Nonsymmorphic Symmetry Research Articles

Tunable Dirac points in a two-dimensional non-symmorphic wallpaper group lattice

Non-symmorphic symmetries protect Dirac nodal lines and cones in lattice systems. Here, we investigate the spectral properties of a two-dimensional lattice belonging to a non-symmorphic group. Specifically, we look at the herringbone lattice, characterized by two sets of glide symmetries applied in two orthogonal directions. We describe the system using a nearest-neighbor tight-binding model containing horizontal and vertical hopping terms. We find two nonequivalent Dirac cones inside the first Brillouin zone along a high-symmetry path. We tune these Dirac cones’ positions by breaking the lattice symmetries using on-site potentials. These Dirac cones can merge into a semi-Dirac cone or unfold along a high-symmetry path. Finally, we perturb the system by applying a dimerization of the hopping terms. We report a flow of Dirac cones inside the first Brillouin zone describing quasi-hyperbolic curves. We present an implementation in terms of CO atoms placed on the top of a Cu(111) surface.

Ni-Intercalation-Induced Topological Nodal Line States and Superconductivity in NiTe

Based on symmetry analyses and first-principles calculations, we report the predictions of topological fermions and superconductivity in Ni-intercalated transition metal chalcogenide NiTe. The self-intercalation of Ni introduces nonsymmorphic symmetry operations, protecting a pair of Dirac points and three intersecting Dirac nodal lines near Fermi energy level (EF), which generate spin-textured surface states across EF. Moreover, the Ni-intercalation strongly enhances electron–phonon coupling in NiTe and makes it an anisotropic Bardeen–Cooper–Schrieffer superconductor with a full superconducting gap and a critical temperature Tc ∼1.5 K. The coexistence of topological surface states and superconductivity implies that NiTe is a potential material platform for exploring topological superconductivity.

Node-line Dirac semimetal manipulated via Kondo mechanism in nonsymmorphic CePt2Si2

Dirac node lines (DNLs) are characterized by Dirac-type linear crossings between valence and conduction bands along one-dimensional node lines in the Brillouin zone (BZ). Spin-orbit coupling (SOC) usually shifts the degeneracy at the crossings thus destroys DNLs, and so far the reported DNLs in a few materials are non-interacting type, making the search for robust interacting DNLs in real materials appealing. Here, via first-principle calculations, we reveal that Kondo interaction together with nonsymmorphic lattice symmetries can drive a robust interacting DNLs in a Kondo semimetal CePt_2Si_2, and the feature of DNLs can be significantly manipulated by Kondo behavior in different temperature regions. Based on the density function theory combining dynamical mean-field theory (DFT+DMFT), we predict a transition to Kondo-coherent state at coherent temperature T_coh= 80 K upon cooling, verified by temperature dependence of Ce-4f self-energy, Kondo resonance peak, magnetic susceptibility and momentum-resolved spectral. Below T_coh, well-resolved narrow heavy-fermion bands emerge near the Fermi level, constructing clearly visualized interacting DNLs locating at the BZ boundary, in which the Dirac fermions have strongly enhanced effective mass and reduced velocity. In contrast, above a crossover temperature T_KS =600 K, the destruction of local Kondo screening drives non-interacting DNLs which are comprised by light conduction electrons at the same location. These DNLs are protected by lattice nonsymmorphic symmetries thus robust under intrinsic strong SOC. Our proposal of DNLs which can be significantly manipulated according to Kondo behavior provides an unique realization of interacting Dirac semimetals in real strongly correlated materials, and serves as a convenient platform to investigate the effect of electronic correlations on topological materials.

Hidden Zeeman-type spin polarization in bulk crystals

Exploring hidden effects that have been overlooked given the nominal global crystal symmetry but are indeed visible in solid-state materials has been a fascinating subject of research recently. Here, we introduce a novel hidden Zeeman-type spin polarization (HZSP) in nonmagnetic bulk crystals with sublattice structures. In the momentum space of these crystals, the doubly degenerate bands formed in a certain plane can exhibit a uniform spin configuration with opposite spin orientations perpendicular to this plane, whereas such degenerate states are spatially separated in a pair of real-space sectors. Interestingly, we find that HZSP can manifest itself in both centrosymmetric and non-centrosymmetric materials. We further demonstrate the important role of nonsymmorphic twofold screw-rotational symmetry played in the formation of HZSP. Moreover, two representative material examples, i.e., centrosymmetric WSe$_2$ and noncentrosymmetric BaBi$_4$O$_7$, are identified to show HZSP via first-principles calculations. Our finding thus not only opens new perspectives for hidden spin polarization research but also significantly broadens the range of materials towards spintronics applications.

Broad-angle coherent perfect absorption-lasing and super-collimation in two-dimensional non-Hermitian photonic crystals.

Coherent perfect absorption-lasing (CPAL) and collimation have been intensively studied for normal and small angle wave incidence. Here, we report a two-dimensional non-Hermitian photonic crystal for broad-angle CPAL and super-collimation. The synergy of a nonsymmorphic glide symmetry of the lattice, gain-loss modulation and an optimization of unit cell induces a parity-time phase transition in the band structure along the Brillouin zone boundary. The transition points, i.e., the exceptional points, form a slab-like contour, with nearly zero dispersion in both real and imaginary parts of the band structure. Such dispersionless band structure significantly enhances the range of incident angle for CPAL and collimation.

Topological Surface States in a Gyroid Acoustic Crystal

The acoustic properties of an acoustic crystal consisting of acoustic channels designed according to the gyroid minimal surface embedded in a 3D rigid material are investigated. The resulting gyroid acoustic crystal is characterized by a spin-1 Weyl and a charge-2 Dirac degenerate points that are enforced by its nonsymmorphic symmetry. The gyroid geometry and its symmetries produce multi-fold topological degeneracies that occur naturally without the need for ad hoc geometry designs. The non-trivial topology of the acoustic dispersion produces chiral surface states with open arcs, which manifest themselves as waves whose propagation is highly directional and remains confined to the surfaces of a 3D material. Experiments on an additively manufactured sample validate the predictions of surface arc states and produce negative refraction of waves at the interface between adjoining surfaces. The topological surface states in a gyroid acoustic crystal shed light on nontrivial bulk and edge physics in symmetry-based compact continuum materials, whose capabilities augment those observed in ad hoc designs. The continuous shape design of the considered acoustic channels and the ensuing anomalous acoustic performance suggest this class of phononic materials with semimetal-like topology as effective building blocks for acoustic liners and load-carrying structural components with sound proofing functionality.

Multiple-symmetry-protected lantern-like nodal walls in lithium-rich compound LiRuO2

Topological semimetals have attracted wide attention due to their potential applications, such as electronic devices and electrocatalysis. Herein, based on the first-principles calculations and symmetry analysis, we first report that ternary compound pnnm-type LiRuO2 is a typical lantern-like nodal wall semimetal. Specifically, without considering spin-orbit coupling (SOC), one-dimensional (1D) two-fold degenerate bands on the ki = ±π (i = x, y) planes form the two-dimensional (2D) topological state (namely, nodal surface) under the constraint of multiple symmetry operations. In addition, the symmetry-enforced nodal network is formed on the kz = ±π planes. Finally, these nodal networks and nodal surfaces are coupled together to form lantern-like nodal walls. Remarkably, these topological states are protected by multiple symmetries, namely, nonsymmorphic two-fold screw-rotational symmetry [S2i (i = x, y)], time-reversal symmetry (T), inversion symmetry (I), glide plane symmetry (σz), and two-fold rotational symmetry (C2x/y). In addition, we further discuss the effect of spin-orbit coupling on the lantern-like nodal walls. We find that even if LiRuO2 contains S2z and T symmetries, these nodal surfaces and nodal networks are still broken. Then, due to the existence of I and T symmetries, Dirac nodal lines and Dirac points are formed in the low-energy region. Therefore, our work indicates that LiRuO2 is an excellent material platform for researching multiple topological states.

Symmetry-enforced planar nodal chain phonons in non-symmorphic materials

Nodal chains in which two nodal rings connect at one point were recently discovered in non-symmorphic electronic systems and then generalized to symmorphic phononic systems. In this work, we identify a new class of planar nodal chains in non-symmorphic phononic systems, where the connecting rings lie in the same plane. The constituting nodal rings are protected by mirror symmetry, and their intersection is guaranteed by the combination of time-reversal and non-symmorphic twofold screw symmetry. The connecting points are fourfold degenerate while those in previous works are twofold degenerate. We found 8 out of 230 space groups that can host the proposed planar nodal chain phonons. Taking wurtzite GaN (space group No. 186) as an example, the planar nodal chain is confirmed by first-principles calculations. The planar nodal chains result in two distinct classes of drumhead surface states on the [10(–1)0] and the [0001] surface Brillouin zones. Our finding reveals a class of planar nodal chains in non-symmorphic phononic systems, expanding the catalog of topological nodal chains and enriching the family of topological surface states.

Bonding and Electronic Nature of the Anionic Framework in LaPd3S4

Double Dirac materials are a topological phase of matter in which a non-symmorphic symmetry enforces greater electronic degeneracy than normally expected – up to eightfold. The cubic palladium bronzes NaPd3O4 and LaPd3S4 are built of Pd3X4 (X = O, S) anionic frameworks that are ionically bonded to A cations (A = Na, La). These materials were recently identified computationally as harboring eightfold fermions. Here we report the preparation of single crystals and electronic properties of LaPd3S4. Measurements down to T = 0.45 K and in magnetic fields up to μ0H = 65 T are consistent with normal Fermi liquid physics of a Dirac metal in the presence of dilute magnetic impurities. This interpretation is further confirmed by analysis of specific heat, magnetization measurements and comparison to density functional theory (DFT) calculations. Through a bonding analysis of the DFT electronic structure of NaPd3O4 and LaPd3S4, we identify the origin of the stability of the anionic Pd3X4 framework at higher electron counts for X = S than X = O, and propose chemical tuning strategies to enable shifting the 8-fold fermion points to the Fermi level.

Homotopic analysis of quantum states in two-dimensional polymorphs by a herringbone lattice model

We report a homotopic analysis of quantum states in two-dimensional (2D) polymorphs. Two typical models, namely the honeycomb-type and Shastry-Sutherland-type models, are continuously connected through a deformable herringbone lattice model having a nonsymmorphic group symmetry. The evolution of Dirac fermions is characterized within the phase space spanned by the deformation parameters. Our tight-binding models, based on single orbitals for each site with and without spin-orbit coupling (SOC), reveal how the SOC convolutes, the manner being important in facilitating the search and design of 2D topological insulators focused on the homotopic feature of symmetry-protected properties independent of specific materials.

Two-dimensional hourglass Weyl nodal loop in monolayer Pb(ClO2)2 and Sr(ClO2)2

The hourglass fermions in solid-state materials have been attracting significant interest recently. However, realistic two-dimensional (2D) materials with hourglass-shaped band structures are still very scarce. Here, through the first-principles calculations, we identify the monolayer Pb(ClO2)2 and Sr(ClO2)2 materials as the new realistic materials platform to realize 2D hourglass Weyl nodal loop. We show that these monolayer materials possess an hourglass Weyl nodal loop circling around the Γ point and Weyl nodal line on the Brillouin zone (BZ) boundary in the absence of spin–orbit coupling (SOC). Through the symmetry analysis, we demonstrate that the hourglass Weyl nodal loop and Weyl nodal line are protected by the nonsymmorphic symmetries, and are robust under the biaxial strains. When we include the SOC, a tiny gap will be opened in the hourglass nodal loop and nodal line, and the nodal line can be transformed into the spin-orbit Dirac points. Our results provide a new realistic material platform for studying the intriguing physics associated with the 2D hourglass Weyl nodal loop and spin-orbit Dirac points.

Multiple Dirac points including potential spin-orbit Dirac points in nonsymmorphic HfGe0.92Te

Looking for new materials with Dirac points has been a fascinating subject of research. Here we report the growth, crystal structure, and band structure of HfGe0.92Te single crystals, featuring three different types of Dirac points. HfGe0.92Te crystalizes in a nonsymmorphic tetragonal space group P4/nmm (No. 129), having square Ge-atom plane with vacancies about 8%. Despite the vacancies on Ge site, the Dirac nodal line composed of conventional Dirac points vulnerable to spin-orbit coupling (SOC) is observed using angle-resolved photoemission spectroscopy, accompanied with the robust Dirac line protected by the nonsymmorphic symmetry against both SOC and vacancies. Specially, spin-orbit Dirac points (SDPs) originated from the surface formed under SOC are hinted to exist according to our experiments and calculations. Quasi-two-dimensional (quasi-2D) characters are observed and further confirmed by angular-resolved magnetoresistance. HfGe0.92Te is a good candidate to explore exotic topological phases or topological properties with three different types of Dirac points and a promising candidate to realize 2D SDPs.

Chiral Majorana fermions in two-dimensional square lattice antiferromagnet with proximity-induced superconductivity

Combination of proximity-induced superconductivity and ferromagnetic exchange field in a two-dimensional square-lattice antiferromagnet with spin–orbit coupling and nonsymmorphic symmetry can induce a topological superconductor phase with chiral Majorana edge states. The lattice model of the Bogoliubov-de Gennes (BdG) Hamiltonian was applied to study the phase diagram of bulks and chiral Majorana edge states in nanoribbons. By numerically studying the phase diagram, we found that the non-uniformity of either the superconducting pairing parameters or the exchange field at the two sublattices is necessary to induce a topological superconductor phase with chiral Majorana edge states. The BdG Chern number of certain topological superconductor phases is ±1 or ±3, such that the corresponding nanoribbons have one or three pairs of chiral Majorana edge states, respectively.

Non-Abelian nonsymmorphic chiral symmetries

The Hofstadter model exemplifies a large class of physical systems characterized by particles hopping on a lattice immersed in a gauge field. Recent advancements on various synthetic platforms have enabled highly controllable simulations of such systems with tailored gauge fields featuring complex spatial textures. These synthetic gauge fields could introduce synthetic symmetries that do not appear in electronic materials. Here, in an $\mathrm{SU}(2)$ non-Abelian Hofstadter model, we theoretically show the emergence of multiple nonsymmorphic chiral symmetries, which combine an internal unitary antisymmetry with fractional spatial translation. Depending on the values of the gauge fields, the nonsymmorphic chiral symmetries can exhibit non-Abelian algebra and protect Kramers quartet states in the bulk band structure, creating general fourfold degeneracy at all momenta. These nonsymmorphic chiral symmetries protect double Dirac semimetals at zero energy, which become gapped into quantum confined insulating phases upon introducing a boundary. Moreover, the parity of the system size can determine whether the resulting insulating phase is trivial or topological. Our work indicates a pathway for creating topology via synthetic symmetries emergent from synthetic gauge fields.

Nonsymmorphic chiral symmetry and solitons in the Rice-Mele model

The Rice-Mele model has two topological and spatially-inversion symmetric phases, namely the Su-Schrieffer-Heeger (SSH) phase with alternating hopping only, and the charge-density-wave (CDW) phase with alternating energies only. The chiral symmetry of the SSH phase is robust in position space, so that it is preserved in the presence of the ends of a finite system and of textures in the alternating hopping. However, the chiral symmetry of the CDW wave phase is nonsymmorphic, resulting in a breaking of the bulk topology by an end or a texture in the alternating energies. We consider the presence of solitons (textures in position space separating two degenerate ground states) in finite systems with open boundary conditions. We identify the parameter range under which an atomically-sharp soliton in the CDW phase supports a localized state which lies within the band gap, and we calculate the expectation value $p_y$ of the nonsymmorphic chiral operator for this state, and the soliton electric charge. As the spatial extent of the soliton increases beyond the atomic limit, the energy level approaches zero exponentially quickly or inversely proportionally to the width, depending on microscopic details of the soliton texture. In both cases, the difference of $p_y$ from one is inversely proportional to the soliton width, while the charge is independent of the width. We investigate the robustness of the soliton level in the presence of disorder and sample-to-sample parameter variations, comparing with a single soliton level in the SSH phase with an odd number of sites.

Tunable Weyl half-semimetals in two-dimensional iron-based materials MFeSe (M=Tl,In,Ga)

The layered iron chalcogenide materials have attracted considerable attention recently for their exotic superconductivity at relatively high temperatures. Topological phases are, however, seldom proposed in these materials. Based on density-functional theory calculations together with symmetry analysis, 100% spin-polarized Weyl semimetals, namely Weyl half semimetals (WHSMs), are predicted in two-dimensional (2D) TlFeSe and GaFeSe monolayers, built based on FeSe monolayers. The acquired Weyl fermions are protected by a nonsymmorphic symmetry. Dissimilarly, the InFeSe monolayer is found to be a quantum anomalous Hall (QAH) insulator with a large band gap (403 meV). By tuning the magnetization direction, the monolayers can vary from a WHSM to a QAH insulator or vice versa. The phase-transition mechanism is analyzed by using an effective $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ model. Our work provides a pathway to carry out the fascinating 2D WHSMs and the QAH effect in one material which will have promising applications in not only spintronics but also topological microelectronics.

Origin of nonsymmorphic bosonization formulas in generalized antiferromagnetic Kitaev spin- 12 chains from a renormalization-group perspective

Recently, in the Luttinger liquid phase of the one-dimensional generalized antiferromagnetic Kitaev spin-1/2 model, it has been found that the Abelian bosonization formulas of the local spin operators only respect the exact discrete nonsymmorphic symmetry group of the model, not the emergent U(1) symmetry. In this work, we perform a renormalization group (RG) study to provide explanations for the origin of the U(1) breaking terms in the bosonization formulas. We find that the lack of U(1) symmetry originates from the wave-function renormalization effects in the spin operators along the RG flow induced by the U(1) breaking interactions in the microscopic Hamiltonian. In addition, the RG analysis can give predictions to the signs and order of magnitudes of the coefficients in the bosonization formulas. Our work is helping to understand the rich nonsymmorphic physics in one-dimensional Kitaev spin models.

Nonsymmorphic P21/m‐MoTe2: Novel 2D Topological Materials

Nonsymmorphic symmetries play a significant role in the emergence of topological nontrivial phases in some 2D materials. Herein, a new 2D material P21/m‐MoTe2 with nontrivial topological properties and nonsymmorphic symmetries is proposed. With the help of first‐principles calculations, it is found that it is dynamically stable, and has relatively high energetic stability. When spin–orbit coupling (SOC) is ignored, this material is a semimetal with two intersects appearing on the high‐symmetry line Γ–Y. Detailed symmetry analysis reveal that they are protected by the screw operator . After SOC is turned on, it turns into a topological insulator with a global bandgap of 94 meV. The large gap ensures that the topological nontrivial phase is robust to an external perturbation. The discovery of nonsymmorphic P21/m‐MoTe2 expands the family of transition metal dichalcogenides and provides a new platform to study topological phases in 2D nonsymmorphic materials.

Two-dimensional Dirac nodal line state protected against spin-orbit coupling in MoTe monolayer

We theoretically study a kind of protected double-nodal line state in two-dimensional (2D) transition metal monochalcogenide MoTe by using symmetry analysis and first-principles calculation. Among this kind of materials, there are various intriguing electronic properties, such as the long-range Coulomb interaction. The 2D layer four-fold degeneracy nodal line material is susceptible to the perturbation of strong spin-orbit coupling (SOC). We find that the Dirac degeneracy of two low-energy bands around the S point of the Brillouin zone is robust against SOC in P21/m-MoTe. The degeneracy nodal lines are protected by nonsymmorphic screw-rotational crystal symmetry and time-reversal symmetry. Due to the protection of symmetries, the Dirac double node line states under SOC are different from the accidental band touching. Furthermore, several isolog structures of P21/m-MoTe are also discussed. We find that the fourfold degeneracy nodal line state of MoSe also is robust against SOC, similar to MoTe.

Third-order topological insulators with wallpaper fermions in Tl4PbTe3 and Tl4SnTe3

Nonsymmorphic symmetries open up horizons of exotic topological boundary states and even generalize the bulk–boundary correspondence, which, however, the third-order topological insulator in electronic materials are still unknown. Here, by means of the symmetry analysis and k · p models, we uncover the emergence of long-awaited third-order topological insulators and the wallpaper fermions in space group I4/mcm (No.140). Based on this, we present the hourglass fermion, fourfold-degenerate Dirac fermion, and Möbius fermion in the (001) surface of Tl4XTe3 (X = Pb/Sn) with a nonsymmorphic wallpaper group p4g. Remarkably, 16 helical corner states reside on eight corners in Kramers pair, rendering the real electronic material of third-order topological insulators. More importantly, a time-reversal polarized octupole polarization is defined to uncover the nontrivial third-order topology, as is implemented by the 2nd and 3rd order Wilson loop calculations. Our results could considerably broaden the range of wallpaper fermions and lay the foundation for future experimental investigations of third-order topological insulators.