Even-odd effects of Heisenberg chains on long-range interaction and entanglement

A strongly coupled Heisenberg chain provides an important channel for quantum communication through its many-body ground state. Yet, the nature of the effective interactions and the ability to mediate long-range entanglement differs significantly for chains of opposite parity. Here, we contrast the characters of even- and odd-size chains when they are coupled to external qubits. Additional parity effects emerge in both cases, depending on the positions of the attached qubits. Some striking results include (i) the emergence of maximal entanglement and (ii) spin-chain Ruderman-Kittel-Kasuya-Yosida interactions for qubits attached to an even chain, and (iii) the ability of chains of either parity to mediate qubit entanglement that is undiminished by distance.

Figure 1

(Color online) (a) Schematics of two qubits, $A$ and $B$, weakly attached to nodes $i=2$ and $j=7$ of a chain with $N$ spins. The qubit perturbation affects the low-energy manifolds of (b) an odd-size chain and (c) an even-size chain. (Left: unperturbed. Right: perturbed.) Here $D_1$, $D_2$, and $Q$ denote low-lying doublet and quadruplet states, respectively. $S$ denotes a singlet and $T$ a triplet. $\Delta$ is the ground energy gap of the chain. $\delta$ is equal to $\left|J_{A,i}^{\ast }\right|=\left|J_{B,j}^{\ast }\right|$ for an odd-size chain and $\left|J_{A,B}^{\ast }\left(i,j\right)\right|$ for an even-size chain.

Figure 2

(Color online) (a) Local magnetic moment $m_i/{\mu }_B=⟨0_c|{\sigma }_{iz}|0_c⟩$ of an odd-size chain with $N=9$. (b) Concurrence $C_{i,j}$ of all possible spin pairs when qubits $A$ and $B$ are coupled to sites 1 and 9 of the chain with $N=9$.

Figure 3

(Color online) Concurrence between qubits $A$, $B$, and the central spin $C$ as a function of $\lambda$, as defined in Eq. (5).

Figure 4

(Color online) Concurrence $C_{i,j}$ of arbitrary spin pairs in the ground state of two even-size geometries $\left(N=8\right)$. (a) Qubits $A$ and $B$ are attached to sites 1 and 8 of an open chain. (b) $A$ and $B$ are attached to sites 1 and 4 of a ring.

Figure 5

(Color online) (a) Spin-spin correlation $⟨0_c|{\sigma }_{1x}{\sigma }_{jx}|0_c⟩$ for the ground state of the even-size chain with $N=10$. (b) The SC-RKKY interaction $J_{A,B}^{\ast }\left(1,j\right)$ normalized by $\left|J_{A,B}^{\ast }\left(1,1\right)\right|$. The labels “Exact” and “Approx.” correspond to Eqs. (7, 8), respectively.

Figure 6

(Color online) An engineered spin superlattice with ferromagnetic or antiferromagnetic couplings. Strongly coupled three spins (denoted by thick lines) in a dotted box act as a single effective spin. Couplings between effective spins and spins can be either ferromagnetic (“F,” red line) or antiferromagnetic (“A,” blue line), depending on the attachment site.