Nonsymmorphic Symmetry Research Articles

Nodal phases in non-Hermitian wallpaper crystals

Symmetry and non-Hermiticity play pivotal roles in photonic lattices. While symmetries, such as parity-time (PT) symmetry, have attracted ample attention, more intricate crystalline symmetries have been neglected in comparison. Here, we investigate the impact of the 17 wallpaper space groups of two-dimensional crystals on non-Hermitian band structures. We show that the non-trivial space group representations enforce degeneracies at high symmetry points and dictate their dispersion away from these points. In combination with either T or PT, the symmorphic p4 mm symmetry and the non-symmorphic p2mg, p2gg, and p4gm symmetries protect exceptional chains intersecting at the pertinent high symmetry points.

Coexistence of electron and phonon topology in conjunction with quantum transport device modeling

The escalating research in the field of topology necessitates an understanding of the underlying rich physics behind the materials possessing unique features of non-trivial topology in both electronic and phononic states. Due to the interaction between electronic quasiparticles and spin degrees of freedom, the realization of magnetic topological materials has opened up a new frontier with unusual topological phases, however, these are rarely reported alongside phononic quasiparticle excitations. In this work, by first-principles calculations and symmetry analysis, the intermetallic ferromagnetic compounds MnGaGe and MnZnSb with the coexistence of exceptional topological features in the electronic and phononic states are proposed. These compounds host nodal surface on ky=π plane in bulk Brillouin zone in the electronic and phononic spectra protected by the combination of time-reversal symmetry and nonsymmorphic two-fold screw-rotation symmetry. In the former case, a spin-polarized nodal surface is present in the majority and minority spin channels and found to be robust to ground-state magnetic polarization. The presence of nodal line features is analyzed in both the quasiparticle spectra, whose non-trivial nature is confirmed by the Berry phase calculation. The incorporation of spin–orbit coupling in the electron spectra introduces distinctive characteristics in the transport properties, facilitating the emergence of anomalous Hall conductivity through Berry curvature in both bulk and monolayer. Furthermore, the monolayer has been proposed as a two-terminal device model to investigate the quantum transport properties using the non-equilibrium Green’s function approach. This superlative combination of observations and modeling sets the path for a greater level of insight into the behavior and aspects of topological materials at the atomic scale.

Non-Hermitian skin effect enforced by nonsymmorphic symmetries

Non-Hermitian skin effect enforced by nonsymmorphic symmetries

Anomalies in the Dirac bands in the proximity of correlated electrons.

Dirac fermions, particles with zero rest mass, are observed in topological materials and are believed to play a key role in the exotic phenomena in fundamental science and the advancement of quantum technology. Most of the topological systems studied so far are weakly correlated systems and the study of their properties in the presence of electron correlation is an interesting emerging area of research, where the electron correlation is expected to enhance the effective mass of the particles. Here, we studied the properties of Dirac bands in a non-symmorphic layered Kondo lattice system, CeAgSb2, employing high-resolution angle-resolved photoemission spectroscopy and first-principles calculations. In addition to the Dirac cones due to non-symmorphic symmetry, this material hosts Dirac fermions in the squarenet layer in the proximity of a strongly correlated Ce layer exhibiting Kondo behavior. Experimental results reveal crossings of the highly dispersive linear bands at the Brillouin zone boundary due to non-symmorphic symmetry. In addition, there are anisotropic Dirac cones constituted by the squarenet Sb 5p states forming a diamond-shaped nodal line. These Dirac bands are linear in a wide energy range with an unusually high slope. Interestingly, near the local Ce 4f bands, these bands exhibit a change in the slope akin to the formation of a 'kink' observed in other materials due to electron-phonon coupling. The emergence of such exotic properties in proximity to strongly correlated electronic states has significant implications in the study of complex quantum materials including unconventional superconductors.

Angle-resolved photoemission study of NbGeSb with non-symmorphic symmetry

We investigate the electronic structure of NbGeSb with non-symmorphic symmetry. We employ angle-resolved photoemission spectroscopy (ARPES) to observe and identify the bulk and surface states over the Brillouin zone. By utilizing high-energy photons, we identify the bulk Fermi surface and bulk nodal line along the direction X–R, while the Fermi surface of the surface state is observed by using low-energy photons. We observe the splitting of surface bands away from the high-symmetry point X¯ . The density functional theory calculations on bulk and 1 to 5-layer slab models, as well as spin textures of NbGeSb, verify that the band splitting could be attributed to the Rashba-like spin–orbit coupling caused by space-inversion-symmetry breaking at the surface. These splitted surface bands cross with each other, forming two-dimensional Weyl-like crossings that are protected by mirror symmetry. Our findings provide insights into the two-dimensional topological and symmetry-protected band inversion of surface states.

Pressure-induced superconductivity in the nonsymmorphic topological insulator KHgAs

Recently, topological insulators (TIs) KHgX (X = As, Sb, Bi) with hourglass-shaped dispersion have attracted great interest. Different from the TIs protected by either time-reversal or mirror crystal symmorphic symmetry tested in previous experiments, these materials were proposed as the first material class whose band topology relies on nonsymmorphic symmetries. As a result, KHgX shows many exotic properties, such as hourglass-shaped electronic channels and three-dimensional doubled quantum spin Hall effects. To date, high-pressure experimental studies on these nonsymmorphic TIs are minimal. Here, we carried out high-pressure electrical measurements up to 55 GPa, together with high-pressure X-ray diffraction measurements and high-pressure structure prediction on KHgAs. We found a pressure-induced semiconductor-metal transition between ~16 and 20 GPa, followed by the appearance of superconductivity with a Tc of ~3.5 K at approximately 21 GPa. The superconducting transition temperature was enhanced to a maximum of ~6.6 K at 31.8 GPa and then slowly decreased until 55 GPa. Furthermore, three high-pressure phases within 55 GPa were observed, and their crystal structures were established. Our results showed the high-pressure phase diagram of KHgAs and determined the pressure-induced superconductivity in nonsymmorphic TIs. Thus, our study can be used to facilitate further research on superconductivity and topologically nontrivial features protected by nonsymmorphic symmetries.

Pushing the limits of complete omnidirectional bandgaps in 2D nonsymmorphic single-phase phononic crystals

We expand the limits of complete omnidirectional bandgaps (neither P- nor S-waves can propagate) for high symmetry two-dimensional phononic crystal (PnC) designs. We reveal an extremely large 124% complete omnidirectional bandgap and demonstrate the possibility of creating custom, mechanically robust PnCs with improved characteristics via simple geometric changes to known PnC designs. The findings are experimentally validated, proving that it is feasible to achieve extreme ultrasonic attenuation using the nonsymmorphic p4gm symmetry group design, for both P- and S-waves, which significantly outperforms symmorphic p4mm-group PnC designs. We shed light on the high attenuation properties of the p4gm PnC for S-waves (which is rarely explored experimentally), revealing differences between the mode types. Practical insight into the design of PnCs with improved acoustic properties for potential applications in the field of vibration isolation, most notably when S-wave elimination is vital, is discussed.

Interplay between altermagnetism and nonsymmorphic symmetries generating large anomalous Hall conductivity by semi-Dirac points induced anticrossings

Interplay between altermagnetism and nonsymmorphic symmetries generating large anomalous Hall conductivity by semi-Dirac points induced anticrossings

An anisotropic van der Waals dielectric for symmetry engineering in functionalized heterointerfaces

Van der Waals dielectrics are fundamental materials for condensed matter physics and advanced electronic applications. Most dielectrics host isotropic structures in crystalline or amorphous forms, and only a few studies have considered the role of anisotropic crystal symmetry in dielectrics as a delicate way to tune electronic properties of channel materials. Here, we demonstrate a layered anisotropic dielectric, SiP2, with non-symmorphic twofold-rotational C2 symmetry as a gate medium which can break the original threefold-rotational C3 symmetry of MoS2 to achieve unexpected linearly-polarized photoluminescence and anisotropic second harmonic generation at SiP2/MoS2 interfaces. In contrast to the isotropic behavior of pristine MoS2, a large conductance anisotropy with an anisotropy index up to 1000 can be achieved and modulated in SiP2-gated MoS2 transistors. Theoretical calculations reveal that the anisotropic moiré potential at such interfaces is responsible for the giant anisotropic conductance and optical response. Our results provide a strategy for generating exotic functionalities at dielectric/semiconductor interfaces via symmetry engineering.

Solvable model for discrete time crystal enforced by nonsymmorphic dynamical symmetry

Solvable model for discrete time crystal enforced by nonsymmorphic dynamical symmetry

Two-dimensional Dirac phonons without/with inversion symmetry

In a recent study, two-dimensional Dirac phonons that are protected by nonsymmorphic symmetries in spinless systems were systematically investigated. However, the focus of this study was on the classification of Dirac phonons. To address the gap in the research on the topological features of 2D Dirac phonons based on their effective models, we classified the 2D Dirac phonons into two classes: without or with inversion symmetry, thereby clarifying the minimal symmetry requirements for enforcing 2D Dirac points. Based on symmetry analysis, we discovered that screw symmetries, together with time-reversal symmetry, play an essential role in the existence of Dirac points. To validate this result, we constructed the k·p model to describe the Dirac phonons and discussed their topological features accordingly. We found that a 2D Dirac point could be considered as a composition of two 2D Weyl points with opposite chirality. Furthermore, we provided two concrete materials to demonstrate our findings. Overall, our work provides a more detailed study of 2D Dirac points in spinless systems and clarifies their topological features.

Frequency-dependent Faraday and Kerr rotation in anisotropic nonsymmorphic Dirac semimetals

We calculate the frequency-dependent longitudinal and Hall conductivities and the Faraday and Kerr rotation angles for a single sheet of anisotropic Dirac semimetal protected by nonsymmorphic symmetry in the presence of a Zeeman term coupling to the out-of-plane component of the spin. While the Zeeman term causes a rotation of the plane of polarization of the light, the anisotropy causes the appearance of an elliptically polarized component in an initially linearly polarized beam. The two effects can be combined in a single complex Faraday rotation angle. At the zero-frequency limit, we find a finite value of the Faraday rotation angle, which is given by $2{\ensuremath{\alpha}}_{F}$, where ${\ensuremath{\alpha}}_{F}$ is the effective fine structure constant associated with the velocity of the linearly dispersing Dirac fermions. We also find a logarithmic enhancement of the Faraday (and Kerr) rotation angles as the frequency of the light approaches the absorption edge associated with the Zeeman-induced gap. While the enhancement is reduced by impurity scattering, it remains significant for an attainable level of material purity. These results indicate that two-dimensional Dirac materials protected by nonsymmorphic symmetry are responsive to Zeeman couplings and can be used as platforms for magneto-optic applications, such as the realization of polarization-rotating devices.

Catalog of noninteracting tight-binding models with two energy bands in one dimension

We classify Hermitian tight-binding models describing noninteracting electrons on a one-dimensional periodic lattice with two energy bands. To do this, we write a generalized Rice-Mele model with two orbitals per unit cell, including all possible complex-valued long-range hoppings consistent with Hermicity. We then apply different forms of time-reversal, charge-conjugation and chiral symmetry in order to constrain the parameters, resulting in an array of possible models in different symmetry classes. For each symmetry class, we define a single, canonical form of the Hamiltonian and identify models that are related to the canonical form by an off-diagonal unitary transformation in the atomic basis. The models have either symmorphic or nonsymmorphic nonspatial symmetries (time $T$, chiral and charge-conjugation). The nonsymmorphic category separates into two types of state of matter: an insulator with a $\mathbb{Z}_2$ topological index in the absence of nonsymmorphic time-reversal symmetry or, in the presence of nonsymmorphic time-reversal symmetry, a metallic state. The latter is an instance of Kramer's degeneracy with one degeneracy point in the Brillouin zone as opposed to no degeneracy points in symmorphic systems with $T^2 = 1$ and two in symmorphic systems with $T^2 = - 1$.

Second-order topological corner states in two-dimensional elastic wave metamaterials with nonsymmorphic symmetries

This paper investigates the higher-order topological behavior of elastic wave metamaterials composed of L-shaped pillars attached to a plate. In this system, the multi-dimensional topological transitions can be developed for physical transfer among the corner, edge, and bulk elastic wave modes by tuning the rotation angle of the L-shaped pillars. Different from the four degenerate corner states in the symmorphic system, these corner states in this metamaterial evolve into two doublets because of the lack of mirror symmetries. With the energy highly trapped at the corners, it thus can immediately realize elastic resonators/sensors with high-quality factors. Besides, the coupling effect between corner and edge states is analyzed in the multi-dimensional topological system. The edge–corner coupling effect could be applied to relevant fields, such as high-sensitive detection, elastic energy harvesting, and optomechanical sensing.

Coexistence of Dirac points and nodal chains in photonic metacrystal.

Gapless topological phases, i.e. topological semimetals, come in various forms such as Weyl/Dirac semimetals, nodal line/chain semimetals, and surface-node semimetals. However, the coexistence of two or more topological phases in a single system is still rare. Here, we propose the coexistence of Dirac points and nodal chain degeneracies in a judiciously designed photonic metacrystal. The designed metacrystal exhibits nodal line degeneracies lying in perpendicular planes, which are chained together at the Brillouin zone boundary. Interestingly, the Dirac points, which are protected by nonsymmorphic symmetries, are located right at the intersection points of nodal chains. The nontrivial Z2 topology of the Dirac points is revealed by the surface states. The Dirac points and nodal chains are located in a clean frequency range. Our results provide a platform for studying the connection between different topological phases.

Controlling Dzyaloshinskii-Moriya interaction in a centrosymmetric nonsymmorphic crystal

Presence of the Dzyaloshinskii-Moriya (DM) interaction in limited noncentrosymmetric materials leads to novel spin textures and exotic chiral physics. The emergence of DM interaction in centrosymmetric crystals could greatly enrich material realization. Here we show that an itinerant centrosymmetric crystal respecting a nonsymmorphic space group is a new platform for the DM interaction. Taking the P4/nmm space group as an example, we demonstrate that the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction induces the DM interactions, in addition to the Heisenberg exchange and the Kaplan-Shekhtman-Entin-wohlman-Aharony (KSEA) interaction. The direction of DM vector depends on the positions of magnetic atoms in the real space, and the amplitude depends on the location of the Fermi surface in the reciprocal space. The diversity stems from the position-dependent site groups and the momentum-dependent electronic structures guaranteed by the nonsymmorphic symmetries. Our study unveils the role of the nonsymmorphic symmetries in affecting magnetism, and suggests that the nonsymmorphic crystals can be promising platforms to design magnetic interactions.

Topological electride phase of sodium at high pressures and temperatures

Ab initio evolutionary structure searches coupled with quasiharmonic calculations predict that the insulating Na hP4 phase transitions to a novel P63/m phase between 200 GPa at 150 K, and 350 GPa at 1900 K. P63/m Na is a topological semimetal with a Dirac nodal surface that is protected by a non-symmorphic symmetry, S2z . It is characterized by localized non-nuclear charge within 1D honeycomb channels and 0D cages rendering it an electride. These results highlight the complexity of warm dense sodiums electronic structure and free energy landscape that emerges at conditions where ionic cores overlap.

Symmetry‐Enforced 2D Hourglass Phononic Nodal Net

Topological nodal‐line semimetals have received extensive attention in recent years, and they exhibit a variety of shapes under various crystallographic symmetry protections, including nodal rings, nodal chains, and nodal knots. By using symmetry analysis and an effective force constant model, it is proposed that a class of 2D hourglass nodal nets protected by nonsymmorphic symmetry can exist in the space group , where the glide operations play a crucial role in the formation of the nets. Further effective model analysis shows that the connect points of the nets are Dirac points that are forced by the symmetries. Based on first‐principle calculations, it is predicted that this nodal net can be realized in the phonon dispersion of 3D Sn. The Berry phase and nontrivial surface states further support its nontrivial topological characteristics. Herein, the studies expand the nodal structure in the study of nodal line in phononic systems and offer a solid foundation for subsequent experimental investigation on this challenging topological state.

Nonsymmorphic symmetry protected spin-orbit semi-Dirac fermions in two dimensions

Semi-Dirac fermions (SDFs) in two-dimensional (2D) systems, which simultaneously exhibit linear and quadratic dispersion around them, are a bridge linking linear and quadratic Dirac cones. However, the so-called SDFs in the reported 2D materials are doubly degenerate (namely semi-Weyl fermions), and are vulnerable against spin-orbital coupling (SOC). Here, we propose a 2D SOC-robust SDF which arises from nonsymmorphic symmetries. Unlike the known 2D SDFs, due to the presence of inversion and time reversal symmetries ($\mathcal{I}\mathcal{T}$), each crossing band is doubly degenerate, making the SDF of fourfold degeneracy. By high throughput screening, we find that 26 candidate 2D materials (such as ${\mathrm{CuCN}}_{2}$, ${\mathrm{CaI}}_{2}$, TlF) can hold SDFs, which belong to layer groups 40 ($pmam$), 43 ($pbaa$), and 45 ($pbma$), respectively. Furthermore, the symmetry protection mechanism, associated Fermi arc edge states, the catalytic performance, and the topological phase transitions under symmetry breaking of SDFs are revealed, providing a way for understanding 2D SOC-robust SDFs. Overall, our work proposes a class of topological phases, and provides a platform to study their fascinating physical effects.

Transport properties of CrP

CrP has many exotic physical properties due to a four-fold degenerate band crossing at the Y point of the Brillouin zone, which is protected by the nonsymmorphic symmetry of the space group. We carried out the heat capacity, electrical and thermal transport measurements on CrP and extracted the electron thermal conductivity. Due to the difference in energy and momentum relaxation time during electron–phonon inelastic scattering, the normalized Lorentz number decreases below about 160 K. Below 25.6 K, the normalized Lorentz number begins to recover, which is due to the dominance of elastic scattering between electrons and defects at low temperatures.