Symmetry and band topology in low-dimensional crystals

Symmetry and band topology in low-dimensional crystals

1. Milan Damnjanović, SANU, NanoLab, Fizički fakultet, Univerzitet u Beogradu

During past several years topological band theory is rapidly developing, becoming most propulsive theoretical field in condensed matter. It combines different mathematical techniques to analyse and apply symmetry: besides group theory, graph and K-theory are normally employed. In contrast from the three-dimensional crystals, Q1D and Q2D systems are less elaborated. Hence, in this lecture we give an overview of the subject, and then present some new results about the low-dimensional systems.
A consequence of the crystal periodicity along Þ (Þ=1,2,3) directions is that symmetry group of crystal has translational Þ-dimensional subgroup (lattice). Combining the other Euclidean symmetry with this subgroup, one gets line, layer and space groups. Each of these groups stratify Euclidean space into orbits with various stabilizers (group fixing a particular point), being a point groups. Collection of the orbits with the same stabilizers is called stratum. Different orbits, and therefore different strata exhaust the space. Taking a point from each orbit one gets fundamental domain, a finite part of space, while typical representative of each stratum is called Wyckoff position, and it may be understood as a contraction of each stratum to a point. Adding the arrows from special to more general neighbouring strata, one gets fundamental domain graph (FDG). This transforms the set of strata into the partially ordered set, with the corresponding stabilizers ordered by super-to-subgroup relation.
In general, a system consists of the ions positioned in the points rPp of the orbits P of the symmetry group. For each point a finite-dimensional space VPpis associated, determined by a set of orbitals defining quantum tight-binding space. The only physical condition is that such space is invariant under the corresponding stabilizer group, i.e. that the stabilizer action is a representation P(FP) of the stabilizer in this orbit space. Using this, an induced representation DP(G)=P(FP)G can be formed, and generally for the whole system D(G)=PDP(G). All such representations have important property when restricted to translational subgroup TÞ: each irreducible representation ∆(k)( TÞ) of the translational group TÞ occurs the same number of times (frequency number f=f k); wave vector k takes values from the Brillouin zone, which is p-dimensional torus.
This is the very origin of the Bloch theory: to each point k of the torus, vector subspace Vk of the dimension f is associated. Hamiltonian reduces within each of them, and the eigenvalues and eigenvectors van be found separatel, givin the spectrum of Hamiltonian in the form of energy bands over Brillouin zone.
The action of the group in the Brillouin zone is such that translations fix each point, meaning that the stratification (similar to the case of Euclidean space) gives infinite stabilizers Fk (with full TÞ as their subgroup), while the orbits are finite. Stratification gives irreducible domain (instead of fundamental domain) with different strata partially ordered by connectivity, equivalent to super-to-subgroup relation of stabilizers. Eventually, contraction of strata to points gives the irreducible domain graph. Induction procedure, from the (allowed) irreducible representations of stabilizers results in the complete set unitary irreducible representations of G.
Finally, we can combine results related to Euclidean space (band representations) and Brillouin zone (bands and irreducible representations). While each representation is in a unique way (unique frequency numbers) decomposed onto irreducible components, band representations are a subset which can be generated by integer combination of a finite number of them, elementary band representation (EBR). These are those which are induced from the irreducible representations of the maximal stabilizers. Indeed, it can be shown that such set generates all other band representations. Still, there may be cases of the exceptional representations, when different maximal strata give the same EBR. This is generalized to arbitrary exceptional band representation, which can be induced from two different Wyckoff positions. These situations indicate that the Wannier centers of the electrons is not clearly defined, i.e. that the centers depend on the model Hamiltonian, and may be between ions (in contrast to the expectation).
Also, there are situations of pairs A and B of band representations which when subtracted (A-B) is a representation itself (the frequency numbers are non-negative), but cannot be induced from any stabilizer. This is indication of fragile topology (in accordance with K-theory).
Finally, the obvious restrictions on the band representations (equal dimension of spaces in each k-point, compatibility and monodromy relations) leave a band structures space of the dimension BS. On the other hand, elementary band representations form an (invariant abelian) subgroup of this space, with atomic limits (AI). The corresponding factor group is called group of symmetry indicators, SI=BS/AI. Their cosets are spanned by stable topological structures.
Generally, all these quantities are calculated for 3D crystals (including time reversal symmetry). We discuss here Q1D and Q2D structures. For Q2D and layer groups result are partly known, and here we discuss their completition. For Q1D systems and line groups there are only few results in literature. It is known that group of symmetry indicators is trivial, so there is no stable not fragile topological bands. Only in the context of Wannier localization there are new results: complete list of exceptional band representations, indicating all the cases of possible obstructed atomic limits.

Ključne reči :

Tematska oblast: SIMPOZIJUM A - Nauka materije, kondenzovane materije i fizika čvrstog stanja

Datum: 08.07.2024.

Contemporary Materials 2024 - Savremeni Materijali

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