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 Θ: Θϕ(x,y,z)≡ϕ∗(x,y,z), hereafter, ϕ(x,y,z) represents an arbitrary wave function for single electron.
(2) 2-fold rotational operator R2: R2ϕ(x,y,z)≡ϕ(−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 R3: R3ϕ(→r)≡ϕ(A→r), where A=(cos2π3−sin2π30sin2π3cos2π30001) →r=(x,y,z) and we choose the lattice site as the origin point o of the coordinate.
(4) Mirror symmetry operator Π: Πϕ(x,y,z)≡ϕ(x,y,−z).
Second-quantization language:
(1) TR symmetry operator T: TCi↑T−1=Ci↓,TCi↓T−1=−Ci↑, where C=a,b are the annihilation operators referred to the two sublattices of graphene.
(2) 2-fold rotational operator P2: P2a(x,y)P−12≡b(−x,−y),P2b(x,y)P−12≡a(−x,−y), P2 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 P3: P3C(→x)P−13≡C(A→x),→x=(x,y),C=a,b, where A=(cos2π3−sin2π3sin2π3cos2π3) and P3 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 iλ∑≪ij≫vijC†iσzCj should be invariant under M while the Rashba term iλR∑<ij>C†i(σ×pij)zCj will not be.
(2)Here mirror operation is just reflection in one of the three spatial axes (i.e. (x,y,z)→(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