Entanglement versus chaos in disordered spin chains

We use a Heisenberg spin-1∕2 chain to investigate how chaos and localization may affect the entanglement of pairs of qubits. To measure how much entangled a pair is, we compute its concurrence, which is then analyzed in the delocalized (localized) and in the chaotic (nonchaotic) regimes. Our results indicate that chaos reduces entanglement and that entanglement decreases in the region of strong localization. In the transition region from a chaotic to a nonchaotic regime localization increases entanglement. We also show that entanglement is larger for strongly interacting qubits (nearest neighbors) than for weakly interacting qubits (next and next-next neighbors).

Figure 1

(Color online) We show the maximum concurrence for several pairs of qubits with the same level spacing. Circles indicate that the selected pairs correspond to the nearest-neighbor qubits $(n,n+1)$, squares give the next-nearest neighbors $(n,n+2)$, and triangles correspond to the next-next-nearest neighbors $(n,n+3)$. We set $d=100$ and $J_{XY}=1$. Top left, $J_Z=0$; top right, $J_Z=J_{XY}$; middle left, $J_Z=10J_{XY}$; middle right, $J_Z=100J_{XY}$; bottom left, $J_Z=159J_{XY}$; and bottom right, $J_Z=327J_{XY}$.

Figure 2

Top: dependence of $⟨\overline{N_{pc}}⟩$ on $d∕J$. The $\overline{N_{pc}}$ corresponds to an average over the 924 eigenstates for each $d∕J$ and $⟨\overline{N_{pc}}⟩$ comes from the next average over 20 different sequences of random numbers. Bottom: dependence of $⟨\eta ⟩$ on $d∕J$. Here again, $⟨⟩$ indicates an average over 20 different sequences of 12 Gaussian random numbers. We set $J=1$.

Figure 3

(Color online) The top gives the dependence of $⟨C_{\max }⟩$ on $d∕J$ and the bottom gives $⟨\overline{C}⟩$ for nearest-neighbor qubits $N$. In both, the solid and black line represents the pair of qubits 1-2, the dotted and red line gives the pair 3-4, the dashed and blue gives the pair 6-7, and the dot-dashed and green represents the qubits 10-11.

Figure 4

(Color online) Top: maximum concurrence for next-nearest neighbor qubits $\left(NN\right)$. The solid and black line gives the pair 1-3, the dotted and red is for the qubits 2-4, the dashed and blue represents 3-5, and the dot-dashed and green gives the pair 4-6. Bottom: maximum concurrence for next-next-nearest neighbor qubits $\left(NNN\right)$. The solid and black line gives the pair 2-5, the dotted and red is for the qubits 4-7, the dashed and blue represents 5-8, and the dot-dashed and green gives the pair 7-10.