Analytical calculation of topological invariants for four-band systems
Keskinen, Panu (2021)
2021
Teknis-luonnontieteellinen DI-ohjelma - Master's Programme in Science and Engineering
Tekniikan ja luonnontieteiden tiedekunta - Faculty of Engineering and Natural Sciences
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Hyväksymispäivämäärä
2021-05-19
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:tuni-202104273851
https://urn.fi/URN:NBN:fi:tuni-202104273851
Tiivistelmä
In this thesis topological insulators are examined. Topology is a subfield of mathematics, which studies the properties of geometric objects under smooth deformations. We can define topological invariants for these objects, which remain constant under the deformations. Topological insulators are fundamentally different from conventional insulators. The bulk of the system acts as an insulator, but they have conducting surface states. The conductivity of the surface is quantized, which is related to the value of the topological invariant.
The main goal of this thesis is the analytical calculation of topological invariants for four-band systems. Performing the calculations analytically would give deeper insight into the properties of the system.
In the first part of the thesis, the theoretical background behind conventional insulators is recapped. The general Hamiltonian in the second quantization formalism is introduced. Then topological insulators and their topological invariants are examined.
The second part focuses on two-band Chern insulators. This type of system is described by a Hamiltonian, which is a Hermitian 2 × 2 matrix. It can be expressed as a linear combination of the identity matrix and the three Pauli matrices. The topological invariant corresponding to the Chern insulator is the Chern number, which can be analytically calculated. The invariant is integer-valued, where zero corresponds with the topologically trivial phase.
The next section focuses on four-band Z2 invariants. The Hamiltonian of such a system is a Hermitian 4 × 4 matrix. The basis is formed by the identity matrix, 5 Dirac gamma matrices and their 10 commutators. If the Hamiltonian can be turned into a block diagonal form, it reduces to two uncoupled Chern insulators. This allows the analytical calculation of the Z2 invariant, since it can be deduced by comparing the Chern numbers of these two systems. This invariant can only have a value of 0 or 1, which correspond to the trivial and topological phase.
The system can not generally be block diagonalized, but it is possible in certain special cases. These correspond to cases, where the Hamiltonian only has either time reversal symmetry or inversion symmetry breaking terms, which are mutually anticommuting.
The methods outlined before are applied to a few example systems. These include the KaneMele model, Bi2Se3, diamond and the BHZ model. For each system, the Z2 invariant is calculated and plotted as a function of some parameter of the Hamiltonian such that topological and trivial phases emerge.
The main goal of this thesis is the analytical calculation of topological invariants for four-band systems. Performing the calculations analytically would give deeper insight into the properties of the system.
In the first part of the thesis, the theoretical background behind conventional insulators is recapped. The general Hamiltonian in the second quantization formalism is introduced. Then topological insulators and their topological invariants are examined.
The second part focuses on two-band Chern insulators. This type of system is described by a Hamiltonian, which is a Hermitian 2 × 2 matrix. It can be expressed as a linear combination of the identity matrix and the three Pauli matrices. The topological invariant corresponding to the Chern insulator is the Chern number, which can be analytically calculated. The invariant is integer-valued, where zero corresponds with the topologically trivial phase.
The next section focuses on four-band Z2 invariants. The Hamiltonian of such a system is a Hermitian 4 × 4 matrix. The basis is formed by the identity matrix, 5 Dirac gamma matrices and their 10 commutators. If the Hamiltonian can be turned into a block diagonal form, it reduces to two uncoupled Chern insulators. This allows the analytical calculation of the Z2 invariant, since it can be deduced by comparing the Chern numbers of these two systems. This invariant can only have a value of 0 or 1, which correspond to the trivial and topological phase.
The system can not generally be block diagonalized, but it is possible in certain special cases. These correspond to cases, where the Hamiltonian only has either time reversal symmetry or inversion symmetry breaking terms, which are mutually anticommuting.
The methods outlined before are applied to a few example systems. These include the KaneMele model, Bi2Se3, diamond and the BHZ model. For each system, the Z2 invariant is calculated and plotted as a function of some parameter of the Hamiltonian such that topological and trivial phases emerge.