Table 1 Quantized Berry phases of a self-closed band along a loop Γ protected by different types of twofold DAS point-group symmetries G.
From: Symmetry-protected topological exceptional chains in non-Hermitian crystals
Symmetry type | Loop | Other conditions | Quantized Berry phases (\(0\,{{\mbox{or}}}\,\,\pi \,\,{{{{{{\mathrm{mod}}}}}}}\,\,2\pi\)) | Examples |
|---|---|---|---|---|
G | \(\Gamma =\hat{g}{\Gamma }^{-1}\) | none | \({\theta }^{{{{{{{{\rm{LR}}}}}}}}}={\theta }^{{{{{{{{\rm{RL}}}}}}}}}={\theta }^{{{{{{{{\rm{LL}}}}}}}}}={\theta }^{{{{{{{{\rm{RR}}}}}}}}}=\arg (p(0)p(\pi ))\) | \({{{{{{{\mathcal{P}}}}}}}}\), M |
G-† | \(\Gamma =\hat{g}{\Gamma }^{-1}\) | two intersections of Γ and ΠG are in the exact phase | \({{{{{{{\rm{Re}}}}}}}}[{\theta }^{{{{{{{{\rm{LR}}}}}}}}}]={{{{{{{\rm{Re}}}}}}}}[{\theta }^{{{{{{{{\rm{RL}}}}}}}}}]=\frac{1}{2}({\theta }^{{{{{{{{\rm{LL}}}}}}}}}+{\theta }^{{{{{{{{\rm{RR}}}}}}}}})=\arg (\tilde{p}(0)\tilde{p}(\pi ))\) | M-† |
\(G{{{{{{{\mathcal{T}}}}}}}}\) | \(\Gamma =-\hat{g}\Gamma\) | Γ is in the exact phase | \({{{{{{{\rm{Re}}}}}}}}[{\theta }^{{{{{{{{\rm{LR}}}}}}}}}]={{{{{{{\rm{Re}}}}}}}}[{\theta }^{{{{{{{{\rm{RL}}}}}}}}}]={\theta }^{{{{{{{{\rm{LL}}}}}}}}}={\theta }^{{{{{{{{\rm{RR}}}}}}}}}\) | \({{{{{{{\mathcal{PT}}}}}}}}\), \({C}_{2}{{{{{{{\mathcal{T}}}}}}}}\) |
\(G{{{{{{{\mathcal{T}}}}}}}}\,{{\mbox{-}}}\,{{{\dagger}}}\) | \(\Gamma =-\hat{g}\Gamma\) | none | \({\theta }^{{{{{{{{\rm{LR}}}}}}}}}={\theta }^{{{{{{{{\rm{RL}}}}}}}}}=\frac{1}{2}({\theta }^{{{{{{{{\rm{LL}}}}}}}}}+{\theta }^{{{{{{{{\rm{RR}}}}}}}}})\) | \({{{{{{{\mathcal{PT}}}}}}}}\,{{\mbox{-}}}\,{{{\dagger}}}\) |
