CORRELATION: MANY-BODY RECONSTRUCTION OF FLAT BANDS

The strongly reduced bandwidth endows MATBG with pronounced electronic correlation that profoundly affects its electronic behavior. In this section, we highlight the key features of correlated electronic behavior and discuss the underlying many-body physics associated with symmetry breaking, phase transition, and electron localization.

Correlated insulators and flavor symmetry breaking

Owing to the spin and valley degrees of freedom in graphene, complete filling of each degenerate valence and conduction flat band in MATBG requires a total of four electrons or holes per moiré unit cell. The band-filling factor ν, defined as the number of electrons and holes per moiré unit cell, ranges from -4 when all flat bands are empty, to 0 corresponding to charge neutrality with the valence flat bands filled, and up to 4 when all flat bands are filled. According to single-particle Bloch band theory, a partially filled valence or conduction flat band (-4 < ν < 0 or 0 < ν < 4) is expected to be metallic. However, experimental results show pronounced gapped states at filling factor ν = ±2 and ν = 3, providing solid evidence of electronic correlation in MATBG [5,7,8]. Generally, if the kinetic energy of electrons is less than the on-site Coulomb interaction in a spin-degenerate Bloch band, the nearest electron hopping is forbidden, giving rise to an antiferromagnetic Mott insulating state at half filling. Like a Mott insulator, the suppressed Fermi velocity of the flat band in MATBG reduces kinetic energy to a scale comparable with the Coulomb interaction which is inversely proportional to the moiré wavelength. Consequently, correlated insulating states appear at these integer fillings.

Electronic correlation in MATBG drives spontaneous spin and valley flavor symmetry breaking. Adding carriers to the system, the spin and valley flavors are populated not equally but through a sequence, forming a cascade of phase transitions. Within the flat bands, spectroscopic measurements of STM (Fig. 3a) reveal sharp transitions at every integer filling, indicating a cascade of electronic phase transitions that are attributed to Hubbard sub-bands with lifted spin and valley degeneracies [26]. Thermodynamic measurements that can manifest electron correlation, energy gaps, or phase transitions as anomalies or suppressions in electronic compressibility provide more details on the cascade of phase transitions [26–29]. In scanning single-electron transistor (SET) thermodynamic measurements as shown in Fig. 3b, a striking sequence of sawtooth-like features in compressibility appears near every integer filling [27]. The asymmetric sudden jump and a subsequent gradual decrease in compressibility correspond to a Dirac-like electronic character as near the charge neutrality point (CNP). When the carrier density changes approaching every integer filling, a single flavor takes carriers from other flavors, leaving other flavors unoccupied and forming a ‘reset’ state back to charge neutrality. The reset state is in line with the Landau-level asymmetry reported in magneto-transport measurements [8], where Landau fan from nonzero integer fillings emanate in only one direction away from charge neutrality. These Dirac revival behaviors are also accompanied with negative compressibility—manifested as the pinning of the chemical potential at integer fillings. This behavior has been attributed to flavor Hund’s coupling arising from the combined effect of on-site inter-flavor Coulomb repulsion and inter-site intra-flavor exchange interactions [29]. Notably, such cascade of phase transitions persists to temperatures above the onset of correlated insulating and superconducting states, indicating the flavor symmetry breaking could be the parent state from which correlated insulating states and superconductivity emerge from.

Intervalley coherent ground states

Different from a conventional Mott insulator, the flat bands in MATBG are generally described as multicomponent quantum Hall systems with both spin and valley degeneracies, resembling the zeroth Landau level in graphene [30]. Exchange interaction spontaneously drives the system into a subset of the isospin components with complex quantum valley texture in their wavefunction such as the inter-valley coherence (IVC), valley polarization and valley Hall [31,32]. These symmetry-broken quantum valley textures vary among different charge fillings and are susceptible to strain, magnetic field, displacement field, etc.

Encoded with the symmetry-breaking information, intricate quantum valley textures are intimately associated with various correlated phases in MATBG. In particular, the IVC state which means the spontaneous hybridization between inequivalent K and K’ valleys, is predicted to be a candidate ground state for correlated insulators at even integer fillings and the superconducting states [31,33]. With the atomic resolution, STM can be used to distinguish the quantum valley texture by imaging the atomic-scale spatial distribution of electron states. The LDOS measurements at ν = ±2 shows real-space features that indicate translation and rotation symmetry-broken Kekulé patterns with a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\sqrt 3 \times \sqrt 3$\end{document} super-periodicity on the graphene atomic lattice [25,34], as shown in Fig. 3c. The local magnitude and phase fast Fourier transform (FFT) plots of Kekulé patterns clearly show wavevectors that connect the graphene valleys at K and K’, indicates that the wavefunction at ν = ±2 is a coherent superposition in the two valleys, directly signifying the ground state of IVC. The IVC ground state hosts various candidate orders such as the incommensurate Kekulé spiral (IKS), Kramers intervalley coherent (K-IVC) and time-reversal symmetric intervalley coherent (T-IVC) orders. With a symmetry-based approach to extract local order parameters from STM images, order parameters of IVC bond, IVC sublattice A and IVC sublattice B, decomposed from FFT plots of Kekulé patterns (Fig. 3c), exhibit a long-wavelength moiré-scale modulation. This indicates the IKS order that simultaneously breaks both moiré-scale translation and rotation symmetries as the IVC ground state in the typical strained samples (strain \varepsilon > 0.1%) [25,34–36]. In ultralow-strain samples (\varepsilon < 0.1%) moiré-periodic order parameters show the absence of moiré-scale symmetry breaking, corresponding to the T-IVC ground state.

Remarkably, the Kekulé distortion persists across a range of doping, magnetic fields and temperatures. The coexistence of IVC order with pseudogap behavior and its evolution with carrier density across the superconducting domes suggest that superconductivity may arise from a parent IVC phase, possibly mediated by collective valley-pseudospin fluctuations. Taken together with correlated insulators, these findings provide compelling evidence that intervalley coherent order is a unifying feature of correlated phases in twisted graphene moiré systems, underscoring the central role of the valley degree of freedom in organizing electronic order in graphene flat-band systems.

Localized moments and topological heavy fermions

Soon after the correlated insulators were found in flat bands of MATBG, the microscopic description of its correlation physics has been much sought after. An exotic finding associated with electronic correlation is the localized moments that arise from isospin symmetry-broken ground states. The first signature of localized moments is the Pomeranchuk effect around integer moiré fillings in MATBG [37,38]. Pomeranchuk effect was manifested as the solidification of liquid 3He upon increasing temperature, owing to the large nuclear spin entropy of spatially localized He atoms [39]. As an analogue in MATBG, the Pomeranchuck effect arises from extra entropy of localized and disordered isospin moments at higher temperature which is distinct from the isospin-unpolarized metallic states of itinerant electrons at low-temperature limit. Experimentally, Pomeranchuck effect in MATBG was hinted by the appeared resistive peak of moiré filling ν = -1 at high temperature that implies the localization of electrons [37]. Based on electronic compressibility probes like scanning single electron transistor (SET) and capacitance bridges, thermodynamic measurements by relating the entropy S to the chemical potential μ through Maxwell relations

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clearly revealed the extra entropy around 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${{k}_{\mathrm{B}}}$\end{document} per moiré unit cell for the localized magnetic moments [37,38]. Additional signatures of localized magnetic moments include the sign reversals of the thermopower near the moiré filling ν = ±1, which remain nearly temperature independent from 5 K to 60 K, as well as magnetic-field-induced suppression of the thermopower, resistivity and entropy [40,41].

However, correlated states can exhibit electronic behaviors that are sometimes contradictory to those expected for localized moments. The absence of a thermodynamic gap at most of moiré fillings suggests itinerant electrons are responsible for the strongly correlated phase. In STM measurements, the charge carriers are found to be dispersed around rather than exactly at AA sites of moiré patterns. Most importantly, thermodynamic probes of chemical-potential sensors revealed a cascade of Dirac revival for electrons at each integer filling, pointing to the presence of Dirac-like delocalized electrons [26,27]. On the other hand, the theoretical paradigm of topological Chern band contrasts with the picture of localized moments. Together, these observations indicate that correlated flat bands in MATBG and other related moiré superlattice are governed by a more intricate microscopic mechanism, one that must reconcile the coexistence and competition between itinerant (light) and localized (heavy) electrons.

Theory of topological heavy fermion was proposed to reconcile such disagreement: the flat band described by Bistrtzer-MacDonald model is renormalized by the topology and interactions into a hybridization of localized f orbitals and extended topological metallic conduction c bands at different momenta [42]. Recently, the quantum twisting microscope (QTM) which probes the momentum-resolved tunneling across the junction between monolayer graphene on the tip and the targeted quantum materials on the rotatable substrate, has been developed to be a high-resolution spectroscopic technique to map the band dispersion in momentum space [43,44]. Cryogenic QTM explicitly revealed the topological heavy fermion-like band structure near magic angle: a gapped flat band across most of momentum space while a gapless and dispersive Dirac-like band at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathrm{\Gamma }}$\end{document} point of the mini-Brillouin zone [45] (Fig. 3d). It explained the contradictory characteristics of electronic cascades and Dirac revivals by considering the reshuffling of charge between the localized and delocalized states. To disentangle the hybridized c and f electrons contributions in transport, photo-thermoelectric experiments, which probe the local Seebeck coefficient, provide a direct measure of electron-hole excitation asymmetry across the correlated gap at integer fillings of the moiré unit cell (Fig. 3e) [46,47]. In the framework of topological heavy fermion, the electron and hole excitations correspond to extended c and localized f electrons, respectively. The latter has a diminished lifetime and thus less Seebeck effect contribution, accounting for the observed overall negative S values in the electron-doped region.