Topological Quantum Chemistry

Question: Topological insulators represent only a few hundred of the 200,000 stoichiometric compounds. Why? Is there a problem with the band description of materials?

Chemists and physicists use fundamentally different viewpoints on materials

• physicists: momentum space
• chemists: real space A unifying approach of real and reciprocal space is naturally desirable.

Approach

Compile all possible ways energy bands in a solid can be connected in the BZ to obtain band structures in all non-magnetic space groups.

Crystal symmetries place constraints on allowed connections

• At high symmetry k-points, Bloch functions are classified by irreducible representations of the symmetry group of $\overrightarrow{k_i}$ which also determine the degeneracy
• Away from these points, fewer constraints and degeneracies are lowered
• E.g. $\vec{k}\cdot \vec{p}$ approach allows computation of band structure near high-symmetry points, group theory places constraints on how they can be connected

Use graph theoretic problem of constructing multi-partite graphs

Develop tools to compute way in which real space orbitals determine symmetry character of electronic bands

• Given Wyckoff Positions + orbital symmetry (s, p, d etc.) find symmetry character of all bands throughout BZ

Extend notion of band reps to materials with SOC and/or TRS

• band reps consist of all energy bands + Bloch functions arising from localized orbitals respecting crystal symmetry (+ maybe TRS )
• set of band reps $\subset$ set of groups of bands obtained from this new graph theory

Identify set of ==elementary band represenations== (EBRs) corresponding to the smallest sets of bands that can be derived from Wannier Functions (10,403 of them $\forall$ space groups, Wyckoff positions, and orbitals)

• If the number of e-’s is fraction of number of connected bands (Connectivity) that form EBR, then system is symmetry enforced semi-metal (Semimetals)
• largest number of connected bands in EBR is 24, smallest fraction of filled bands in a semi-metal is 1/24
• If number of connected bands is smaller than total number of bands in EBR, then momentum space description exists but a localized Wannier one does not and we have a topological obstruction
• Topological Insulators are those materials with bands not in the list of elementary components but that are in the graph enumeration

Graph Theory and band structure

Consider $D$ dimensional crystal invariant under the action of a space group with elements $\{R|\vec{d}\}\in G$ where $R$ is a rotation or rotoinversion $\vec{d}$ is a translation such that $\{R|\vec{d}\}\vec{x}=R\vec{x}+\vec{d}$ . The Bravais lattice of a space group is generated by a set of linearly independent translations $\{E|{\vec{t}}_i\}$ , $i=1,\cdots ,D$ where $E$ is the identity.

Each $\vec{k}$ in the BZ remains invariant up to translation by reciprocal lattice vector under action of symmetry elements in its little group $G_{\vec{k}}\subset G$.The Bloch wavefunctions $|u_{n\vec{k}}⟩$ transform under a sum of irreps of $G_{\vec{k}}$

• Bands at high symmetry points will have non-accidental degeneracies equal to dimension of the representations
• Spinless and SOC absent systems have ordinary linear reps
• Systems with SOC have double-valued reps

References

• Bradlyn et.al, Nature, 20 Jul 2017