Session F03: Topology and Interactions in Quantum Materials

Machine learning large-scale simulation and band-mixing fractional quantum anomalous Hall effect in twisted MoTe2

The emergence of topological flat bands in long-wavelength moiré superlattices provides exciting opportunities to realize the lattice analogs of both the integer and fractional quantum Hall effects without external magnetic fields. Recently, optical and direct transport evidence of both integer and fractional quantum anomalous Hall effects have been reported in twisted MoTe2. Here we investigate the moiré band structures and the strong correlation effects in twisted MoTe2 for a wide range of twist angles, employing a combination of various techniques. We first develop the neural network mapping of lattice potential and density functional Hamiltonian under various van der Waals corrections, using massive local training datasets and transfer learning. Using machine-learning accelerated large-scale first principles calculations, we obtain angle dependent topological bands and realistic continuum modeling descriptions down to small twist angles. Furthermore, we explore the phase diagrams and transition of the system through continuum model exact diagonalization. Our multi-band exact diagonalization analysis reveals significant band-mixing effects and the strong competition between charge density wave orders and fractional quantum anomalous Hall states.   

Beyond GNNs: Enhanced Topological Classification, Discovery, and Prediction

Topological materials have transformative potential across various technological areas, and there has been significant interest in accurately predicting these properties through machine learning, especially using Graph Neural Networks (GNNs). These graph models, and to a larger extent the Message Passing Neural Network variants, have been shown to work well for many applications. However, they face critical limitations in simultaneously modeling local and global atomic interactions, capturing mid-sized structural motifs, and approximating relevant classes of functions due to their bounded expressive power.

To address these challenges, we designed new representations based on isometry invariants of periodic point sets, and improved model architectures inspired by folklore variants of the Weisfeiler-Lehman graph isomorphism tests. These modifications have demonstrated substantial enhancements on benchmark materials datasets. In this talk, we report on the state-of-the-art models we trained for predicting topological classifications and highlight the practical insights and applications gained from our models.   

Topological Zero-Energy Domain-Wall States in Generalized Su-Schrieffer-Heeger and Kitaev Chains Beyond the Topological Classification

Conventional belief is that the Altland-Zirnbauer tenfold classification table for topological insulators and superconductors determines the existence of zero-energy topological states. We demonstrate zero-energy topological domain-wall states in classes thought to be forbidden using representative one-dimensional generalized Su-Schrieffer-Heeger (SSH) and Kitaev chains. The tight-binding and topological field theory indicates zero-energy domain-wall states robust to disorder emerges. A low-energy effective approach and SU(N) transformations simplifies the effective Hamiltonians into a form equivalent to SSH and Kitaev chains. Finally, we show the Berry curvature for the generalized SSH and Kitaev chains and that the respective Berry phase difference of neighboring domains are quantized. The quantized Berry phase difference indicates a general bulk-boundary principle, protecting zero-energy topological states.   

Quantum Geometry and Band Topology in Interacting Systems

Quantum geometry (or band geometry) and band topology describe, respectively, the local and global properties of the Bloch electron wavefunctions in quantum materials. While extensive research has been conducted on these topics at the single-particle level, the interplay between band geometry/topology and interactions is much less explored. In this talk, I will mainly cover two recent developments in this direction. First, I will discuss the significant contributions from the band geometry to the electron-phonon coupling constants in graphene and MgB₂. Then, I will talk about fractional Chern insulators (FCIs), which are induced by Coulomb interaction in fractionally filled (nearly-)flat Chern bands; FCIs have recently been observed in twisted MoTe₂ and graphene-hBN superlattices, and I will mainly address how to explain various experimental phenomena simultaneously. I will also cover other related aspects and future research directions.   

The quantum geometric origin of capacitance in insulators

Recent years have seen a significant focus on topological insulators famous for their unique electronic Hilbert space properties, often referred to as quantum geometry. Among these properties, the quantum metric is highly notable for its relevance in multiband superconductors among numerous other applications. Consequently, there is a growing demand to comprehend the features of the quantum metric and develop methods for its measurement.

We establish that this quantity can be routinely accessed through quasi-adiabatic AC transport probes. Specifically, we suggest that the quantum metric quantifies the oscillator strength of the electric dipole moment induced by the AC electric field, and serves as a main ingredient in determining the magnitude of the dielectric constant. This insight allows us to predict: (i) the quantization of the capacitance in Landau levels, (ii) an enhanced bulk capacitance in simple Chern insulators, and (iii) the sensitivity of the capacitance to the twist angle in twisted bilayer graphene.   

Unveiling topological phase complex through structurally tunning bismuth halides

The synergy of geometry, symmetry, and topology has profound implications especially in the study of topological materials. While strong topological insulators are not rare experimentally, the realizations of weak and higher-order topological insulators in solids are extremely challenging. Our recent studies have shown that quasi-one-dimensional Bi₄X₄ (X = Br, I) is an ideal system to overcome all such challenges, because of its weak interlayer coupling, large intrinsic gap, band simplicity near Fermi level, and multiple clean cleavage surfaces. Here we carry out a systematic study of this fertile candidate system through strategic doping of Br on I sites to tune the bulk crystal structural and achieve seven different structural phases. Strikingly, the intricate stacking manipulation leads to a cascade of topological phases including strong, weak, higher-order, and trivial insulators.  This study suggests that quasi-one-dimensional Bi₄X₄  is a superior semiconducting system for exploring topologically nontrivial phases and their potential applications.   

Orthorhombic bismuth halides: Novel topological insulators

Quasi-one-dimensional bismuth halides, Bi₄Br₄ and Bi₄I₄, are emerging as pivotal platforms for investigating a plethora of topological phases, such as weak and higher-order topological insulators and room-temperature quantum spin Hall insulators. Recent experiments have unveiled new structural phases in Bi₄Br_xI_4-x. By using first-principles calculations and topological band theories, we theoretically show that the new orthorhombic crystal structures host a variety of topological boundary states including Dirac surface states, helical hinge states, and hourglass-like states.    

Three-Dimensional Quantum Anomalous Hall Effect in Magnetic Topological Insulator Trilayers of Hundred-Nanometer Thickness

Magnetic topological states refer to a class of exotic phases in magnetic materials with their non-trivial topological property determined by magnetic spin configurations. An example of such states is the quantum anomalous Hall (QAH) state, which is a zero magnetic field manifestation of the quantum Hall effect. Current research in this direction focuses on QAH insulators with a thickness of less than 10 nm. The thick QAH insulators in the three-dimensional (3D) regime are limited, largely due to inevitable bulk carriers being introduced in thick magnetic TI samples. Here, we employ molecular beam epitaxy (MBE) to synthesize magnetic TI trilayers with a thickness of up to ~106 nm. We find these samples exhibit well-quantized Hall resistance and vanishing longitudinal resistance at zero magnetic field. By varying magnetic dopants, gate voltages, temperature, and external magnetic fields, we examine the properties of these thick QAH insulators and demonstrate the robustness of the 3D QAH effect. The realization of the well-quantized 3D QAH effect indicates that the side surface states of our thick magnetic TI trilayers are gapped and thus do not affect the QAH quantization. The 3D QAH insulators of hundred-nanometer thickness provide a promising platform for the exploration of fundamental physics, including axion physics and image magnetic monopole, and the advancement of electronic and spintronic devices to circumvent Moore’s law.   

Majorana representation for topological edge states in Dirac fermion with non-quantized Berry phase

Research on the Dirac fermions has expanded to encompass systems that are described by three-level Hamiltonians, including α-T₃ lattice realized in Hg_1−xCd_xTe at a critical doping x = 0.17 [Nat. Phys. 10, 233 (2014), PRB 92, 035118 (2015)]. The α-T₃ lattice is constructed by connecting an additional atom to one of two atoms in each unit cell of the honeycomb lattice, with relative hopping strength 0 ≤ α ≤ 1. By tuning α, the α-T₃ lattice interpolates graphene (pseudospin S=1/2) and dice lattice (S=1) for α=0 and α=1, respectively. This is followed by a transition from diamagnetic to paramagnetic orbital susceptibility as a manifestation of the continuous change of the Berry phase γ from π to 0 [PRL 112, 026402 (2014)]. Thus, it is interesting to study whether topologically protected edge states exist in α-T₃ ribbons, where γ is no longer quantized. However, the bulk-boundary correspondence is yet to be formulated.

Here, we investigate the bulk-boundary correspondence in zigzag ribbon of α-T₃ lattice. By imposing boundary conditions on the wavefunction, we analytically determine the ranges of edge states in the Brillouin zone. We find that the zigzag ribbon can be mapped into a variant of the Su-Schrieffer-Heeger chain [PR Research 4, 013185 (2022)] by unitary transforms of the Hamiltonian. Finally, we prove that the topologically trivial and non-trivial phases of the zigzag ribbon (indicated by the absence and presence of the edge states, respectively), can be distinguished from the Majorana representation for the eigenstate [PRB 104, 195131 (2021)], where the Z₂ invariant is defined from the azimuthal winding number of the eigenstate on the Bloch sphere.    

Spectral localizer in trivial metals as topological insulator zero-modes

Metals and topological insulators have in common that they cannot be described by exponentially localized wave-functions. Here we establish a relationship between these two seemingly unrelated observations. The connection is explicit in the low-lying states of the spectral localizer of trivial metals, an operator that measures the obstruction to finding localized eigenstates. The low-lying spectrum of the spectral localizer of metals is determined by the zero-mode solutions of Dirac fermions with a varying mass parameter. We use this observation, valid in any dimension, to determine the difference between the localizer spectrum of trivial and topological metals, and conjecture the spectrum of the localizer for fractional quantum Hall edges. Because the localizer is a local real-space operator, it may be used as a tool to differentiate between non-crystalline topological and trivial metals, and to characterize strongly correlated systems, for which local topological markers are scarce.