Boundary Effects in the Critical Scaling of Entanglement Entropy in 1D Systems

We present exact diagonalization and density matrix renormalization group results for the entanglement entropy of critical spin-$1/2$ $XXZ$ chains. We find that open boundary conditions induce an alternating term in both the energy density and the entanglement entropy which are approximately proportional, decaying away from the boundary with a power law. The power varies with anisotropy along the $XXZ$ critical line and is corrected by a logarithmic factor, which we calculate analytically, at the isotropic point. A heuristic resonating valence bond explanation is suggested.

Figure 1

$E_A(L/2,L)$ computed by DMRG (keeping $m=512$ states) at the SU(2) point up to $L=1000$ for the nearest neighbor model $J_2/J_1=0$ (○) and at the critical point $J_2/J_1=0.241167$ (□) (a). Full lines are fits, indicated on the main panel. Zooms on large $L$ regions are shown in (b) and (c): (b) absence of log corrections for $J_2/J_1=0.241167$; (c) excellent agreement between Eq. (7) and DMRG.

Figure 2

A particular valence bond state. The number of valence bonds crossing each link, $N(x)$, is given.

Figure 3

Results from exact diagonalization of $XX$ chains with $L=2000$ spins $1/2$. EE $\mathcal{S}(x,2000)$, plotted vs the subsystem size $x$ for PBC (upper symbols) and OBC (lower symbols); the insets shows zooms around the chain center. The full lines are fits: Eq. (10) with $s_1=0.726067$ for the PBC case whereas in the OBC case, some uniform and staggered terms $[0.055-(-1{)}^x]/\frac{L}{\pi }\sin (\frac{\pi x}{L})$ have been added to Eq. (1).

Figure 4

Linear behavior of the alternating part of the EE, ${\mathcal{S}}_A$ as a function of the alternating energy density, $-E_A$, both computed using DMRG on critical open $XXZ$ chains of size $200\le L\le 1000$ for a few values of the anisotropy $\Delta$. Data from ED at $\Delta =0$ are also shown for $L=2000$. Dashed lines are linear fits of the form ${\mathcal{S}}_A=-\alpha E_A$ (see text). The inset (a) is a zoom close to 0, showing data for $\Delta =-3/4$ and $-1/2$. The inset (b) shows the prefactor $\alpha$ vs $\Delta$ extracted from the numerical data (circles), for a larger set of values of $\Delta$, which is compared with $\pi /2v=\mu /\sin \mu$, $\mu ={cos}^{-1}\Delta$.

Figure 5

Comparison between the alternating part of EE ${\mathcal{S}}_A$ and $E_A$ [Eq. (3)] from DMRG with $m=512$ states, for Heisenberg models. Power law decay of ${\mathcal{S}}_A(L/2,L)+\alpha E_A(L/2,L)$ drawn in a log-log plot, with $\alpha =1$ for the nearest neighbor chain ($J_2=0$) and $\alpha =1.00169$ at the critical second neighbor coupling $J_2=0.241167$. Lines are power law fits: $\sim L^{-2.56}$ for $J_2=0.241167$ and $L^{-2.59}$ for $J_2=0$.