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We report two theoretical discoveries for $mathbb{Z}_2$-topological metals and semimetals. It is shown first that any dimensional $mathbb{Z}_2$ Fermi surface is topologically equivalent to a Fermi point. Then the famous conventional no-go theorem, which was merely proven before for $mathbb{Z}$ Fermi points in a periodic system without any discrete symmetry, is generalized to that the total topological charge is zero for all cases. Most remarkably, we find and prove an unconventional strong no-go theorem: all $mathbb{Z}_2$ Fermi points have the same topological charge $ u_{mathbb{Z}_2} =1$ or $0$ for periodic systems. Moreover, we also establish all six topological types of $mathbb{Z}_2$ models for realistic physical dimensions.
Topological phases of matter lie at the heart of physics, connecting elegant mathematical principles to real materials that are believed to shape future electronic and quantum computing technologies. To date, studies in this discipline have almost ex
We present a many-body exact diagonalization study of the $mathbb{Z}_2$ and $mathbb{Z}_4$ Josephson effects in circuit quantum electrodynamics architectures. Numerical simulations are conducted on Kitaev chain Josephson junctions hosting nearest-neig
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