# Spin-up - edge of breaking - Auto Craft Engineering Ltd

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Real experimental systems, however, are far from the idealized picture presented above in which spin-up and spin-down electrons are not coupled. A very important achievement was the realization that the quantum spin Hall state remains to be non-trivial even after the introduction of spin-up spin-down scattering, [4] which destroys the quantum spin Hall effect. In a separate paper, Kane and Mele introduced a topological Z 2 {\displaystyle \mathbb {Z} _{2}} invariant which characterizes a state as trivial or non-trivial band insulator (regardless if the state exhibits or does not exhibit a quantum spin Hall effect). Further stability studies of the edge liquid through which conduction takes place in the quantum spin Hall state proved, both analytically and numerically that the non-trivial state is robust to both interactions and extra spin-orbit coupling terms that mix spin-up and spin-down electrons. Such a non-trivial state (exhibiting or not exhibiting a quantum spin Hall effect) is called a topological insulator , which is an example of symmetry protected topological order protected by charge conservation symmetry and time reversal symmetry. (Note that the quantum spin Hall state is also a symmetry protected topological state protected by charge conservation symmetry and spin- S z {\displaystyle S_{z}} conservation symmetry. We do not need time reversal symmetry to protect quantum spin Hall state. Topological insulator and quantum spin Hall state are different symmetry protected topological states. So Topological insulator and quantum spin Hall state are different states of matter.)

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