Bipartite entanglement and entropic boundary law in lattice spin systems

We investigate bipartite entanglement in spin-$1∕2$ systems on a generic lattice. For states that are an equal superposition of elements of a group $G$ of spin flips acting on the fully polarized state ${\mid 0⟩}^{\otimes n}$, we find that the von Neumann entropy depends only on the boundary between the two subsystems $A$ and $B$. These states are stabilized by the group $G$. A physical realization of such states is given by the ground state manifold of the Kitaev’s model on a Riemann surface of genus $\mathfrak{g}$. For a square lattice, we find that the entropy of entanglement is bounded from above and below by functions linear in the perimeter of the subsystem $A$ and is equal to the perimeter (up to an additive constant) when $A$ is convex. The entropy of entanglement is shown to be related to the topological order of this model. Finally, we find that some of the ground states are absolutely entangled, i.e., no partition has zero entanglement. We also provide several examples for the square lattice.

Figure 1

A system of spins on an 2D irregular lattice; a typical plaquette and star are denoted by $p$ and $s$, respectively. The subsystem $A$ contains all the spins within the thick boundary. The operators $B_p$ for the plaquettes situated inside the boundary act only on the spins of the subsystem $A$. The $B_p$ of the outside plaquettes that share a link with the boundary act on both subsystems $A$ and $B$ and there is one such plaquette operator for each link in the boundary of $A$. Hence the entropy is $S=L_{\partial A}-n_c$ ($n_c$ is the number of constraints; see text).

Figure 2

(Color online) A $k\times k$ square lattice of the torus; opposite boundaries are identified. The end of an open $z$ string anticommutes with the star based at the site where the string is. Similarly, the end of an open $x$ string anticommutes with the plaquette on the square where it ends. The figure also shows that a star on the lattice corresponds to a plaquette in the dual lattice and vice versa.

Figure 3

(Color online) A lattice region consisting of all the spins inside or crossed by a loop (thick line) on the dual lattice. The ${\mathrm{\Sigma }}_A$ stars based on the sites inside the region $A$ (red diamonds) act exclusively on the subsystem $A$ and the ${\mathrm{\Sigma }}_B$ stars based on the black sites act only on the subsystem $B$. There are ${\mathrm{\Sigma }}_{AB}$ stars (white dots) acting on both subsystems $A$ and $B$. The total number of stars is $n_s={\mathrm{\Sigma }}_A+{\mathrm{\Sigma }}_B+{\mathrm{\Sigma }}_{AB}=n-n_p+2-2\mathfrak{g}$, where $n$ $\left(n_p\right)$ is the number of spins (plaquettes) of the lattice.

Figure 4

(Color online) The subsystems $A$ [thick (blue) spins] used in calculating the entropy $S$: (a) the spin chain; (b) the vertical ladder; (c) the cross; (d) all the vertical spins.

Figure 5

(Color online) A region $A$ of the lattice obtained by taking all the spins [thin (black) lines] inside or crossed by a loop [thick (red) line]. ${\mathrm{\Sigma }}_{A,B}$ are the number of sites whose stars act exclusively on $A$ [diamonds (red) sites] and respectively, $B$ [solid (blue) circles]; $d_{A,B}=2^{{\mathrm{\Sigma }}_{A,B}}$. The number of sites with stars acting on both subsystems [open (white) sites] is ${\mathrm{\Sigma }}_{AB}=n_1+n_2+n_3$; $n_i$ is the number of [open (white)] sites having $i$ nearest neighbors inside $A$. Area conservation implies ${\mathrm{\Sigma }}_A+{\mathrm{\Sigma }}_B+{\mathrm{\Sigma }}_{AB}=k^2$. The entropy is $S={\mathrm{\Sigma }}_{AB}-1$. (a) If the boundary is a convex loop (i.e., a rectangle), ${\mathrm{\Sigma }}_{AB}$ is equal to the boundary length $L_{\partial A}$ (in lattice units), since in this case $n_2=n_3=0$; the entropy is $S=L_{\partial A}-1$. (b) If the boundary is an arbitrary loop, $L_{\partial A}=n_1+2n_2+3n_3$, hence the entropy is $S=L_{\partial A}-n_2-2n_3-1$; in the figure $n_2=1$ [open (white) triangle] and $n_3=1$ [open (white) square].