# How to understand the entanglement in a lattice fermion system?

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 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 $V_s$?

• Good question. The issue of entanglement in fermionic Fock space has been studied in a number of works. As far as I know there is no unique definition of entanglement; see this nice review for more details. The map between the fermion and spin picture can be achieved using a Jordan-Wigner transformation, which is a bit more subtle than the simple idenfication you alluded to. When going between spin and fermion pictures, local operators get transformed to highly non-local operators in general. – Mark Mitchison Oct 12 '13 at 23:03
• @ Mark Mitchison Thanks for your suggested review. – Kai Li Oct 13 '13 at 7:43
• Here is a very nice answer by Prof.Wen on overflow: physicsoverflow.org/6359/… – Kai Li Apr 6 '15 at 10:05
• This article may be relevant: nature.com/articles/srep20603 – Tarek Jun 8 '16 at 14:59

Mark Mitchison is right. The concept of entanglement in systems of indistinguishable particles is more controversial than it is in the case of systems composed of distinguishable subsystems. You need to define first what do you mean by it when it comes, for example, to fermions. Do you mean entanglement between particles (connected with single Slater determinants), modes, pairing of states or whether a given state can be written as a convex combination of Gaussian states or sth completely else. You also should specify do you want to consider fermionic state with a ﬁxed number of fermions (and then use the criteria from here) or just to ﬁx the parity of the fermionic state and not the number of fermions, obtaining e.g. Gaussian states. This is also important, because even though physical states have a fixed number of fermions, Gaussian fermionic states are important approximations to physically non-trivial states, such as the superconducting BCS state. Of course then, the super-selection rule should also play a role somehow.

And about your question, you can find a nice definition of short-ranged entanglement in topological insulators in Sec. II of http://arxiv.org/pdf/1004.3835v2.pdf

• @ Piotr Ćwikliński Thanks for your wonderful explanation. – Kai Li Nov 17 '13 at 10:00

An approach to define the entanglement between identical particles, is to use the so-called geometric idea.

For example, the generic wave function of a fermionic system is not a Slater determinant. However, since the simplest wave function for a fermionic system is a Slater determinant, we can quantify the entanglement between the identical fermions by studying how close the wave function is to a Slater determinant.

This approach is taken in the paper

http://journals.aps.org/pra/abstract/10.1103/PhysRevA.89.012504

They have a numerical algorithm to construct the best Slater approximation of an arbitrary fermionic wave function. By the best, they mean the overlap between the Slater determinant and the target wave function is maximized.