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I encountered the notion of $\mathbb{Z}_2$ symmetry in an article. Can someone give a definition?

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  • $\begingroup$ If you give the article, can give a more specific explanation rather than the general definition of group action. $\endgroup$
    – AHusain
    Jul 13, 2019 at 13:46
  • $\begingroup$ @AHusain This one : arxiv.org/abs/1812.01041 $\endgroup$
    – cnada
    Jul 13, 2019 at 14:05

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There is an operator $P$ such that $P^2$ is the identity and commutes with the Hamiltonians.

In this case $P=\prod \sigma_x$.

This is a $\mathbb{Z}_2$ because $P$ and the identity form a group isomorphic to the integers modulo 2 (odds and evens)

Identity corresponds to evens, $P$ to odds. $P^2=1$ corresponds to odd plus odd equals even.

If there were some other operators commuting with the Hamiltonians, then would be some other kind of symmetry, not $\mathbb{Z}_2$. But in this case, it is not that complicated.

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  • $\begingroup$ Is not the $\mathbb{Z}_2$ symmetry refering more to $P= \prod \sigma_x$ in this case? $\endgroup$
    – cnada
    Jul 14, 2019 at 12:30
  • $\begingroup$ Yes, typo. Edited $\endgroup$
    – AHusain
    Jul 14, 2019 at 14:17

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