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We show that introducing spin-singlet or spin-triplet superconductivity into twisted bilayer graphene induces higher-order topological superconductivity. $C_{2z}T$-protected corner states of Majorana Kramers pairs appear at the boundary between domains with opposite signs of pairing, and zero modes materialize in Abrikosov vortices. The topology of the superconducting phase originates from the anomaly [1] -- the absence of a lattice support -- of the single-valley band structure of twisted bilayer graphene, which is protected by $C_{2z}T$ and the particle-hole symmetry $\cal P$. We prove that any pairing (spin-singlet or spin-triplet) term preserving valley-U(1), spin-SU(2), time-reversal, $C_{2z}T$, and $\cal P$ must drive the system into a higher-order topological superconductor phase. Here spin-SU(2) is the global spin-SU(2) for the singlet pairing and a combination of two SU(2)'s in the two valleys for the triplet pairing. Using a Dirac Hamiltonian, we demonstrate the existence of corner modes and confirm this with numerical calculations. These corner states are stable even if the approximate particle-hole symmetry $\cal P$ is weakly broken, which is true in experimental setups. Finally, we suggest an experiment to detect the topological superconductivity: by observing the fractional Josephson effect in a TBG-TSC Josephson system., 6+31 pages, 3+8 figures

We study the correlated insulating phases of twisted bilayer graphene (TBG) in the absence of lattice strain at integer filling $\nu=\pm3$. Using the self-consistent Hartree-Fock method on a particle-hole symmetric model and allowing translation symmetry breaking terms, we obtain the phase diagram with respect to the ratio of $AA$ interlayer hopping $(w_0)$ and $AB$ interlayer hopping $(w_1)$. When the interlayer hopping ratio is close to the chiral limit ($w_0/w_1 \lesssim 0.5$), a quantum anomalous Hall state with Chern number $\nu_c = \pm 1$ can be observed consistent with previous studies. Around the realistic value $w_0/w_1 \approx 0.8$, we find a spin and valley polarized, translation symmetry breaking, state with $C_{2z}T$ symmetry, a charge gap and a doubling of the moir\'e unit cell, dubbed the $C_{2z}T$ stripe phase. The real space total charge distribution of this $C_{2z}T$ stripe phase in the flat band limit does not have modulation between different moir\'e unit cells, although the charge density in each layer is modulated, and the translation symmetry is strongly broken. Other symmetries, including $C_{2z}$, $C_{2x}$ and particle-hole symmetry $P$, and the topology of the $C_{2z}T$ stripe phase are also discussed in detail. We observed braiding and annihilation of the Dirac nodes by continuously turning on the order parameter to its fully self-consistent value, and provide a detailed explanation of the mechanism for the charge gap opening despite preserving $C_{2z}T$ and valley $U(1)$ symmetries. In the transition region between the quantum anomalous Hall phase and the $C_{2z}T$ stripe phase, we find an additional competing state with comparable energy corresponding to a phase with a tripling of the moir\'e unit cell., Comment: 39 pages, 33 figures

Electronic structure calculations indicate that the Sr2FeSbO6 double perovskite has a flat-band set just above the Fermi level that includes contributions from ordinary sub-bands with weak kinetic electron hopping plus a flat sub-band that can be attributed to the lattice geometry and orbital interference. To place the Fermi energy in that flat band, electron doped samples with formulas Sr2-xLaxFeSbO6 (0 < x < 0.3) were synthesized and their magnetism and ambient temperature crystal structures determined by high-resolution synchrotron X-ray powder diffraction. All materials appear to display an antiferromagnetic-like maximum in the magnetic susceptibility, but the dominant spin coupling evolves from antiferromagnetic to ferromagnetic on electron doping. Which of the three sub-bands or combinations is responsible for the behavior has not been determined., 31 pages, 8 figures

Topological materials discovery has emerged as an important frontier in condensed matter physics. While theoretical classification frameworks have been used to identify thousands of candidate topological materials, experimental determination of materials' topology often poses significant technical challenges. X-ray absorption spectroscopy (XAS) is a widely-used materials characterization technique sensitive to atoms' local symmetry and chemical bonding, which are intimately linked to band topology by the theory of topological quantum chemistry (TQC). Moreover, as a local structural probe, XAS is known to have high quantitative agreement between experiment and calculation, suggesting that insights from computational spectra can effectively inform experiments. In this work, we leverage computed X-ray absorption near-edge structure (XANES) spectra of more than 10,000 inorganic materials to train a neural network (NN) classifier that predicts topological class directly from XANES signatures, achieving F$_1$ scores of 89% and 93% for topological and trivial classes, respectively. Additionally, we obtain consistent classifications using corresponding experimental and computational XANES spectra for a small number of measured compounds. Given the simplicity of the XAS setup and its compatibility with multimodal sample environments, the proposed machine learning-augmented XAS topological indicator has the potential to discover broader categories of topological materials, such as non-cleavable compounds and amorphous materials, and may further inform field-driven phenomena in situ, such as magnetic field-driven topological phase transitions.

Bloch's theorem is the centerpiece of topological band theory, which itself has defined an era of quantum materials research. However, Bloch's theorem is broken by a perpendicular magnetic field, making it difficult to study topological systems in strong flux. For the first time, moir\'e materials have made this problem experimentally relevant, and its solution is the focus of this work. We construct gauge-invariant irreps of the magnetic translation group at $2\pi$ flux on infinite boundary conditions, allowing us to give analytical expressions in terms of the Siegel theta function for the magnetic Bloch Hamiltonian, non-Abelian Wilson loop, and many-body form factors. We illustrate our formalism using a simple square lattice model and the Bistritzer-MacDonald Hamiltonian of twisted bilayer graphene, obtaining reentrant ground states at $2\pi$ flux under the Coulomb interaction.

Additional material for the article "All topological bands of all nonmagnetic stoichiometric materials”, published in Science on 20 May 2022.

The discovery of new catalysts that are efficient and sustainable is a major research endeavor for many industrial chemical processes. This requires an understanding and determination of the catalytic origins, which remains a challenge. Here, a novel method to identify the position of active sites based on searching for crystalline symmetry-protected obstructed atomic insulators (OAIs) that have metallic surface states is described. The obstructed Wannier charge centers (OWCCs) in OAIs are pinned by symmetries at some empty Wyckoff positions so that surfaces that accommodate these sites are guaranteed to have metallic obstructed surface states (OSSs). It is proposed and confirmed that the OSSs are the catalytic activity origins for crystalline materials. The theory on 2H-MoTe2, 1T′-MoTe2, and NiPS3 bulk single crystals is verified, whose active sites are consistent with the calculations. Most importantly, several high-efficiency catalysts are successfully identified just by considering the number of OWCCs and the symmetry. Using the real-space-invariant theory applied to a database of 34 013 topologically trivial insulators, 1788 unique OAIs are identified, of which 465 are potential high-performance catalysts. The new methodology will facilitate and accelerate the discovery of new catalysts for a wide range of heterogeneous redox reactions.

Interactions among electrons and the topology of their energy bands can create novel quantum phases of matter. Most topological electronic phases appear in systems with weak electron-electron interactions. The instances where topological phases emerge only as a result of strong interactions are rare, and mostly limited to those realized in the presence of intense magnetic fields. The discovery of flat electronic bands with topological character in magic-angle twisted bilayer graphene (MATBG) has created a unique opportunity to search for new strongly correlated topological phases. Here we introduce a novel local spectroscopic technique using a scanning tunneling microscope (STM) to detect a sequence of topological insulators in MATBG with Chern numbers C = $\pm$ 1, $\pm$ 2, $\pm$ 3, which form near $\nu$ = $\pm$ 3, $\pm$ 2, $\pm$ 1 electrons per moir\'e unit cell respectively, and are stabilized by the application of modest magnetic fields. One of the phases detected here (C = +1) has been previously observed when the sublattice symmetry of MATBG was intentionally broken by hexagonal boron nitride (hBN) substrates, with interactions playing a secondary role. We demonstrate that strong electron-electron interactions alone can produce not only the previously observed phase, but also new and unexpected Chern insulating phases in MATBG. The full sequence of phases we observed can be understood by postulating that strong correlations favor breaking time-reversal symmetry to form Chern insulators that are stabilized by weak magnetic fields. Our findings illustrate that many-body correlations can create topological phases in moir\'e systems beyond those anticipated from weakly interacting models., Comment: 17 pages, 4 figures

Magic-angle twisted bilayer graphene (MATBG) exhibits a rich variety of electronic states, including correlated insulators, superconductors, and topological phases. Understanding the microscopic mechanisms responsible for these phases requires determining the interplay between electron-electron interactions and quantum degeneracy due to spin and valley degrees of freedom. Signatures of strong electron-electron correlations have been observed at partial fillings of the flat electronic bands in recent spectroscopic measurements. Transport experiments have shown changes in the Landau level degeneracy at fillings corresponding to an integer number of electrons per moir\'e unit cell. However, the interplay between interaction effects and the degeneracy of the system is currently unclear. Using high-resolution scanning tunneling microscopy (STM), we observed a cascade of transitions in the spectroscopic properties of MATBG as a function of electron filling. We find distinct changes in the chemical potential and a rearrangement of the low-energy excitations at each integer filling of the moir\'e flat bands. These spectroscopic features are a direct consequence of Coulomb interactions, which split the degenerate flat bands into Hubbard sub-bands. We find these interactions, the strength of which we can extract experimentally, to be surprisingly sensitive to the presence of a perpendicular magnetic field, which strongly modifies the spectroscopic transitions. The cascade of transitions we report here characterizes the correlated high-temperature parent phase from which various insulating and superconducting ground-state phases emerge at low temperatures in MATBG.

Dirac and Weyl semimetals both exhibit arc-like surface states. However, whereas the surface Fermi arcs in Weyl semimetals are topological consequences of the Weyl points themselves, the surface Fermi arcs in Dirac semimetals are not directly related to the bulk Dirac points, raising the question of whether there exists a topological bulk-boundary correspondence for Dirac semimetals. In this work, we discover that strong and fragile topological Dirac semimetals exhibit one-dimensional (1D) higher-order hinge Fermi arcs (HOFAs) as universal, direct consequences of their bulk 3D Dirac points. To predict HOFAs coexisting with topological surface states in solid-state Dirac semimetals, we introduce and layer a spinful model of an s–d-hybridized quadrupole insulator (QI). We develop a rigorous nested Jackiw–Rebbi formulation of QIs and HOFA states. Employing ab initio calculations, we demonstrate HOFAs in both the room- (α) and intermediate-temperature (α″) phases of Cd3As2, KMgBi, and rutile-structure (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \beta ^{\prime} $$\end{document}β′-) PtO2., The existence of a topological bulk-boundary correspondence for Dirac semimetals has remained an open question. Here, Wieder et al. predict one-dimensional hinge states originating from bulk three-dimensional Dirac points in solid-state Dirac semimetals, revealing condensed matter Dirac fermions to be higher-order topological.

We present a two-step method specifically tailored for band structure calculation of the small-angle moir\'{e}-pattern materials which contain tens of thousands of atoms in a unit cell. In the first step, the self-consistent field calculation for ground state is performed with $O(N)$ Krylov subspace method implemented in OpenMX. Secondly, the crystal momentum dependent Bloch Hamiltonian and overlap matrix are constructed from the results obtained in the first step and only a small number of eigenvalues near the Fermi energy are solved with shift-invert and Lanczos techniques. By systematically tuning two key parameters, the cutoff radius for electron hopping interaction and the dimension of Krylov subspace, we obtained the band structures for both rigid and corrugated twisted bilayer graphene structures at the first magic angle ($\theta=1.08^\circ$) and other three larger ones with satisfied accuracy on affordable costs. The band structures are in good agreement with those from tight binding models, continuum models, plane-wave pseudo-potential based $ab~initio$ calculations, and the experimental observations. This efficient two-step method is to play a crucial role in other twisted two-dimensional materials, where the band structures are much more complex than graphene and the effective model is hard to be constructed.

We consider the extent to which symmetry eigenvalues reveal the topological character of bands. Specifically, we compare distinct atomic limit phases (band representations) that share the same irreducible representations (irreps) at all points in the Brillouin zone and, therefore, appear equivalent in a classification based on eigenvalues. We derive examples where such "irrep-equivalent" phases can be distinguished by a quantized Berry phase or generalization thereof. These examples constitute a generalization of the Su-Schrieffer-Heeger chain: neither phase is topological, in the sense that localized Wannier functions exist, yet there is a topological obstruction between them. We refer to two phases as "Berry obstructed atomic limits" if they have the same irreps, but differ by Berry phases. This is a distinct notion from eigenvalue obstructed atomic limits, which differ in their symmetry irreps at some point in the Brillouin zone. We compute exhaustive lists of elementary band representations that are irrep-equivalent, in all space groups, with and without time-reversal symmetry and spin-orbit coupling, and use group theory to derive a set of necessary conditions for irrep-equivalence. Finally, we conjecture, and in some cases prove, that irrep-equivalent elementary band representations that are not equivalent can be distinguished by a topological invariant., 18 pages, plus tables and Supplementary Material

A general theoretical technique is introduced to identify materials that host flat bands. Applying topological quantum chemistry provides the generating bases for these flat bands in all space groups. Exotic phases of matter can emerge from the interplay between strong electron interactions and non-trivial topology. Materials that have non-dispersing bands in their electronic band structure, such as twisted bilayer graphene, are prime candidates for strongly interacting physics. However, existing theoretical models for obtaining these 'flat bands' in crystals are often too restrictive for experimental realizations. Here we present a generic theoretical technique for constructing perfectly flat bands from bipartite crystalline lattices. Our prescription encapsulates and generalizes the various flat-band models in the literature and is applicable to systems with any orbital content, with or without spin-orbit coupling. Using topological quantum chemistry, we build a complete topological classification in terms of symmetry eigenvalues of all the gapped and gapless flat bands. We also derive criteria for the existence of symmetry-protected band touching points between the flat and dispersive bands, and identify the gapped flat bands as prime candidates for fragile topological phases. Finally, we show that the set of all perfectly flat bands is finitely generated and construct the corresponding bases for all 1,651 Shubnikov space groups.

While the chiral linear Luttinger liquid is integrable via bosonization, its non-linear counterpart does not admit for an analytic solution. In this work, we find a sub-extensive number of exact eigenstates for a large family of density-density interaction terms. These states are embedded in a continuum of strongly-correlated excited states. The real-space entanglement entropy of some exact states scales logarithmically with system size, while that of others has volume-law scaling. We introduce momentum-space entanglement as an unambiguous differentiator between these exact states and the remaining excited states. With regard to momentum space, the exact states behave as bona fide quantum many body scars: they exhibit identically zero momentum-space entanglement, while typical eigenstates behave thermally. We corroborate this finding by a level statistics analysis. Furthermore, we detail the general formalism for systematically finding all interaction terms and associated exact states, and present a number of infinite exact state sequences extending to arbitrarily high energies. Unlike many previous examples of quantum many body scars, the exact states uncovered here do not lie at equidistant energies, and do not follow from a special operator algebra. Instead, they are uniquely enabled by the interplay of Fermi statistics and chirality., Comment: 27 pages, 10 figures, accepted manuscript

Magnetic topological materials represent a class of compounds with properties that are strongly influenced by the topology of their electronic wavefunctions coupled with the magnetic spin configuration. Such materials can support chiral electronic channels of perfect conduction, and can be used for an array of applications, from information storage and control to dissipationless spin and charge transport. Here we review the theoretical and experimental progress achieved in the field of magnetic topological materials, beginning with the theoretical prediction of the quantum anomalous Hall effect without Landau levels, and leading to the recent discoveries of magnetic Weyl semimetals and antiferromagnetic topological insulators. We outline recent theoretical progress that has resulted in the tabulation of, for the first time, all magnetic symmetry group representations and topology. We describe several experiments realizing Chern insulators, Weyl and Dirac magnetic semimetals, and an array of axionic and higher-order topological phases of matter, and we survey future perspectives.

We study the conditions for Bloch bands to be spanned by symmetric and strictly compact Wannier states that have zero overlap with all lattice sites beyond a certain range. Similar to the characterization of topological insulators in terms of an algebraic (rather than exponential) localization of Wannier states, we find that there may be impediments to the compact localization even of topologically "trivial" obstructed atomic insulators. These insulators admit exponentially-localized Wannier states centered at unoccupied orbitals of the crystalline lattice. First, we establish a sufficient condition for an insulator to have a compact representative. Second, for $\mathcal{C}_2$ rotational symmetry, we prove that the complement of fragile topological bands cannot be compact, even if it is an atomic insulator. Third, for $\mathcal{C}_4$ symmetry, our findings imply that there exist fragile bands with zero correlation length. Fourth, for a $\mathcal{C}_3$-symmetric atomic insulator, we explicitly derive that there are no compact Wannier states overlapping with less than $18$ lattice sites. We conjecture that this obstruction generalizes to all finite Wannier sizes. Our results can be regarded as the stepping stone to a generalized theory of Wannier states beyond dipole or quadrupole polarization., 6 pages, 4 figures, supplement included

Magic-angle (θ=1.05°) twisted bilayer graphene (MATBG) has shown two seemingly contradictory characters: the localization and quantum-dot-like behavior in STM experiments, and delocalization in transport experiments. We construct a model, which naturally captures the two aspects, from the Bistritzer-MacDonald (BM) model in a first principle spirit. A set of local flat-band orbitals (f) centered at the AA-stacking regions are responsible to the localization. A set of extended topological semimetallic conduction bands (c), which are at small energetic separation from the local orbitals, are responsible to the delocalization and transport. The topological flat bands of the BM model appear as a result of the hybridization of f and c electrons. This model then provides a new perspective for the strong correlation physics, which is now described as strongly correlated f electrons coupled to nearly free c electrons-we hence name our model as the topological heavy fermion model. Using this model, we obtain the U(4) and U(4)×U(4) symmetries of Refs. [1-5] as well as the correlated insulator phases and their energies. Simple rules for the ground states and their Chern numbers are derived. Moreover, features such as the large dispersion of the charge ±1 excitations [2,6,7], and the minima of the charge gap at the Γ_{M} point can now, for the first time, be understood both qualitatively and quantitatively in a simple physical picture. Our mapping opens the prospect of using heavy-fermion physics machinery to the superconducting physics of MATBG.

Twisted bilayer graphene (TBG) is remarkable for its topological flat bands, which drive strongly-interacting physics at integer fillings, and its simple theoretical description facilitated by the Bistritzer-MacDonald Hamiltonian, a continuum model coupling two Dirac fermions. Due to the large moir\'e unit cell, TBG offers the unprecedented opportunity to observe reentrant Hofstadter phases in laboratory-strength magnetic fields near $25$T. This Letter is devoted to magic angle TBG at $2\pi$ flux where the magnetic translation group commutes. We use a newly developed gauge-invariant formalism to determine the exact single-particle band structure and topology. We find that the characteristic TBG flat bands reemerge at $2\pi$ flux, but, due to the magnetic field breaking $C_{2z} \mathcal{T}$, they split and acquire Chern number $\pm1$. We show that reentrant correlated insulating states appear at $2\pi$ flux driven by the Coulomb interaction at integer fillings, and we predict the characteristic Landau fans from their excitation spectrum. We conjecture that superconductivity can also be re-entrant at $2\pi$ flux., Comment: (See also "Observation of re-entrant correlated insulators and interaction driven Fermi surface reconstructions at one magnetic flux quantum per moire unit cell in magic-angle twisted bilayer graphene" by Das et al.)

Topological materials discovery has evolved at a rapid pace over the past 15 years following the identification of the first nonmagnetic topological insulators (TIs), topological crystalline insulators (TCIs), and 3D topological semimetals (TSMs). Most recently, through complete analyses of symmetry-allowed band structures - including the theory of Topological Quantum Chemistry (TQC) - researchers have determined crystal-symmetry-enhanced Wilson-loop and complete symmetry-based indicators for nonmagnetic topological phases, leading to the discovery of higher-order TCIs and TSMs. The recent application of TQC and related methods to high-throughput materials discovery has revealed that over half of all of the known stoichiometric, solid-state, nonmagnetic materials are topological at the Fermi level, over 85% of the known stoichiometric materials host energetically isolated topological bands, and that just under $2/3$ of the energetically isolated bands in known materials carry the stable topology of a TI or TCI. In this Review, we survey topological electronic materials discovery in nonmagnetic crystalline solids from the prediction of the first 2D and 3D TIs to the recently introduced methods that have facilitated large-scale searches for topological materials. We also discuss future venues for the identification and manipulation of solid-state topological phases, including charge-density-wave compounds, magnetic materials, and 2D few-layer devices., Comment: Final version, 27 pages, 7 figures

We analytically compute the scanning tunneling microscopy (STM) signatures of integer-filled correlated ground states of the magic angle twisted bilayer graphene (TBG) narrow bands. After experimentally validating the strong-coupling approach at $\pm 4$ electrons/moir\'e unit cell, we consider the spatial features of the STM signal for 14 different many-body correlated states and assess the possibility of Kekul\'e distortion (KD) emerging at the graphene lattice scale. Remarkably, we find that coupling the two opposite graphene valleys in the intervalley-coherent (IVC) TBG insulators does not always result in KD. As an example, we show that the Kramers IVC state and its nonchiral $\mathrm{U} \left( 4 \right)$ rotations do not exhibit any KD, while the time-reversal-symmetric IVC state does. Our results, obtained over a large range of energies and model parameters, show that the STM signal and Chern number of a state can be used to uniquely determine the nature of the TBG ground state., Comment: 8+80 pages, 2+65 figures. New version matches the published version

Flat-band superconductivity has theoretically demonstrated the importance of band topology to correlated phases. In two dimensions, the superfluid weight, which determines the critical temperature through the Berezinksii-Kosterlitz-Thouless criteria, is bounded by the Fubini-Study metric at zero temperature. We show this bound is nonzero within flat bands whose Wannier centers are obstructed from the atoms - even when they have identically zero Berry curvature. Next, we derive general lower bounds for the superfluid weight in terms of momentum space irreps in all 2D space groups, extending the reach of topological quantum chemistry to superconducting states. We find that the bounds can be naturally expressed using the formalism of real space invariants (RSIs) that highlight the separation between electronic and atomic degrees of freedom. Finally, using exact Monte Carlo simulations on a model with perfectly flat bands and strictly local obstructed Wannier functions, we find that an attractive Hubbard interaction results in superconductivity as predicted by the RSI bound beyond mean-field. Hence, a nonzero superfluid weight constitutes a nontrivial bulk property that distinguishes obstructed bands from trivial bands in the presence of interactions., 5+31 pages, 3+2 figures, 2 tables

For over 100 years, the group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry. However, the group theory of crystals with commensurate magnetic order has remained incomplete for the past 70 years, due to the complicated symmetries of magnetic crystals. In this work, we complete the 100-year-old problem of crystalline group theory by deriving the small corepresentations, momentum stars, compatibility relations, and magnetic elementary band corepresentations of the 1,421 magnetic space groups (MSGs), which we have made freely accessible through tools on the Bilbao Crystallographic Server. We extend Topological Quantum Chemistry to the MSGs to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids – Magnetic Topological Quantum Chemistry (MTQC). Using MTQC, we derive the complete set of symmetry-based indicators of electronic band topology, for which we identify symmetry-respecting bulk and anomalous surface and hinge states., The band topology of nonmagnetic crystals can be characterized by Topological Quantum Chemistry (TQC), whereas the band topology of magnetic crystals remains unexplored. Here, the authors extend TQC to the magnetic space groups to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids.

The Hamiltonian of the magic-angle twisted symmetric trilayer graphene (TSTG) can be decomposed into a twisted-bilayer-graphene- (TBG-) like flat band Hamiltonian and a high-velocity Dirac fermion Hamiltonian. We use Hartree-Fock mean field approach to study the projected Coulomb interacting Hamiltonian of TSTG developed in C\ifmmode \u{a}\else \u{a}\fi{}lug\ifmmode \u{a}\else \u{a}\fi{}ru et al. [Phys. Rev. B 103, 195411 (2021)] at integer fillings $\ensuremath{\nu}=\ensuremath{-}3,\phantom{\rule{0.222222em}{0ex}}\ensuremath{-}2,\phantom{\rule{0.222222em}{0ex}}\ensuremath{-}1$, and 0 measured from charge neutrality. We study the phase diagram with ${w}_{0}/{w}_{1}$, the ratio of $AA$ and $AB$ interlayer hoppings, and the displacement field, which introduces an interlayer potential $U$ and hybridizes the TBG-like bands with the Dirac bands. At small $U$, we find the ground states at all fillings $\ensuremath{\nu}$ are in the same phases as the tensor products of a Dirac semimetal with the filling $\ensuremath{\nu}$ TBG insulator ground states, which are spin-valley polarized at $\ensuremath{\nu}=\ensuremath{-}3$, and fully (partially) intervalley coherent at $\ensuremath{\nu}=\ensuremath{-}2,0$ ($\ensuremath{\nu}=\ensuremath{-}1$) in the flat bands. An exception is $\ensuremath{\nu}=\ensuremath{-}3$ with ${w}_{0}/{w}_{1}\ensuremath{\gtrsim}0.7$, which possibly becomes a metal with competing orders at small $U$ due to charge transfers between the Dirac and flat bands. At strong $U$ where the bandwidths exceed interactions, all the fillings $\ensuremath{\nu}$ enter a metal phase with small or zero valley polarization and intervalley coherence. Lastly, at intermediate $U$, semimetal or insulator phases with zero intervalley coherence may arise for $\ensuremath{\nu}=\ensuremath{-}2,\phantom{\rule{0.222222em}{0ex}}\ensuremath{-}1,\phantom{\rule{0.222222em}{0ex}}0$. Our results provide a simple picture for the electron interactions in TSTG systems, and reveal the connection between the TSTG and TBG ground states.

We study the quantum dynamics of a simple translation invariant, center-of-mass (CoM) preserving model of interacting fermions in one dimension (1D), which arises in multiple experimentally realizable contexts. We show that this model naturally displays the phenomenology associated with fractonic systems, wherein single charges can only move by emitting dipoles. This allows us to demonstrate the rich Krylov fractured structure of this model, whose Hilbert space shatters into exponentially many dynamically disconnected subspaces. Focusing on exponentially large Krylov subspaces, we show that these can be either be integrable or non-integrable, thereby establishing the notion of Krylov-restricted thermalization. We analytically find a tower of integrable Krylov subspaces of this Hamiltonian, all of which map onto spin-1/2 XX models of various system sizes. We also discuss the physics of the non-integrable subspaces, where we show evidence for weak Eigenstate Thermalization Hypothesis (ETH) restricted to each non-integrable Krylov subspace. Further, we show that constraints in some of the thermal Krylov subspaces cause the long-time expectation values of local operators to deviate from behavior typically expected from translation-invariant systems. Finally, we show using a Schrieffer-Wolff transformation that such models naturally appear as effective Hamiltonians in the large electric field limit of the interacting Wannier-Stark problem, and comment on connections of our work with the phenomenon of Bloch many-body localization.

The discovery of Quantum Many-Body Scars (QMBS) both in Rydberg atom simulators and in the Affleck-Kennedy-Lieb-Tasaki (AKLT) spin-1 chain model, have shown that a weak violation of ergodicity can still lead to rich experimental and theoretical physics. In this review, we provide a pedagogical introduction to and an overview of the exact results on weak ergodicity breaking via QMBS in isolated quantum systems with the help of simple examples such as the fermionic Hubbard model. We also discuss various mechanisms and unifying formalisms that have been proposed to encompass the plethora of systems exhibiting QMBS. We cover examples of equally-spaced towers that lead to exact revivals for particular initial states, as well as isolated examples of QMBS. Finally, we review Hilbert Space Fragmentation, a related phenomenon where systems exhibit a richer variety of ergodic and non-ergodic behaviors, and discuss its connections to QMBS., 52 pages, 13 figures. Review article, comments welcome. v2: Expanded discussion with several additional illustrations, references added

The crystal symmetry of a material dictates the type of topological band structures it may host, and therefore symmetry is the guiding principle to find topological materials. Here we introduce an alternative guiding principle, which we call 'quasi-symmetry'. This is the situation where a Hamiltonian has an exact symmetry at lower-order that is broken by higher-order perturbation terms. This enforces finite but parametrically small gaps at some low-symmetry points in momentum space. Untethered from the restraints of symmetry, quasi-symmetries eliminate the need for fine-tuning as they enforce that sources of large Berry curvature will occur at arbitrary chemical potentials. We demonstrate that a quasi-symmetry in the semi-metal CoSi stabilizes gaps below 2 meV over a large near-degenerate plane that can be measured in the quantum oscillation spectrum. The application of in-plane strain breaks the crystal symmetry and gaps the degenerate point, observable by new magnetic breakdown orbits. The quasi-symmetry, however, does not depend on spatial symmetries and hence transmission remains fully coherent. These results demonstrate a class of topological materials with increased resilience to perturbations such as strain-induced crystalline symmetry breaking, which may lead to robust topological applications as well as unexpected topology beyond the usual space group classifications., Nat. Phys. (2022)

Moir\'e superlattices in two-dimensional (2D) van der Waals (vdW) heterostructures provide 20 an efficient way to engineer electron band properties. The recent discovery of exotic quantum phases and their interplay in twisted bilayer graphene (tBLG) has built this moir\'e system one of the most renowned condensed matter platforms (1-10). So far the studies of tBLG has been mostly focused on the lowest two flat moir\'e bands at the first magic angle {\theta}m1 ~ 1.1{\deg}, leaving high-order moir\'e bands and magic angles largely unexplored. Here we report 25 an observation of multiple well-isolated flat moir\'e bands in tBLG close to the second magic angle {\theta}m2 ~ 0.5{\deg}, which cannot be explained without considering electron-election interactions. With high magnetic field magneto-transport measurements, we further reveal a qualitatively new, energetically unbound Hofstadter butterfly spectrum in which continuously extended quantized Landau level gaps cross all trivial band-gaps. The 30 connected Hofstadter butterfly strongly evidences the topologically nontrivial textures of the multiple moir\'e bands. Overall, our work provides a new perspective for understanding the quantum phases in tBLG and the fractal Hofstadter spectra of multiple topological bands.