magic-angle twisted trilayer graphene (TTG), revealing unconventional nodal-gap superconductivity. Utilizing radio-frequency reflectometry techniques to measure the kinetic inductive response of superconducting TTG coupled to a microwave resonator, we find a linear temperature dependence of ρ s at low temperatures and nonlinear Meissner effects in the current-bias dependence, both indicating nodal structures in the superconducting order parameter. Furthermore, the doping dependence shows a linear correlation between the zero-temperature ρ s and the superconducting transition temperature T c , reminiscent of Uemura’s relation in cuprates, suggesting phase-coherence-limited superconductivity. Our results provide strong evidence for nodal superconductivity in TTG and put strong constraints on the mechanisms of these graphene-based superconductors. Measurements of the superfluid stiffness in twisted trilayer graphene reveal unconventional nodal-gap superconductivity, where the superconducting transition is controlled by phase fluctuations rather than Cooper-pair breaking.

Flat-band moiré graphene systems are a quintessential platform for investigating correlated phases of matter. Various interaction-driven ground states have been proposed, but despite extensive experimental effort, there has been little direct evidence that distinguishes between various phases, in particular near the charge neutrality point. Here we probe the fine details of the density of states and the effects of Coulomb interactions in alternating-twist trilayer graphene by imaging the local thermodynamic quantum oscillations with a nanoscale scanning superconducting quantum interference device. We find that the charging self-energy due to occupied electronic states is most important in explaining the high-carrier-density physics. At half-filling of the conduction flat band, we observe ferromagnetic-driven symmetry breaking, suggesting that it is the most robust mechanism in the hierarchy of phase transitions. Near charge neutrality, where exchange energy dominates over charging self-energy, we find a nematic semimetal ground state, which is theoretically favoured over gapped states in the presence of heterostrain. In this semimetallic phase, the flat-band Dirac cones migrate towards the mini-Brillouin zone centre, spontaneously breaking the threefold rotational symmetry. Our low-field local quantum oscillation technique can be used to explore the ground states of many strongly interacting van der Waals systems. Our understanding of the phase diagram of twisted graphene structures is incomplete. Now, twisted trilayer graphene is examined using a technique that locally images quantum oscillations and shows that a nematic semimetal is favoured at low density.

Fractional Chern insulators (FCIs) are lattice analogues of fractional quantum Hall states that may provide a new avenue towards manipulating non-Abelian excitations. Early theoretical studies 1 – 7 have predicted their existence in systems with flat Chern bands and highlighted the critical role of a particular quantum geometry. However, FCI states have been observed only in Bernal-stacked bilayer graphene (BLG) aligned with hexagonal boron nitride (hBN) 8 , in which a very large magnetic field is responsible for the existence of the Chern bands, precluding the realization of FCIs at zero field. By contrast, magic-angle twisted BLG 9 – 12 supports flat Chern bands at zero magnetic field 13 – 17 , and therefore offers a promising route towards stabilizing zero-field FCIs. Here we report the observation of eight FCI states at low magnetic field in magic-angle twisted BLG enabled by high-resolution local compressibility measurements. The first of these states emerge at 5 T, and their appearance is accompanied by the simultaneous disappearance of nearby topologically trivial charge density wave states. We demonstrate that, unlike the case of the BLG/hBN platform, the principal role of the weak magnetic field is merely to redistribute the Berry curvature of the native Chern bands and thereby realize a quantum geometry favourable for the emergence of FCIs. Our findings strongly suggest that FCIs may be realized at zero magnetic field and pave the way for the exploration and manipulation of anyonic excitations in flat moiré Chern bands. A study using local compressibility measurements reports fractional Chern insulator states at low magnetic field in magic-angle twisted bilayer graphene, and establishes the applied magnetic field as a means to tune the Berry curvature distribution.

A central goal of quantum condensed matter physics is to understand, realize, and control phases of matter that exhibit macroscopic quantum phenomena. Moiré materials offer an unprecedented ability to do so through hosting strongly interacting electrons in topological bands. This setting was previously restricted to the FQHE, where electrons under massive magnetic fields split into new particles that carry a fraction of the electron's charge. This newly central experimental setting demands new theoretical tools that are applicable to strongly interacting topological bands. Existing theories are, naturally, specific to the only prior existing example, the lowest Landau level associated with the traditional fractional quantum Hall effect, and by their nature rule out several phases of matter including superconductivity. In this thesis, we develop strong coupling theories in the topological setting and use them make predictions on the interacting physics of twisted graphene systems. In Chapter 1, we will show how to analytically predict fractionalization in topological bands without relying on mimicking the lowest Landau level. Chapter 2 will compare and contrast a class of twisted graphene systems using a variety of theoretical tools. In Chapter 3, we report on a theoretical framework that accesses Mott physics in the topological bands of TBG. Mott physics, a key ingredient of high temperature superconductors, is typically studied in bands without topology. We report on qualitatively new phenomena that emerge from the combination of Mott physics and band topology.

Twisted bilayer graphene (TBG) has elements in common with two paradigmatic examples of strongly correlated physics: quantum Hall physics and Hubbard physics. On one hand, TBG hosts flat topological Landau-level-like bands which exhibits quantum anomalous Hall effects. On the other hand, these bands have concentrated charge density and show signs of extensive entropy resembling local moments. The combination of these features leads to a question: can decoupled moments emerge in an isolated topological band, despite the lack of exponentially localized Wannier states? In this work, we answer the question affirmatively by proposing a minimal model for these bands in TBG that combines topology and charge concentration at the AA sites, leading to analytic wavefunctions that closely approximate those of the BM model with realistic parameters. Importantly, charge concentration also leads to Berry curvature concentration at \\(\\), generating a small parameter \\(s\\) that yields analytic tractability. We show that, rather surprisingly, the model hosts nearly decoupled flavor moments without any extra degrees of freedom. These moments are non-local due to topology-enforced power-law tails, yet have parametrically small overlap. We further develop a diagrammatic expansion in which the self energy can be computed exactly to leading order in \\(s^2\\) in the fluctuating moment regime. At charge neutrality, we find a \"Mott semimetal\", with large flavor entropy and a Mott gap everywhere in the BZ except for the vicinity of the \\(\\) point. Away from neutrality, the Mott semimetal gaps out in a spectrally imbalanced manner, with one Mott band having zero \\(Z_k\\) at the \\(\\) point. The model accurately reproduces results from finite temperature thermodynamic measurements, leads to new experimental predictions, and resolves the problem of the emergence of Hubbard physics in isolated topological bands.

Fractional Chern insulators realize the remarkable physics of the fractional quantum Hall effect (FQHE) in crystalline systems with Chern bands. The lowest Landau level (LLL) is known to host the FQHE, but not all Chern bands are suitable for realizing fractional Chern insulators (FCI). Previous approaches to stabilizing FCIs focused on mimicking the LLL through momentum space criteria. Here instead we take a real-space perspective by introducing the notion of vortexability. Vortexable Chern bands admit a fixed operator that introduces vortices into any band wavefunction while keeping the state entirely within the same band. Vortexable bands admit trial wavefunctions for FCI states, akin to Laughlin states. In the absence of dispersion and for sufficiently short ranged interactions, these FCI states are the ground state -- independent of the distribution of Berry curvature. Vortexable bands are much more general than the LLL, and we showcase a recipe for constructing them. We exhibit diverse examples in graphene-based systems with or without magnetic field, and with any Chern number. A special class of vortexable bands is shown to be equivalent to the momentum space \"trace condition\" or \"ideal band condition\". In addition, we also identify a more general form of vortexability that goes beyond this criterion. We introduce a modified measure that quantifies deviations from general vortexability which can be applied to generic Chern bands to identify promising FCI platforms.

We introduce a systematic expansion tailored to systems with strong local interactions and capable of computing response functions, including finite DC transport, analytically. The expansion is controlled by a small parameter \\(s^2\\) that measures the area of the momentum space region where kinematics of the theory is concentrated. In real space, this corresponds to single-particle or correlated hopping terms with amplitudes that decay over a length scale \\(1/s\\) and scale in magnitude as \\(s^2\\) in two dimensions. In the limit \\(s^2 1\\), long, self-avoiding tunneling paths dominate over paths revisiting the same site. This enables systematic controlled calculations of various physical quantities. We illustrate the method with three applications. (i) A Hubbard model with concentrated dispersion: we analytically obtain spectral broadening which scales as \\(s^2\\) and identify a high-temperature bad metal with \\(T\\)-linear resistivity coexisting with parametrically long-lived quasiparticles, as well as an intermediate-temperature \"thermal FL*\" with a small hole pocket that coexists with thermally disordered fluctuating local moments, all within a single controlled framework. (ii) A correlated-hopping model with interesting electron-trion dynamics. (iii) A model of Chern bands with concentrated Berry curvature, motivated by twisted bilayer graphene, which realizes a Mott semimetal where we compute the broadening for the electron and trion spectral functions. At the end, we discuss how our approach paves the way to addressing various challenging questions in strongly correlated systems and outline its various generalizations.

We consider a family of twisted graphene multilayers consisting of \\(n\\)-untwisted chirally stacked layers, e.g., AB, ABC, etc, with a single twist on top of \\(m\\)-untwisted chirally stacked layers. Upon neglecting both trigonal warping terms for the untwisted layers and the same sublattice hopping between all layers, the resulting models generalize several remarkable features of the chiral model of twisted bilayer graphene (CTBG). In particular, they exhibit a set of magic angles which are identical to those of CTBG at which a pair of bands (i) are perfectly flat, (ii) have Chern numbers in the sublattice basis given by \\( (n,-m)\\) or \\( (n + m - 1, -1)\\) depending on the stacking chirality, and (iii) satisfy the trace condition, saturating an inequality between the quantum metric and the Berry curvature, and thus realizing ideal quantum geometry. These are the first higher Chern bands that satisfy (iii) beyond fine-tuned models or combinations of Landau levels. We show that ideal quantum geometry is directly related to the construction of fractional quantum Hall model wavefunctions. We provide explicit analytic expressions for the flat band wavefunctions at the magic angle in terms of the CTBG wavefunctions. We also show that the Berry curvature distribution in these models can be continuously tuned while maintaining perfect quantum geometry. Similar to the study of fractional Chern insulators in ideal \\(C = 1\\) bands, these models pave the way for investigating exotic topological phases in higher Chern bands for which no Landau level analog is available.

The rise of moiré materials has led to experimental realizations of integer and fractional Chern insulators in small or vanishing magnetic fields. At the same time, a set of minimal conditions sufficient to guarantee a Abelian fractional state in a flat band were identified, namely \"ideal\" or \"vortexable\" quantum geometry. Such vortexable bands share essential features with the lowest Landau level, while excluding the need for more fine-tuned aspects such as flat Berry curvature. A natural and important generalization is to ask if such conditions can be extended to capture the quantum geometry of higher Landau levels, particularly the first (1LL), where non-Abelian states at \\( = 1/2,2/5\\) are known to be competitive. The possibility of realizing these states at zero magnetic field , and perhaps even more exotic ones, could become a reality if we could identify the essential structure of the 1LL in Chern bands. In this work, we introduce a precise definition of 1LL quantum geometry, along with a figure of merit that measures how closely a given band approaches the 1LL. We apply the definition to identify two models with 1LL structure -- a toy model of double bilayer twisted graphene and a more realistic model of strained Bernal graphene.