We investigate the interplay between charge density wave (CDW) order and topological nodal-line states in square-net materials. Our Ginzburg-Landau theory predicts a CDW instability that generically opens a gap at the Fermi energy while preserving the nodal-line crossing. However, as the Fermi level approaches the nodal line, the density of states at the nodal line decreases, eventually disappearing as the CDW vector Q goes to zero. Exactly at Q = 0, the order parameter explicitly breaks the glide symmetry protecting the nodal line, which allows a gap to open. Yet, for small but finite Q, the nodal line may vanish within experimental resolution even when the glide symmetry is preserved. Our results provide a consistent explanation for recent experimental observations.

Strong correlation, in concert with symmetry and topology, engenders novel gapless phases of matter, though only a tip of the iceberg has been seen. An exemplary framework is provided by Weyl-Kondo semimetals, in which Weyl fermions develop through crystalline symmetry constraints on the emergent low-energy heavy-fermion excitations. This paradigm has opened up new opportunities to explore correlated topologies without a noninteracting counterpart, but fully realizing this potential requires a large base of candidate materials. Here we confront the challenge on both fronts by studying heavy fermion systems with hexagonal space groups. This family contains a large number of chiral nonsymmorphic crystal structures that promote Weyl degeneracies and, in addition, feature geometric frustration in the $f$-electron magnetism. Our calculations for the heavy fermion states identify Weyl-Kondo semimetals with chiral or achiral Weyl nodes in the respective structural classes. We also develop a new search strategy for the difficult case of strongly correlated materials, using a combination of materials database, symmetry classification and experiments, and propose as candidate topological heavy fermion systems the chiral CePt$_2$B and achiral Ce$_2$NiGe$_3$ and Ce$_6$Co$_{2-δ}$Si$_3$. Our findings raise the prospect for strongly correlated metallic topology in the unusual setting of exotic quantum magnetism.

Recent observations of superconductivity in twisted bilayer WSe_2 (tWSe_2) have motivated theoretical proposals for unconventional pairing mechanisms. A central question is whether band topology plays an essential role in the system's correlation physics. In this letter, we develop a first-principles-based description of the top moiré valence bands in tWSe_2. Using density functional theory (DFT) calculations, we identify the bands in the relevant range of twist angles to be topologically non-trivial, with the top valence bands carrying Chern numbers C=(+1,+1) for the K valley. In order to treat the strong correlation physics, we construct compact molecular orbitals directly from the DFT wave functions through a partial Wannierization procedure and with the guidance of spinful C_3z symmetry representations. This yields a localized f orbital together with a complementary topological c orbital, allowing us to extract hopping and hybridization amplitudes from first principles. The resulting parameters provide an ab initio benchmark for the effective Hamiltonian. Our work establishes a foundation for understanding superconductivity in moiré TMDs and highlights tWSe_2 as a promising platform for exploring topological superconductivity.

Despite significant progress in the study of higher-order topological insulators (HOTIs), the chiral C_4𝒯-protected HOTI has remained elusive in electronic materials, where C_4𝒯 denotes the product of a four-fold rotation and time-reversal symmetry. We show that altermagnetism, a recently discovered form of collinear magnetism, provides a new route to realize this elusive phase. Specifically, we construct a microscopic model that combines a three-dimensional topological insulator with a collinear d-wave altermagnet on a Lieb lattice. Through analytical and numerical calculations, we show that the magnetism shifts and gaps the surface Dirac cones to produce the desired chiral hinge channels. Finally, we identify promising material classes to realise our proposal. Our results establish collinear altermagnetism as a route to intrinsic chiral higher-order topology and open a new path toward the discovery of C_4𝒯-protected HOTIs in real materials.

We investigate the effect of remote band mixing on the stability of fractional Chern insulators in a family of models that approximate continuum descriptions of moiré materials. Our results suggest that the experimentally observed asymmetry between filling fractions ν=1/3 and ν=2/3 in twisted MoTe_2 originates from a competition between a fractional Chern insulator, an electron Wigner crystal, and a hole Wigner crystal. In the absence of band mixing, the leading instability at ν= 1/3 is the electron crystal, whereas at ν= 2/3 the main competing phase is the hole crystal. Remote band mixing substantially lowers the energy of the electron crystal but has only a weak effect on the hole crystal. Consequently, it destabilizes the fractional Chern insulator at ν=1/3 more strongly than at ν=2/3. This mechanism also provides an explanation for the emergence of re-entrant integer quantum anomalous Hall states in moiré MoTe_2 for fillings ν>1/2.

Bernal-stacked bilayer graphene modulated by a superlattice potential is a highly tunable system predicted to realize isolated topological flat bands. In this work we calculate the band structure and quantum geometry of bilayer graphene subject to both triangular and square superlattices, across a wide range of gate voltages. We identify the parameter regime that optimizes the "single-particle indicators" for the stability of a fractional Chern insulator (FCI) when a topological flat band is partially filled. Our results guide the experimental realization of an FCI in this platform.

Robust flavor-polarized phases are a striking hallmark of many flat-band moiré materials. In this work, we trace the origin of this spontaneous polarization to a previously overlooked quantum-geometric quantity: the quantum-geometric dipole. Analogous to how the quantum metric governs the spatial spread of wavepackets, we show that the quantum-geometric dipole sets the characteristic size of particle-hole excitations, e.g. magnons in a ferromagnet, which in turn boosts their gap and stiffness. Indeed, the larger the particle-hole separation, the weaker the mutual attraction, and the stronger the excitation energy. In topological bands, this energy enhancement admits a lower bound within the single-mode approximation, highlighting the crucial role of topology in flat-band ferromagnetism. We illustrate these effects in microscopic models, emphasizing their generality and relevance to moiré materials. Our results establish the quantum-geometric dipole as a predictive geometric indicator for ferromagnetism in flat bands, a crucial prerequisite for topological order.

Monolayer 1T'-WS2 is predicted to be a two-dimensional topological insulator, but its intrinsic electronic properties are masked by strong interlayer coupling in its metallic and superconducting bulk parent phase, 2M-WS2. Isolating monolayers by mechanical exfoliation is also hindered by this coupling, preventing experimental examination of monolayer properties. Here we show that 2M-WS2 undergoes amine intercalation through a simple wet-chemical reaction, yielding superlattices in which the 1T' layers are structurally preserved but electronically decoupled by neutral molecular spacers. Intercalation expands the interlayer spacing from 0.5 to 1-4 nm and reconstructs the stacking while preserving the intralayer 1T' framework. Controlled (de)intercalation reversibly switches the system between a superconducting metal and an insulator with an activation gap matching that of the isolated monolayer. Density functional theory indicates that the electronically decoupled layers retain the nontrivial Z2 topology of the monolayer. Chiral amine intercalation further induces chiroptical activity in WS2 electronic transitions. Overall, the successful intercalation challenges the long-held view that group VIB dichalcogenides are inert toward neutral-molecule intercalation and demonstrates molecular intercalation as a general chemical route for realizing monolayer-like topological-insulator physics and enabling chiral van der Waals superlattices in bulk single crystals.

We investigate the many-body ground states in a family of fractionally filled bands where the Berry curvature fluctuations can be tuned while maintaining ideal quantum geometry. We numerically find that the neutral gap of the fractional Chern insulator (FCI) ground state decreases as the Berry curvature becomes less homogeneous, ultimately driving an instability to a charge density wave. We further extend our analysis to bands perturbed away from the ideal limit and give examples where a less ideal band geometry results in a more stable FCI phase. To explain our findings, we apply the single-mode approximation to the ground-state wave functions of the ideal band, from which we obtain analytic expressions for the FCI many-body gap in the limit of weak Berry curvature variations. Finally, we make a connection between our results and experimentally relevant systems where FCIs have been observed.

We propose a new pathway to the quantized anomalous Hall effect (QAHE) by coupling an altermagnet to a topological crystalline insulator (TCI). The former gaps the topological surface states of the TCI, thereby realizing the QAHE in a robust and switchable platform with near- vanishing magnetization. We demonstrate the feasibility of this approach by studying a slab of the TCI SnTe coupled to an altermagnetic RuO2 layer. Our first-principles calculations reveal that the d-wave altermagnetism in RuO2 induces a 7 meV gap to the Dirac surface states on the (110) surface of SnTe, producing a finite anomalous Hall effect. Our approach generalizes to broader classes of altermagnetic materials and TCIs, thereby providing a family of topological altermagnetic heterostructures with small or vanishing magnetization that support nontrivial Chern numbers. Our results highlight a promising new topological platform with great tunability and applications to spintronics.

We present a geometric formulation of optical, thermoelectric, and thermal linear response in clean, zero temperature band insulators based on a single object: a generalized time-dependent quantum geometric tensor (g-tQGT) built from correlations of projected particle and heat polarization operators. Within this framework, the AC transport tensors admit compact expressions that make their geometric content explicit. The response splits into a Berry curvature contribution that remains finite in the DC limit and a frequency correction governed by the quantum metric, implying geometry driven effects even in topologically trivial insulators. At equal times, the g-tQGT recovers the usual integrated QGT and yields energy-weighted thermal analogs whose antisymmetric parts are fixed by orbital and heat magnetization. Importantly, in the thermal channel, a thermal quantum geometric tensor is obtained. Casting the theory in a Hilbert-Schmidt inner product form yields a bound on the trace of the thermal QGT, an uncertainty relation on the projected polarization operators and a purely geometric upper bound on the finite-time accumulated response. The latter is used in the optical channel to derive a geometric upper bound on the electric current. Finally, time derivatives of the g-tQGT are used to generate a hierarchy of generalized thermoelectric and thermal sum rules, and bounds on these sum rules are obtained. These bounds are used to find inequalities between different physical objects such as the optical mass, susceptibility functions and magnetizations.

We develop a symmetry indicator framework to efficiently predict the topology of superlattice-induced minibands with spin-orbit coupling. Our algorithm requires input only from the parent material before the superlattice is applied. The simplification arises by assuming a perturbatively weak superlattice potential; however, our results extend beyond the perturbative regime as long as the superlattice-induced gaps remain open. We first consider a time-reversal- and inversion-symmetric system subject to a weak superlattice potential and derive a compact formula for the Z2 invariant of the lowest miniband. We then extend to time-reversal breaking systems and compute the Chern number. We apply our theory to selected transition-metal dichalcogenides, HgTe/CdTe quantum wells, and thin films of three-dimensional topological insulators and Dirac semimetals. We find topological superlattice bands can arise even from nontopological materials, broadening the pool of candidates for realizing topological flat bands. Our theory predicts which geometry and periodicity of superlattice will yield topological bands for a given material, providing a clear guiding principle for designing topological superlattice heterostructures.

Quantum geometry plays a fundamental role in many aspects of condensed matter physics. Among its central objects are the Berry curvature and the quantum metric – quantities that, while distinct, are intertwined through geometric constraints. In this article, we survey recent progress in understanding when and how this bound is saturated, with particular emphasis on the emergence of momentum-space holomorphicity of Bloch states. These developments highlight a profound connection between certain ideal Bloch bands and the Hilbert space structure of the lowest Landau level. We elucidate this relationship through a review of quantum Hall physics in both homogeneous and spatially varying magnetic fields, and conclude by exploring its implications for the search for fractionalized phases in emerging platforms, including moiré materials.

Graphene-based multilayer systems serve as versatile platforms for exploring the interplay between electron correlation and topology, thanks to distinctive low-energy bands marked by significant quantum metric and Berry curvature from graphene's Dirac bands. Here, we investigate Mott physics and local spin moments in Dirac bands hybridized with a flat band of localized orbitals in functionalized graphene. Via hybridization control, a topological transition is realized between two symmetry-distinct site-selective Mott states featuring local moments in different Wyckoff positions, with a geometrically enforced metallic state emerging in between. We find that this geometrically controlled real-space switching of local moments and associated metal-insulator physics may be realized through proximity coupling of epitaxial graphene on SiC(0001) with group IV intercalants, where the Mott state faces geometrical obstruction in the large-hybridization limit. Our Letter shows that chemically functionalized graphene provides a correlated electron platform, very similar to the topological heavy fermions in graphene moiré systems but at significantly enhanced characteristic energy scales.

Moiré materials host fractional Chern insulators and electron crystals in close proximity, but the mechanism selecting between them remains an open question. We address this competition in Chern bands with ideal but momentum-dependent quantum geometry – Aharonov-Casher bands. We present an ansatz wave function for generalized Wigner crystals and, by comparing its energy to that of the competing Laughlin-like state, map out the phase diagram at filling fraction ν=1/m as a function of the degree of geometric non-uniformity. Our work identifies quantum geometry-controlled zero point fluctuations of the charge density of the generalized Wigner crystal as the mechanism controlling its relative stability, implying a kind of quantum Lindemann criterion for the crystal-liquid phase boundary.

The discovery of the axial amplitude mode-commonly referred to as the Higgs mode-in charge density wave systems, such as rare-earth tritellurides, indicates the presence of a hidden order. A theoretical study proposed that this axial Higgs mode arises from a hidden orbital texture of the charge density wave, which produces a ferroaxial charge order. However, experimental evidence for the specific hidden order has been lacking. Here, we demonstrate a ferroaxial order of electronic origin throughout the rare-earth tritellurides. In ErTe3 and HoTe3, which exhibit two distinct charge density waves with different ordering temperatures, a detailed investigation shows that the high-temperature charge order phase breaks translational, rotational and all vertical as well as diagonal mirror symmetries. Furthermore, this phase produces an axial Higgs mode and an axial electronic gap. By contrast, the low-temperature phase breaks only translational symmetry and gives rise to a scalar Higgs mode. Notably, both phases preserve the space inversion and time-reversal symmetries. These findings are consistent with a ferroaxial phase driven by coupled orbital and charge orders, highlighting the role of Higgs modes in revealing hidden orders in systems with intertwined charge density waves.

The discovery of superconductivity and correlated electronic phases in twisted bilayer WSe$_2$ (Xia et al., Nature 2024; Guo et al., Nature 2025) has generated considerable excitement. Accompanying the superconductivity and a correlated insulator phase is the Kondo-lattice-like phenomenology in transport properties. Here we consider how such phenomenology can develop when the combination of the active bands are topological. We advance a unique construction of compact molecular orbitals through a partial Wannierization that is symmetry preserving. The resulting Anderson lattice model provides the basis for a microscopic understanding of the experimental observation, including the involved energy scales. Our approach may apply to a broad range of settings where topology and correlations interplay.

Van der Waals heterostructures promise on-demand designer quantum phases through control of monolayer composition, stacking, twist angle, and external fields. Yet, experimental efforts have been narrowly focused, leaving much of this vast moiré landscape unexplored and potential promises unrealized. Here, we present a scalable workflow for high-throughput characterization of twisted homobilayers and apply it to K-valley semiconductors. Combining small-scale density functional theory with perturbation theory, we efficiently extract moiré band gaps, valley Chern numbers, magic angles, and the threshold for lattice relaxation. Beyond this rapid high-throughput characterization, we parameterize a continuum model for each material, which provides a starting point for more detailed study. Our survey delivers an actionable map for systematic exploration of correlated and topological phases in moiré homobilayers, and identifies promising new platforms: chromium-based transition metal dichalcogenides for high-temperature quantum anomalous Hall effects, transition metal nitride halides for intertwined superconducting and moiré physics, and atomically thin III-V semiconductors for room-temperature-scale moiré effects.

Two-dimensional materials subject to long-wavelength modulations have emerged as novel platforms to study topological and correlated quantum phases. In this article, we develop a versatile and computationally inexpensive method to predict the topological properties of materials subjected to a superlattice potential by combining degenerate perturbation theory with the method of symmetry indicators. In the absence of electronic interactions, our analysis provides a systematic rule to find the Chern number of the superlattice-induced miniband starting from the harmonics of the applied potential and a few material-specific coefficients. Our method also applies to anomalous (interaction-generated) bands, for which we derive an efficient algorithm to determine all Chern numbers compatible with a self-consistent solution to the Hartree-Fock equations. Our approach gives a microscopic understanding of the quantum anomalous Hall insulators recently observed in rhombohedral graphene multilayers.

One of the most striking signatures of Weyl fermions is their surface Fermi arcs. Less known is that Fermi arcs can also be localized at internal twin boundaries where two Weyl materials of opposite chirality meet. In this work, we derive constraints on the topology and connectivity of these "internal Fermi arcs." We show that internal Fermi arcs can exhibit transport signatures and propose two probes: quantum oscillations and a quantized chiral magnetic current. We propose merohedrally twinned B20 materials as candidates to host internal Fermi arcs, verified through both model and ab initio calculations. Our theoretical investigation sheds lights on the topological features and motivates experimental studies into the intriguing physics of internal Fermi arcs.