Materials from Topological Quantum Chemistry Maia G. Vergniory Kyoto October 2017 Nature 547, 298--305 (2017), Phys. Rev. E 96, 023310 (2017) , Journal of Applied Crystallography 50 (5) 2017, arXiv: 1709.01935, arXiv:1709.01937
Topological Insulators and Topological Semimetals Topological Insulators / Dirac Fermions Topological protection from time reversal or some crystal symmetry Topological Semimetals / Weyl, Dirac and “beyond” Fermions (3fold, 6fold and 8fold crossings)
NonSymmorphic Symmetries Bring In New Phenomena z a c X Hg z Y T y 100 Γ Z x c /2 Surface States in KHgSb k K x a k y One glide plane allows for the presence of L k z b 010 M Hourglass-like fermions on the surface a Γ 2 A K a a 100 X U 1 U y Γ X Hg Γ Z Z x K 010 Surface States in Sr2Pb3, a Dirac Nonsymmorphic insulator 4-fold degeneracy surface state at the M point with Two glide planes (a) 1.5 1 Energy(eV) 3,6,8-degeneracies (3 can also be realized E F 0 with symmorphic), nodal chains, etc Energy(eV) -1 -2 Γ M K Γ A L H A
⎬ Non-predictive classification of Topological Bands Open questions: ? 1. How do we know the classification is complete? ? Given an orbital content on a material on a lattice, what are the topological phases? 2. How can we find topological materials? 200000 materials in ICSD database: 100 time reversal topological insulators Set of measure zero… 10 mirror Chern insulators Are topological materials that esoteric? 15 Weyl semimetals 15 Dirac semimetals 3 Non-Symmorphic topological insulators Group Graph theory theory We propose a classification that captures all crystal symmetries and has predictive power Chemistry
Recall: a space group is a set of symmetries that defines a crystal structure in 3D Ingredients: { • unit lattice translations ( 𝚮 3 ) 230 • point group operations (rotations, reflections) Space-Groups • non-symmorphic (screw, glide) • orbitals • atoms in some lattice positions Image: 1605.06824 Ma et al How do we go from real space orbitals sitting on lattice sites to electronic bands (without a Hamiltonian)? ELEMENTARY BAND REPRESENATIONS Zak PRB 26 (1982)
Elementary Band Representations (building blocks) Band Representation (BR) : set of bands linked to a localized orbital respecting all the crystal symmetries. They relate electrons on site to momentum space description. Elementary BR : smallest set of bands cannot be decomposed in elementary bands Physical Elementary R : when EBR also respects TR symmetry Composite BR: A BR which is not elementary is a “composite” (P)EBRs are connected along the BZ 2 0 -2 Energy (eV) -4 -6 -8 -10 K M L H A Γ Γ Zak PRB 26 (1982)
｛ Induction of a (P)EBR: Example of the honeycomb lattice Lets consider the generators of 2D P6mm : {C 2 ,C 3 ,m 11 } Lattice vectors: Lattice site: Wyckoff 2b, spinfull p z q e 1 = √ 3/2 x +1/2 y e 1 p z e 2 = √ 3/2 x -1/2 y e 2 Site-symmetry group, G q , leaves q invariant Cosset decomposition of a Space Group : n G = ∪ (g ) (G q ⋉ 𝚮 3 ) g ∉ G q , 𝛃 𝛃 𝛃 =1
Induction of a (P)EBR: Example of the honeycomb lattice Consider one lattice site: q G = ∪ (g ) (G q ⋉ 𝚮 3 ) 𝛃 𝛃 p z (2) (1) e 1 e 2 Site-symmetry group, G q , leaves q invariant {C 3 |01}, {m 11 |00} ≈ C 3v (1) Orbitals at q transform under a rep, 𝝇 , of G q C 3 e 2 {C 2 |?} {C 3 |01} {m 11 |00}
Induction of a (P)EBR: Example of the honeycomb lattice Consider one lattice site: q G = ∪ (g ) (G q ⋉ 𝚮 3 ) 𝛃 𝛃 p z (2) (1) e 1 e 2 Site-symmetry group, G q , leaves q invariant {C 3 |01}, {m 11 |00} ≈ C 3v (1) Orbitals at q transform under a rep, 𝝇 , of G q C 3 e 2 {C 2 |?} {C 3 |01} {m 11 |00}
Induction of a (P)EBR: Example of the honeycomb lattice Consider one lattice site: q G = ∪ (g ) (G q ⋉ 𝚮 3 ) 𝛃 𝛃 p z (2) (1) e 1 e 2 Site-symmetry group, G q , leaves q invariant {C 3 |01}, {m 11 |00} ≈ C 3v (1) Orbitals at q transform under a rep, 𝝇 , of G q Rep E C 3 M E 2 1 0 -2 Γ 6 Character table for the double-valued representation of C 3v
Induction of a (P)EBR: Example of the honeycomb lattice Consider one lattice site: q G = ∪ (g ) (G q ⋉ 𝚮 3 ) 𝛃 𝛃 p z (2) (1) e 1 e 2 Site-symmetry group, G q , leaves q invariant {C 3 |01}, {m 11 |00} ≈ C 3v (1) Orbitals at q transform under a rep, 𝝇 , of G q ︷ Elements of space group g ∉ G q (cosset representatives) move sites (2) in an orbit “Wyckoff position” {C 2 |00},{E|00} q’ q Wyckoff multiplicity: 2 orbit of q
Induction of a (P)EBR: Example of the honeycomb lattice Γ 6 induced in C 6v electron bands sitting at pz orbitals in Wyckoff 2b in Wall paper group 17 𝝇 G = 𝝇 ↑ G Cosset representative g: {C 2 |00},{E|00} h ∈ G, generators of honeycomb lattice: C 2 ,C 3 , σ 𝝇 i 𝜷 ,j 𝜸 (h) = 𝝇 ij ( g 𝜷𝜸 ) -1 g 𝜷𝜸 = g 𝜷 {E|t 𝜷𝜸 }hg 𝜸 k 𝝇 G (h) =e -( k · t 𝜷𝜸 ) 𝝇 ij ( g 𝜷𝜸 ) dimension of this band representations = connectivity in the Brillouin zone
Subduction in k space: IRREPS at points, lines ( 𝝇 ↑ G) ↓ G k Restricting to the little group at k to find irreps at each k point (subduction) -> all bands connected All 10403 decompositions now tabulated on the Bilbao Crystallographic Server By construction, a band representation has an atomic limit, and all atomic limits yield a band representation Recall: Topological bands CANNOT Have Maximally Localized Wannier Functions…
Why are Elementary Band Representations Important? 1) Bands in ρ G are connected (this phase can always realized) in the Brillouin zone 2) Bands in ρ G are not connected: at least one topological band Disconnected (P)EBR = set of disconnected bands that connected form an (P)EBR
Why are Elementary Band Representations Important? 1) Bands in ρ G are connected (this phase can always realized) in the Brillouin zone 2) Bands in ρ G are not connected: at least one topological band Disconnected (P)EBR = set of disconnected bands that connected form an (P)EBR Our definition of a topological band = anything that is not a band representation
Obstructed atomic limit Orbital hybridization BR are induced from localized molecular orbitals, away from the atoms In terms of EBRs? 1 st limit CBR: 𝝉 v ↑ G a ⊕ 𝝉 c ↑ G a EBR 2 Composite BR EBR 1 ⎬ 2 nd limit CBR: 𝝇 v ↑ G m ⊕ 𝝇 c ↑ G m 1 st limit: orbitals lie in the atomic sites 2 nd limit: orbitals do not coincide with the atoms
Obstructed atomic limit Orbital hybridization BR are induced from localized molecular orbitals, away from the atoms In terms of EBRs? 1 st limit CBR: 𝝉 v ↑ G a ⊕ 𝝉 c ↑ G a EBR 2 𝜃 ↑ G a ≈ 𝝉 v ⊕ 𝝉 c Composite BR 𝜃 ↑ G m ≈ 𝝇 v ⊕ 𝝇 c EBR 1 ⎬ 2 nd limit CBR: 𝝇 v ↑ G m ⊕ 𝝇 c ↑ G m 1 st limit: orbitals lie in the atomic sites 2 nd limit: orbitals do not coincide with the atoms This is a “chemical bonding” transition (ex: from week to a strong covalent bonding) N. Read Phys.Rev. B (2017), W. A. Benalcazar Science (2017)
TQC statement All sets of bands induced from symmetric, localized orbitals, are topologically trivial by design. Zak PRB 26 (1982)
TQC statement NOT All sets of bands induced from symmetric, localized orbitals, are topologically trivial by design. NOT Zak PRB 26 (1982)
Elementary Band Representations (reciprocal space) Global information about band structure: enumerate all EBRs 1. Maximal k-vectors and path 2. Compatibility relations 3. Graph theory: identification of disconnected bands Zak PRB 26 (1982)
1. Maximal k-vectors and paths For all the 203 SG: maximal k-vectors + minimal set non-redundant connections (1) k vector in a manifold is maximal if its little co-group it’s not a subgroup of another manifold of vectors k’ (in general coincides with high-symmetry k-vector) P4/ncc Maximal (first BZ) k -vec mult. Coordinates Little co-group TR Ŵ 1 (0 , 0 , 0) 4 /mmm ( D 4 h ) yes yes Z 1 (0 , 0 , 1 / 2) 4 /mmm ( D 4 h ) yes yes M 1 (1 / 2 , 1 / 2 , 0) 4 /mmm ( D 4 h ) yes yes A 1 (1 / 2 , 1 / 2 , 1 / 2) 4 /mmm ( D 4 h ) yes yes R 2 (0 , 1 / 2 , 1 / 2) mmm ( D 2 h ) yes yes X 2 (0 , 1 / 2 , 0) mmm ( D 2 h ) yes yes 2 (0 , 0 ,w ) , 0 < w < 1 / 2 4 mm ( C 4 v ) no no � V 2 (1 / 2 , 1 / 2 ,w ) , 0 < w < 1 / 2 4 mm ( C 4 v ) no no W 4 (0 , 1 / 2 ,w ) , 0 < w < 1 / 2 mm 2( C 2 v ) no no 4 ( u,u, 0) , 0 < u < 1 / 2 mm 2( C 2 v ) no no � S 4 ( u,u, 1 / 2) , 0 < u < 1 / 2 mm 2( C 2 v ) no no � 4 (0 ,v, 0) , 0 < v < 1 / 2 mm 2( C 2 v ) no no U 4 (0 ,v, 1 / 2) , 0 < v < 1 / 2 mm 2( C 2 v ) no no Y 4 ( u, 1 / 2 , 0) , 0 < u < 1 / 2 mm 2( C 2 v ) no no T 4 ( u, 1 / 2 , 1 / 2) , 0 < u < 1 / 2 mm 2( C 2 v ) no no D 8 ( u,v, 0) , 0 < u < v < 1 / 2 m ( C s ) no no E 8 ( u,v, 1 / 2) , 0 < u < v < 1 / 2 m ( C s ) no no C 8 ( u,u,w ) , 0 < u < w < 1 / 2 m ( C s ) no no B 8 (0 ,v,w ) , 0 < v < w < 1 / 2 m ( C s ) no no F 8 ( u, 1 / 2 ,w ) , 0 < u < w < 1 / 2 m ( C s ) no no GP 16 ( u,v,w ) , 0 < u < v < w < 1 / 2 1(1) no no Physical Review E 96 (2), 023310
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