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=                       Topological insulator                        =
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                             Introduction
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A topological insulator is a material with non-trivial
symmetry-protected topological order that behaves as an insulator in
its interior but whose surface contains conducting states, meaning
that electrons can only move along the surface of the material.
However, having a conducting surface is not unique to topological
insulators, since ordinary band insulators can also support conductive
surface states. What is special about topological insulators is that
their surface states are symmetry-protectedZheng-Cheng Gu and
Xiao-Gang Wen
[https://arxiv.org/abs/0903.1069 Tensor-Entanglement-Filtering
Renormalization Approach and Symmetry-Protected Topological Order]
Phys. Rev. B80, 155131 (2009).
by particle number conservation and time-reversal symmetry. In
two-dimensional (2D) systems, this ordering is analogous to a
conventional electron gas subject to a strong external magnetic field
causing electronic excitation gap in the sample bulk and metallic
conduction at the boundaries or surfaces.

The distinction between 2D and 3D topological insulators is
characterized by the Z-2 topological invariant, which defines the
ground state. In 2D, there is a single Z-2 invariant distinguishing
the insulator from the quantum spin-Hall phase whiles in 3D, there are