Let us take the famous Kane-Mele(KM) model as our starting point.

Due to the time-reversal(TR), 2-fold rotational(or 2D space inversion), 3-fold rotational and mirror symmetries of the honeycomb lattice system, we can derive the intrinsic spin-orbit(SO) term. Further more, if we apply a spatially uniform electric field perpendicular to the 2D lattice(now the mirror symmetry is broken), a (extra) Rashba-type SO term will emerge.

To present my question more clearly, I will first give a more detailed description of the above symmetry operations in both first- and second- quantization formalism. In the follows, a 3D Cartesian coordinate has been set up where the 2D lattice lies in the $xoy$ plane.

**First-quantization language:**

(1) TR symmetry operator $\Theta:$ $\Theta\phi(x,y,z)\equiv \phi^*(x,y,z)$, hereafter, $\phi(x,y,z)$ represents an **arbitrary wave function** for single electron.

(2) 2-fold rotational operator $R_2:$ $R_2\phi(x,y,z)\equiv \phi(-x,-y,z)$, where we choose the middle point of the nearest-neighbour bond as the origin point $o$ of the coordinate.

(3) 3-fold rotational operator $R_3:$ $R_3\phi(\vec{r} )\equiv \phi(A\vec{r})$, where $A=\begin{pmatrix}
\cos\frac{2\pi}{3}& -\sin\frac{2\pi}{3}& 0\\
\sin\frac{2\pi}{3}& \cos\frac{2\pi}{3} & 0\\
0& 0 & 1
\end{pmatrix}$ $\vec{r}=(x,y,z)$ and we choose the lattice site as the origin point $o$ of the coordinate.

(4) **Mirror symmetry operator $\Pi:$ $\Pi\phi(x,y,z)\equiv \phi(x,y,-z)$.**

**Second-quantization language:**

(1) TR symmetry operator $T:$ $TC_{i\uparrow}T^{-1}=C_{i\downarrow}, TC_{i\downarrow}T^{-1}=-C_{i\uparrow}$, where $C=a,b$ are the annihilation operators referred to the two sublattices of graphene.

(2) 2-fold rotational operator $P_2:$ $P_2a(x,y)P_2^{-1}\equiv b(-x,-y), P_2b(x,y)P_2^{-1}\equiv a(-x,-y)$, $P_2$ is unitary and we choose the middle point of the nearest-neighbour bond as the origin point $o$ of the coordinate.

(3) 3-fold rotational operator $P_3:$ $P_3C(\vec{x})P_3^{-1}\equiv C(A\vec{x}), \vec{x}=(x,y),C=a,b$, where $A=\begin{pmatrix}
\cos\frac{2\pi}{3}& -\sin\frac{2\pi}{3}\\
\sin\frac{2\pi}{3}& \cos\frac{2\pi}{3} \\
\end{pmatrix}$ and $P_3$ is unitary, we choose the lattice site as the origin point $o$ of the coordinate.

(4) **Mirror symmetry operator $M:$ ????**

as you see, that's what I want to ask: how to define the mirror symmetry operator $M$ **in terms of second-quantization language** for this 2D lattice system? Or maybe there is no well defined $M$ for this model? Thanks in advance.

**Remarks:**

(1) A direct way to verify your definition of $M$ being correct or not is as follows: The intrinsic SO term $i\lambda\sum_{\ll ij \gg }v_{ij}C_i^\dagger\sigma_zC_j$ should be invariant under $M$ while the Rashba term $i\lambda_R\sum_{<ij>}C_i^\dagger \left ( \mathbf{\sigma}\times\mathbf{p}_{ij}\right )_zC_j$ will not be.

(2)Here **mirror** operation is just reflection in one of the three spatial axes (i.e. $(x,y,z)\rightarrow (x,y,-z)$), not the “parity” operation in the context of "CPT symmetry" in field theory.

This post imported from StackExchange Physics at 2014-03-09 08:46 (UCT), posted by SE-user K-boy