Strongly correlated quantum matter exhibits a rich variety of remarkable properties, but the organizing principles that underlie the behavior remain to be established. Graphene heterostructures, which can host narrow moire electron bands that amplify the correlation effect, represent a new setting to make progress on this overarching issue. In such correlated moire systems, an insulating state is a prominent feature of the phase diagram and may hold the key to understanding the basic physics. Here we advance the notion of a fragile insulator, a correlation-driven insulating state that is on the verge of a delocalization transition into a bad metal. Using a realistic multiorbital Hubbard model as a prototype for narrow band moire systems, we realize such a fragile insulator and demonstrate a nematic order in this state as well as in the nearby bad metal regime. Our results are consistent with the observed electronic anisotropy in the graphene moire systems and provide a natural understanding of what happens when the insulator is tuned into a bad metal. We propose the fragile insulator and the accompanying bad metal as competing states at integer fillings that analogously anchor the overall phase diagram of the correlated moire systems and beyond.
Strongly correlated quantum matter exhibits a rich variety of remarkable properties, but the organizing principles that underlie the behavior remain to be established. Graphene heterostructures, which can host narrow moire electron bands that amplify the correlation effect, represent a new setting to make progress on this overarching issue. In such correlated moire systems, an insulating state is a prominent feature of the phase diagram and may hold the key to understanding the basic physics. Here we advance the notion of a fragile insulator, a correlation-driven insulating state that is on the verge of a delocalization transition into a bad metal. Using a realistic multiorbital Hubbard model as a prototype for narrow band moire systems, we realize such a fragile insulator and demonstrate a nematic order in this state as well as in the nearby bad metal regime. Our results are consistent with the observed electronic anisotropy in the graphene moire systems and provide a natural understanding of what happens when the insulator is tuned into a bad metal. We propose the fragile insulator and the accompanying bad metal as competing states at integer fillings that analogously anchor the overall phase diagram of the correlated moire systems and beyond.
Moiré materials have emerged as a rich platform for exploring strong correlation effects in low dimensions, with twisted bilayer graphene (TBG) as a paradigmatic example. To distill the essential ingredients driving moiré-induced phases, a simplified one-dimensional analog -- a two-leg ladder with spatially modulated interleg hopping and a uniform magnetic flux -- was recently introduced. This model, which we refer to as the moiré ladder, features a nearly flat lowest-energy band in a suitable parameter regime, capturing the band-flattening mechanism of TBG. We investigate the ground-state phase diagram of the moiré ladder using a combination of bosonization and density matrix renormalization group (DMRG) techniques, and systematically disentangle the respective roles of the flux and the hopping modulation. At half filling, previous numerical work identified a metal-insulator transition at finite interaction strength and an unexpected ferromagnetic ground state. Revisiting this, we show that the metal-insulator transition can be understood perturbatively within bosonization, governed by the number of Fermi points. In contrast, the ferromagnetic correlations are nonperturbative and require both flux and spatial modulation -- neither alone is sufficient. We extend our analysis to other fillings: one-quarter, three-quarters, slightly above half filling (half filling plus two electrons), and slightly below half filling (half filling minus two electrons). At moderate interactions, we observe ferromagnetism below half filling and antiferromagnetism above; at stronger interactions, ferromagnetism dominates across all studied fillings. Crucially, the analysis demonstrates that periodic interleg hopping alone does not engender new correlated phases; the magnetic flux is essential for the observed unconventional behavior.
The dynamics of out-of-time order correlators is studied in strongly correlated electron systems. This certain type of correlation function is a measure for the process of quantum information scrambling which is linked to the fundamental processes of entanglement growth and thermalization in quantum systems. Particularly interesting are quantum chaotic systems where the scrambling rate specifies the exponential growth of out-of-time order correlators. Whereas the scrambling rate is well defined theoretically, its interpretation in a physical context is still under debate. Furthermore, its role for physical effects and quantities which can be measured in experiments is little understood and deserves further investigation. In the first part of this work, scrambling rates are determined analytically and are analyzed for interacting electrons in graphene and electrons in a disordered metal coupled to a dissipative bath of phonons. A resemblance between the scrambling rate and the single-particle scattering rates is observed. In addition, a proposed link between quantum chaotic dynamics of out-of-time order correlators and the dynamics of fluctuations of observables is investigated. This principle could be used to determine scrambling rates in experiments. Subsequently, possible electron orders are investigated in the system of small-angle twisted-bilayer graphene in the second part of this work. Here, the interplay between the moiré interference pattern and the interlayer hybridization due to a finite interlayer tunneling of electrons causes the bandwidth of the so-called moiré bands to depend on the twist-angle. For twist-angles in the vicinity of the magic-angle, interaction effects are strongly enhanced and a series of strong correlation effects is expected. A minimal low-energy model of interacting electrons is derived where relevant electron-electron interaction processes are identified. This model is analyzed with respect to possible electronic ground states as a function of the twist-angle. A tendency towards a formation of nematic states is observed in a weak-coupling regime, whereas various Mott-insulating states are found for strong couplings.
Twisted van der Waals heterostructures have latterly received prominent attention for their many remarkable experimental properties and the promise that they hold for realizing elusive states of matter in the laboratory. We propose that these systems can, in fact, be used as a robust quantum simulation platform that enables the study of strongly correlated physics and topology in quantum materials. Among the features that make these materials a versatile toolbox are the tunability of their properties through readily accessible external parameters such as gating, straining, packing and twist angle; the feasibility to realize and control a large number of fundamental many-body quantum models relevant in the field of condensed-matter physics; and finally, the availability of experimental readout protocols that directly map their rich phase diagrams in and out of equilibrium. This general framework makes it possible to robustly realize and functionalize new phases of matter in a modular fashion, thus broadening the landscape of accessible physics and holding promise for future technological applications.
In this work, we investigate a continuum model of two-dimensional electrons interacting via long-range Coulomb forces and subject to a triangular periodic potential at filling factor ν=2. Using neural quantum states based on message-passing neural networks optimized with variational Monte Carlo, we determine the ground-state phase diagram as a function of periodic potential strength and Zeeman field. Although the noninteracting system is expected to form a conventional band insulator, we show that strong correlations can qualitatively reshape the insulating state and, in part of the phase diagram, suppress the band insulator in favor of a correlated metallic phase, with a first-order transition separating the two regimes. Moreover, the metal--insulator transition exhibits a strong dependence on spin polarization: the unpolarized system becomes insulating at substantially weaker periodic potentials through the formation of an s-wave molecular insulator, where opposite-spin electrons bind near each potential minimum, whereas the fully polarized system requires a significantly deeper potential and instead forms a symmetry-broken orbital nematic molecular crystal. This spin-dependent molecular transition gives rise to discontinuous magnetization curves, metamagnetic transitions, and several multicritical points in the phase diagram, together with clear signatures in the density profiles, real-space correlation functions, and structure factors. To further elucidate the role of the long-range character of the Coulomb interaction, we also study short-range repulsive interactions, which can be realized with ultracold-atom platforms. We find that the qualitative structure of the many-body phase diagram remains unchanged, demonstrating the robustness of our results across different interaction models.
Inspired by the rich physics of twisted 2D bilayer moiré systems, we study Coulomb interacting systems subjected to two overlapping finite 1D lattice potentials of unequal periods through exact numerical diagonalization. Unmatching underlying lattice periods lead to a 1D bichromatic "moiré" superlattice with a large unit cell and consequently a strongly flattened band, exponentially enhancing the effective dimensionless electron-electron interaction strength and manifesting clear signatures of enhanced Mott gaps at discrete fillings. An important nonperturbative finding is a remarkable fine-tuning effect of the precise lattice commensuration, where slight variations in the relative lattice periods may lead to a suppression of the correlated insulating phase, in qualitative agreement with the observed fragility of the correlated insulating phase in twisted bilayer graphene. Our predictions, which should be directly verifiable in bichromatic optical lattices, establish that the competition between interaction and incommensuration is a key element of the physics of moiré superlattices.
Motivated by recent experimental studies that have found signatures of a correlated insulator phase and tuning superconductivity in twisted bilayer graphene, we study the temperature-dependent conductivity, the spin correlation and the superconducting pairing correlation within a two-orbital Hubbard model on an emergent honeycomb lattice. Evaluation of the temperature dependence of the conductivity demonstrates that there is a metal-insulator transition, and the Mott phase at strong coupling is accompanied by antiferromagnetic order. The electronic correlation drives a d+id superconducting pairing to be dominant over a wide filling region. All of the dc conductivity, the spin correlation and the superconductivity are suppressed as the interlayer coupling strength increases, and the critical Uc for metal-insulator transition is also reduced. Our intensive numerical results reveal that the TBG should be a uniquely tunable platform for exploring correlated states.