Effective cutting of a quantum spin chain by bond impurities

Spin chains are promising media for short-haul quantum communication. Their usefulness is manifested in all those situations where stationary information carriers are involved. In the majority of the communication schemes relying on quantum spin chains, the latter are assumed to be finite in length, with well-addressable end-chain spins. In this paper we propose that such a configuration could actually be achieved by a mechanism that is able to effectively cut a spin ring through the insertion of bond defects. We then show how suitable physical quantities can be identified as figures of merit for the effectiveness of the cut. We find that, even for modest strengths of the bond defect, a ring is effectively cut at the defect site. In turn, this has important effects on the amount of correlations shared by the spins across the resulting chain, which we study by means of a scattering-based mechanism of a clear physical interpretation.

Figure 1

A ring of interacting spin-$1/2$ particles, all coupled through an $XX$ model, includes a bond defect: While all spin pairs $(n,n+1)$ with $n\ne -1/2$ are mutually interacting with strength $J$, the pair $(-1/2,1/2)$ experiences the strength $j$. The spins are all subjected to a homogeneous magnetic field $h$.

Figure 2

Correlators $\langle {\widehat{\sigma }}_n^x{\widehat{\sigma }}_{n+1}^x\rangle$ (top) and $\langle {\widehat{\sigma }}_n^z{\widehat{\sigma }}_{n+1}^z\rangle$ (bottom) for $j=0,0.5,0.8,1.5,2,11$ (corresponding to increasing absolute values for $n+1/2$ even). The straight lines correspond to the correlators in the PBC case, $j=1$. For $j=11$ the data are indistinguishable from the OBC limit.

Figure 3

The nearest-neighbor correlation functions $\langle {\widehat{\sigma }}_n^x{\widehat{\sigma }}_{n+1}^x\rangle$ and $\langle {\widehat{\sigma }}_n^z{\widehat{\sigma }}_{n+1}^z\rangle$ corresponding to the second and third bond after the defect, vs $j$. The $xx$ ($zz$) correlators take positive (negative) values; their absolute value increases (decreases) with $j$ for $n=3/2$ ($n=5/2$). The dashed lines show that the third-bond correlators at $j\to \infty$ behave as the second-bond correlators of the open chain, i.e., $j=0$.

Figure 4

QD and CC (normalized with respect to the $j=1$ value) plotted vs $j$ for the two spins at sites $\pm 3/2$, i.e., sitting at opposite sides of the impurity. Cutting the chain affects both quantum and classical correlations in an essentially identical way. Inset: Log-log plot of QD and CC (here indistinguishable) vs $j$, showing that they obey a $j^{-2}$ scaling law, which is in fact independent of the distance of the sites.

Figure 5

Nearest-neighbor concurrence $C_{n,n+1}$ for $j=6$ vs the bond index $n+\frac{1}{2}$. Panels (a) and (b) are for $h=0.5,\frac{1}{\sqrt{2}}$ respectively. The straight dashed line shows the value of the concurrence at $j=1$. The magnetic field sets the periodicity of the one- and two-point spin correlators, which enter the concurrence, to $p=3,4$ respectively.

Figure 6

Fidelity $\mathcal{F}\left(\right|\Sigma \rangle ,\rho )$ between the reduced state $\rho$ of our model and the pure state $|\Sigma \rangle$ of a linear chain with the same number of spins by varying the coupling strength $j>1$ at different values of the magnetic field $h=0,\frac{1}{2},\frac{1}{\sqrt{2}},1$ [panels (a), (b), (c), and (d) respectively]. We have taken $M=10,100,1000$ in all panels. The dashed line shows the behavior of the function $1-1/j^2$, which matches the thermodynamic limit of the state fidelity at large magnetic fields.