Entanglement measures are useful tools in characterizing otherwise unknown quantum phases and indicating transitions between them. Here we examine the concurrence and entanglement entropy in quantum spin chains with random long-range couplings, spatially decaying with a power-law exponent $\alpha$. Using the strong disorder renormalization group (SDRG) technique, we find by analytical solution of the master equation a strong disorder fixed point, characterized by a fixed point distribution of the couplings with a finite dynamical exponent, which describes the system consistently in the regime $\alpha >\frac{1}{2}$. A numerical implementation of the SDRG method yields a power-law spatial decay of the average concurrence, which is also confirmed by exact numerical diagonalization. However, we find that the lowest-order SDRG approach is not sufficient to obtain the typical value of the concurrence. We therefore implement a correction scheme which allows us to obtain the leading-order corrections to the random singlet state. This approach yields a power-law spatial decay of the typical value of the concurrence, which we derive both by a numerical implementation of the corrections and by analytics. Next, using numerical SDRG, the entanglement entropy (EE) is found to be logarithmically enhanced for all $\alpha$, corresponding to a critical behavior with an effective central charge $c=\ln \left(2\right)$, independent of $\alpha$. This is confirmed by an analytical derivation. Using numerical exact diagonalization (ED), we confirm the logarithmic enhancement of the EE and a weak dependence on $\alpha$. For a wide range of partition size $l$, the EE fits a critical behavior with a central charge close to $c=1$, which is the same as for the clean Haldane-Shastry model with a power-law-decaying interaction with $\alpha =2$. Only for small $l\ll L$, in a range which increases with the number of spins $N$, we find deviations which are rather consistent with the strong disorder fixed point central charge $c=\ln \left(2\right)$. Furthermore, we find using ED that the concurrence shows power-law decay, albeit with smaller power exponents than obtained by SDRG. We also present results obtained with DMRG and find agreement with ED for sufficiently small $\alpha <2$, whereas for larger $\alpha$ DMRG tends to underestimate the entanglement entropy and finds a faster decaying concurrence.

中文翻译：

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具有长距离反铁磁相互作用的无序量子自旋链的纠缠性质

纠缠措施是有用的工具，可用于表征否则为未知的量子相并指示它们之间的过渡。在这里，我们研究了量子自旋链中具有随机长距离耦合，并随幂律指数在空间上衰减的并发和纠缠熵。$\alpha$。通过使用强无序重归一化组（SDRG）技术，我们通过主方程的解析解找到了一个强无序固定点，其特征是具有有限动态指数的联轴器的不动点分布，这在系统范围内一致地描述了系统$\alpha >\frac{\mathrm{1个}}{2}$。SDRG方法的数字实现产生平均并发的幂律空间衰减，这也由精确的数字对角线确定。但是，我们发现最低阶SDRG方法不足以获取并发的典型值。因此，我们实现了一种校正方案，该方案允许我们获得对随机单重态的前导校正。这种方法产生并发典型值的幂律空间衰减，我们通过校正的数值实现和分析得出该衰减。接下来，使用数值SDRG，发现对所有熵的纠缠熵（EE）$\alpha$，对应于具有有效中央电荷的临界行为 $C=\ln （2）$， 独立于 $\alpha$。通过分析推导证实了这一点。使用数值精确对角线化（ED），我们确认EE的对数增强和对$\alpha$。适用于各种分区大小$升$，EE适合关键行为，其中心电荷接近 $C=\mathrm{1个}$，这与具有幂律衰减作用的干净Haldane-Shastry模型相同 $\alpha =2$。只为小$升\ll \mathrm{大号}$，在随旋转数增加的范围内 $ñ$，我们发现与强无序定点中心电荷相当一致的偏差 $C=\ln （2）$。此外，我们发现使用ED并发会显示幂律衰减，尽管功率指数比SDRG获得的功率指数小。我们还介绍了用DMRG获得的结果，并发现与ED的协议足够小$\alpha <2$，而较大的 $\alpha$ DMRG倾向于低估纠缠熵，并发现更快的衰减并发。