Table 1 Summary of monopole transformations on square (staggered flux state) and honeycomb lattices, where Φ1/2/3 are spin singlet monopoles (Φ± = Φ1 ∓ i Φ2) and Φ4/5/6 are spin triplet monopoles
From: Unifying description of competing orders in two-dimensional quantum magnets
Lattice | T 1 | T 2 | R x | Rotation | \({\cal{T}}\) | Note | |
|---|---|---|---|---|---|---|---|
Square | \({{\Phi}}_{\mathrm{{1}}}^{\mathrm{\dag}}\) | \({{\Phi }}\) 1 | \(-{{\Phi }}\) 1 | \(-{{\Phi }}\) 1 | \(-{{\Phi }}\) 3 | \({\mathrm{\Phi }}_1^\dag\) | \({\mathrm{Re}}[{\mathrm{\Phi }}_1]\) as \(\bar \psi \tau ^3\psi\) |
Square | \({\mathbf{\Phi}}_{\mathrm{{2}}}^{\mathrm{\dag}}\) | \(-{{\Phi }}\) 2 | \(-{{\Phi }}\) 2 | \(-{{\Phi }}\) 2 | \(-{{\Phi }}\) 2 | \(- {\mathrm{\Phi }}_2^\dag\) | \({\mathrm{Im}}[{\mathrm{\Phi }}_2]\) trivial |
Square | \({{\Phi }}_3^\dag\) | \(-{{\Phi }}\) 3 | \({{\Phi }}\) 3 | \({{\Phi }}\) 3 | \({{\Phi }}\) 1 | \({{\Phi }}_3^{\dag}\) | \({\mathrm{Re}}[{\mathrm{\Phi }}_3]\) as \(\bar \psi \tau ^1\psi\) |
Square | \({{\Phi }}_{4/5/6}^\dag\) | \(-{{\Phi }}\) 4/5/6 | \(-{{\Phi }}\) 4/5/6 | \({{\Phi }}\) 4/5/6 | \({{\Phi }}\) 4/5/6 | \(- {\mathrm{\Phi }}_{4/5/6}^\dag\) | \({\mathrm{Re}}[{\mathrm{\Phi }}_{4/5/6}]\) as \(\bar \psi \tau ^2 \otimes \sigma ^{1/2/3}\psi\) |
Honeycomb | \({\mathrm{\Phi }}_ + ^\dag\) | \({\mathrm{{e}}}^{ - i\frac{{2\pi }}{3}}{\mathrm{\Phi }}_ + ^\dag\) | \({\mathrm{{e}}}^{ - i\frac{{2\pi }}{3}}{\mathrm{\Phi }}_ + ^\dag\) | \({{\Phi }}\) + | \({\mathrm{{e}}}^{i\frac{\pi }{3}}{\mathrm{\Phi }}_ - ^\dag\) | \({{\Phi }}\) + | \({\mathrm{Im}}[{\mathrm{\Phi }}_1]\) as \(\bar \psi \tau ^1\psi\) |
Honeycomb | \({\mathrm{\Phi }}_ - ^\dag\) | \({\mathrm{{e}}}^{i\frac{{2\pi }}{3}}{\mathrm{\Phi }}_ - ^\dag\) | \({\mathrm{{e}}}^{i\frac{{2\pi }}{3}}{\mathrm{\Phi }}_ - ^\dag\) | \({{\Phi }}\) − | \({\mathrm{{e}}}^{i\frac{\pi }{3}}{\mathrm{\Phi }}_ + ^\dag\) | \({{\Phi }}\) − | \({\mathrm{Im}}[{\mathrm{\Phi }}_2]\) as \(\bar \psi \tau ^2\psi\) |
Honeycomb | \({\mathbf{\Phi }}_{\mathrm{{3}}}^{\mathrm{\dag}}\) | \({\mathrm{\Phi }}_3^\dag\) | \({\mathrm{\Phi }}_3^\dag\) | \({{\Phi }}\) 3 | \({\mathrm{\Phi }}_3^\dag\) | \({{\Phi }}\) 3 | \({\mathrm{Re}}[{\mathrm{\Phi }}_3]\) trivial |
Honeycomb | \({\mathrm{\Phi }}_{4/5/6}^\dag\) | \({\mathrm{\Phi }}_{4/5/6}^\dag\) | \({\mathrm{\Phi }}_{4/5/6}^\dag\) | \(-{{\Phi }}\) 4/5/6 | \(- {\mathrm{\Phi }}_{4/5/6}^\dag\) | \(-{{\Phi }}\) 4/5/6 | \({\mathrm{Im}}[{\mathrm{\Phi }}_{4/5/6}]\) as \(\bar \psi \tau ^3 \otimes \sigma ^{1/2/3}\psi\) |
