Session K41: Geometry of Moire' Chern Bands and Fractional Hall States at Zero Magnetic Field

Vortexable Chern Bands and Fractional Chern Insulators in Moire Graphene and Transition Metal Dichalcogenides

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 Chern bands emerge naturally in chiral twisted graphene, and fractional Chern insulators were subsequently observed experimentally. Recently, zero-field fractional Chern insulators, and potentially a zero-field composite Fermi liquid, were also observed in the nearly-vortexable twisted MoTe_2. New and exciting nearly-vortexable platforms are also appearing, including periodically strained graphene and helically twisted graphene.   

Magic-angle helical trilayer graphene.

The moiré patterns generated by twisting van der Waals materials has given rise to a new regime of physics in which electronic interactions and quantum geometry are at the

forefront. The unprecedented degree of tunability in these devices has led to the experimental realization of a remarkably diverse set of physical phenomena. While the

majority of studies have focused on two-layer materials, going to three or more layers vastly increases the space of possibilities, which is just beginning to be explored.

I will discuss magic-angle helical trilayer graphene (HTG), a deceptively simple structure consisting of three graphene layers with identical twist angles relative to one another.  I will show how lattice relaxation, topological flat bands, and strong interactions come together to make HTG a uniquely rich platform for realizing strongly correlated topological states and for exploring their phase transitions.   

Origin of flat bands and topology in twisted transition metal dichalcogenides homobilayers.

We explore the chiral limit of massive Dirac field theories applicable to moire materials and uncover the similar origin of flat bands in both twisted bilayer graphene and twisted transition metal dichalcogenides (TMD) homobilayers. We find that the non-trivial topology of the flat bands is due to a layer-orbit coupling, the analog of spin-orbit coupling for relativistic Dirac fermions, which is non-perturbative on the moire scale, and allows for the flat bands to be exactly flat. We compare the flat band in moire TMDs to the Landau levels of a TMD monolayer under a magnetic field, and discuss the remarkable robustness of the flat bands to certain types of disorder.   

Recent Developments in Fractional Chern Insulators.

Fractional Chern insulators (FCIs), initially conceptualized as lattice counterparts of fractional quantum Hall (FQH) states in the absence of a magnetic field, stand as a captivating frontier in condensed matter physics, illustrating a complex interplay between robust electron-electron interactions, band topology, and quantum geometry. This presentation provides an overview of recent key developments in the realm of FCIs, synthesizing cutting-edge theoretical advancements and remarkable experimental findings.

After over a decade of theoretical exploration, engineered moiré superlattice materials have recently emerged as an experimental platform for FCI physics. Milestone experimental results include weak-field FQH states in magic angle twisted bilayer graphene on boron nitride, originating from lattice FCIs rather than continuum Landau level states. Even more spectacular recent observations have identified authentic FCIs exhibiting a (zero-field) fractional quantum anomalous Hall effect in both twisted transition metal dichalcogenides and multilayer graphene.

The primary focus of this presentation is to elucidate the distinctions between moiré Chern bands and Landau levels, arising due to symmetry-breaking competing states as a consequence of the underlying quantum geometry in the moiré lattice setting. This platform also opens intriguing avenues for numerous new phenomena beyond Landau level physics, including FCIs in bands with higher Chern numbers, high-temperature FCIs, fractional anomalous Hall crystals combining charge density wave (CDW) order with anyon excitations, non-Abelian genons accompanying topological lattice defects, spin singlet and multicomponent FCIs, as well as a plethora of competing ordered single and multi-band states. This offers an exciting prospect where fundamental theory evolves through direct interaction with experiments, with first principles calculations serving as a bridge linking these two realms.   

Observation of fractional quantum anomalous Hall effect

The synergy of topology, strong correlations, and spontaneous symmetry breaking can give rise to exotic quantum states of matter. A celebrated example is the rationally designed flat Chern band, which hosts fractional Chern insulators. These insulators can exhibit the fractional quantum anomalous Hall effect(FQAHE), which is the lattice analog of the fractional quantum Hall effect at zero magnetic field without Landau level formation. In this talk, I will present the experimental observation of high-temperature FQAHE in twisted MoTe₂ bilayer, using combined magneto-optical spectroscopy and transport measurements. Additionally, I will present an anomalous Hall state around the filling factor of -1/2, the leading candidate for which is a composite Fermi liquid state. Notably, the states associated with the FQAH and associated effects can be electrically tuned into a variety of topologically trivial phases. These results mark a promising direction for the investigation of charge fractionalization and anyon statistics at zero magnetic field and elevated temperature.