TOPOLOGY EMERGENT FROM CORRELATED FLAT BANDS

Topology characterized as Berry curvature and Chern number for electronic bands has long been the focus of research on quantum materials. Crystalline graphene has exhibited topological behaviors, such as quantum Hall effect under magnetic field [48,49] or Berry curvature hotspot for valley electrons under broken inversion symmetry [50,51]. Though the Haldane model and Kane-Mele model were first proposed based on monolayer graphene [52,53], the topologically nontrivial Chern band at zero magnetic field has not been realized in graphene system until the advent of MATBG moiré superlattice.

Orbital Chern insulators

By breaking the C2zT symmetry, the charge neutrality point of MATBG is gapped to form the degenerate conduction and valence valley-projected Chern bands. In case of MATBG alignment with the hexagonal boron nitride substrate, the broken C2z sublattice symmetry endows the valley-projected flat band with a Chern number C = ±1, which is related to two different valleys K and K’ by time-reversal symmetry. The topological Chern flat bands can be regarded as copies of Landau levels at a finite out-of-plane magnetic field. At moiré filling of ν = ±1 and ν = ±3, exchange interactions drive the system to be a valley-polarized Chern insulators and a net Chern number for the whole system is equal to one (Fig. 4a) [9,10,54]. Consequently, the transverse Hall resistance Rxy is quantized to be h/e2 and longitudinal resistance Rxx vanishes at zero magnetic field. Both the transverse Hall resistance and longitudinal resistance feature pronounced hysteresis with respect to out-of-plane magnetic field, with the coercive field at the order of 0.1 T (Fig. 4b). This is the so-called quantum anomalous Hall (QAH) effect. The energy gap of the QAH insulator has been found to exceed the Curie temperature, and the Hall resistance remains quantized to within 0.1% of the von Klitzing constant at zero magnetic field, persists to temperatures of several Kelvin [10]. This contrasts with the low temperature and small energy gap of QAH states in magnetic-doped topological insulators (MTIs), where magnetic disorders severely impact the performance of QAH state [55]. However, experimental realization of QAH in hBN-aligned MATBG has so far been limited to a small number of devices [10,54,56], highly depending on the stacking configuration. The absence or fragility of QAH states in hBN-aligned MATBG might originate from the incommensurability between graphene-graphene and graphene-hBN moiré patterns, which generates effective potential disorders that suppress the global percolation of local Chern numbers [57]. Thereby, QAH states are suggested to emerge in a narrow twist-angle window in which two moiré patterns become commensurate.

Figure 4.: Nontrivial topology of flat band in twisted graphene moiré superlattice: (a) Schematics of band structure at full filling ν = 4 and ν = 3 [10]. The net Chern number at ν = 3 is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{C}{\rm net}} = 1$\end{document}. (b) Quantized anomalous Hall effect characterized by longitudinal resistance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{R}{xx}}$\end{document} and Hall resistance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{R}{xy}}$\end{document} at filling factor ν = 3 [10]. The arrows indicate sweeping directions of magnetic field. (c) Mapping plot of magnetization density m. The black dash lines indicate the edges of the sample [56]. The unit of m is Bohr magneton per moiré unit cell (u.c.). (d) Magnetization jump across the Chern insulator gap in twisted monolayer-bilayer graphene [60]. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Delta {{R}{yx}} = R{yx}^{{{B}_ \downarrow }} - R_{yx}^{{{B}_ \uparrow }}$\end{document} is the Hall resistance difference between different sweeping directions of magnetic field. (e) Fractional Chern insulators (FCIs) in MATBG [12]. The top panel shows the local inverse compressibility \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathrm{d}}\mu /{\mathrm{d}}n$\end{document} as a function of magnetic field B and filling factor ν. In the bottom panel, charge density waves and FCIs are illustrated by the light blue and orange lines, respectively._

Different from MTIs where the large spin magnetization of several Bohr magneton \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\mu }B}$\end{document} per atom is dominate and orbital magnetization is negligible, graphene is not an intrinsically magnetic material and preserves vanishing spin-orbit coupling. As a result, orbital contribution to the magnetic moment can be comparable to the spin in graphene systems. The valley degree of freedom in graphene offers time-reversal-symmetry-related electron species that is separated from the spin. By driving the valley-projected flat bands into a valley-polarized QAH insulator at moiré fillings ν = ±1 and ν = ±3 with electron interactions, the net nonzero orbital moment is obtained and primarily contributes to the whole magnetization. This is why QAH state in twisted graphene superlattice is also known as “orbital Chern insulators” [56,58]. The orbital ferromagnetism can be assured by measuring the magnetization. For the spin and valley projected flat band, if ferromagnetism is from spin moment, the magnetization density m should be 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\mu }{\mathrm{B}}}$\end{document} per moiré unit cell area. While for orbital moment, the current loop on the moiré length scale generates orbital magnetization of several Bohr magnetons. Magnetometry of nano superconducting quantum interference device (SQUID) scanning local probe has high magnetic field sensitivity (15 nT/Hz1/2) and spatial resolution (10-nm scale), providing an accurate measure of magnetization density m. Experiments indicate a magnetization density m of 2–4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\mu }{\mathrm{B}}}$\end{document} per moiré unit cell (Fig. 4c), in good agreement with the prediction of orbital moment [56]. This result corresponds to magnetization density of 1.8 × 10−4 to 3.6 × 10−4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\mu }{\mathrm{B}}}$\end{document} per carbon atom, which is distinct from MTIs with a value of several \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\mu }_{\mathrm{B}}}$\end{document} per atom.

In a spin or orbital Chern insulator, the topologically protected edge states also contribute to the magnetization. When the chemical potential μ is changed by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathrm{\delta }}\mu$\end{document}, edge-state contribution of magnetization \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathrm{\delta }}M$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathrm{\delta }}M/{\mathrm{\delta }}\mu = C{\mathrm{e}}/2\pi \hbar $\end{document}, where C is the Chern number, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathrm{e}}$\end{document} is the elementary charge, ℏ is the reduced Planck constant. Across the Chern gap \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{E}_{\mathrm{\Delta }}}$\end{document}, the magnetization jumps by

(4) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{eqnarray*}\frac{{\Delta M}}{{{{\mu }_{\mathrm{B}}}/{{A}_{uc}}}} = \frac{{C{{m}_{\mathrm{e}}}{{A}_{uc}}{{E}_{\mathrm{\Delta }}}}}{{\pi {{\hbar }^2}}}, \end{eqnarray*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{A}{uc}}$\end{document} is the area of unit cell, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{m}{\mathrm{e}}}$\end{document} is the electron mass [59]. In MTIs, the spin magnetization is around 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\mu }{\mathrm{B}}}/{{A}{uc}}$\end{document}. As the Chern gap \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{E}{\mathrm{\Delta }}}$\end{document} is on the order of few millielectronvolts, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{E}{\mathrm{\Delta }}} \ll {{\hbar }^2}/{{m}{\mathrm{e}}}{{A}{uc}}$\end{document} since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{A}{uc}}$\end{document} of MTIs is on the atomic scale. The edge-state contribution of magnetization jumping Δ M across the Chern gap is negligible. However, for orbital Chern insulators in moiré superlattice, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{A}{uc}}$\end{document} is the large moiré unit cell area at the order of 100 nm2, therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{E}{\mathrm{\Delta }}}$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\approx {{\hbar }^2}/{{m}{\mathrm{e}}}{{A}{uc}}$\end{document}. This results in Δ M comparable to the orbital magnetization of several \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{\mu }{\mathrm{B}}}/{{A}_{uc}}$\end{document}. Such additional prominent magnetization from edge states in orbital Chern insulators can reverse the sign of magnetization when the chemical potential is swept across the Chern gap with electrostatic gating. Transport and local magnetic imaging probed such electric-field-induced nonvolatile reversal of magnetic states (Fig. 4d) [60], paving the way for application of nonvolatile and ultralow-power magnetic memory devices.

Fractional Chern insulators

The discovery of Chern insulators in twisted graphene superlattice invoked the immense interest to realize FCIs by fractionalizing the flat Chern bands. FCIs are the lattice analogues of fractional quantum Hall (FQH) states, characterized by spontaneous breaking of both lattice translational and time-reversal symmetries [61–66]. Like FQH states, FCIs could host quasi-particle excitations of anyons that obey fractional statistics. When topological Chern band is gapped at its partially filling ν due to electron–electron interactions, FCIs appear if the parent Chern number C is also fractionalized into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{C}\nu } = \nu C$\end{document}. Therefore, the formation of FCIs requires a particular quantum geometry of uniform Berry curvature distribution in the parent flat Chern bands. The trials of realizing zero-magnetic-field FCIs in a variety of twisted graphene systems came to failures, as the competed topological-trivial charge density wave states with zero Chern number are mostly favored [12]. Magnetic field is demonstrated to be of critical importance to flatten the Berry curvature distribution in MATBG, leading to the emergence of FCIs in a finite out-of-plane magnetic field (Fig. 4e). Recent progresses in experiments have realized fractional quantum anomalous Hall (FQAH) states—the form of FCIs at zero magnetic field, in other moiré superlattice systems like twisted MoTe2 [67–70] and rhombohedral graphene/hBN superlattice [71]. Flat bands of twisted MoTe2 and rhombohedral graphene/hBN superlattice have Chern numbers of C = ± 1. At zero magnetic field, FQAH states exhibit quantized Hall resistance plateaus of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{R}{xy}} = \frac{h}{{\nu {{e}^2}}}$\end{document} at fractional fillings of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\nu = \frac{2}{3},\frac{3}{5},$\end{document} etc., resembling the Jain sequence of FQH states [72].

Topological electronic crystals

At fractional filling ν of flat bands, Coulomb interaction or an underlying staggered potential can break the continuous or discrete translational symmetry, driving the two-dimensional electron gas (2DEG) into electronic crystal phases such as Wigner crystals or charge density wave (CDW) states. In topologically nontrivial moiré superlattices, if time-reversal symmetry is spontaneously broken, these electronic crystal states may additionally acquire anomalous Hall responses characterized by nonzero Chern numbers [54,73–75].

In contrast to FCIs where the Hall resistance is fractional and directly tied to the filling ν (i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{R}{xy}} = \frac{h}{{\nu C{{{\mathrm{e}}}^2}}}$\end{document}), the Chern number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{C}\nu }$\end{document} of topological electronic crystal (TEC) states is not universally fixed by ν C, and can exhibit a variety of behaviors depending on the interplay between interactions and band topology. Recent experiments in graphene moiré systems have revealed several representative cases: (ⅰ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{C}\nu }$\end{document} coincides with an integer value consistent with ν C when the parent band carries a higher Chern number (C\not= 1), such that interaction-driven symmetry breaking gaps out the system without fractionalization (e.g. in twisted monolayer-bilayer graphene [73]); (ⅱ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{C}\nu }$\end{document} remains equal to the parent-band Chern number C over a finite range of filling, despite the presence of additional carriers (e.g. in MATBG in finite magnetic field [1] and hBN-aligned rhombohedral multilayer graphene [74]); and (ⅲ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{C}_\nu }$\end{document} deviates substantially from that of the parent band, reflecting a strong reconstruction of the underlying topology (e.g. in twisted bilayer-trilayer graphene [75] and hBN-aligned rhombohedral multilayer graphene [76]).

The second case bears resemblance to the reentrant quantum Hall effect (RQHE) observed at high magnetic fields, where the Hall conductance returns to that of a nearby integer quantum Hall state upon partial filling of higher Landau levels. In conventional RQHE, this behavior is understood as the coexistence of an incompressible integer quantum Hall liquid with a topologically trivial pinned electron solid (such as a Wigner crystal, bubble phase, or stripe phase) formed by the excess carriers [77–79]. By analogy, TEC states in this regime may be interpreted as a coexistence of an integer QAH background and an additional, topologically trivial electronic crystal [74].

However, recent theoretical works suggest a more intrinsic possibility: even in the absence of an external magnetic field, a Wigner crystal formed in a topological band can itself spontaneously break time-reversal symmetry and develop a nonzero integer Chern number, provided that the parent band hosts nontrivial quantum geometry of concentrated Berry curvature [80–84]. This phase, referred to as an anomalous Hall crystal (AHC), can be viewed as a zero-field analog of a Hall crystal [85]. Importantly, the Chern number of an AHC is not simply inherited from the parent band, but instead emerges from the interplay between electron–electron interactions and band topology, corresponding to the third case described above. In general, AHC states are expected to arise at both continuous incommensurate and commensurate fillings of the underlying moiré patterns where electronic interactions—not the moiré potential—drive the spontaneous breaking of translational symmetry. Nevertheless, when the moiré potential plays a significant role, generalized AHC states at discrete commensurate fillings may also be stabilized [75].