The Z2 classification of dimensional reduced Hopf insulators

Area of Science:

• Condensed matter physics
• Topological materials science

Background:

• Hopf insulators are defined by a topological invariant, the Hopf index, distinct from Chern numbers used in conventional topological insulators.
• Unlike topological insulators, Hopf insulators lack symmetry protection, presenting unique theoretical challenges.

Purpose of the Study:

• To investigate the existence of a new topological invariant for two-dimensional (2D) systems with vanishing Chern numbers.
• To construct a model Hamiltonian exhibiting this novel topological property.
• To analyze the topological edge modes associated with this new classification.

Main Methods:

• Dimensional reduction of three-dimensional (3D) Hopf insulators to two dimensions.
• Construction of a specific 2D model Hamiltonian with a vanishing Chern number.
• Numerical calculation of edge modes in the constructed 2D model.

Main Results:

• Demonstration of a new Z(2) topological index for 2D Hamiltonians with zero Chern number.
• Successful construction of a model Hamiltonian realizing this Z(2) topological phase.
• Numerical evidence of topological edge modes protected by the Z(2) index.

Conclusions:

• The study establishes a new Z(2) classification for a class of 2D topological materials, extending the concept of Hopf insulators.
• The findings reveal the existence of symmetry-inde Hopf insulators in two dimensions.
• Numerical results confirm the theoretical predictions regarding protected edge modes.