Nonsymmorphic Space Group Symmetry Research Articles

Symmetry-protected full-space persistent spin texture in two-dimensional materials

Persistent spin texture (PST) is an interesting phenomenon that a material maintains a uniform spin configuration in the momentum space. Recently, it was proposed that the PST can be enforced by the nonsymmorphic space group symmetry at certain high symmetry points in the Brillouin zone [Tao et al., Nat. Commun. 9, 2763 (2018)]. In this Letter, through group theory analysis, we report that the PST could occur in the whole Brillouin zone of a two-dimensional material, as long as it keeps $xy$ mirror (or glide-plane) symmetry but breaks the combination of inversion symmetry and time-reversal symmetry. By performing first-principles calculations, we demonstrate that this full-space persistent spin texture (FPST) can occur in both nonmagnetic material (e.g., monolayer ${\mathrm{MoS}}_{2}$) and magnetic material (e.g., superlattice ${\mathrm{LaGa}}_{0.5}{\mathrm{Mn}}_{0.5}{\mathrm{O}}_{3}$). Furthermore, our $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ Hamiltonian analysis demonstrates that the FPST is robust for nondegenerate bands with an energy gap. The FPST proposed in this work may lead to promising spintronic applications as it may result in an extremely long spin-relaxation time.

Monolayer RhB4 : Half-auxeticity and almost ideal spin-orbit Dirac point semimetal

Structural-property relationship, the connection between materials' structures and their properties, is central to the materials research. Especially at reduced dimensions, novel structural motifs often generate unique physical properties. Motivated by a recent work reporting a novel half-auxetic effect in monolayer ${\mathrm{PdB}}_{4}$ with a hypercoordinated structure, here we extensively explore similar two-dimensional (2D) transition-metal boride structures $M{\mathrm{B}}_{4}$ with $M$ covering $3d$ and $4d$ elements. Our investigation screens out one stable candidate, the monolayer ${\mathrm{RhB}}_{4}$. We find that monolayer ${\mathrm{RhB}}_{4}$ also shows half-auxeticity, i.e., the material always expands in a lateral in-plane direction in response to an applied strain in the other direction, regardless of whether the strain is positive or negative. We show that this special mechanical character is intimately tied to the hypercoordinated structure with the $MC\text{\textcopyright{}}{\mathrm{B}}_{8}$ structural motif. Furthermore, regarding electronic properties, monolayer ${\mathrm{RhB}}_{4}$ is found to be an example of an almost ideal 2D spin-orbit Dirac-point semimetal. The low-energy band structure is clean, with a pair of fourfold degenerate Dirac points robust under spin-orbit coupling located close to the Fermi level. These Dirac points are $enforced$ by the nonsymmorphic space-group symmetry which is also determined by the lattice structure. Our work deepens the fundamental understanding of structural-property relationship in reduced dimensions. The half-auxeticity and the spin-orbit Dirac points will make monolayer ${\mathrm{RhB}}_{4}$ a promising platform for nanomechanics and nanoelectronics applications.

RRuB2 (R=Y, Lu) , topological superconductor candidates with hourglass-type Dirac ring

Topological properties and topological superconductivity in real materials have attracted intensive experimental and theoretical attention recently. Based on symmetry analysis and first-principles electronic structure calculations, we predict that $R$RuB$_{2}$ ($R$=Y, Lu) are not only topological superconductor (TSC) candidates, but also own the hybrid hourglass-type Dirac ring which is protected by the nonsymmorphic space group symmetry. Due to the band inversion around the time-reversal invariant $\Gamma$ point in the Brillouin zone,$R$RuB$_{2}$ also have Dirac topological surface states (TSSs). More importantly, their TSSs on the (010) surface are within the band gap of bulk and cross the Fermi level, which form single Fermi surfaces. Considering the fact that both YRuB$_{2}$ and LuRuB$_{2}$ are superconductors with respective superconducting transition temperatures ($T_c$) of 7.6 K and 10.2 K, the superconducting bulks will likely induce superconductivity in the TSSs via the proximity effect. The ternary borides $R$RuB$_{2}$ may thus provide a very promising platform for studying the properties of topological superconductivity and hourglass fermions in the future experiments.

Weyl-Kondo semimetals in nonsymmorphic systems

There is considerable current interest to explore electronic topology in strongly correlated metals, with heavy fermion systems providing a promising setting. Recently, a Weyl-Kondo semimetal phase has been concurrently discovered in theoretical and experimental studies. The theoretical work was carried out in a Kondo lattice model that is time-reversal invariant but inversion-symmetry breaking. In this paper, we show in some detail how nonsymmorphic space-group symmetry and strong correlations cooperate to form Weyl nodal excitations with highly reduced velocity and pin the resulting Weyl nodes to the Fermi energy. A tilted variation of the Weyl-Kondo solution is further analyzed here, following the recent consideration of such effect in the context of understanding a large spontaneous Hall effect in Ce$_3$Bi$_4$Pd$_3$ (Dzsaber et al., arXiv:1811.02819). We discuss the implications of our results for the enrichment of the global phase diagram of heavy fermion metals, and for the space-group symmetry enforcement of topological semimetals in other strongly correlated settings.

A 2D nonsymmorphic Dirac semimetal in a chemically modified group-VA monolayer with a black phosphorene structure.

A symmetry-protected 2D Dirac semimetal has attracted intense interest for its intriguing material properties. Here, we report a 2D nonsymmorphic Dirac semimetal state in a chemically modified group-VA 2D puckered structure. Based on first-principles calculations, we demonstrate the existence of 2D Dirac fermions in a one-side modified phosphorene structure in two different types: one with a Dirac nodal line (DNL) structure for light elements with negligible spin-orbit coupling (SOC) and the other having an hourglass band protected by a nonsymmorphic symmetry for heavy elements with strong SOC. In the absence of SOC, the DNL exhibits an anisotropic behavior and unique electronic properties, such as constant density of states. The Dirac node is protected from gap opening by the nonsymmorphic space group symmetry. In the presence of SOC, the DNL states split and form an hourglass-shaped dispersion due to the broken inversion symmetry and the Rashba SOC interaction. Moreover, around certain high symmetry points in the Brillouin zone, the spin orientation is enforced to be along a specific direction. We construct an effective tight-binding model to characterize the 2D nonsymmorphic Dirac states. Our result provides a promising material platform for exploring the intriguing properties of essential nodal-line and nodal-point fermions in 2D systems.

Persistent spin texture enforced by symmetry

Persistent spin texture (PST) is the property of some materials to maintain a uniform spin configuration in the momentum space. This property has been predicted to support an extraordinarily long spin lifetime of carriers promising for spintronics applications. Here, we predict that there exists a class of noncentrosymmetric bulk materials, where the PST is enforced by the nonsymmorphic space group symmetry of the crystal. Around certain high symmetry points in the Brillouin zone, the sublattice degrees of freedom impose a constraint on the effective spin–orbit field, which orientation remains independent of the momentum and thus maintains the PST. We illustrate this behavior using density-functional theory calculations for a handful of promising candidates accessible experimentally. Among them is the ferroelectric oxide BiInO3—a wide band gap semiconductor which sustains a PST around the conduction band minimum. Our results broaden the range of materials that can be employed in spintronics.

Two-dimensional spin-orbit Dirac point in monolayer HfGeTe

Dirac points in two-dimensional (2D) materials have been a fascinating subject of research, with graphene as the most prominent example. However, the Dirac points in existing 2D materials, including graphene, are vulnerable against spin-orbit coupling (SOC). Here, based on first-principles calculations and theoretical analysis, we propose a new family of stable 2D materials, the HfGeTe-family monolayers, which represent the first example to host so-called spin-orbit Dirac points (SDPs) close to the Fermi level. These Dirac points are special in that they are formed only under significant SOC, hence they are intrinsically robust against SOC. We show that the existence of a pair of SDPs are dictated by the nonsymmorphic space group symmetry of the system, which are very robust under various types of lattice strains. The energy, the dispersion, and the valley occupation around the Dirac points can be effectively tuned by strain. We construct a low-energy effective model to characterize the Dirac fermions around the SDPs. Furthermore, we find that the material is simultaneously a 2D $\mathbb{Z}_2$ topological metal, which possesses nontrivial $\mathbb{Z}_2$ invariant in the bulk and spin-helical edge states on the boundary. From the calculated exfoliation energies and mechanical properties, we show that these materials can be readily obtained in experiment from the existing bulk materials. Our result reveals HfGeTe-family monolayers as a promising platform for exploring spin-orbit Dirac fermions and novel topological phases in two-dimensions.

Interplay of nonsymmorphic symmetry and spin-orbit coupling in hyperkagome spin liquids: Applications to Na4Ir3O8

Na$_4$Ir$_3$O$_8$ provides a material platform to study three-dimensional quantum spin liquids in the geometrically frustrated hyperkagome lattice of Ir$^{4+}$ ions. In this work, we consider quantum spin liquids on hyperkagome lattice for generic spin models, focusing on the effects of anisotropic spin interactions. In particular, we classify possible $\mathbb{Z}_2$ and $U(1)$ spin liquid states, following the projective symmetry group analysis in the slave-fermion representation. There are only three distinct $\mathbb{Z}_2$ spin liquids, together with 2 different $U(1)$ spin liquids. The non-symmorphic space group symmetry of hyperkagome lattice plays a vital role in simplifying the classification, forbidding "$\pi$-flux" or "staggered-flux" phases in contrast to symmorphic space groups. We further prove that both $U(1)$ states and one $Z_2$ state among all 3 are symmetry-protected gapless spin liquids, robust against any symmetry-preserving perturbations. Motivated by the "spin-freezing" behavior recently observed in Na$_4$Ir$_3$O$_8$ at low temperatures, we further investigate the nearest-neighbor spin model with dominant Heisenberg interaction subject to all possible anisotropic perturbations from spin-orbit couplings. We found a $U(1)$ spin liquid ground state with spinon fermi surfaces is energetically favored over $Z_2$ states. Among all spin-orbit coupling terms, we show that only Dzyaloshinskii-Moriya (DM) interaction can induce spin anisotropy in the ground state when perturbing from the isotropic Heisenberg limit. Our work paves the way for a systematic study of quantum spin liquids in various materials with a hyperkagome crystal structure.

Graphene-like Dirac states and quantum spin Hall insulators in square-octagonalMX2(M=Mo, W;X=S, Se, Te) isomers

We studied the square-octagonal lattice of the transition metal dichalcogenide MX$_2$ (with $M$=Mo, W; $X$=S, Se and Te), as an isomer of the normal hexagonal compound of MX$_2$. By band structure calculations, we observe the graphene-like Dirac band structure in a rectangular lattice of MX$_2$ with nonsymmorphic space group symmetry. Two bands with van Hove singularity points cross each at the Fermi energy, leading to two Dirac cones that locates at opposite momenta. Spin-orbit coupling can open a nontrivial gap at these Dirac points and induce the quantum spin Hall (QSH) phase, the 2D topological insulator. Here, square-octagonal MX$_2$ structures realize the interesting graphene physics, such as Dirac bands and QSH effect, in the transition metal dichalcogenides.

Z2topology in nonsymmorphic crystalline insulators: Möbius twist in surface states

It has been known that an antiunitary symmetry such as time-reversal or charge conjugation is needed to realize ${\mathbit{Z}}_{2}$ topological phases in noninteracting systems. Topological insulators and superconducting nanowires are representative examples of such ${\mathbit{Z}}_{2}$ topological matters. Here we report the ${\mathbit{Z}}_{2}$ topological phase protected by only unitary symmetries. We show that the presence of a nonsymmorphic space group symmetry opens a possibility to realize ${\mathbit{Z}}_{2}$ topological phases without assuming any antiunitary symmetry. The ${\mathbit{Z}}_{2}$ topological phases are constructed in various dimensions, which are closely related to each other by Hamiltonian mapping. In two and three dimensions, the ${\mathbit{Z}}_{2}$ phases have a surface consistent with the nonsymmorphic space group symmetry, and thus they support topological gapless surface states. Remarkably, the surface states have a unique energy dispersion with the M\"obius twist, which identifies the ${\mathbit{Z}}_{2}$ phases experimentally. We also provide the relevant structure in the $K$ theory.