Space-time Inversion Symmetry Research Articles

Real Topological Phonons in 3D Carbon Allotropes.

There has been a significant focus on real topological systems that enjoy space-time inversion symmetry and lack spin-orbit coupling. While the theoretical classification of the real topology has been established, more progress has yet to be made in the materials realization of real topological phononic states in 3D. To address this crucial issue, high-throughput computing is performed to inspect the real topology in the phonon spectrums of the 3D carbon allotropes. Among 1661 carbon allotropes listed in the Samara Carbon Allotrope Database (SACADA), 79 candidates host a phononic real Chern insulating (PRCI) state, 2 candidates host a phononic real nodal line (PRNL) state, 12 candidates host a phononic real Dirac point (PRDP) state, and 10 candidates host a phononic real triple-point pair (PRTPP) state. The PRCI, PRNL, PRTPP, and PRDP states of 27-SG. 166-pcu-h, 1081-SG. 194-42T13-CA, 52-SG. 141-gis, and 132-SG. 191-3,4T157 are exhibited as illustrative examples, and the second-order phononic hinge modes are explored. This study broadens the understanding of 3D topological phonons and expands the material candidates with phononic hinge modes and phononic realtopology.

A topological Hund nodal line antiferromagnet

The interplay of topology, magnetism, and correlations gives rise to intriguing phases of matter. In this study, through state-of-the-art angle-resolved photoemission spectroscopy, density functional theory, and dynamical mean-field theory calculations, we visualize a fourfold degenerate Dirac nodal line at the boundary of the bulk Brillouin zone in the antiferromagnet YMn2Ge2. We further demonstrate that this gapless, antiferromagnetic Dirac nodal line is enforced by the combination of magnetism, space-time inversion symmetry, and nonsymmorphic lattice symmetry. The corresponding drumhead surface states traverse the whole surface Brillouin zone. YMn2Ge2 thus serves as a platform to exhibit the interplay of multiple degenerate nodal physics and antiferromagnetism. Interestingly, the magnetic nodal line displays a d-orbital dependent renormalization along its trajectory in momentum space, thereby manifesting Hund’s coupling. Our findings offer insights into the effect of electronic correlations on magnetic Dirac nodal lines, leading to an antiferromagnetic Hund nodal line.

Semimetallic state transition in Two-Dimensional carbon nitride covalent networks and enhanced electrocatalytic activity for nitrate to ammonia conversion

Single-atom catalysts (SACs), due to their maximum atomic utilization rate, show tremendous potential for application in the electrocatalytic synthesis of ammonia from nitrate. Yet, the development of superior supports that preserve the high selectivity, activity, and stability of SACs remains an imperative challenge. In this work, based on first-principles calculations and tight-binding (TB) model analysis, a new two-dimensional (2D) carbon nitride monolayer, C7N6, is proposed. The C7N6 structure exhibits a strong covalent network, with dynamical, thermal, and mechanical stability. Surprisingly, the structural transition from C9N4 to C7N6 corresponds to a semimetallic state transition. Further symmetry analysis unveils that the Dirac states in C7N6 are protected by space–time inversion symmetry, and the physical origin of the Dirac cone was confirmed using the TB model. Additionally, a non-zero Z2 invariant and significant topological edge states demonstrate its topologically nontrivial nature. Considering the excellent structural and topological properties of C7N6, a three-step screening strategy is designed to identify eligible SACs for electrochemical nitrate reduction reaction (NO3RR), and Ti@C7N6 is identified as possessing the best activity, with the last proton-electron coupling step *NH2→*NH3 being the potential-determining step (PDS), for which the limiting potential is 0.48 V. Moreover, a free energy diagram shows that the *NOH reaction pathway is energetically preferred on Ti@C7N6, and ab initio molecular dynamics (AIMD) calculations at 500 K confirm its good thermal stability. Our study not only provides excellent CN-based support material but also offers theoretical guidance for constructing highly active and selective SACs for nitrate reduction.

3D Carbon Allotropes: Topological Quantum Materials with Obstructed Atomic Insulating Phases, Multiple Bulk‐Boundary Correspondences, and Real Topology

AbstractThe study of topological phases with unconventional bulk‐boundary correspondences and nontrivial real Chern number has garnered significant attention in the topological states of matter. Using the first‐principle calculations and theoretical analysis, a high‐throughput material screening of the 3D obstructed atomic insulators (OAIs) and 3D real Chern insulators (RCIs) based on the Samara Carbon Allotrope Database (SACADA) are performed. Results show that 422 out of 703 3D carbon allotropes are 3D OAIs with multiple bulk‐boundary correspondences, including 2D obstructed surface states (OSSs) and 1D hinge states, which are in 1D and 2Ds lower than the 3D bulk, respectively. The 2D OSSs in these OAIs can be modified when subjected to appropriate boundaries, which benefits the investigation of surface engineering and the development of efficient topological catalysts. These 422 OAIs, which have 2D and 1D boundary states, are excellent platforms for multi‐dimensional topological boundaries research. Remarkably, 138 of 422 OAIs are also 3D RCIs, which show a nontrivial real topology in the protection of spacetime inversion symmetry. This work not only provides a comprehensive list of 3D carbon‐based OAIs and RCIs, but also guides their application in various aspects based on multiple bulk‐boundary correspondences and real topological phases.

Metal-organic framework as high-order topological insulator with protected corner modes

The capacity to design and manipulate the chemical characteristics of metal-organic frameworks (MOFs) presents a promising approach for developing functional materials, particularly with regard to their topological properties. Here, based on first-principles calculations, we predict Ni3(C6S6)2 as a two-dimensional (2D) higher-order topological insulator (HOTI) with protected corner modes, which is the first material example of such topological phase in MOFs reported so far. The non-trivial topology of the material is identified by two different classes of bulk topological invariants: the real Chen number induced by spacetime inversion symmetry, and the fractional corner charge induced by rotational symmetry. The presence of corner states can serve as a diagnostic of its nontrivial topology. Moreover, the HOTI natural is demonstrated to be robust against symmetry perturbations and spin-orbit coupling. Inspired by the topological properties of Ni3(C6S6)2, we can explore regulating organic ligands or metal atoms in metal-organic frameworks to realize other 2D HOTIs, thereby expanding the family of 2D HOTIs. Our findings not only broaden the understanding of physical properties of metal-organic materials, but also extend the study of high-order topological insulators to organic platforms.

Stiefel-Whitney topological charges in a three-dimensional acoustic nodal-line crystal

Band topology of materials describes the extent Bloch wavefunctions are twisted in momentum space. Such descriptions rely on a set of topological invariants, generally referred to as topological charges, which form a characteristic class in the mathematical structure of fiber bundles associated with the Bloch wavefunctions. For example, the celebrated Chern number and its variants belong to the Chern class, characterizing topological charges for complex Bloch wavefunctions. Nevertheless, under the space-time inversion symmetry, Bloch wavefunctions can be purely real in the entire momentum space; consequently, their topological classification does not fall into the Chern class, but requires another characteristic class known as the Stiefel-Whitney class. Here, in a three-dimensional acoustic crystal, we demonstrate a topological nodal-line semimetal that is characterized by a doublet of topological charges, the first and second Stiefel-Whitney numbers, simultaneously. Such a doubly charged nodal line gives rise to a doubled bulk-boundary correspondence—while the first Stiefel-Whitney number induces ordinary drumhead states of the nodal line, the second Stiefel-Whitney number supports hinge Fermi arc states at odd inversion-related pairs of hinges. These results experimentally validate the two Stiefel-Whitney topological charges and demonstrate their unique bulk-boundary correspondence in a physical system.

Spinful Topological Phases in Acoustic Crystals with Projective PT Symmetry.

For the classification of topological phases of matter, an important consideration is whether a system is spinless or spinful, as these two classes have distinct symmetry algebra that gives rise to fundamentally different topological phases. However, only recently has it been realized theoretically that in the presence of gauge symmetry, the algebraic structure of symmetries can be projectively represented, which possibly enables the switch between spinless and spinful topological phases. Here, we report the experimental demonstration of this idea by realizing spinful topological phases in "spinless" acoustic crystals with projective space-time inversion symmetry. In particular, we realize a one-dimensional topologically gapped phase characterized by a 2Z winding number, which features double-degenerate bands in the entire Brillouin zone and two pairs of degenerate topological boundary modes. Our Letter thus overcomes a fundamental constraint on topological phases by spin classes.

Heavy Fermion Meets Antiferromagnetic Sea

CeMnSi exhibits heavy fermion behaviors in an antiferromagnetically ordered state of Mn. The space-time inversion symmetry inherent in the ordered state is key for the unique electron state.

Ideal topological Weyl complex phonons in two dimensions.

The advent of topological phonons has been attracting tremendous attention. However, studies in two-dimensional (2D) systems are limited. Here, we reveal a 2D novel combination of Weyl phonons - a Weyl complex composed of two linear Weyl nodes and one quadratic Weyl node. This Weyl complex consists of crossing points of two specific branches. We show that the coexistence of threefold symmetry - rotation symmetry, inversion symmetry, and time-reversal symmetry - could lead to the presence of the Weyl complex. Based on the symmetry requirement, we further construct the tight-binding model and effective k·p model for characterizing the Weyl complex. Moreover, due to the presence of the spacetime inversion symmetry, the linear and quadratic Weyl nodes feature a quantized (π and 2π) Berry phase, thus defining the corresponding topological charge. Furthermore, Weyl complexes consisting of Weyl points possess an emergent chiral symmetry, an integer topological charge is thus defined. Then, distinguished phenomena for the Weyl complex are studied, in particular, the edge states with three terminals. Our work predicts the presence of this novel 2D topological phase, and provides the symmetry guidance to realize it. Based on the first-principles calculations, we identify an existing material Cu2Si, as a concrete example to demonstrate the presence of the Weyl complex, and also study the phase transition under symmetry breaking.

Phononic Stiefel-Whitney topology with corner vibrational modes in two-dimensional Xenes and ligand-functionalized derivatives

Two-dimensional (2D) Stiefel-Whitney (SW) insulator is a fragile topological state characterized by the second SW class in the presence of space-time inversion symmetry. So far, SWIs have been proposed in several electronic materials but seldom in phononic systems. Here we recognize that a large class of 2D buckled honeycomb crystals termed Xenes and their ligand-functionalized derivatives realize the nontrivial phononic SW topology. The phononic SWIs are identified by a nonzero second SW number $w_2=1$, associated with gaped edge states and robust topological corner modes. Despite the versatility of electronic topological properties in these materials, the nontrivial phononic SW topology is mainly attributed to the double band inversion between in-plane acoustic and out-of-plane optical bands with opposite parities due to the structural buckling of the honeycomb lattice. Our findings not only reveal an overlooked phononic topological property of 2D Xene-related materials, but also afford abundant readily synthesizable material candidates with simple phononic spectra for further experimental studies of phononic SW topology physics.

Generalized fermion doubling theorems: Classification of two-dimensional nodal systems in terms of wallpaper groups

The Nielsen-Ninomiya Theorem has set up a ground rule for the minimal number of the topological points in a Brillouin zone. Notably, in the 2D Brillouin zone, chiral symmetry and space-time inversion symmetry can properly define topological invariants as charges characterizing the stability of the nodal points so that the non-zero charges protect these points. Due to the charge neutralization, the Nielsen-Ninomiya Theorem requires at least two stable topological points in the entire Brillouin zone. However, additional crystalline symmetries might duplicate the points. In this regard, for the wallpaper groups with crystalline symmetries, the minimal number of the nodal points in the Brillouin zone might be more than two. In this work, we determine the minimal numbers of the nodal points for the wallpaper groups in chiral-symmetric and space-time-inversion-symmetric systems separately and provide examples for new topological materials, such as topological nodal time-reversal-symmetric superconductors and Dirac semimetals. This generalized Nielsen-Ninomiya Theorem serves as a guide to search for 2D topological nodal materials and new platforms for twistronics. Furthermore, we show the Nielsen-Ninomiya Theorem can be extended to 2D non-Hermitian systems hosting topologically protected exceptional points and Fermi points for the 17 wallpaper groups and use the violation of the theorem on the surface to classify 3D Hermitian and non-Hermitian topological bulks.

Dirac states in an inclined two-dimensional Su-Schrieffer-Heeger model

We propose to realize Dirac states in an inclined two-dimensional Su-Schrieffer-Heeger model on a square lattice. We show that a pair of Dirac points protected by space-time inversion symmetry appear in the semimetal phase. The locations of these Dirac points are not pinned to any high-symmetry points of the Brillouin zone but are tunable through parameter modulations. Interestingly, the merging of two Dirac points undergoes a topological phase transition that leads to either a weak topological insulator or a nodal-line semimetal. We provide a systematic analysis of these topological phases from both bulk and boundary perspectives combined with symmetry arguments. We also discuss feasible experimental platforms to realize our model.

Euler-obstructed Cooper pairing: Nodal superconductivity and hinge Majorana zero modes

Since the proposal of monopole Cooper pairing in [Phys. Rev. Lett. 120, 067003 (2018)], considerable research efforts have been dedicated to the study of Cooper pairing order parameters constrained (or obstructed) by the nontrivial normal-state band topology at Fermi surfaces in 3D systems. In the current work, we generalize the topologically obstructed pairings between Chern states (like the monopole Cooper pairing) by proposing Euler obstructed Cooper pairing in 3D systems. The Euler obstructed Cooper pairing widely exists between two Fermi surfaces with nontrivial band topology characterized by nonzero Euler numbers; such Fermi surfaces can exist in 3D PT-protected spinless-Dirac/nodal-line semimetals with negligible spin-orbit coupling, where PT is the space-time inversion symmetry. An Euler obstructed pairing channel must have pairing nodes on the pairing-relevant Fermi surfaces, and the total winding number of the pairing nodes is determined by the sum or difference of the Euler numbers on the Fermi surfaces. In particular, we find that when the normal state is time-reversal invariant and the pairing is weak, a sufficiently-dominant Euler obstructed pairing channel with zero total momentum leads to nodal superconductivity. If the Fermi surface splitting is small, the resultant nodal superconductor hosts hinge Majorana zero modes. The possible dominance of the Euler obstructed pairing channel near the superconducting transition and the robustness of the hinge Majorana zero modes against disorder are explicitly demonstrated using effective or tight-binding models. Our work presents the first class of higher-order nodal superconductivity originating from the topologically obstructed Cooper pairing.

Phononic real Chern insulator with protected corner modes in graphynes

Higher-order topological insulators have attracted great research interest recently. Different from conventional topological insulators, higher-order topological insulators do not necessarily require spin-orbit coupling, which makes it possible to realize them in spinless systems. Here, we study phonons in 2D graphyne family materials. By using first-principle calculations and topology/symmetry analysis, we find that phonons in both graphdiyne and $\gamma$-graphyne exhibit a second-order topology, which belongs to the specific case known as real Chern insulator. We identify the nontrivial phononic band gaps, which are characterized by nontrivial real Chern numbers enabled by the spacetime inversion symmetry. The protected phonon corner modes are verified by the calculation on a finite-size nanodisk. Our study extends the scope of higher-order topology to phonons in real materials. The spatially localized phonon modes could be useful for novel phononic applications.

Gapping Fragile Topological Bands by Interactions.

We consider the stability of fragile topological bands protected by space-time inversion symmetry in the presence of strong electron-electron interactions. At the single-particle level, the topological nature of the bands prevents the opening of a gap between them. In contrast, we show that when the fragile bands are half filled, interactions can open a gap in the many-body spectrum without breaking any symmetry or mixing degrees of freedom from remote bands. Furthermore, the resulting ground state is not topologically ordered. Thus, a fragile topological band structure does not present an obstruction to forming a "featureless insulator" ground state. Our construction relies on the formation of fermionic bound states of two electrons and one hole known as "trions." The trions form a band whose coupling to the electronic band enables the gap opening. This result may be relevant to the gapped state indicated by recent experiments in magic angle twisted bilayer graphene at charge neutrality.

Quadratic magnetoelectric effect during field cooling in sputter grown Cr2O3 films

${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}$ is the archetypal magnetoelectric (ME) material, which has a linear coupling between electric and magnetic polarizations. Quadratic ME effects are forbidden for the magnetic point group of ${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}$, due to space-time inversion symmetry. In ${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}$ films grown by sputtering, we find a signature of a quadratic ME effect that is not found in bulk single crystals. We use Raman spectroscopy and magnetization measurements to deduce the removal of space-time symmetry and corroborate the emergence of the quadratic ME effect. We propose that metastable site-selective trace dopants remove the space, time, and space-time inversion symmetries from the original magnetic point group of bulk ${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}$. We include the quadratic ME effect in a model describing the switching process during ME field cooling and estimate the effective quadratic susceptibility value. The quadratic magnetoelectric effect in a uniaxial antiferromagnet is promising for multifunctional antiferromagnetic and magnetoelectric devices that can incorporate optical, strain-induced, and multiferroic effects.

Graphyne as a second-order and real Chern topological insulator in two dimensions

Higher-order topological phases and real topological phases are two emerging topics in topological states of matter, which have been attracting considerable research interest. However, it remains a challenge to find realistic materials that can realize these exotic phases. Here, based on first-principles calculations and theoretical analysis, we identify graphyne, the representative of the graphyne-family carbon allotropes, as a two-dimensional (2D) second-order topological insulator and a real Chern insulator. We show that graphyne has a direct bulk band gap at the three $M$ points, forming three valleys. The bulk bands feature a double band inversion, which is characterized by the real Chern number enabled by the spacetime-inversion symmetry. The real Chern number is explicitly evaluated by both the Wilson loop method and the parity approach. We demonstrate that a nontrivial real Chern number for a 2D system dictates the existence of Dirac-type edge bands and the topological corner states. Furthermore, we find that the topological phase transition in graphyne from the real Chern insulator to a trivial insulator is mediated by a 2D Weyl semimetal phase. The robustness of the corner states against symmetry breaking and possible experimental detection methods are discussed.

Electronic correlations in the normal state of the kagome superconductor KV3Sb5

Recently, intensive studies have revealed fascinating physics, such as charge density wave and superconducting states, in the newly synthesized kagome-lattice materials ${A\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ $(A=\text{K}, \mathrm{Rb}, \mathrm{Cs})$. Despite the rapid progress, fundamental aspects such as the magnetic properties and electronic correlations in these materials have not been clearly understood yet. Here, based on density functional theory plus single-site dynamical mean-field theory calculations, we investigate the correlated electronic structure and magnetic properties of the ${\mathrm{KV}}_{3}{\mathrm{Sb}}_{5}$ family materials in the normal state. We show that these materials are good metals with weak local correlations. The obtained Pauli-like paramagnetism and the absence of local moments are consistent with a recent experiment. We reveal that the band crossings around the Fermi level form three groups of nodal lines protected by the spacetime inversion symmetry, each carrying a quantized $\ensuremath{\pi}$ Berry phase. Our result suggests that the local correlation strength in these materials appears to be too weak to generate unconventional superconductivity, and nonlocal electronic correlation might be crucial in this kagome system.

Switching Spinless and Spinful Topological Phases with Projective PT Symmetry.

A fundamental dichotomous classification for all physical systems is according to whether they are spinless or spinful. This is especially crucial for the study of symmetry-protected topological phases, as the two classes have distinct symmetry algebra. As a prominent example, the spacetime inversion symmetry PT satisfies (PT)^{2}=±1 for spinless/spinful systems, and each class features unique topological phases. Here, we reveal a possibility to switch the two fundamental classes via Z_{2} projective representations. For PT symmetry, this occurs when P inverses the gauge transformation needed to recover the original Z_{2} gauge connections under P. As a result, we can achieve topological phases originally unique for spinful systems in a spinless system, and vice versa. We explicitly demonstrate the claimed mechanism with several concrete models, such as Kramers degenerate bands and Kramers Majorana boundary modes in spinless systems, and real topological phases in spinful systems. Possible experimental realization of these models is discussed. Our work breaks a fundamental limitation on topological phases and opens an unprecedented possibility to realize intriguing topological phases in previously impossible systems.

Tunable magneto-optical effect, anomalous Hall effect, and anomalous Nernst effect in the two-dimensional room-temperature ferromagnet 1T−CrTe2

Utilizing first-principles density functional theory calculations together with group theory analyses, we systematically investigate the spin-order-dependent magneto-optical effect (MOE), anomalous Hall effect (AHE), and anomalous Nernst effect (ANE) in the recently discovered two-dimensional room-temperature ferromagnet $1T\text{\ensuremath{-}}{\mathrm{CrTe}}_{2}$. We find that the spin prefers an in-plane direction by the magnetocrystalline anisotropy energy calculations. The MOE, AHE, and ANE display a period of $2\ensuremath{\pi}/3$ when the spin rotates within the atomic plane, and they are forbidden if a mirror plane perpendicular to the spin direction exists. By reorienting the spin from the in-plane to out-of-plane direction, the MOE, AHE, and ANE are enhanced by around one order of magnitude. Moreover, we establish the layer-dependent magnetic properties of multilayer $1T\text{\ensuremath{-}}{\mathrm{CrTe}}_{2}$ and predict antiferromagnetism and ferromagnetism for bilayer and trilayer $1T\text{\ensuremath{-}}{\mathrm{CrTe}}_{2}$, respectively. The MOE, AHE, and ANE are prohibited in antiferromagnetic bilayer $1T\text{\ensuremath{-}}{\mathrm{CrTe}}_{2}$ due to the existence of the space-time inversion symmetry, whereas all of them are activated in ferromagnetic trilayer $1T\text{\ensuremath{-}}{\mathrm{CrTe}}_{2}$ and are significantly enhanced compared to those of monolayer $1T\text{\ensuremath{-}}{\mathrm{CrTe}}_{2}$. Our results show that the magneto-optical and anomalous transports proprieties of $1T\text{\ensuremath{-}}{\mathrm{CrTe}}_{2}$ can be effectively modulated by altering spin direction and layer number.