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Non-analytic Bloch eigenstates at isolated band degeneracy points exhibit singular behavior in the quantum metric. Here, a description of superfluid weight for zero-energy flat bands in proximity to other high-energy bands is presented, where they together form a singular band gap system. When the singular band gap closes, the geometric and conventional contributions to the superfluid weight as a function of the superconducting gap exhibit different crossover behaviors. The scaling behavior of superfluid weight with the band gap is studied in detail, and the effect on the Berezinskii-KosterlitzThouless (BKT) transition temperature is explored. It is discovered that tuning the singular band gap provides a unique mechanism for enhancing the supercurrent and critical temperature of two-dimensional (2D) superconductors.

The superfluid weight of an isolated flat band in multi-orbital superconductors contains contributions from the band's quantum metric and a lattice geometric term that depends on the orbital positions in the lattice. Since the superfluid weight is a measure of the superconductor's energy fluctuation, it is independent of the lattice geometry, leading to the minimal quantum metric of a band. Here, a perturbation approach is developed to study the superfluid weight and its lattice geometric dependence for composite bands. When all orbitals exhibit uniform pairing, the quantum geometric term contains each band's contribution and an inter-band contribution between every pair of bands in the composite. Based on a band representation analysis, they provide a topological lower bound for the superfluid weight of an isolated composite of flat bands. Using this perturbation approach, an analytical expression of the lattice geometric contribution is obtained. It is expressed in terms of Bloch functions, providing a convenient formula to calculate the superfluid weight for multi-orbital superconductors.

Crystal structure prediction (CSP) stands as a powerful tool in materials science, driving the discovery and design of innovative materials. However, existing CSP methods heavily rely on formation enthalpies derived from density functional theory (DFT) calculations, often overlooking differences between DFT and experimental values. Moreover, material synthesis is intricately influenced by factors such as kinetics and experimental conditions. To overcome these limitations, a novel collaborative approach was proposed for CSP that combines DFT with experimental data, utilizing advanced deep learning models and optimization algorithms. We illustrate the capability to predict formation enthalpies that closely align with actual experimental observations through the transfer learning on experimental data. By incorporating experimental synthesizable information of crystals, our model is capable of reverse engineering crystal structures that can be synthesized in experiments. Applying the model to 17 representative compounds, the results indicate that the model can accurately identify experimentally synthesized structures with high precision. Moreover, the obtained formation enthalpies and lattice constants closely align with experimental values, underscoring the model's effectiveness. The synergistic approach between theoretical and experimental data bridges the longstanding disparities between theoretical predictions and experimental results, thereby alleviating the demand for extensive and costly experimental trials.

In the field of symmetry-protected topological phases, a common wisdom is that the symmetries fix the topological classifications, but they alone cannot determine whether a system is topologically trivial or not. Here, we show that this is no longer true in cases where symmetries are projectively represented. Particularly, the Zak phase, a topological invariant of a one-dimensional system, can be entirely determined by the projective symmetry algebra (PSA). To demonstrate this remarkable effect, we propose a minimal model, termed as flux Su-Schrieffer-Heeger (SSH) model, where the bond dimerization in the original SSH model is replaced by a flux dimerization. We present experimental realization of our flux SSH model in an electric-circuit array, and our predictions are directly confirmed by experimental measurement. Our work refreshes the understanding of the relation between symmetry and topology, opens up new avenues for exploring PSA determined topological phases, and suggests flux dimerization as a novel approach for designing topological crystals.

A band-projection formalism is developed for calculating the superfluid weight in two-dimensional multi-orbital superconductors with an orbital-dependent pairing. It is discovered that, in this case, the band geometric superfluid stiffness tensor can be locally non-positive-definite in some regions of the Brillouin zone. When these regions are large enough or include nodal singularities, the total superfluid weight becomes non-positive-definite due to pairing fluctuations, resulting in the transition of a BCS state to a pair-density wave (PDW). This geometric BCS-PDW transition is studied in the context of two-orbital superconductors, and proof of the existence of a geometric BCS-PDW transition in a generic topological flat band is established.

In nature, active matter, such as worms or dogs, tend to spontaneously form a stable rotational cluster when they flock to the same food source on an unregulated and unconfined surface. In this paper we present an $n$-node flexible active matter model to study the collective motion due to the flocking of individual agents on a two-dimensional surface, and confirm that there exists a spontaneous stable cluster rotation synchronizing with a chirality produced by the alignment of their bodies under the impetus of the active force. A prefactor of 1.86 is obtained for the linear relationship between normalized angular velocity and chirality. The angular velocity of such a rotation is found to be dependent on the individual flexibility, the number of nodes in each individual, and the magnitude of the active force. The conclusions well explain the spontaneous stable rotation of clusters that exists in many flexible active matter, like worms or dogs, when they flock to the same single source.

Identifying novel topological properties of topological quantum states of matter, such as exemplified by the quantized Hall conductance, is a valuable step towards realizing materials with attractive topological attributes that guarantee their imperviousness to realistic imperfections, disorder and environmental disturbances. Is the gravitational coupling coefficient of topological quantum states of matter a promising candidate? Substantially building on well established results for quantum Hall states, using disclinations as tools for controlled creation of pristine spatial curvature free of undesirable artifacts such as would interfere with the electronic motion of interest, herein we report that a large class of lattice topological states of matter exhibit gravitational response, i.e., charge response to intrinsic spatial curvature. This phenomenon is characterized by a topologically quantized coupling constant. Remarkably, the charge-gravity relationship remains linear in the curvature, up to the maximum curvature achievable on the lattice, demonstrating absence of higher order nonlinear response. Our findings facilitate articulating the physical principles underlying the topological quantization of the gravitational coupling constant, in analogy with the insights offered by the Chern number description of the quantized Hall conductance.

The constant potential molecular dynamics simulation method proposed by Siepmann and Sprik and reformulated later by Reed (SR-CPM) has been widely employed to investigate the metallic electrolyte/electrode interfaces, especially for conducting nanochannels with complex connectivity, *e.g.*, carbide-derived carbon or graphene-assembled membrane. This work makes substantial extensions of this seminal SR-CPM approach. First, we introduce two numerical techniques to determine electrode atom charges with an order of magnitude improvement in computational efficiency compared with those widely employed methods. The first numerical technique dramatically accelerates the to calculation of the Ewald interaction matrix $\mathbf{E}$, which takes advantage of the existing highly optimised electrostatic codes. The second technique introduces a new preconditioning technique in the conjugate gradient method to considerably increase the computational efficiency of a linear equation system that determines electrode atomic charges. Our improved SR-CPM implemented in the LAMMPS package can handle extra-large systems, *e.g.*, over 8.1 million electrode atoms. Second, after demonstrating the importance of the electroneutrality constraint, we propose a two-step method to enforce electroneutrality in the following post-treatment step, applicable for matrix and iterative techniques. Third, we propose a solid theoretical analysis for the adjustable parameter $\alpha_i$ (namely the atomic Hubbard-U $U_i^0$), which is arbitrarily selected in many SR-CPM simulation practices. We proposed that the optimised $\alpha_i$ or $U_i^0$ should compensate for the electrical potential/energy discrepancy between the discrete atomistic model and the continuum limit. The analytical and optimal ${\alpha}_i^0$ values are derived for a series of 2D materials.

The quantum anomalous Hall (QAH) effect in magnetic topological insulator (TI) represents a new state of matter originated from the interplay between topology and magnetism. The defining characteristics of the QAH ground state are the quantized Hall resistivity ($\rho_{yx}$) and vanishing longitudinal resistivity ($\rho_{xx}$) in the absence of external magnetic field. A fundamental question concerning the QAH effect is whether it is merely a zero-magnetic-field quantum Hall (QH) effect, or if it can host unique quantum phases and phase transitions that are unavailable elsewhere. The most dramatic departure of the QAH systems from other QH systems lies in the strong magnetic disorders that induce spatially random magnetization. Because disorder and magnetism play pivotal roles in the phase diagram of two-dimensional electron systems, the high degree of magnetic disorders in QAH systems may create novel phases and quantum critical phenomena. In this work, we perform systematic transport studies of a series of magnetic TIs with varied strength of magnetic disorders. We find that the ground state of QAH effect can be categorized into two distinct classes: the QAH insulator and anomalous Hall (AH) insulator phases, as the zero-magnetic-field counterparts of the QH liquid and Hall insulator in the QH systems. In the low disorder limit of the QAH insulator regime, we observe a universal quantized longitudinal resistance $\rho_{xx} = h/e^{2}$ at the coercive field. In the AH insulator regime, we find that a magnetic field can drive it to the QAH insulator phase through a quantum critical point with distinct scaling behaviors from that in the QH phase transition. We propose that the transmission between chiral edge states at domain boundaries, tunable by disorder and magnetic fields, is the key for determining the QAH ground state.

A pair of Dirac points (analogous to a vortex-antivortex pair) associated with opposite topological numbers (with $\pm\pi$ Berry phases) can be merged together through parameter tuning and annihilated to gap the Dirac spectrum, offering a canonical example of a topological phase transition. Here, we report transport studies on thin films of BiSbTeSe$_2$ (BSTS), which is a 3D TI that hosts spin-helical gapless (semi-metallic) Dirac fermion surface states (SS) for sufficiently thick samples, with an observed resistivity close to $h/4e^2$ at the charge neutral point. When the sample thickness is reduced to $\sim$10 nm thick, the Dirac cones from the top and bottom surfaces can hybridize (analogous to a "merging" in the real space) and become gapped to give a trivial insulator. Furthermore, we observe that an in-plane magnetic field can drive the system again towards a metallic behavior, with a prominent negative magnetoresistance (MR, up to $\sim$$-$95\%) and a temperature-insensitive resistivity close to $h/2e^2$ at the charge neutral point. The observation is interpreted in terms of a predicted effect of an in-plane magnetic field to reduce the hybridization gap (which, if small enough, may be smeared by disorder and a metallic behavior). A sufficiently strong magnetic field is predicted to restore and split again the Dirac points in the momentum space, inducing a distinct 2D topological semimetal (TSM) phase with 2 single-fold Dirac cones of opposite spin-momentum windings.

Exploiting novel aspects of the quantum geometry of charged particles in a magnetic field via gauge-invariant variables, we provide tangible connections between the response of quantum Hall fluids to non-uniform electric fields and the characteristic geometry of electronic motion in the presence of magnetic and electric fields. The geometric picture we provide motivates the following conjecture: non-uniform electric fields mimic the presence of spatial curvature. Consequently, the gravitational coupling constant also appears in the charge response to non-uniform electric fields.

An efficient O($N$) divide-conquer (DC) method based on localized natural orbitals (LNOs) is presented for large-scale density functional theories (DFT) calculations of gapped and metallic systems. The LNOs are non-iteratively calculated by a low-rank approximation via a local eigendecomposition of a projection operator for the occupied space. Introducing LNOs to represent the long range region of a truncated cluster reduces the computational cost of the DC method while keeping computational accuracy. A series of benchmark calculations and high parallel efficiency in a multilevel parallelization clearly demonstrate that the O($N$) method enables us to perform large-scale simulations for a wide variety of materials including metals with sufficient accuracy in accordance with development of massively parallel computers.

Quantum anomalous Hall (QAH) effect is a quantum Hall effect that occurs without the need of external magnetic field. A system composed of multiple parallel QAH layers is an effective high Chern number QAH insulator and the key to the applications of the dissipationless chiral edge channels in low energy consumption electronics. Such a QAH multilayer can also be engineered into other exotic topological phases such as a magnetic Weyl semimetal with only one pair of Weyl points. This work reports the first experimental realization of QAH multilayers in the superlattices composed of magnetically doped (Bi,Sb)$_2$Te$_3$ topological insulator and CdSe normal insulator layers grown by molecular beam epitaxy. The obtained multilayer samples show quantized Hall resistance $h/Ne$$^2$, where $h$ is the Planck's constant, $e$ is the elementary charge and $N$ is the number of the magnetic topological insulator layers, resembling a high Chern number QAH insulator.

Topological insulators in the Bi$_2$Se$_3$ family manifest helical Dirac surface states that span the topologically ordered bulk band gap. Recent scanning tunneling microscopy measurements have discovered additional states in the bulk band gap of Bi$_2$Se$_3$ and Bi$_2$Te$_3$, localized at one dimensional step edges. Here numerical simulations of a topological insulator surface are used to explore the phenomenology of edge state formation at the single-quintuple-layer step defects found ubiquitously on these materials. The modeled one dimensional edge states are found to exhibit a stable topological connection to the two dimensional surface state Dirac point.

Quantum anomalous Hall (QAH) effect in magnetic topological insulator (TI) is a novel transport phenomenon in which the Hall resistance reaches the quantum plateau in the absence of external magnetic field. Recently, this exotic effect has been discovered experimentally in an ultrathin film of the Bi2Te3 family TI with spontaneous ferromagnetic (FM) order. An important question concerning the QAH state is whether it is simply a zero-magnetic-field version of the quantum Hall (QH) effect, or if there is new physics beyond the conventional paradigm. Here we report experimental investigations on the quantum phase transition between the two opposite Hall plateaus of a QAH insulator caused by magnetization reversal. We observe a well-defined plateau with zero Hall conductivity over a range of magnetic field around coercivity, consistent with a recent theoretical prediction. The features of the zero Hall plateau are shown to be closely related to that of the QAH effect, but its temperature evolution exhibits quantitative differences from the network model for conventional QH plateau transition. We propose that the chiral edge states residing at the magnetic domain boundaries, which are unique to a QAH insulator, are responsible for the zero Hall plateau. The rich magnetic domain dynamics makes the QAH effect a distinctive class of quantum phenomenon that may find novel applications in spintronics.