Mimicking quantum correlation of a long-range Hamiltonian by finite-range interactions

The quantum long-range extended Ising model possesses several striking features which cannot be observed in the corresponding short-range model. We report that the pattern obtained from the entanglement between any two arbitrary sites of the long-range model can be mimicked by the model having a finite range of interactions provided the interaction strength is moderate. On the other hand, we illustrate that when the interactions are strong, the entanglement distribution in the long-range model does not match the class of a model with a few interactions. We also show that the monogamy score of entanglement is in good agreement with the behavior of pairwise entanglement. Specifically, it saturates when the entanglement in the finite-range Hamiltonian behaves similarly to the long-range model, while it decays algebraically otherwise.

Figure 1

Dispersion relation to prove criticality: ${\omega }_k$ (ordinate) vs $k$ (abscissa) at the critical point $h_c^1=2$ for (a) $\alpha =0.5$ and $h=2$ and (b) $\alpha =1.5$ and $h=2$ in the (a) quasilocal and (b) nonlocal regimes. Different lines indicate different values of $\mathcal{Z}$. The inset indicates the dependence of $k$ on the velocity of the quasiparticles for the LR model and a few-neighbor model. Here $N=256$ (throughout the paper, we consider $N=256$ unless stated otherwise). Both axes are dimensionless.

Figure 2

Dispersion relation to find the critical point: eigenfrequency ${\omega }_k$ (vertical axis) vs $h$ (horizontal axis) at $k=\pi$ for (a) $\alpha =0.5$ and (b) $\alpha =1.5$ in the (a) nonlocal and (b) quasilocal regimes. The value of $h$ for which ${\omega }_k$ becomes zero is the critical point $h_c^2$, in good agreement with Eq. (22). Different lines correspond to different values of $\mathcal{Z}$. Both axes are dimensionless.

Figure 3

Entanglement pattern in (a) the nonlocal regime for $\alpha =0.5$ and (b) the quasilocal regime for $\alpha =1.5$: variation of $E_r$ (ordinate) as a function of $r$ (abscissa). Different symbols represent different coordination number $\mathcal{Z}$. Note that $\mathcal{Z}=255$ indicates the fully connected model. Here $h=2.5$. In (b) the variation of $E_r$ with $r$ follows the law governed by $r^{\beta }$, where $\beta =\{-0.37,-0.43,-0.46,-0.48,-0.53\}$, such that $\beta \approx \alpha$ for the corresponding $\mathcal{Z}=\{5,10,15,20,255\}$. Both axes are dimensionless.

Figure 4

Entanglement trends of a fully connected model, mimicking a few-neighbor model. Two-qubit entanglement $E_r$ ($y$ axis) is plotted vs the variation of $r$ ($x$ axis) for different values of $(\mathcal{Z},h,\alpha )$. For a given $r$ value, equal values of entanglement for different sets of values are observed, which demonstrates that a few-neighbor model can mimic the entanglement of the ground state in the LR model efficiently by adjusting the system parameters accordingly. Both axes are dimensionless.

Figure 5

Effects of phases on entanglement. Two-qubit entanglement $E_r$ (ordinate) is plotted against $r$ (abscissa) in (a) the nonlocal regime for $\alpha =0.2$ and (b)the quasilocal regime for $\alpha =1.5$. Solid, dashed, and dotted lines correspond to different values of $h$. The model is fully connected, i.e., $\mathcal{Z}=N-1$. Counterintuitively, for $\alpha <1$ [in (a)], pairwise entanglement is higher for spins separated by a longer distance $r$ compared to that between nearby sites. Both axes are dimensionless.

Figure 6

Entanglement in the thermodynamic limit: entanglement between first and fifth spins $E_4$ (ordinate) with respect to the system size $N$ (abscissa). Here $\alpha =1.5$ is chosen in the quasilocal regime and $h=2.5$. Both axes are dimensionless.

Figure 7

Functional dependence of $E_r$ on $r$ in the nonlocal regime and disordered phase. Variation of $E_r$ (vertical axis) is plotted vs $r$ (horizontal axis) for different values of $\alpha$ in a fully connected model i.e., $\mathcal{Z}=N-1$, in (a) the ordered phase of the system for $h=1.5$ and (b) the disordered phase of the system for $h=2.9$, both depicting the same features. The solid lines are the numerical fits for the corresponding $\alpha$ values. For small range of $r$, entanglement scales as $r^{-\beta }$, where $\beta \approx \alpha$ in both phases except near the transition regime $\alpha \sim 1$, where entanglement is short ranged. In particular, the numerical fits are (a) $\{r^{-0.28},r^{-0.49},r^{-0.68}\}$ and (b) $\{r^{-0.22},r^{-0.42},r^{-0.62},r^{-0.92}\}$. Both axes are dimensionless.

Figure 8

Monogamy score as an indicator for the distribution of entanglement: monogamy score ${\delta }_{C^2}$ (ordinate) vs $\mathcal{Z}$ (abscissa). Different lines indicate different values of $h$ in (a) the nonlocal regime for $\alpha =0.5$ and (b) the quasilocal regime for $\alpha =1.5$. Both axes are dimensionless.

Figure 9

Monogamy score with coordination number: ${\delta }_{C^2}$ (vertical axis) vs $\mathcal{Z}$ (horizontal axis) for (a) $h=1.5$ and (b) $h=2.5$, i.e., two different phases of the system.. Different kinds of lines represent different values of $\alpha$. Both axes are dimensionless.