Table 2 Real-synthetic symmetries and harmonic generation selection rules in (2+1)D symmetry broken systems.
From: Selection rules in symmetry-broken systems by symmetries in synthetic dimensions
\(\hat{X}\) | \({\hat{{{{{{\rm{\zeta }}}}}}}}_{\hat{X}}\left({{{{{\boldsymbol{Q}}}}}}\right)\) | Harmonic generation selection rule |
\(\hat{T}\) | \(\bar{{{{{{\boldsymbol{Q}}}}}}}\) | \({E}_{{nx}}^{\left({abcd}\right)},{E}_{{ny}}^{\left({abcd}\right)}\) \(\in{\mathbb{R}}\) |
\(\hat{Q}\) | \(-\bar{{{{{{\boldsymbol{Q}}}}}}}\) | \({E}_{{nx}}^{\left({abcd}\right)},{E}_{{ny}}^{\left({abcd}\right)}\in {i}^{1+a+b+c+d}{\mathbb{R}}\) |
\(\hat{G}\) | \({\left(-{{{{{\bf{1}}}}}}\right)}^{{{{{{\bf{1}}}}}}+{{{{{\boldsymbol{s}}}}}}}\bar{{{{{{\boldsymbol{Q}}}}}}}\) | \({E}_{{nx}}^{\left({klhj}\right)},{E}_{{ny}}^{\left({klhj}\right)}\in {i}^{n+1+\left(s+1\right)(a+b+c+d)}{\mathbb{R}}\) |
\({\hat{Z}}_{y}\) | \(\left(\begin{array}{cc}{\left(-1\right)}^{s+1} & 0\\ 0 & {\left(-1\right)}^{s}\end{array}\right)\left(\begin{array}{c}{q}_{x}\\ {q}_{y}\end{array}\right)\) | \(n+\left(s+1\right)\left(a+c\right)+s\left(b+d\right)=2q\Rightarrow\) \({E}_{{nx}}^{\left({abcd}\right)}=0\) \(n+\left(s+1\right)\left(a+c\right)+s\left(b+d\right)=2q+1\Rightarrow\) \({E}_{{ny}}^{\left({abcd}\right)}=0\) |
\({\hat{D}}_{y}\) | \(\left(\begin{array}{cc}-1 & 0\\ 0 & 1\end{array}\right)\left(\begin{array}{c}{\bar{q}}_{x}\\ {\bar{q}}_{y}\end{array}\right)\) | \({E}_{{nx}}^{\left({abcd}\right)}\in {i}^{1+a+c}{\mathbb{R}}\) \({E}_{{ny}}^{\left({abcd}\right)}\in {i}^{a+c}{\mathbb{R}}\) |
\({\hat{H}}_{y}\) | \(\left(\begin{array}{cc}{\left(-1\right)}^{s+1} & 0\\ 0 & {\left(-1\right)}^{s}\end{array}\right)\left(\begin{array}{c}{\bar{q}}_{x}\\ {\bar{q}}_{y}\end{array}\right)\) | \({E}_{{nx}}^{\left({abcd}\right)}\in {i}^{n+1+\left(s+1\right)\left(a+c\right)+s\left(b+d\right)}{\mathbb{R}}\) \({E}_{{ny}}^{\left({abcd}\right)}\in {i}^{n+\left(s+1\right)\left(a+c\right)+s\left(b+d\right)}{\mathbb{R}}\) |
\({\hat{C}}_{{NM}}\) | \({e}^{-\frac{i2\pi s}{N}}{{\hat{R}}^{({{{{{\boldsymbol{Q}}}}}})}}_{N,M}{\cdot }{{{{{\boldsymbol{Q}}}}}}\) | \({E}_{{Rn}}^{\left({abcd}\right)}\) is forbidden unless \({mod}\left(n-M\left(a-b-c+d\right)-s\left(a+b-c-d\right)-M,N\right)=0\) \({E}_{{Ln}}^{\left({klhj}\right)}\) is forbidden unless \({mod}\left(n-M\left(a-b-c+d\right)-s\left(a+b-c-d\right)+M,N\right)=0\) |
\({\hat{e}}_{{NM}}\) | \({e}^{-\frac{i2\pi s}{N}}{{\hat{L}}^{{{{{{\boldsymbol{(}}}}}}{{{{{\boldsymbol{Q}}}}}}{{{{{\boldsymbol{)}}}}}}}}_{1/b}\cdot {{\hat{R}}^{\left({{{{{\boldsymbol{Q}}}}}}\right)}}_{N,M}\cdot {{\hat{L}}^{{{{{{\boldsymbol{(}}}}}}{{{{{\boldsymbol{Q}}}}}}{{{{{\boldsymbol{)}}}}}}}}_{b}{\cdot}{{{{{\boldsymbol{Q}}}}}}\) | \({E}_{-n}^{\left({abcd}\right)}\) is forbidden unless \({mod}(n-M\left(a-b-c+d\right)-s\left(a+b-c-d\right)-M,N)=0\) \({E}_{+n}^{\left({abcd}\right)}\) is forbidden unless \({mod}\left(n-M\left(a-b-c+d\right)-s\left(a+b-c-d\right)+M,N\right)=0\) |
