Topological insulator is a fermion system with only *short-ranged entanglement*, what does the *entanglement* mean here?

For example, the Hilbert space $V_s$ of a lattice $N$ spin-1/2 system is $V_s=V_1\otimes V_2\otimes...\otimes V_N$, where $V_i$ is the Hilbert space of the spin on site $i$. And the meaning of an entanglement state belongs to $V_s$ is clear — a state which can not be written as a direct tensor product of the $N$ single spin states.

Now consider a spinless fermion system lives on the same lattice as spin-1/2, in the 2nd quantization framework, the fermion operators $c_i,c_j$ on different lattices $i,j$ do not commute with each other and the Hilbert space $V_f$ of the fermion system can not be written as a direct product of $N$ single fermion Hilbert spaces. **Thus, how to understand the entanglement in this fermion system?**

Mathematically, we can make a **natural** linear bijective map between $V_f$ and $V_s$, simply say, just let $\mid 0\rangle=\mid \downarrow\rangle,\mid 1\rangle=\mid \uparrow\rangle$. **Thus, can we understand the entanglement of a fermion state in $V_f$ through its corresponding spin state in $V_s$?**

This post imported from StackExchange Physics at 2014-03-09 08:41 (UCT), posted by SE-user K-boy