Nonsymmorphic Symmetry Research Articles

Composite Dirac semimetals

Weak topological insulators and Dirac semimetals are gapped and nodal phases with distinct topological properties, respectively. Here, we propose a novel topological phase that exhibits features of both and is dubbed composite Dirac semimetal (CDSM). In its bulk, the CDSM has a pair of Dirac points and a pair of bands inverted along a high-symmetry path. At side surfaces, a pair of Fermi arcs connecting the projected Dirac points coexist with a pair of Fermi loops traversing the surface Brillouin zone. A nonsymmorphic symmetry dictates degeneracies between the Fermi arcs and the Fermi loops. We characterize the CDSM by multiple topological invariants and show that, under a transition without breaking any symmetry, it deforms into a topological crystalline insulator hosting two pairs of surface Fermi loops. We demonstrate the CDSM in two models and predict its realization in the KAuTe-family materials.

Directional massless Dirac fermions in a layered van der Waals material with one-dimensional long-range order

One or a few layers of van der Waals (vdW) materials are promising for applications in nanoscale electronics. Established properties include high mobility in graphene, a large direct gap in monolayer MoS2, the quantum spin Hall effect in monolayer WTe2 and so on. These exciting properties arise from electron quantum confinement in the two-dimensional limit. Here, we use angle-resolved photoemission spectroscopy to reveal directional massless Dirac fermions due to one-dimensional confinement of carriers in the layered vdW material NbSi0.45Te2. The one-dimensional directional massless Dirac fermions are protected by non-symmorphic symmetry, and emerge from a stripe-like structural modulation with long-range translational symmetry only along the stripe direction as we show using scanning tunnelling microscopy. Our work not only provides a playground for investigating further the properties of directional massless Dirac fermions, but also introduces a unique component with one-dimensional long-range order for engineering nano-electronic devices based on heterostructures of vdW materials.

Multiple band crossings and Fermi surface topology: Role of double nonsymmorphic symmetries in MnP-type crystal structures

We use relativistic ab initio methods combined with model Hamiltonian approaches to analyze the normal-phase electronic and structural properties of the recently discovered WP superconductor. Remarkably, the outcomes of such study can be employed to set fundamental connections among WP and the CrAs and MnP superconductors belonging to the same space group. One of the key features of the resulting electronic structure is represented by the occurrence of multiple band crossings along specific high-symmetry lines of the Brillouin zone. In particular, we demonstrate that the eightfold band degeneracy obtained along the $\mathit{SR}$ path at $({k}_{x},{k}_{y})=(\ensuremath{\pi},\ensuremath{\pi})$ is due to inversion time-reversal invariance and a pair of nonsymmorphic symmetries. The presence of multiple degenerate Fermi points along the $\mathit{SR}$ direction constrains the topology of the Fermi surface, which manifests distinctive marks when considering its evolution upon band filling variation. If the Fermi level crosses the bands along the $\mathit{SR}$ line as it happens at the nominal filling of the MnP, these Fermi surfaces are open or closed Fermi pockets. Moving the relative position of the Fermi level away from the eightfold degenerate bands as for the WP and CrAs compounds, the electronic changeover exhibits a simultaneous modification of the Fermi-surface dimensionality and topology. Four two-dimensional (2D) Fermi-surface sheets are centered around the $\mathit{SR}$ line with a corrugated profile along the ${k}_{z}$ direction. Moreover, we show that the spin-orbit interaction determines a selective removal of the band degeneracy and, consequently, a splitting of the quasi-2D Fermi sheets, as it happens in WP. Finally, we comment on the connections between our results and recent experimental and theoretical proposals about the triplet superconductivity in this class of compounds.

Growth and Strain Engineering of Trigonal Te for Topological Quantum Phases in Non-Symmorphic Chiral Crystals

Strained trigonal Te has been predicted to host Weyl nodes supported by a non-symmorphic chiral symmetry. Using low-pressure physical vapor deposition, we systematically explored the growth of trigonal Te nanowires with naturally occurring strain caused by curvature of the wires. Raman spectra and high mobility electronic transport attest to the highly crystalline nature of the wires. Comparison of Raman spectra for both straight and curved nanowires indicates a breathing mode that is significantly broader and shifted in frequency for the curved wires. Strain induced by curvature during growth therefore may provide a simple pathway to investigate topological phases in trigonal Te.

Hourglass Fermion in Two-Dimensional Material.

The hourglass fermion, as an exotic quasiparticle protected by nonsymmorphic symmetry, has excited great research interest recently. However, its bulk counterpart in two-dimensional (2D) solid-state materials has seldom been studied. In this Letter, we propose a 2D rectangular lattice made of p_{x} and p_{y} orbitals with glide mirror symmetry but without inversion symmetry to realize the hourglass fermion. The glide mirror symmetry guarantees a Dirac nodal line, while the Rashba spin-orbital coupling splits it into two Weyl nodal lines and generates two pairs of hourglass fermion located at the glide mirror plane. Furthermore, based on first principles calculations, we predict a surface-supported 2D material Bi/Cl-SiC(111) to realize our proposal, making a huge-bandwidth hourglass cone. Moreover, the hourglass fermion exhibits a spin-momentum locking spin texture and also sustains a giant spin Hall conductivity. Our results demonstrate a general routine for designing an hourglass fermion in 2D materials, which will be easily extended to other surfaces with different adatoms and lattice symmetries.

Magnetic correlations in the pressure-induced superconductor CrAs investigated by As75 nuclear magnetic resonance

We report $^{75}$As-NMR results for CrAs under pressure, which shows superconductivity adjoining a helimagnetically ordered state. We successfully evaluated the Knight shift from the spectrum, which is strongly affected by the quadrupole interaction. The Knight shift shows the remarkable feature that the uniform spin susceptibility increases toward low temperatures in the paramagnetic state. This is in sharp contrast to CrAs at ambient pressure, and also to cuprates and Fe pnictides, where antiferromagnetic correlations are dominant. Superconductivity emerges in CrAs under unique magnetic correlations, which probably originate in the three-dimensional zigzag structure of its nonsymmorphic symmetry.

Floquet Second-Order Topological Insulators from Nonsymmorphic Space-Time Symmetries.

We propose a systematic way of constructing Floquet second-order topological insulators (SOTIs) based on time-glide symmetry, a nonsymmorphic space-time symmetry that is unique in Floquet systems. In particular, we are able to show that the static enlarged Hamiltonian in the frequency domain acquires reflection symmetry, which is inherited from the time-glide symmetry of the original system. As a consequence, one can construct a variety of time-glide symmetric Floquet SOTIs using the knowledge of static SOTIs. Moreover, the time-glide symmetry only needs to be implemented approximately in practice, enhancing the prospects of experimental realizations. We consider two examples, a 2D system in class AIII and a 3D system in class A, to illustrate our ideas, and then present a general recipe for constructing Floquet SOTIs in all symmetry classes.

Unconventional topological phase transition in non-symmorphic material KHgX (X\u2009=\u2009As, Sb, Bi)

Traditionally topological phase transition describes an evolution from topological trivial to topological nontrivial state. Originated from the non-symmorphic crystalline symmetry, we propose in this work an unconventional topological phase transition scheme between two topological nontrivial insulating states mediated by a Dirac gapless state, differing from the traditional topological phase transition. The KHgX (X = As, Sb, Bi) family is the first experimentally realized topological non-symmorphic crystalline insulator (TNCI), where the topological surface states are characterized by the Mobius-twisted connectivity. Based on first-principles calculations, we present a topological insulator–metal transition from TNCI into a Dirac semimetal (DSM) via applying an external pressure on KHgX. We find an unusual mirror Chern number Cm = −3 for the DSM phase of KHgX in the non-symmorphic crystal structure, which is topologically distinct from the traditional DSM such as Na3Bi and Cd3As2. Furthermore, we predict a new TNCI phase in KHgX via symmetry breaking. The topological surface states in this new TNCI phase display zigzag connectivity, different from the unstressed one. Our results offer a comprehensive study for understanding how the topological surface states evolve from a quantum phase transition in non-symmorphic system.

Z_{4} Topological Superconductivity in UCoGe.

Topological nonsymmorphic crystalline superconductivity (TNCS) is an intriguing phase of matter, offering a platform to study the interplay between topology, superconductivity, and nonsymmorphic crystalline symmetries. Interestingly, some of TNCSs are classified into Z_{4} topological phases, which have unique surface states referred to as a Möbius strip or an hourglass, and they have not been achieved in symmorphic superconductors. However, material realization of Z_{4} TNCS has never been known, to the best of our knowledge. Here, we propose that the paramagnetic superconducting phase of UCoGe under pressure is a promising candidate of Z_{4}-nontrivial TNCS enriched by glide symmetry. We evaluate Z_{4} invariants of UCoGe by deriving the formulas relating Z_{4} invariants to the topology of Fermi surfaces. Applying the formulas and previous abinitio calculations, we clarify that three odd-parity representations out of four are Z_{4}-nontrivial TNCS, whereas the other is also Z_{2}-nontrivial TNCS. We also discuss possible Z_{4} TNCS in CrAs and related materials.

Efficient method to compute Z4 indices with glide symmetry and applications to the Möbius materials CeNiSn and UCoGe

We propose an efficient method to numerically evaluate $\mathbb{Z}_4$-indices of M\"obius/hourglass topological phases with glide symmetry. Our method directly provides $\mathbb{Z}_4$-indices in the lattice Brillouin zone while the existing method requires careful observations of momentum dependent Wannier charge centers. As applications, we perform systematic computation of $\mathbb{Z}_4$-indices for M\"obius materials, $\mathrm{CeNiSn}$ and $\mathrm{UCoGe}$. In particular, our analysis elucidates that $\mathrm{UCoGe}$ shows strong M\"obius superconductivity for the $A_u$- or $B_{3u}$-representation whose topology has not been fully characterized. Furthermore, obtained phase diagrams reveal novel topological gapless excitations in the bulk which are protected by nonsymmorphic glide symmetry. We observe these gapless excitations with glide symmetry by doping holes into the superconducting phase of the $B_{1u}$- or $B_{3u}$-representation.

Hourglass Weyl loops in two dimensions: Theory and material realization in monolayer GaTeI family

Nodal loops in two-dimensional (2D) systems are typically vulnerable against spin-orbit coupling (SOC). Here, we explore 2D systems with a type of doubly degenerate nodal loops that are robust under SOC and feature an hourglass type dispersion. We present symmetry conditions for realizing such hourglass Weyl loops, which involve nonsymmorphic lattice symmetries. Depending on the symmetry, the loops may exhibit different patterns in the Brillouin zone. Based on first-principles calculations, we identify the monolayer GaTeI-family materials as a realistic material platform to realize such loops. These materials host a single hourglass Weyl loop circling around a high-symmetry point. Interestingly, there is also a spin-orbit Dirac point enabled by an additional screw axis. We show that the hourglass Weyl loop and the Dirac point are robust under a variety of applied strains. By breaking the screw axis, the Dirac point can be transformed into a second Weyl loop. Furthermore, by breaking the glide mirror, the hourglass Weyl loop and the spin-orbit Dirac point can both be transformed into a pair of spin-orbit Weyl points. Our work offers guidance and realistic material candidates for exploring fascinating physics of several novel 2D emergent fermions.

Strong spin-orbit coupling and Dirac nodal lines in the three-dimensional electronic structure of metallic rutile IrO2

Using high-resolution angle-resolved photoemission spectroscopy and ab initio calculation, we have studied the bulk and surface electronic structure of metallic rutile $5d$ transition metal oxide ${\mathrm{IrO}}_{2}$ that harbors both edge and corner sharing Ir-O octahedrons. We observe strong modulation of the band structure by spin-orbit coupling (SOC). The measured band structure is well reproduced by our ab initio calculation without band renormalization, suggesting the absence of the SOC-enhanced correlation effect in ${\mathrm{IrO}}_{2}$. In accordance with the calculation, we visualize two types of Dirac nodal lines (DNLs) protected by mirror symmetry and nonsymmorphic crystal symmetry, respectively. SOC gaps the first type of DNLs, which may contribute largely to the strong spin Hall effect. The second type of DNLs at the edges of Brillouin zone, however, remain intact against SOC. Our results not only provide important insights into the exotic transport properties of ${\mathrm{IrO}}_{2}$, but also shed light on the understanding of the role of SOC in the iridate family.

Topological phases of nonsymmorphic crystals: Shastry-Sutherland lattice at integer filling

Motivated by intertwined crystal symmetries and topological phases, we study the possible realization of topological insulators in nonsymmorphic crystals at integer fillings. In particular, we consider spin-orbit-coupled electronic systems of two-dimensional crystal Shastry-Sutherland lattices at integer filling where the gapless line degeneracy is protected by glide reflection symmetry. Based on a simple tight-binding model, we investigate how the topological insulating phase is stabilized by breaking nonsymmorphic symmetries but in the presence of time reversal symmetry and inversion symmetry. In addition, we also discuss the regime where Dirac semimetal is stabilized, having nontrivial ${Z}_{2}$ invariants even without spin-orbit coupling. Our study can be extended to more general cases where all lattice symmetries are broken and we also discuss possible application to topological Kondo insulators in nonsymmorphic crystals where crystal symmetries can be spontaneously broken as a function of the Kondo coupling.

Composite topological nodal lines penetrating the Brillouin zone in orthorhombic AgF2

It has recently been found that nonsymmorphic symmetries can bring many exotic band crossings. Here, based on symmetry analysis, we predict that materials with time-reversal symmetry in the space group of Pbca (No. 61) possess rich symmetry-enforced band crossings, including nodal surfaces, fourfold degenerate nodal lines and hourglass Dirac loops, which appear in triplets as ensured by the cyclic permutation symmetry. We take Pbca AgF2 as an example in real systems and studied its band structures with ab initio calculations. Specifically, in the absence of spin-orbit coupling (SOC), besides the above-mentioned band degeneracies, this system features a nodal chain and a nodal armillary sphere penetrating the Brillouin zone (BZ). While with SOC, we find a new configuration of the hourglass Dirac loop/chain with the feature traversing the BZ, which originates from the splitting of a Dirac loop confined in the BZ. Furthermore, guided by the bulk-surface correspondence, we calculated the surface states to explore these bulk nodal phenomena. The evolution of these interesting nodal phenomena traversing the BZ under two specific uniaxial strains is also discussed.

Fermions and bosons in nonsymmorphic PdSb2 with sixfold degeneracy

PdSb2 is a candidate for hosting 6-fold-degenerate exotic fermions (beyond Dirac and Weyl fermions).The nontrivial band crossing protected by the nonsymmorphic symmetry plays a crucial role in physical properties. We have grown high-quality single crystals of PdSb2 and characterized their physical properties under several stimuli (temperature, magnetic field, and pressure). While it is a diamagnetic Fermi-liquid metal under ambient pressure, PdSb2 exhibits a large magnetoresistance with continuous increase up to 14 T, which follows the Kohler's scaling law at all temperatures. This implies one-band electrical transport, although multiple bands are predicted by first principles calculations. By applying magnetic field along the [111] direction, de Haas-van Alphen oscillations are observed with frequency of 102 T. The effective mass is nearly zero (0.045m0) with the Berry phase close to {\pi}, confirming that the band close to the R point has a nontrivial character. Under quasihydrostatic pressure (p), evidence for superconductivity is observed in the resistivity below the critical temperature Tc. The dome-shaped Tc versus p is obtained with maximum Tc~2.9 K. We argue that the formation of Cooper pairs (bosons) is the consequence of the redistribution of the 6-fold-degenerate fermions under pressure.

Nonsaturating large magnetoresistance in the high carrier density nonsymmorphic metal CrP

The band structure of high carrier density metal CrP features an interesting crossing at the Y point of the Brillouin zone. The crossing, which is protected by the nonsymmorphic symmetry of the space group, results in a hybrid, semi-Dirac-like energy-momentum dispersion relation near Y. The linear energy-momentum dispersion relation along Y-$\Gamma$ is reminiscent of the observed band structure in several semimetallic extremely large magnetoresistance (XMR) materials. We have measured the transverse magnetoresistance of CrP up to 14 T at temperatures as low as $\sim$ 16 mK. Our data reveal a nonsaturating, quadratic magnetoresistance as well as the behaviour of the so-called `turn-on' temperature in the temperature dependence of resistivity. Despite the difference in the magnitude of the magnetoresistance and the fact that CrP is not a semimetal, these features are qualitatively similar to the observations reported for XMR materials. Thus, the high-field electrical transport studies of CrP offer the prospect of identifying the possible origin of the nonsaturating, quadratic magnetoresistance observed in a wide range of metals.

Topological bulk band structures of the hourglass and Dirac nodal-loop types in Ce Kondo systems: CeNiSn, CeRhAs, and CeRhSb

We have investigated the electronic structures of Ce Kondo systems, CeNiSn, CeRhAs, and CeRhSb, which have attracted recent attention as promising candidates of ``M\obius Kondo insulators.'' In the conventional density functional theory calculations, all three systems are obtained to be semimetallic, while only CeRhAs opens a full gap in the band calculation incorporating the more elaborate (modified Becke-Jones) exchange-correlation potential. Intriguingly, for all three systems, we have obtained the hourglass-type bulk band crossings around $\mathbf{k}=S$, which produce Dirac nodal-loop structures enclosing S in the [100] Brillouin zone boundary. Such topologically protected hourglass-type band structures arise from the spin-orbit coupling on top of the nonsymmorphic crystal symmetry $(Pnma)$ of these systems. Our finding suggests that Ce Kondo insulators would display temperature $(T)$ -dependent topological phase transitions from a potential M\obius Kondo insulator at low $T$ to a Dirac nodal-loop semimetal at high $T$.

Uncovering and tailoring hidden Rashba spin\u2013orbit splitting in centrosymmetric crystals

Hidden Rashba and Dresselhaus spin splittings in centrosymmetric crystals with subunits/sectors having non-centrosymmetric symmetries (the R-2 and D-2 effects) have been predicted theoretically and then observed experimentally, but the microscopic mechanism remains unclear. Here we demonstrate that the spin splitting in the R-2 effect is enforced by specific symmetries, such as non-symmorphic symmetry in the present example, which ensures that the pertinent spin wavefunctions segregate spatially on just one of the two inversion-partner sectors and thus avoid compensation. We further show that the effective Hamiltonian for the conventional Rashba (R-1) effect is also applicable for the R-2 effect, but applying a symmetry-breaking electric field to a R-2 compound produces a different spin-splitting pattern than applying a field to a trivial, non-R-2, centrosymmetric compound. This finding establishes a common fundamental source for the R-1 effect and the R-2 effect, both originating from local sector symmetries rather than from the global crystal symmetry per se.

Edge states in a two-dimensional nonsymmorphic semimetal

Dirac materials have unique transport properties, partly due to the presence of surface states. A new type of Dirac materials, protected by non-symmorphic symmetries was recently proposed by Young and Kane [1]. By breaking of time reversal or inversion symmetry one can split the Dirac cones into Weyl nodes. The later are characterized by local Chern numbers, that makes them two-dimensional analogs of Weyl semimetals. We find that the formation of the Weyl nodes is accompanied by an emergence of one-dimensional surface states, similar to Fermi arcs in Weyl semimetals and edge states in two-dimensional graphene. We explore these states for a quasi-one-dimensional non-symmorphic ribbon. The type and strength of applied deformation control the location and Weyl nodes and their composition. This determines the properties of emerging edge states. The sensitivity of these edge states to the external deformations makes non-symmorphic materials potentially useful as a new type of electromechanical sensors.

Multi-loop node line states in ternary MgSrSi-type crystals

Node line band-touchings protected by mirror symmetry (named as m-NLs), the product of inversion and time reversal symmetry S = PT (named as s-NLs), or nonsymmorphic symmetry are nontrivial topological objects of topological semimetals in the Brillouin Zone. In this work, we screened a family of MgSrSi-type crystals using first principles calculations, and discovered that more than 70 members are node-line semimetals. A new type of multi-loop structure was found in AsRhTi that a s-NL touches robustly with a m-NL at some “nexus point”, and in the meanwhile a second m-NL crosses with the s-NL to form a Hopf-link. Unlike the previously proposed Hopf-link formed by two s-NLs or two m-NLs, a Hopf-link formed by a s-NL and a m-NL requires a minimal three-band model to characterize its essential electronic structure. The associated topological surface states on different surfaces of AsRhTi crystal were also obtained. Even more complicated and exotic multi-loop structure of NLs were predicted in AsFeNb and PNiNb. Our work may shed light on search for exotic multi-loop node-line semimetals in real materials.