Elementary Band Representations Research Articles

From topological nodal-line semimetals to quantum spin Hall insulators in tetragonal SnX monolayers (X=F, Cl, Br, I)

Abstract Two-dimensional (2D) topological materials have recently garnered significant interest due to their profound physical properties and promising applications for future quantum nanoelectronics. Achieving various topological states within one type of materials is, however, seldom reported. Based on first-principles calculations and tight-binding models, we investigate topological electronic states in a novel family of 2D halogenated tetragonal stanene (T-SnX, X = F, Cl, Br, I). All the four monolayers are found being unusual topological nodal-line semimetals (NLSs), protected by a glide mirror symmetry. When spin-orbit coupling (SOC) is turned on, T-SnF and TSnCl are still ascertained as topological NLSs due to the remaining band inversion, primarily composed of Sn pxy orbitals, while T-SnBr and T-SnI become quantum spin Hall insulators. The phase transition is ascribed to moving up in energy of Sn s orbitals and increasing of SOC strengths. The topology origin in the materials is uniformly rationalized through elementary band representations. The robust and diverse topological states found in the 2D T-SnX monolayers position them as an excellent material platform for the development of innovative topological electronics.

Topological characterization of nonmagnetic one-dimensional crystals – elementary band representations for line groups

Topological characterization of nonmagnetic one-dimensional crystals – elementary band representations for line groups

Identifying and Constructing Complex Magnon Band Topology.

Magnetically ordered materials tend to support bands of coherent propagating spin wave, or magnon, excitations. Topologically protected surface states of magnons offer a new path toward coherent spin transport for spintronics applications. In this work we explore the variety of topological magnon band structures and provide insight into how to efficiently identify topological magnon bands in materials. We do this by adapting the topological quantum chemistry approach that has used constraints imposed by time reversal and crystalline symmetries to enumerate a large class of topological electronic bands. We show how to identify physically relevant models of gapped magnon band topology by using so-called decomposable elementary band representations, and in turn discuss how to use symmetry data to infer the presence of exotic symmetry enforced nodal topology.

Topology invisible to eigenvalues in obstructed atomic insulators

We consider the extent to which symmetry eigenvalues reveal the topological character of bands. Specifically, we compare distinct atomic limit phases (band representations) that share the same irreducible representations (irreps) at all points in the Brillouin zone and, therefore, appear equivalent in a classification based on eigenvalues. We derive examples where such "irrep-equivalent" phases can be distinguished by a quantized Berry phase or generalization thereof. These examples constitute a generalization of the Su-Schrieffer-Heeger chain: neither phase is topological, in the sense that localized Wannier functions exist, yet there is a topological obstruction between them. We refer to two phases as "Berry obstructed atomic limits" if they have the same irreps, but differ by Berry phases. This is a distinct notion from eigenvalue obstructed atomic limits, which differ in their symmetry irreps at some point in the Brillouin zone. We compute exhaustive lists of elementary band representations that are irrep-equivalent, in all space groups, with and without time-reversal symmetry and spin-orbit coupling, and use group theory to derive a set of necessary conditions for irrep-equivalence. Finally, we conjecture, and in some cases prove, that irrep-equivalent elementary band representations that are not equivalent can be distinguished by a topological invariant.

IrRep: Symmetry eigenvalues and irreducible representations of ab initio band structures

We present IrRep – a Python code that calculates the symmetry eigenvalues of electronic Bloch states in crystalline solids and the irreducible representations under which they transform. As input it receives bandstructures computed with state-of-the-art Density Functional Theory codes such as VASP, Quantum Espresso, or Abinit, as well as any other code that has an interface to Wannier90. Our code is applicable to materials in any of the 230 space groups and double groups preserving time-reversal symmetry with or without spin-orbit coupling included, for primitive or conventional unit cells. This makes IrRep a powerful tool to systematically analyze the connectivity and topological classification of bands, as well as to detect insulators with non-trivial topology, following the Topological Quantum Chemistry formalism: IrRep can generate the input files needed to calculate the (physical) elementary band representations and the symmetry-based indicators using the ▪ routine of the Bilbao Crystallographic Server. It is also particularly suitable for interfaces with other plane-waves based codes, due to its flexible structure. Program summaryProgram Title:IrRepCPC Library link to program files::https://doi.org/10.17632/yznd2ky9r6.1Developer's repository link:https://github.com/stepan-tsirkin/irrepLicensing provisions: GPLv3Programming language: PythonNature of problem: Symmetry properties of electronic band structures in solids are tightly related to their topological features. This relation is set mathematically by the formalisms of Topological Quantum Chemistry [1,2] and symmetry-based indicators of topology [3], but their application requires knowledge of the irreducible representations of the bands. Therefore, a code to calculate irreducible representations of ab initio bands is essential for a systematic theoretical search and classification of topological materials.Solution method:IrRep reads wave functions from files generated by VASP, Abinit, or Quantum Espresso (also files written as input for Wannier90), determines the space group and symmetry operations via the spglib library, evaluates the eigenvalues of the symmetry operations for the selected bands, and applies Group Theory to determine their irreducible representations.

Elementary band representations for the single-particle Green's function of interacting topological insulators

We discuss the applicability of elementary band representations (EBRs) to diagnose spatial- and time-reversal-symmetry protected topological phases in interacting insulators in terms of their single-particle Green's functions. We do so by considering an auxiliary non-interacting system $H_{\textrm{T}}(\mathbf{k}) = -G^{-1}(0, \mathbf{k})$, known as the topological Hamiltonian, whose bands can be labeled by EBRs. This labeling is stable if neither (i) the gap in the spectral function at zero frequency closes, (ii) the Green's function has a zero at zero frequency or (iii) the Green's function breaks a protecting symmetry. We demonstrate the use of EBRs applied to the Green's function on the one-dimensional Su-Schrieffer-Heeger model with Hubbard interactions, which we solve by exact diagonalization for a finite number of unit cells. Finally, the use of EBRs for the Green's function to diagnose so-called symmetry protected topological phases is discussed, but remains an open question.

Elementary band representations for (double)-line groups

Quantized topological invariants characterizing topological phases in quasi-1D materials are usually considered only on the basis of spatial inversion parity eigenvalues. However, symmetry of quasi-1D systems is far more complex and their complete topological characterisation can be obtained only on the basis of elementary band representations (EBRs) for the relevant symmetry groups. We derive complete sets of inequivalent EBRs for line groups (LG), the symmetry groups of all quasi-1D systems with either translational or helical periodicity. Besides, we determine also EBRs for double-LGs, accounting for spin degree of freedom. In order to illustrate applicability of the results obtained, we analyze electronic-band topology of a chiral single-wall carbon nanotube, using EBRs for relevant (double)-LG and discuss Su-Schrieffer–Heeger model from EBR-perspective.

Electronic Band Topology of Monoclinic MoS2 Monolayer: Study Based on Elementary Band Representations for Layer Groups

A monoclinic monolayer is demonstrated to be a topological semimetal (spinless), or zero‐gap semiconductor (spinful, without spin–orbit coupling), or topologically trivial insulator (finite spin–orbit coupling). The latter contradicts a previous prediction that exhibits quantum‐spin Hall (QSH) effect, and the reported value of ‐invariant, calculated from inversion‐parity values at four Brillouin‐zone high‐symmetry points. Namely, as the electronic states of two of these points are double degenerate and transform according to irreducible representations carrying no parity, the band topology of is investigated using elementary band representations (EBRs) for layer groups (LGs). Novel subroutines, which output EBRs and Wilson loop operators for low‐dimensional systems, are incorporated into POLSym code. Based on the calculated band structure decomposition onto EBRs of the relevant symmetry group and Wilson loop eigen‐spectra, it is revealed that valence‐band Wannier functions do not break the symmetry, and that Wannier centers are localized within a unit cell. Moreover, the QSH state is proved to be not realizable within ‐phase group‐VI transition‐metal dichalcogenides, because EBRs for the relevant symmetry groups are topologically trivial. However, it is predicted that among compounds that form other monoclinic structure, with symmorphic LG symmetry, there may exist topologically nontrivial phases, including the QSH state.

Electronic structures and topological properties in nickelates Ln n+1Ni n O2n+2.

After the significant discovery of the hole-doped nickelate compound Nd0.8Sr0.2NiO2, analyses of the electronic structure, orbital components, Fermi surfaces and band topology could be helpful to understand the mechanism of its superconductivity. Based on first-principle calculations, we find that Ni n}{}3d_{x^2-y^2} states contribute the largest Fermi surface. The n}{}Ln 5d_{3z^2-r^2} states form an electron pocket at Γ, while 5dxy states form a relatively bigger electron pocket at A. These Fermi surfaces and symmetry characteristics can be reproduced by our two-band model, which consists of two elementary band representations: B1g@1a ⊕ A1g@1b. We find that there is a band inversion near A, giving rise to a pair of Dirac points along M-A below the Fermi level upon including spin-orbit coupling. Furthermore, we perform density functional theory based Gutzwiller (DFT+Gutzwiller) calculations to treat the strong correlation effect of Ni 3d orbitals. In particular, the bandwidth of n}{}3d_{x^2-y^2} has been renormalized largely. After the renormalization of the correlated bands, the Ni 3dxy states and the Dirac points become very close to the Fermi level. Thus, a hole pocket at A could be introduced by hole doping, which may be related to the observed sign change of the Hall coefficient. By introducing an additional Ni 3dxy orbital, the hole-pocket band and the band inversion can be captured in our modified model. Besides, the nontrivial band topology in the ferromagnetic two-layer compound La3Ni2O6 is discussed and the band inversion is associated with Ni n}{}3d_{x^2-y^2} and La 5dxy orbitals.

Systematic study of stacked square nets: From Dirac fermions to material realizations

Non-symmorphic symmetries protect Dirac line nodes in square net materials. This phenomenon has been most prominently observed in ZrSiS. Here, we systematically study the symmetry-protected nodal fermions that result from different ways of embedding the square net into a larger unit cell. Surprisingly, we find that a nonsymmorphic space group is not a necessary condition for a filling enforced semimetal: symmorphic space groups can also host nodal fermions that are enforced by band folding and electron count, that is, a combination of a particular structural motif combined with electron filling. We apply the results of this symmetry analysis to define an algorithm, which we utilize to find square net materials with nodal fermions in specific symmorphic space groups. We highlight one result of this search, the compound ThGeSe, which has not been discussed before in the context of nodal fermions. Finally, we discuss how band folding can impose constraints on band connectivity beyond the connectivity of single elementary band representations.

Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry

We present a general methodology towards the systematic characterization of crystalline topological insulating phases with time reversal symmetry (TRS).~In particular, taking the two-dimensional spinful hexagonal lattice as a proof of principle we study windings of Wilson loop spectra over cuts in the Brillouin zone that are dictated by the underlying lattice symmetries.~Our approach finds a prominent use in elucidating and quantifying the recently proposed ``topological quantum chemistry" (TQC) concept.~Namely, we prove that the split of an elementary band representation (EBR) by a band gap must lead to a topological phase.~For this we first show that in addition to the Fu-Kane-Mele $\mathbb{Z}_2$ classification, there is $C_2\mathcal{T}$-symmetry protected $\mathbb{Z}$ classification of two-band subspaces that is obstructed by the other crystalline symmetries, i.e.~forbidding the trivial phase. This accounts for all nontrivial Wilson loop windings of split EBRs \textit{that are independent of the parameterization of the flow of Wilson loops}.~Then, we show that while Wilson loop winding of split EBRs can unwind when embedded in higher-dimensional band space, two-band subspaces that remain separated by a band gap from the other bands conserve their Wilson loop winding, hence revealing that split EBRs are at least "stably trivial", i.e. necessarily non-trivial in the non-stable (few-band) limit but possibly trivial in the stable (many-band) limit.~This clarifies the nature of \textit{fragile} topology that has appeared very recently.~We then argue that in the many-band limit the stable Wilson loop winding is only determined by the Fu-Kane-Mele $\mathbb{Z}_2$ invariant implying that further stable topological phases must belong to the class of higher-order topological insulators.

Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice

In this work, we examine the topological phases that can arise in triangular lattices with disconnected elementary band representations. We show that, although these phases may be "fragile" with respect to the addition of extra bands, their topological properties are manifest in certain nontrivial holonomies (Wilson loops) in the space of nontrivial bands. We introduce an eigenvalue index for fragile topology, and we show how a nontrivial value of this index manifests as the winding of a hexagonal Wilson loop; this remains true even in the absence of time-reversal or sixfold rotational symmetry. Additionally, when time-reversal and twofold rotational symmetry are present, we show directly that there is a protected nontrivial winding in more conventional Wilson loops. Crucially, we emphasize that these Wilson loops cannot change without closing a gap to the nontrivial bands. By studying the entanglement spectrum for the fragile bands, we comment on the relationship between fragile topology and the "obstructed atomic limit" of B. Bradlyn et al., Nature 547, 298--305 (2017). We conclude with some perspectives on topological matter beyond the K-theory classification.

No-go theorem for topological insulators and high-throughput identification of Chern insulators

For any symmorphic magnetic space group $G$, it is proven that topological band insulators with vanishing first Chern numbers cannot have a groundstate composed of a single, energetically-isolated band. This no-go statement implies that the minimal dimension of tight-binding Hamiltonians for such topological insulators is four if the groundstate is stable to addition of trivial bands, and three if the groundstate is unstable. A sure-fire recipe is provided to design models for Chern and unstable topological insulators by splitting elementary band representations; this recipe, combined with recently-constructed Bilbao tables on such representations, can be systematized for mass identification of topological materials. All results follow from our theorem which applies to any single, isolated energy band of a $G$-symmetric Schr\"odinger-type or tight-binding Hamiltonian: for such bands, being topologically trivial is equivalent to being a band representation of $G$.

Topology of Disconnected Elementary Band Representations.

Elementary band representations are the fundamental building blocks of atomic limit band structures. They have the defining property that at partial filling they cannot be both gapped and trivial. Here, we give two examples-one each in a symmorphic and a nonsymmorphic space group-of elementary band representations realized with an energy gap. In doing so, we explicitly construct a counterexample to a claim by Michel and Zak that single-valued elementary band representations in nonsymmorphic space groups with time-reversal symmetry are connected. For each example, we construct a topological invariant to explicitly demonstrate that the valence bands are nontrivial. We discover a new topological invariant: a movable but unremovable Dirac cone in the "Wilson Hamiltonian" and a bent-Z_{2} index.

Building blocks of topological quantum chemistry: Elementary band representations

The link between chemical orbitals described by local degrees of freedom and band theory, which is defined in momentum space, was proposed by Zak several decades ago for spinless systems with and without time-reversal in his theory of "elementary" band representations. In Nature 547, 298-305 (2017), we introduced the generalization of this theory to the experimentally relevant situation of spin-orbit coupled systems with time-reversal symmetry and proved that all bands that do not transform as band representations are topological. Here, we give the full details of this construction. We prove that elementary band representations are either connected as bands in the Brillouin zone and are described by localized Wannier orbitals respecting the symmetries of the lattice (including time-reversal when applicable), or, if disconnected, describe topological insulators. We then show how to generate a band representation from a particular Wyckoff position and determine which Wyckoff positions generate elementary band representations for all space groups. This theory applies to spinful and spinless systems, in all dimensions, with and without time reversal. We introduce a homotopic notion of equivalence and show that it results in a finer classification of topological phases than approaches based only on the symmetry of wavefunctions at special points in the Brillouin zone. Utilizing a mapping of the band connectivity into a graph theory problem, which we introduced in Nature 547, 298-305 (2017), we show in companion papers which Wyckoff positions can generate disconnected elementary band representations, furnishing a natural avenue for a systematic materials search.

Double crystallographic groups and their representations on the Bilbao Crystallographic Server

A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The double crystallographic groups are required in the study of physical systems whose Hamiltonian includes spin-dependent terms. In the symmetry analysis of such systems, instead of the irreducible representations of the space groups, it is necessary to consider the single- and double-valued irreducible representations of the double space groups. The new section includes databases of symmetry operations (DGENPOS) and of irreducible representations of the double (point and space) groups (REPRESENTATIONS DPGandREPRESENTATIONS DSG). The toolDCOMPRELprovides compatibility relations between the irreducible representations of double space groups at differentkvectors of the Brillouin zone when there is a group–subgroup relation between the corresponding little groups. The programDSITESYMimplements the so-called site-symmetry approach, which establishes symmetry relations between localized and extended crystal states, using representations of the double groups. As an application of this approach, the programBANDREPcalculates the band representations and the elementary band representations induced from any Wyckoff position of any of the 230 double space groups, giving information about the properties of these bands. Recently, the results ofBANDREPhave been extensively applied in the description of and the search for topological insulators.

Comments on the decomposition of the regular representation of crystallographic space groups into band representations

We discuss the decomposition of the regular representation of crystallographic space groups into elementary band representations. It is shown that the decomposition is in general not unique. In particular, we discuss some orthorhombic space groups in detail.

Connectivity of energy bands in crystals

It is shown that in crystals with nonsymmorphic space groups all energy bands corresponding to elementary band representations are composite and connected; i.e., these bands have several branches, and there are enough contact points among them so that one can travel continuously through all of them. The concept of elementary band representations is explained. The proof is essentially based on the property of monodromy occurring for families of representations of nonsymmorphic space groups.

Band representations and duals of the nonsymmorphic crystallographic plane groups pg and p4gm

Duals of the groups pg and p4gm are constructed. (1) The group pg has only one band representation (an elementary one) equivalent to the regular representation. There exists a quasiband representation close to it, of multiplicity two, which separates the branches of the band. Finite groups which approximate pg are discussed. (2) Two elementary band representations associated with the Wyckoff position b of the group p4gm are shown to be nonequivalent although they have the same decomposition in unirreps.