In Kitaevs's paper (Anyons in an exactly solved model and beyond) section 2.1-2.2, he seems to be extending the Hilbert space of a multi-spin system using Majorana operators. More specifically, if there are only 2 spins/vertices, the original Hilbert space would be $\mathfrak{H}\otimes\mathfrak{H}$ where $\mathfrak{H}\cong \mathbb{C}^2$, and each Hilbert space $\mathfrak{H}$ would be replaced with its fermionic Fock space $\mathfrak{F}$ so that the Pauli spins $\sigma^\alpha$ can be written as Majorana operators (extended in a natural way). My initial guess would be that the overall extended space would just be $\mathfrak{F}\otimes \mathfrak{F}$. However, this doesn't seem to make sense since in this case the naturally extended Majorana operator on different vertices would commute with each instead of anti-commute, e.g., $c_1 \equiv c\otimes I$ commutes with $c_2 = I\otimes c$ where $c$ is a Majoranan operator on $\mathfrak{F}$.
So what would the natural extended space be?
Best Answer
The extended Hilbert space would be a tensor product, but the Majorana operators have to act "non-locally" on this space. To define the action of the Majorana operators on this space you have to choose some (arbitrary) ordering of the sites $1, \cdots, N$, and define $$c_i = \left[\bigotimes_{j=1}^{i-1} (-1)^F\right] \otimes c \otimes \left[ \bigotimes_{j=i+1}^N I \right]$$ where $(-1)^{F}$ is the fermion parity at a given site. This has the property that the $c_i$'s anti-commute at different sites. The non-local strings of fermion parities will cancel out in the spin operators, since they are always the product of two Majorana operators at the same site.