Boundary condition analysis of first and second order topological insulators

Area of Science:

• Condensed Matter Physics
• Quantum Mechanics
• Materials Science

Background:

• Topological insulators are materials with conducting surface states and insulating bulk.
• Understanding boundary phenomena is crucial for characterizing topological phases.
• Dirac fermion models on lattices are key theoretical frameworks for studying topological insulators.

Purpose of the Study:

• To analytically investigate boundary conditions in Dirac fermion lattice models.
• To determine the dispersion relations of edge and hinge states.
• To explore the role of Hamiltonian symmetry in constraining boundary conditions and its relation to topological properties.

Main Methods:

• Analytical solutions of boundary conditions for Dirac fermion lattice models.
• Derivation of dispersion relations for edge and hinge states.
• Investigation of Hamiltonian symmetries and their impact on boundary conditions.

Main Results:

• Obtained dispersion relations for edge and hinge states by solving boundary conditions.
• Clarified that Hamiltonian symmetry imposes constraints on boundary conditions.
• Demonstrated an edge-hinge analog of the bulk-edge correspondence.

Conclusions:

• The study provides analytical insights into boundary phenomena in topological insulators.
• Hamiltonian symmetry plays a critical role in defining boundary conditions.
• A novel bulk-edge correspondence analog between gapped edge states and gapless hinge states was established.