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The application of topology, a branch of mathematics, to the study of electronic states in crystalline materials has had a revolutionary impact on the field of condensed matter physics. For example, the development of topological band theory has delivered new approaches and tools to characterize the electronic structure of materials, resulting in the discovery of new phases of matter with exotic properties. In the framework of topological band theory, the crossings between energy levels of electrons are characterized by topological invariants, which predict the presence of topological boundary states. Given the frequency of energy level crossings on the potential energy surface in molecules, the applicability of these concepts to molecular systems could be of great interest for our understanding of reaction dynamics. However, challenges arise due to differing quantum mechanical descriptions of solids and molecules. Out work aims to bridge the gap between topological band theory and molecular chemistry. We propose that the Euler Class, a topological invariant, can be used to categorize and analyse the distribution of nonadiabatic couplings on the potential energy surface. To exemplify this connection, we introduce a model system with two distinct regimes that are characterized by different values of the Euler Class, yet identical potential energy surfaces. Contrary to expectations set by the Born-Oppenheimer approximation, we propose that these two regimes don't exhibit identical dynamics, due to a qualitatively distinct distribution of nonadiabatic couplings.

A quantitative description of the excited electronic states of point defects and impurities is crucial for understanding materials properties, and possible applications of defects in quantum technologies. This is a considerable challenge for computational methods, since Kohn-Sham density-functional theory (DFT) is inherently a ground state theory, while higher-level methods are often too computationally expensive for defect systems. Recently, embedding approaches have been applied that treat defect states with many-body methods, while using DFT to describe the bulk host material. We implement such an embedding method, based on Wannierization of defect orbitals and the constrained random-phase approximation approach, and perform systematic characterization of the method for three distinct systems with current technological relevance: a carbon dimer replacing a B and N pair in bulk hexagonal BN (C$_{\text{B}}$C$_{\text{N}}$), the negatively charged nitrogen-vacancy center in diamond (NV$^-$), and an Fe impurity on the Al site in wurtzite AlN ($\text{Fe}_{\text{Al}}$). For C$_{\text{B}}$C$_{\text{N}}$ we show that the embedding approach gives many-body states in agreement with analytical results on the Hubbard dimer model, which allows us to elucidate the effects of the DFT functional and double-counting correction. For the NV$^-$ center, our method demonstrates good quantitative agreement with experiments for the zero-phonon line of the triplet-triplet transition. Finally, we illustrate challenges associated with this method for determining the energies and orderings of the complex spin multiplets in $\text{Fe}_{\text{Al}}$.

Here we report an ultrafast optical spectroscopic study of the nodal-line semimetal ZrSiTe. Our measurements reveal that, converse to other compounds of the family, the sudden injection of electronic excitations results in a strongly coherent response of an $A_{1g}$ phonon mode which dynamically modifies the distance between Zr and Te atoms and Si layers. "Frozen phonon" DFT calculations, in which band structures are calculated as a function of nuclear position along the phonon mode coordinate, show that large displacements along this mode alter the material's electronic structure significantly, forcing bands to approach and even cross the Fermi energy. The incoherent part of the time domain response reveals that a delayed electronic response at low fluence discontinuously evolves into an instantaneous one for excitation densities larger than $3.43 \times 10^{17}$ cm$^{-3}$. This sudden change of the dissipative channels for electronic excitations is indicative of an ultrafast Lifshitz transition which we tentatively associate to a change in topology of the Fermi surface driven by a symmetry preserving $A_{1g}$ phonon mode.

Drumhead surface states that link together loops of nodal lines arise in Dirac nodal-line semimetals as a consequence of the topologically non-trivial band crossings. We used low-temperature scanning tunneling microscopy and Fourier-transformed scanning tunneling spectroscopy to investigate the quasiparticle interference (QPI) properties of ZrSiTe. Our results show two scattering signals across the drumhead state resolving the energy-momentum relationship through the occupied and unoccupied energy ranges it is predicted to span. Observation of this drumhead state is in contrast to previous studies on ZrSiS and ZrSiSe, where the QPI was dominated by topologically trivial bulk bands and surface states. Furthermore, we observe a near $\mathbf{k} \rightarrow -\mathbf{k}$ scattering process across the $\Gamma$-point, enabled by scattering between the spin-split drumhead bands in this material.

In some topological insulators, such as graphene and WTe$_2$, band inversion originates from chemical bonding and space group symmetry, in contrast to materials such as Bi$_2$Se$_3$, where the band inversion derives from relativistic effects in the atoms. In the former, band inversion is susceptible to changes of the chemical environment, e.g. by defects, while the latter are less affected by defects due to the larger energy scale associated with atomic relativistic effects. Motivated by recent experiments, we study the effect of Te-vacancies and Te-adatoms on the electronic properties of WTe$_2$. We find that the Te-vacancies have a formation energy of $2.21$ eV, while the formation energy of the Te-adatoms is much lower with $0.72$ eV. The vacancies strongly influence the band structure and we present evidence that band inversion is already reversed at the nominal composition of WTe$_{1.97}$. In contrast, we show that the adatoms do not change the electronic structure in the vicinity of the Fermi level and thus the topological properties. Our findings indicate that Te-adatoms should be present in thin films that are grown in a Te-rich environment, and we suggest that they have been observed in scanning tunneling microscopy experiments.

Nodal-line semimetals are topologically non-trivial states of matter featuring band crossings along a closed curve, i.e. nodal-line, in momentum space. Through a detailed analysis of the electronic structure, we show for the first time that the normal state of the superconductor NaAlSi, with a critical temperature of $T_{\rm c} \approx$ 7 K, is a nodal-line semimetal, where the complex nodal-line structure is protected by non-symmorphic mirror crystal symmetries. We further report on muon spin rotation experiments revealing that the superconductivity in NaAlSi is truly of bulk nature, featuring a fully gapped Fermi-surface. The temperature-dependent magnetic penetration depth can be well described by a two-gap model consisting of two $s$-wave symmetric gaps with $\Delta_1 =$ 0.6(2) meV and $\Delta_2 =$ 1.39(1) meV. The zero-field muon experiment indicates that time-reversal symmetry is preserved in the superconducting state. Our observations suggest that notwithstanding its topologically non-trivial band structure, NaAlSi may be suitably interpreted as a conventional London superconductor, while more exotic superconducting gap symmetries cannot be excluded. The intertwining of topological electronic states and superconductivity renders NaAlSi a prototypical platform to search for unprecedented topological quantum phases.

We study the electronic structure of the nodal line semimetal ZrSiTe both experimentally and theoretically. We find two different surface states in ZrSiTe - topological drumhead surface states and trivial floating band surface states. Using the spectra of Wilson loops, we show that a non-trivial Berry phase that exists in a confined region within the Brillouin Zone gives rise to the topological drumhead-type surface states. The $\mathbb{Z}_2$ structure of the Berry phase induces a $\mathbb{Z}_2$ 'modular arithmetic' of the surface states, allowing surface states deriving from different nodal lines to hybridize and gap out, which can be probed by a set of Wilson loops. Our findings are confirmed by \textitab-initio calculations and angle-resolved photoemission experiments, which are in excellent agreement with each other and the topological analysis. This is the first complete characterization of topological surface states in the family of square-net based nodal line semimetals and thus fundamentally increases the understanding of the topological nature of this growing class of topological semimetals.

In this paper, a topological classification of molecules and their chemical reactions is proposed on a single particle level . We consider zero-dimensional electronic Hamiltonians in a real-space tight-binding basis with spinless time-reversal symmetry and an additional spatial reflection symmetry. The symmetry gives rise to a perplectic structure and suggests a $\mathbb{Z}_2$ invariant in form of a pfaffian, which can be captured by an entanglement cut. We apply our findings to a class of chemical reactions studied by Woodward and Hoffmann, where a reflection symmetry is preserved along a one-dimensional reaction path and argue that the topological classification should contribute to the rate constants of these reactions. More concretely, we find that a reaction takes place experimentally whenever the reactants and products can be adiabatically deformed into each other, while reactions that require a change of topological invariants have not been observed experimentally.

A monolayer of WTe$_2$ has been shown to display quantum spin Hall (QSH) edge modes persisting up to 100~K in transport experiments. Based on density-functional theory calculations and symmetry-based model building including the role of correlations and substrate support, we develop an effective electronic model for WTe$_2$ which fundamentally differs from other prototypical QSH settings: we find that the extraordinary robustness of quantum spin Hall edge modes in WTe$_2$ roots in a glide symmetry due to which the topological gap opens away from high-symmetry points in momentum space. While the indirect bulk gap is much smaller, the glide symmetry implies a large direct gap of up to 1~eV in the Brillouin zone region of the dispersing edge modes, and hence enables sharply boundary-localized QSH edge states depending on the specific boundary orientation.

We predict a magnetic Weyl semimetal in the inverse Heusler Ti2MnAl, a compensated ferrimagnet with a vanishing net magnetic moment and a Curie temperature of over 650 K. Despite the vanishing net magnetic moment, we calculate a large intrinsic anomalous Hall effect (AHE) of about 300 S/cm. It derives from the Berry curvature distribution of the Weyl points, which are only 14 meV away from the Fermi level and isolated from trivial bands. Different from antiferromagnets Mn3X (X= Ge, Sn, Ga, Ir, Rh, and Pt), where the AHE originates from the non-collinear magnetic structure, the AHE in Ti2MnAl stems directly from the Weyl points and is topologically protected. The large anomalous Hall conductivity (AHC) together with a low charge carrier concentration should give rise to a large anomalous Hall angle. In contrast to the Co-based ferromagnetic Heusler compounds, the Weyl nodes in Ti2MnAl do not derive from nodal lines due to the lack of mirror symmetries in the inverse Heusler structure. Since the magnetic structure breaks spin-rotation symmetry, the Weyl nodes are stable without SOC. Moreover, because of the large separation between Weyl points of opposite topological charge, the Fermi arcs extent up to 75% of the reciprocal lattice vectors in length. This makes Ti2MnAl an excellent candidate for the comprehensive study of magnetic Weyl semimetals. It is the first example of a material with Weyl points, large anomalous Hall effect and angle despite a vanishing net magnetic moment.

Very recently, the half-metallic compound Co$_3$Sn$_2$S$_2$ was predicted to be a magnetic WSM with Weyl points only 60 meV above the Fermi level ($E_F$). Owing to the low charge carrier density and large Berry curvature induced,Co$_3$Sn$_2$S$_2$ possesses both a large anomalous Hall conductivity (AHC) and a large anomalous Hall angle (AHA), which provide strong evidence for the existence of Weyl points in Co$_3$Sn$_2$S$_2$. In this work, we theoretically studied the surface topological feature of Co$_3$Sn$_2$S$_2$ and its counterpart Co$_3$Sn$_2$Se$_2$. By cleaving the sample at the weak Sn--S/Se bonds, one can achieve two different surfaces terminated with Sn and S/Se atoms, respectively. The resulting Fermi arc related states can range from the energy of the Weyl points to $E_F$--0.1 eV in the Sn-terminated surface. Therefore, it should be possible to observe the Fermi arcs in angle-resolved photoemission spectroscopy (ARPES) measurements. Furthermore, in order to simulate quasiparticle interference (QPI) in scanning tunneling microscopy (STM) measurements, we also calculated the joint density of states (JDOS) for both terminals. This work would be helpful for a comprehensive understanding of the topological properties of these two magnetic WSMs and further ARPES and STM measurements.

Since the discovery of the anomalous Hall effect (AHE), the anomalous Hall conductivity (AHC) has been thought to be zero when there is no net magnetization. However, the recently found relation between the intrinsic AHE and the Berry curvature predicts other possibilities, such as a large AHC in non-colinear antiferromagnets with no net magnetization but net Berry curvature. Vice versa, the AHE in principle could be tuned to zero, irrespective of a finite magnetization. Here, we experimentally investigate this possibility and demonstrate that, the symmetry elements of Heusler magnets can be changed such that the Berry curvature and all the associated properties are switched while leaving the magnetization unaffected. This enables us to tune the AHC from 0 \Omega-1cm-1 up to 1600 \Omega-1cm-1 with an exceptionally high anomalous Hall angle up to 12 %, while keeping the magnetization same. Our study shows that the AHC can be controlled by selectively changing the Berry curvature distribution, independent of the magnetization.

We have found a ferromagnetic Weyl semimetal (WSM) in half metallic Co$_3$Sn$_2$Se$_2$. The three pairs of Weyl points near Fermi level (E$_F$) are derived from nodal lines gapped by spin-orbit coupling (SOC). Though the Weyl points are 0.11 eV above the charge neutral point, Fermi arc related states in the cleaved surface can range from E$_F$ -0.15 to E$_F$ +0.11 eV in energy space, due to the surface bands dispersion. Hence, Weyl points related physics should be detected by surface measurements, such as ARPES and STM. Because of the large Berry curvature deriving from the gapped nodal lines and Weyl points, the anomalous Hall conductivity of Co$_3$Sn$_2$Se$_2$ can keep above 620 S/cm in a large energy window. Beside magnetic WSM, Co$_3$Sn$_2$Se$_2$ also possesses a Z$_2$ topological semimetal phase as the temperature is above Curie temperature. Owing to the quasi two-dimensional lattice structure, it is also possible to realize quantum anomalous Hall effect in its 2D limit via the size effect. Therefore, Co$_3$Sn$_2$Se$_2$ provides a good platform for the interplay of different topological states and magnetic states.

The quantum anomalous Hall effect (QAHE) and magnetic Weyl semimetals (WSMs) are topological states induced by intrinsic magnetic moments and spin-orbit coupling. Their similarity suggests the possibility of achieving the QAHE by dimensional confinement of a magnetic WSM along one direction. In this study, we investigate the emergence of the QAHE in the two-dimensional (2D) limit of magnetic WSMs due to finite size effects in thin films and step-edges. We demonstrate the feasibility of this approach with effective models and real materials. To this end, we have chosen the layered magnetic WSM Co$_3$Sn$_2$S$_2$, which features a large anomalous Hall conductivity and anomalous Hall angle in its 3D bulk, as our material candidate. In the 2D limit of Co$_3$Sn$_2$S$_2$ two QAHE states exist depending on the stoichiometry of the 2D layer. One is a semimetal with a Chern number of 6, and the other is an insulator with a Chern number of 3. The latter has a band gap of 0.05 eV, which is much larger than that in magnetically doped topological insulators. Our findings naturally explain the existence of chiral states in step edges of bulk Co$_3$Sn$_2$S$_2$ which habe been reported in a recent experiment at $T = 4K$ and present a realistic avenue to realize QAH states in thin films of magnetic WSMs.

Various Co2 based Heusler compounds are predicted to be Weyl materials. These systems with broken symmetry possess a large Berry curvature, and introduce exotic transport properties. The present study on epitaxially grown Co2TiSn films is an initial approach to understand and explore this possibility. The anomalous Hall effect in the well-ordered Co2TiSn films has been investigated both experimentally and theoretically. The measured Hall conductivity is in good agreement to the calculated Berry curvature. Small deviations between them are due to the influence of skew scattering on the Hall effect. From theoretical point of view, the main contribution to the anomalous Hall effect originates from slightly gapped nodal lines, due to a symmetry reduction induced by the magnetization. It has been found that only part of the nodal lines contributed near to the anomalous Hall conductivity at a fixed Fermi energy which can be explained from a magnetic symmetry analysis. Furthermore, from hard x-ray photoelectron spectroscopy measurements, we establish the electronic structure in the film that is comparable to the theoretical density of states calculations. The present results provide deeper insight into the spintronics from the prospect of topology.

Recent interest in topological semimetals has lead to the proposal of many new topological phases that can be realized in real materials. Next to Dirac and Weyl systems, these include more exotic phases based on manifold band degeneracies in the bulk electronic structure. The exotic states in topological semimetals are usually protected by some sort of crystal symmetry and the introduction of magnetic order can influence these states by breaking time reversal symmetry. Here we show that we can realize a rich variety of different topological semimetal states in a single material, $\rm CeSbTe$. This compound can exhibit different types of magnetic order that can be accessed easily by applying a small field. It allows, therefore, for tuning the electronic structure and can drive it through a manifold of topologically distinct phases, such as the first nonsymmorphic magnetic topological material with an eight-fold band crossing at a high symmetry point. Our experimental results are backed by a full magnetic group theory analysis and ab initio calculations. This discovery introduces a realistic and promising platform for studying the interplay of magnetism and topology.

Weyl and Dirac fermions have created much attention in condensed matter physics and materials science. Recently, several additional distinct types of fermions have been predicted. Here, we report ultra-high electrical conductivity in MoP at low temperature, which has recently been established as a triple point Fermion material. Here we show that the electrical resistivity is 6 n-ohm cm at 2 K with a large mean free path of 11 microns. de Haas-van Alphen oscillations reveal spin splitting of the Fermi surfaces. In contrast to noble metals with similar conductivity and number of carriers, the magnetoresistance in MoP does not saturate up to 9 T at 2 K. Interestingly, the momentum relaxing time of the electrons is found to be more than 15 times larger than the quantum coherence time. This difference between the scattering scales shows that momentum conserving scattering dominates in MoP at low temperatures.

We introduce the notion of a band-inverted, topological semimetal in two-dimensional nonsymmorphic crystals. This notion is materialized in the monolayers of MTe$_2$ (M $=$ W, Mo) if spin-orbit coupling is neglected. We characterize the Dirac band touching topologically by the Wilson loop of the non-Abelian Berry gauge field. An additional feature of the Dirac cone in monolayer MTe$_2$ is that it tilts over in a Lifshitz transition to produce electron and hole pockets, a type-II Dirac cone. These pockets, together with the pseudospin structure of the Dirac electrons, suggest a unified, topological explanation for the recently-reported, non-saturating magnetoresistance in WTe$_2$, as well as its circular dichroism in photoemission. We complement our analysis and first-principle bandstructure calculations with an $\textit{ab-initio}$-derived-derived tight-binding model for the WTe$_2$ monolayer.

We study the two-dimensional (2D) Hubbard model using exact diagonalization for spin-1/2 fermions on the triangular and honeycomb lattices decorated with a single hexagon per site. In certain parameter ranges, the Hubbard model maps to a quantum compass model on those lattices. On the triangular lattice, the compass model exhibits collinear stripe antiferromagnetism, implying $d$-density wave charge order in the original Hubbard model. On the honeycomb lattice, the compass model has a unique, quantum disordered ground state that transforms nontrivially under lattice reflection. The ground state of the Hubbard model on the decorated honeycomb lattice is thus a 2D fermionic symmetry-protected topological phase. This state -- protected by time-reversal and reflection symmetries -- cannot be connected adiabatically to a free-fermion topological phase.

Based on the interplay of theory and experiment, a large new family of filled group 9 (Co, Rh and Ir) skutterudites is designed and synthesized. The new materials fill the empty cages in the structures of the known binary CoSb3, RhSb3 and IrSb3 skutterudites with alkaline, alkaline earth, and rare earth atoms to create compounds of the type AyB4X12; A atoms fill the cages to a fraction y, B are the group 9 transition metals, and X is a mixture of electronegative main group elements chosen to achieve chemical stability by adjusting the electron counts to electron-precise values. Forty-three new compounds are reported, antimony-tin and phosphorous-silicon based, with 63 compositional variations presented. The new family of compounds is large and general. The results described here can be extended to the synthesis of hundreds of new group 9 filled skutterudites.

Design principles and novel predictions of new 3D Dirac semimetals are presented, along with the context of currently known materials. Current materials include those based on a topological to trivial phase transition, such as in TlBiSe$_{2-x}$S$_x$ and Hg$_{1-x}$Cd$_x$Te, Bi$_{1-x}$Sb$_x$, Bi$_{2-x}$In$_x$Se$_3$, and Pb$_{1-x}$Sn$_x$Se. Some more recently revealed materials, Na$_3$Bi and Cd$_3$As$_2$, are 3D Dirac semimetals in their native composition. The different design principles presented each yield novel predictions for new candidates. For Case I, 3D Dirac semimetals based on charge balanced compounds, BaAgBi, SrAgBi, YbAuSb, PtBi$_2$ and SrSn$_2$As$_2$ are identified as candidates. For Case II, 3D Dirac semi-metals in analogy to graphene, BaGa$_2$ is identified as a candidate, and BaPt and Li$_2$Pt are discussed. For Case III, 3D Dirac semi-metals based on glide planes and screw axes, TlMo$_3$Te$_3$ and the AMo$_3$X$_3$ family in general (A=K, Na, In, Tl, X=Se,Te) as well as the Group IVb trihalides such as HfI$_3$ are identified as candidates. Finally we discuss conventional intermetallic compounds with Dirac cones, and identify Cr$_2$B as a potentially interesting material.

Motivated by the concept of Möbius aromatics in organic chemistry, we extend the recently introduced concept of fragile Mott insulators (FMI) to ring-shaped molecules with repulsive Hubbard interactions threaded by a half-quantum of magnetic flux ($hc/2e$). In this context, a FMI is the insulating ground state of a finite-size molecule that cannot be adiabatically connected to a single Slater determinant, i.e., to a band insulator, provided that time-reversal and lattice translation symmetries are preserved. Based on exact numerical diagonalization for finite Hubbard interaction strength $U$ and existing Bethe-ansatz studies of the one-dimensional Hubbard model in the large-$U$ limit, we establish a duality between Hubbard molecules with $4n$ and $4n+2$ sites, with $n$ integer. A molecule with $4n$ sites is an FMI in the absence of flux but becomes a band insulator in the presence of a half-quantum of flux, while a molecule with $4n+2$ sites is a band insulator in the absence of flux but becomes an FMI in the presence of a half-quantum of flux. Including next-nearest-neighbor-hoppings gives rise to new FMI states that belong to multidimensional irreducible representations of the molecular point group, giving rise to a rich phase diagram.

We analyze the superconductors TiNCl and ZrNCl from a local bonding perspective. Although TiNCl crystallizes in an orthorhombic structure and ZrNCl crystallizes in a hexagonal structure, both compounds show significant structural similarities, for example that both consist of layered metal-Nitrogen networks. The local bonding in those two structures is very similar, giving rise to a dispersive conduction band mostly consisting of metal-d -states. Upon doping both compounds show structural changes, which lead to short metal-metal distances, indicating a bonding interaction that might be important for the appearance of superconductivity in these systems. We furthermore draw analogies to other superconductors that are close to a charge density wave instability around a d 1 -configuration and offer a different perspective on this class of superconductors, which show non-BCS-like superconductivity.

We report the observation of superconductivity in the CuIr2Se4 spinel induced by partial substitution of Pt for Ir. The optimal doping level for superconductivity in Cu(Ir1-xPtx)2Se4 is x = 0.2, where Tc is 1.76 K. A superconducting Tc vs. composition dome is established between the metallic, normal conductor CuIr2Se4 and semiconducting CuIrPtSe4. Electronic structure calculations show that the optimal Tc occurs near the electron count of a large peak in the calculated electronic density of states and that CuIrPtSe4 is a band-filled insulator. Characterization of the superconducting state in this heavy metal spinel through determination of ∆C/\gammaTc, indicates that it is BCS-like. The relatively high upper critical field at the optimal superconducting composition (Hc2(0) = 3.2 T) is much larger than that reported for analogous rhodium spinels and is comparable to or exceeds the Pauli field (mu0Hp), suggesting that strong spin orbit coupling may influence the superconducting state. Further, comparison to doped CuIr2S4 suggests that superconductivity in iridium spinels is not necessarily associated with the destabilization of a charge-ordered spin-paired state through doping.

We report the discovery of weak topological insulators by ab initio calculations in a honeycomb lattice. We propose a structure with an odd number of layers in the primitive unit-cell as a prerequisite for forming weak topological insulators. Here, the single-layered KHgSb is the most suitable candidate for its large bulk energy gap of 0.24 eV. Its side surface hosts metallic surface states, forming two anisotropic Dirac cones. Though the stacking of even-layered structures leads to trivial insulators, the structures can host a quantum spin Hall layer with a large bulk gap, if an additional single layer exists as a stacking fault in the crystal. The reported honeycomb compounds can serve as prototypes to aid in the finding of new weak topological insulators in layered small-gap semiconductors.