Entanglement scaling in fermion chains with a localization-delocalization transition and inhomogeneous modulations

We study the scaling of logarithmic negativity between adjacent subsystems in critical fermion chains with various inhomogeneous modulations through numerically calculating its recently established lower and upper bounds. For random couplings, as well as for a relevant aperiodic modulation of the couplings, which induces an aperiodic singlet state, the bounds are found to increase logarithmically with the subsystem size, and both prefactors agree with the predicted values characterizing the corresponding asymptotic singlet state. For the marginal Fibonacci modulation, the prefactors in front of the logarithm are different for the lower and the upper bound and vary smoothly with the strength of the modulation. In the delocalized phase of the quasiperiodic Harper model, the scaling of the bounds of the logarithmic negativity and that of the entanglement entropy are compatible with the logarithmic scaling of the homogeneous chain. At the localization transition, the scaling of the above entanglement characteristics holds to be logarithmic, but the prefactors are significantly reduced compared to those of the translationally invariant case, roughly by the same factor.

Figure 1

Upper and lower bounds for the entanglement negativity in the homogeneous fermion chain. The subsystem sizes are up to $\ell =1024$ in the case of the upper bound and $\ell =500000$ in the case of the lower bound. The lower bound shows log-periodic oscillations; the borders of the periods are indicated by arrows.

Figure 2

(a) Lower (LB) and upper (UB) bound of the entanglement negativity in the random model, as a function of the system size $L$, for fixed ratios $\ell /L=1/8$ and $\ell /L=1/4$. (b) The corresponding effective prefactors defined in Eq. (26). The horizontal line indicates the asymptotic value $ln\left(2\right)/6=0.1155\cdots$ predicted by the SDRG method.

Figure 3

Lower and upper bound of the entanglement negativity for the RAM, as a function of the system size $L$. Inset: The ratios of the prefactors ${\kappa }_{u,l}\left(r\right)$ obtained for different values of $r$ and ${\kappa }^{\mathrm{RAM}}=c_{\mathrm{eff}}^{\mathrm{RAM}}/6=0.07432\cdots$ predicted by the SDRG method.

Figure 4

Prefactors of the lower and upper bounds as a function of the modulation strength $r$ for the Fibonacci modulation. The horizontal line indicates the singlet-state limit, ${lim}_{r\to 0}{c}_{\mathrm{eff}}^{\mathrm{FM}}\left(r\right)/6=0.1327...$. In the inset, the dependence of the upper (open symbols) and lower (solid symbols) bounds on $L$ for two different values of the modulation strength are shown. The straight line has a slope characteristic of the singlet-state limit.

Figure 5

Entanglement entropy of the Harper model. (a) Entanglement entropy in the extended phase $h=0.5$ (upper data with crosses) and in the critical point $h=1.0$ (lower data with circles) as a function of $ln\ell$ calculated in systems of different size $L=2F_n$. The straight lines have slopes 0.33 and 0.26. (b) Entanglement entropy plotted against $h$ for various system sizes. The system and subsystem sizes were chosen to be $L=2F_n$ and $\ell =F_{n-4}$, respectively. The inset shows the entanglement entropy in the localized phase as a function of the logarithm of the localization length $l_{\mathrm{loc}}=1/lnh$. The green line has a slope 0.26.

Figure 6

The upper and lower bounds of the entanglement negativity of the Harper model. (a) Dependence on the system size in the delocalized phase ($h=0.5$) and at the critical point ($h=1$). The slopes of fitted lines from top to bottom are 0.25 and 0.212. (b) Dependence on the localization length in the localized phase. The black line with slope 0.21 is drawn to guide the eye. The inset shows the upper and lower bounds as a function of $L$ for $h=1.5$.