Fractional Quantum Anomalous Hall Effect and
Fractional Chern Insulators

Spontaneous symmetry breaking and more recently entanglement are two cornerstones of quantum matter. We introduce the notion of anisotropic entanglement ordered phases, where the spatial profile of spin-pseudospin entanglement spontaneously lowers the four-fold rotational symmetry of the underlying crystal to a two-fold one, while the charge density retains the full symmetry. The resulting phases, which we term entanglement smectic and entanglement stripe, exhibit a rich Goldstone mode spectrum and set of phase transitions as a function of underlying anisotropies. We discuss experimental consequences of such anisotropic entanglement phases distinguishing them from more conventional charge or spin stripes. Our discussion of this interplay between entanglement and spontaneous symmetry breaking focuses on multicomponent quantum Hall systems realizing textured Wigner crystals, as may occur in graphene or possibly also in moire systems, highlighting the rich landscape and properties of possible entanglement ordered phases.

Tight-binding models are ubiquitous in condensed matter physics. Fundamental properties of many materials can be understood from simple toy models of electrons hopping between orbitals on a lattice. In particular, crystallographic symmetries are crucial in determining the degeneracies of band structures at high symmetry points in the Brillouin zone. These degeneracies fundamentally affect the properties of the material that is being modelled, ranging from the allowed optical transitions to symmetry-indicated topological invariants. Getting a full list of crystal symmetries of a material is therefore crucial when trying to predict or understand its physical properties. In this poster, we systematically study the symmetries of lattices in 3 dimensions and their implications for the corresponding tight-binding models. We uncover a landscape of tight-binding models related by crystal symmetries, and apply these findings to understand the recently uncovered altermagnetic nature of RuO2 and MnTe.

Helical twisted trilayer graphene (HTTG), characterized by three layers of graphene with same successive twists, is an unique tunable platform for realizing a variety of correlated and topological phases. It exhibits a supermoire with domains centered around stacking points ABA or BAB, where two well separated low energy bands appear with Chern numbers +/-(2, -1) forming a Chern mosaic. When the twists are tuned to a ’magic-angle’, these bands flatten perfectly at the chiral limit, with large degeneracies at the zero energy. We show that HTTG retains such precise flatness of the low energy bands in the chiral limit even when a perpendicular magnetic field is applied. By a mapping of the zero-energy wavefunctions with those of the lowest Landau level, we identify the analytical forms of the zero-modes at finite magnetic fields. Furthermore, we find topological phase transitions involving gap closings and openings, at fields corresponding to unit and half-flux per unit cell, leading to higher Chern number bands. Such transitions happen at the supermoire scale which alter with each other when the direction of the field is reversed. Due to large moire length scale, these transitions at strong flux limit can become experimentally accessible.

Two-dimensional hole gas (2DHG) system exists the competition of quantum Hall effect (QHE) liquid and Wigner solid (WS) solid under high magnetic field (B) and low temperature (T). We report non-linear differential conductance around $\nu$=1 and $nu$=1/3 states in GaAs/AlGaAs quantum well (QW) 2DHGs, which in accord with the feature of type-II CFWS. Particularly, the applied electric field affects the magnetoconductance diagram in a way similar to the temperature, which is known as “E-T duality”. According to our analysis, this “E-T duality” is consistent with the Berezinskii-Kosterlitz-Thouless (BKT) theory, which is well-known in two-dimensional phase transitions.

The bosonic fractional Chern insulator is intricately connected to quantum spin liquids. Thus, we investigate the Read-Rezayi fractional quantum Hall states for $k=2$ (Moore-Read) and $k=3$. The non-abelian excitations of the $k=3$-state are Fibonacci anyons and thus provide a platform for universal quantum computing. Following the analogy of the Calogero-Sutherland model, we construct parent continuum Hamiltonians for these states. Based on the identification of fractional quantum Hall states with correlation functions of CFTs, we use the Belavin–Polyakov–Zamolodchikov equation for the corresponding primary fields to find destruction operators with the ground state in their kernels and combine them to form parent Hamiltonians.

Quantum Monte Carlo simulations are applied to study quantum antiferromagnets on a hexagonal lattice under three-axis stress. We consider the triangular geometry, where strain induces dimerization of the exchange couplings around the three corners of the triangle, thereby disrupting the antiferromagnetic order. The antiferromagnetic regions continuously shrink with increasing strain, and for the same strain intensity, the precise numerical results yield much smaller antiferromagnetic regions than predicted by linear spin-wave theory. We also extract the local magnetization using numerical analytical continuation for the Heisenberg case and find no evidence of pseudo-Landau levels. However, it is currently unclear whether the absence of pseudo-Landau levels is a consequence of the numerical analytical continuation itself. Therefore, the existence of pseudo-Landau levels for magnons generated by three-axis strain in the Heisenberg Hamiltonian remains an open question. Our theoretical results are closely related to two-dimensional van der Waals antiferromagnets and hold promise for experimental realization.

A recent experiment has reported the first observation of a zero-field fractional Chern insulator (FCI) phase in twisted bilayer MoTe2 moiré superlattices. The experimental observation is at an unexpected large twist angle 3.7◦ and calls for a better understanding of the FCI in real materials. In this work, we perform large-scale density functional theory calculation for the twisted bilayer MoTe2, and find that lattice reconstruction is crucial for the appearance of an isolated flat Chern band. The existence of the FCI state at ???? = −2∕3 is confirmed by exact diagonalization. We establish phase diagrams with respect to the twist angle and electron interaction, which reveal an optimal twist angle of 3.5◦ for the observation of FCI. We further demonstrate that an external electric field can destroy the FCI state by changing band geometry and show evidence of the ???? = −3∕5 FCI state in this system. Our research highlights the importance of accurate single particle band structure in the quest for strong correlated electronic states and provides insights into engineering fractional Chern insulator in moiré superlattices.

Laughlin states have recently been constructed on fractal lattices, and the charge and braiding statistics of the quasiholes were used to confirm that these states have Laughlin-type topology. Here we investigate density, correlation, and entanglement properties of the states on a fractal lattice derived from a Sierpinski triangle with the purpose of identifying similarities and differences compared to two-dimensional systems and with the purpose of investigating whether various probes of topology work for fractal lattices. Similarly to two-dimensional systems, we find that the connected particle-particle correlation function decays roughly exponentially with the distance between the lattice sites measured in the two-dimensional plane, but the values also depend on the local environment. Contrary to two-dimensional systems, we find that the entanglement entropy does not follow the area law if one defines the area to be the number of nearest-neighbor bonds that cross the edge of the selected subsystem. Considering bipartitions with two bonds crossing the edge, we find a close to logarithmic scaling of the entanglement entropy with the number of sites in the subsystem. This also means that the topological entanglement entropy cannot be extracted using the Kitaev-Preskill or the Levin-Wen methods. Studying the entanglement spectrum for different bipartitions, we find that the number of states below the entanglement gap is robust and the same as for Laughlin states on two-dimensional lattices. Reference: PHYSICAL REVIEW B 105, 085152 (2022)

The dynamical response of a quantum spin liquid upon injecting a hole is a pertinent open question. In experiments, the hole spectral function, measured momentum-resolved in angle-resolved photoemission spectroscopy (ARPES) or locally in scanning tunneling microscopy (STM), can be used to identify spin liquid materials. In this study, we employ tensor network methods to simulate the time evolution of a single hole doped into the Kitaev spin-liquid ground state and reveal two fundamentally different scenarios. For ferromagnetic spin couplings, the spin liquid is highly susceptible to hole doping: a Nagaoka ferromagnet forms dynamically around the doped hole, even at weak coupling. By contrast, in the case of antiferromagnetic spin couplings, the hole spectrum demonstrates an intricate interplay between charge, spin, and flux degrees of freedom, best described by a parton mean-field ansatz of fractionalized holons and spinons. Moreover, we find a good agreement of our numerical results to the analytically solvable case of slow holes. Our results demonstrate that dynamical hole spectral functions provide rich information on the structure of fractionalized quantum spin liquids.

We investigate the global many-body phase diagram in twisted bilayer MoTe2, using continuum model multiband exact diagonalization. From previous large-scale first-principles calculations, two groups of continuum model parameters have been proposed from D2 and dDsC corrections, respectively. Compared to the experimental phase diagram, we note the dDsC continuum model well captures the υ=1integer, υ=-2/3 fractional quantum anomalous Hall states, and υ=-1/3 charge density wave states simultaneously. While for the continuum model from D2 correction, we observe a Mott ferroelectric state at υ=1 for dielectric constant ϵ<10 at experimental twist angles 〖3.0〗^∘<θ<〖4.0〗^∘. More importantly, in the D2 continuum model, υ=-1/3 and υ=-2/3 are both fractional quantum anomalous Hall states at ϵ=10, and charge density wave states at ϵ=5.

Recent observations of the fractional anomalous quantum Hall effect in moir\'e materials have reignited the interest in fractional Chern insulators (FCIs). The chiral limit in which analytic Landau level-like single particle states form an "ideal" Chern band and local interactions lead to Laughlin-like FCIs at $1/3$ filling, has been very useful for understanding these systems by relating them to continuum Landau levels. We show, however, that, even in the idealized chiral limit, a fluctuating quantum geometry leads to strongly broken symmetries and a phenomenology very different from that of Landau levels. In particular, particle-hole symmetry is strongly violated and e.g. at $2/3$ filling an emergent interaction driven Fermi liquid state with no Landau level counterpart is energetically favoured.

Our study delves into the effect of lattice relaxation on the moiré band structures of twisted bilayer MoTe2 , implemented by large-scale first-principles calculations and transfer learning neural network. Throughout our study, we have incorporated two van der Waals correction methods: the Grimme D2 method and a density-dependent energy correction. Notably, the latter method demonstrates a continuous evolution of bandwidth with respect to twist angles. Our findings reveal the critical role of in-plane lattice displacements, which generate substantial pseudomagnetic fields, reaching up to 250 T. Building on these insights, we have developed a comprehensive continuum model with a single set of parameters for a wide range of twist angles, providing a useful starting point for many-body simulation.

While electronic topological phases in moiré heterostructures of transition metal dichalcogenides (TMDs) have been confirmed experimentally, exotic magnetic orders have not been reported so far. In this work, we investigate TMD moiré bilayers at $n=3/4$ filling of the moiré flat band, where they can develop a kagome charge order. We derive an effective spin model for the resulting localized spins and find that its further neighbor spin interactions can be much less suppressed than the corresponding electron hopping strength. Using density matrix renormalization group simulations, we study its phase diagram and, for realistic model parameters relevant for WSe$_2$/WS$_2$, we show that this material can realize the exotic chiral spin liquid phase and the highly debated kagome spin liquid. Our work thus demonstrates that the frustration and strong interactions present in TMD heterobilayers provide an exciting platform to study spin liquid physics.

The Laughlin and Moore-Read states can be formulated as infinite-dimensional-matrix product states. The construction allows us to generalize the Laughlin and Moore-Read states to arbitrary lattices embedded in two dimensions and to find few-body, non-local, exact parent Hamiltonians for the states. We numerically and analytically study the properties of the Laughlin state on fractal lattices with different Hausdorff dimensions. We find that the states support anyons with fractional statistics as expected for the Laughlin state, but the states do not necessarily follow the area law. Finally, we optimize local Hamiltonians to have maximal ground state overlap with the Laughlin state. References: Phys. Rev. Research 2, 023401 (2020) Phys. Rev. B 105, 085152 (2022) J. Stat. Mech. 2023, 053103 (2023) Phys. Rev. A 107, 063315 (2023)

Braiding non-Abelian anyons in topologically ordered systems has been proposed as a possible route towards topologically protected quantum computing. While recent experiments based on various platforms have made significant progress towards this goal, coherent control over individual anyonic excitations has still not been achieved today. At the same time, progress in cold-atom quantum simulators resulted in the realization of a two-boson $\nu=1/2$-Laughlin state, a paradigmatic fractional quantum Hall state hosting Abelian anyonic quasi-holes. Here we show that cold atoms in quantum gas microscopes are a suitable platform to create and manipulate these quasi-holes. First, we show that a Laughlin state of eight bosons can be realized by connecting small patches accessible in experiments. Next, we demonstrate that two cross-shaped pinning potentials are sufficient to create two quasi-holes in this Laughlin state. Starting with these two quasi-holes we numerically perform an adiabatic exchange procedure, and reveal their semionic braiding statistics for various exchange paths, thus clarifying the topological nature of these excitations. Finally, we propose an experimentally feasible interferometry protocol to probe the braiding phase in quantum gas microscopes, using a two-level impurity immersed in the fractional quantum Hall fluid. We conclude that braiding experiments of Abelian anyons are now within reach in cold-atom quantum simulators, providing a crucial step towards the long-standing goal of non-Abelian anyon braiding with local coherent control.

We suggest a simple chirality probe in TR invariant Weyl and Dirac semimetals by nonlinear Hall response. Chiral anomaly effect arising in parallel electric and magnetic fields causes Weyl cones of different chirality to become shifted in energy space with respect to each other which leads to chirally asymmetric intranode relaxation times. Due to this asymmetry, electrical currents excited by external electric field do not perfectly compensate each other. We predict that this effect could be also observed in circular dichroism measurements.

We present a hybrid Floquet-Trotter approach for the quantum simulation of Kitaev's toric-code Hamiltonian, implementable in superconducting circuits. Our method exploits the commutativity of different terms in Kitaev’s Hamiltonian and achieves the required four-spin interactions in a nonperturbative way. Building on this result, we further present protocols for preparing topologically ordered ground states with high fidelity and to simulate the transition into the spin liquid phase. We discuss opportunities to use periodic driving to mimic non-Abelian behaviour in the Abelian model.

The formalism for composite fermions, initially developed for bosons at filling factor $ \nu $ = 1 [1, 2], has been a cornerstone in understanding the fractional quantum Hall effect (FQHE). We derive the Dirac composite fermion theory for a half-filled Landau level from first principles [3]. In this talk, we introduce a variant of dipole representation for composite fermions in a half-filled Landau level, taking into account the symmetry under exchange of particles and holes. This is implemented by a special constraint on composite fermion and composite hole degree of freedom (of an enlarged space), that makes the resulting composite particle, dipole, a symmetric object. Our investigation focuses on an effective Hamiltonian that commutes with the constraint in physical space while preserving boost invariance at the Fermi level. Our calculations [4] of the Fermi liquid parameter F2 demonstrate remarkable agreement with previous numerical investigations [5]. Furthermore, the quantum Hall bilayer at filling factor $ \nu $=1 represents a competition of Bose-Einstein condensation (BEC) at small distances between layers and fermionic condensation, which influence grows with distance and results in two separate Fermi liquid states of underlying quasiparticles at very large (infinite) distance. The most intriguing question is whether at intermediate distances between layers a special, distinct phase exists, or a single transition occurs, with possibility that this happens at infinite distance. Here, using a dipole representation of fermionic quasiparticles, we find a support for the latter scenario: for a large, relevant interval BEC condensation, identified as a Cooper s-wave pairing of dipole quasiparticles, wins over Cooper p-wave pairing and s−wave excitonic pairing of the same quasiparticles [6]. [1] N. Read, Phys. Rev. B 58, 16262 (1998). [2] Z. Dong and T. Senthil, Phys. Rev. B 102, 205126 (2020). [3] D. Gočanin, S. Predin, M. D. Ćirić, V. Radovanović, and M. Milovanović, Phys. Rev. B 104, 115150 (2021). [4] S. Predin, A. Knežević, and M. V. Milovanović, Phys. Rev. B 107, 155132 (2023). [5] K. Lee, J. Shao, E.-A. Kim, F. D. M. Haldane, and E. H. Rezayi, Phys. Rev. Lett. 121, 147601 (2018). [6] S. Predin, and M. Milovanovic, S. Predin and M. V. Milovanovic, Phys. Rev. B 108, 155129, (2023).

In the last few decades, the coupling between layers in two-dimensional correlated electronic system has been one of the most interesting topics in the field of condensed matter physics [1-3]. InAs/GaSb quantum well (QW) is one of the most interesting layer-layer correlated systems. Thanks to the unique inverted band structure, InAs/GaSb bilayer systems can simultaneously hosts spatially closed 2DEG and 2DHG in InAs and GaSb QW, respectively. In recent years, many exotic quantum phases have been observed in this system [4-6]. Particularly, excitonic topological order found in InAs/GaSb system is of great interest, since the excitons carry 1/2 fractional topological charge similar to composite fermion in fractional quantum Hall system [6]. The moat band dispersion of exciton is a prerequisite for excitonic topological order. Besides imbalanced carrier density of electrons and holes [6], theoretical analysis found that Rashba spin-orbital coupling can also help to form moat band of excitons [7]. In this work, we show some preliminary experiment results that the Rashba spin-orbital coupling can stabilize excitonic topological order in InAs/GaSb bilayer system. References: [1] J. Eisenstein and A. MacDonald, Nature 432, 691 (2004). [2] X. Liu, K. Watanabe, T. Taniguchi, B. I. Halperin, and P. Kim, Nature Physics 13, 746 (2017). [3] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Nature 556, 43 (2018). [4] L. Du, I. Knez, G. Sullivan, and R.-R. Du, Physical Review Letters 114, 096802 (2015). [5] L. Du, X. Li, W. Lou, G. Sullivan, K. Chang, J. Kono, and R.-R. Du, Nature Communications 8, 1971 (2017). [6] R. Wang, T. A. Sedrakyan, B. Wang, L. Du, and R.-R. Du, Nature 619, 57 (2023). [7] C. Wu, Modern Physics Letters B, 23, 1 (2009)

Quantized dynamics is essential for natural processes and technological applications alike. It has been shown that quantized particle transport in Thouless pumps is not restricted to the limit of slow driving in non-Hermitian Floquet systems [1]. The degree of topological protection provided by non-Hermiticity in such systems is of great interest. Here we conduct a comparative study of the robustness of directional transport in the presence of static random disorder in two periodically driven systems – quantum ratchet [2] and fast Thouless pumps [1]. In the case of the ratchet, quantized transport relies on a resonant effect that requires fine-tuning of driving parameters. On the contrary, directional transport in Thouless pumping is of a topological origin and observed for a range of driving frequencies considering the closed cycle in parameter space. Our experimental implementation of the models is realized by evanescently coupled dielectric-loaded surface-plasmon polariton waveguide arrays based on the mathematical identity between the coupled mode theory equations and the discrete Schrödinger equation in the tight-binding approximation. In the present joint experimental and theoretical study, we analyzed the effect of topological protection on directional transport by gradually introducing identical on-site disorder distribution to both systems. We demonstrated that topologically protected directional transport in the case of a Thouless pump is able to sustain static disorder while in the case of a quantum ratchet, the high-strength disorder results in a complete directionality breakdown. References [1] Z. Fedorova, H. Qiu, S. Linden, J. Kroha, Observation of topological transport quantization by dissipation in fast Thouless pumps, Nat Commun 11, 3758 (2020). [2] Z. Fedorova, C. Dauer, A. Sidorenko, S. Eggert, J. Kroha, S. Linden, Dissipation engineered directional filter for quantum ratchets, Phys. Rev. Research 3(1), 013260 (2021).

Multipoles are fundamental quantities in the context of (higher order) topological insulators. However, it is known that definitions of bulk multipoles have some difficulties in general dimensions. Indeed, the well-known Resta formula for dipoles gives vanishing dipole moments in higher dimensions. Also, a similar formula for quadrupoles does not work for thermodynamically large systems even in two dimensions. In this study, we propose new many-body multipole indices which have several advantages over the previously proposed ones [1]. Our argument is based on an introduction of appropriate magnetic fields and point group symmetries. The magnetic fields can twist the spatial symmetries and thus probe anomalous responses to point group operations, which leads to the multipole indices in interacting systems. With the new indices, we can prove the bulk-boundary correspondence (or the filling anomaly) in presence of interactions. These analytical discussions are fully supported by numerical calculations. [1] Yasuhiro Tada and Masaki Oshikawa, "Many-body multipole index and bulk-boundary correspondence", arXiv:2302.00800, to be published in Phys. Rev. B.

The fractional quantum Hall (FQH) effect has become one of the most studied phenomena in condensed matter physics for the past 40 years. One classic approach studying these systems is to compute their thermal Hall conductance (THC) since it can be a quantized quantity for certain FQH states and they do not depends on the details of the system. The quantized value of THC, which is an integer or fractional value in units of $K_0$ ($K_0 = \frac{\pi^2 k_B^2}{3h}$), can be used to determine whether the FQH states is Abelian or non-Abelian states. This quantization however can only be robust if the edge of the system is modelled by using the chiral Luttinger liquid ($\chi$LL) under the linear dispersion. In experiments, this model may break down due to the nonlinearity of the confinement potential and the finite temperature effect. In this work, we analytically derive the THC of FQH states (i) with finite-size/low-temperature correction and (ii) under general dispersion relation. This can be accomplished by proper mathematical tools including integer partition generating function, Mellin transformation and asymptotic expansion. We conjectured that the THC can only be a universal quantity if the system is under linear dispersion. The non-universal corrections can provide guidance for a reasonable error range of THC measurements in realistic experiments.

We propose a frustration-free model for the Moore-Read quantum Hall state on sufficiently thin cylinders with circumferences $\lesssim 7$ magnetic lengths. While the Moore-Read Hamiltonian involves complicated long-range interactions between triplets of electrons in a Landau level, our effective model is a simpler one-dimensional chain of qubits with deformed Fredkin gates. We show that the ground state of the Fredkin model has high overlap with the Moore-Read wave function and accurately reproduces the latter's entanglement properties. Moreover, we demonstrate that the model captures the dynamical response of the Moore-Read state to a geometric quench, induced by suddenly changing the anisotropy of the system. We elucidate the underlying mechanism of the quench dynamics and show that it coincides with the linearized bimetric field theory. The minimal model introduced here can be directly implemented as a first step towards quantum simulation of the Moore-Read state, as we demonstrate by deriving an efficient circuit approximation to the ground state and implementing it on IBM quantum processor.

Antiferromagnetic topological insulators (AFMTIs) with gapless edge states represent a novel class of topological states for spintronics applications. Understanding principles behind symmetry protection and exploring AFMTIs with desirable properties, manipulable by external stimuli, are crucial. In this study, through first-principles calculations and symmetry analysis, we investigate the topological properties of monolayer SrMn2Bi2, demonstrating their sensitivity to the magnetic configuration. When the system is an out-of-plane antiferromagnetic ground state, we observe a gapless helical edge state protected by the mirror plane combined with time reversal symmetry. In the ferromagnetic state, the system resides in the quantum anomalous Hall phase, and the topology is trivial for the in-plane magnetization. We show that the topological properties can be efficiently manipulated by strain. Additionally, we emphasize that constructing proper Wannier functions which obey symmetry constraints in key for avoiding the prediction of spurious states in the surface spectra. Our work not only provides an ideal candidate for AFMTIs, but also guides the symmetry analysis of magnetic topological materials using Wannier functions.

Recently, a new time-reversal-symmetry breaking excitonic ground state with long-range quantum entanglement called ETO state has been discovered in shallow inverted InAs/GaSb electron-hole bilayer systems. It is attributed to frustration induced by imbalance of electron/hole density or spin polarization and a bosonic moat band with exotic topological order at zero magnetic field. Theoretically, moat band structure is likely to result in (fractional) quantum anomalous Hall effect (FQAH) or the so-called fractional Chern insulators, which is very encouraging in this field. After thorough low-temperature electrical transport experiment of this new state, we suggest that only thermal transport can distinguish this FQAH state from other topological order because excitons do not carrier charges but carry energy. Its thermal conductance is predicted to be the same as that of 1/2 bosonic FQH state. Our future experiment is going to explore the thermal transport nature of ETO state.

Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase. In this project, we derived a set of analytical results for the energy gap of the graviton modes with two-body and three-body Hamiltonians in both the long-wavelength and the thermodynamic limit. These allow us to construct model Hamiltonians for the graviton modes in different FQH phases and to elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) with respect to the graviton modes and their corresponding ground states. Numerical results of the Laughlin $\nu = 1/5$ and the Gaffnian $\nu = 2/5$ phases confirm that for gapped phases, low-lying neutral excitations can undergo a "phase transition" even when the ground state is invariant. The compressibility of the Gaffnian phase, the possibility of multiple graviton modes, the transition from the graviton modes to the "hollow-core" modes, and the chirality of graviton modes are discussed in detail, as well as their experimental consequences.

We investigate the moir\'e band structures and the strong correlation effects in twisted bilayer MoTe$_2$ for a wide range of twist angles, employing a combination of various techniques. Using large-scale first principles calculations, we pinpoint realistic continuum modeling parameters, subsequently deriving the maximally localized Wannier functions for the top three moir\'e bands. Simplifying our model with reasonable assumptions, we obtain a minimal two-band model, encompassing Coulomb repulsion, correlated hopping, and spin exchange. Our minimal interaction models pave the way for further exploration of the rich many-body physics in twisted MoTe$_2$. Furthermore, we explore the phase diagrams of the system through Hartree-Fock approximation and exact diagonalization. Our two-band exact diagonalization analysis underscores significant band-mixing effects in this system, which enlarge the optimal twist angle for fractional quantum anomalous Hall states.

Topological bands with a flat dispersion host strongly correlated states with or without intrinsic topological orders. The kinetic part of the system is trivial and the system is dominated by the interaction. At a first glance, electrons do not have any preference to occupy in the Brillouin zone. Despite the featureless kinetic dispersion, topological bands are usually equipped with nontrivial band geometry. We show that the nonuniform band geometry gives rise to emergent Fermi surfaces and it leads to a general particle-hole asymmetry. The electrons tend to fill regions in the Brillouin zone where their quantum distance is shorter. The emergent Fermi surface transforms the strongly interacting problem to a weakly interacting one. This dictates the low-energy physics and serves as a guiding principle for potential symmetry-breaking states. We show that in moiré materials, the quantum distance can be well approximated by a local quantity called the quantum metric. From this simple quantity, we can deduce what phases are favoured in different moiré systems at fractional fillings.

Fractional quantum Hall states are stabilized in two-dimensional systems with applied magnetic field. They are promising platforms for realizing kinetically constrained dynamics. Particles restricted to one Landau level behave as if they were on a one-dimensional lattice with effective dipole-conserving interactions. In this work, we investigate the rich quantum dynamics of this system on a torus. We find that in the thick torus limit, the system relaxes with hydrodynamic tails governed by subdiffusive fracton hydrodynamics. When decreasing the thickness of the torus, the dynamics slow down and display long prethermalization behaviors. We propose to use these signatures to characterize the dynamics of fractional quantum Hall states in quantum simulators of ultra-cold atoms. A key experimental signature of their hydrodynamics behavior can be found in the fluctuations of the number of particles in a subregion of the system. Furthermore, we provide a connection with experiments, by studying the full (non-projected) evolution in the basic constituent of our model. Our analysis shows that quantum dynamics of fractional quantum Hall states is particularly rich due to an effective dipole conservation, entailing Hilbert space fragmentation and fracton hydrodynamics.

Fractional Chern insulators (FCIs) generalize the conventional fractional quantum Hall effect from continuum two-dimensional electron gases to lattice setups. We propose twisted double bilayer graphene (TDBG) as a versatile platform for the realization of FCIs without the need of a magnetic field. The conduction band of TDBG can carry Chern number $C=1$ and $C=2$, which is readily controlled by tuning the vertical gate potential and the twist angle. By extensive exact diagonalization, we explore the many-body phase diagram of the system at band filling $\nu=1/3$, $2/5$, and $1/5$. Remarkably, we find compelling numerical evidence of various FCIs in different regions of band Chern number, including spin-valley polarized states and spin singlet Halperin states. We try to understand the stability of these states by considering the energetics and quantum geometry of the topological flat band.