# How can pseudospin be a vector? (Graphene)

In graphene science, I don't understand how one interprets pseudospin as a vector. I thought 'pseudospin' was the vector of Pauli matrices. So how can it be a vector that one can plot for example in the image below? Moreover Pauli matrices contain just numbers $(1,i)$ so how can it change direction?

I think I must have missed the point entirely, perhaps someone can nudge me in the right direction to understanding this. • Thanks for the comment, the image appears fine for me and on a friends computer who's not logged in. I'll try again in any case. Sorry though, for a visual reference just google 'pseudospin graphene'. – Tom Nov 2 '14 at 13:17
• It shows up now. I'm not sure what I'm looking at, though - what are the $E$, $k_x$ and $k_y$ directions? – ACuriousMind Nov 2 '14 at 13:20
• The blue arrows are real spin and the red arrows are pseudospin. To be honest I'm not sure! The question stands without the image though. I stole it from this paper if you have access: nature.com/nphys/journal/v7/n1/pdf/nphys1869.pdf – Tom Nov 2 '14 at 13:23

## 1 Answer

In order to appreciate the periodicity of graphene one has to recognize that it consists out of two interpenetrating hexagonal Bravais sublattices, A and B, which together make up the honeycomb lattice. The two sublattices are like two degrees of freedom, and the electron can have an amplitude to be on sublattice A, and an amplitude to be on the sublattice B. The presence of the sublattice basis $\{|A\rangle,|B\rangle\}$ resembles the case with spin one-half particle, where an electron can be in spin-up state $|\uparrow\rangle$, spin-down state $|\downarrow\rangle$, or any superposition of these. The sublattice basis is therefore called a pseudospin.

Now, recall that the superposition of the form

\begin{equation} \cos\frac{\theta}{2}|\uparrow\rangle+\sin\frac{\theta}{2}e^{i\phi}|\downarrow\rangle \end{equation}

represents the general state where the orientation of the spin is given through the polar angle $\theta$, and the azimuthal angle $\phi$ of the cylindrical coordinate system. For instance, $\theta=\pi/2,\phi=0$ is the state of a particle whose spin is pointing in $+x$ direction, while $\theta=\pi/2,\phi=\pi/2$ is the state of a particle whose spin is pointing in $+y$ direction. Therefore, the orientation of a 3D vector with fixed length can be encoded through two amplitudes, and this is why the pseudospin in graphene can be thought of as a vector.

Also, it seems you may have misinterpreted the vector of Pauli matrices $\boldsymbol{\sigma}=(\sigma_x,\sigma_y)$ in the Hamiltonian of graphene ($H=\hbar v_F\boldsymbol{\sigma}\cdot\boldsymbol{k}$) as pseudospin. This is an operator, and not pseudospin itself. Finally, the valley degree of freedom is sometimes referred to as valley pseudospin, more often as valley isospin, and most often as just valley, and should not be confused with sublattice pseudospin.