Critical entanglement of $XXZ$ Heisenberg chains with defects

We study the entanglement properties of anisotropic open spin one-half Heisenberg chains with a modified central bond. The entanglement entropy between the two half-chains is calculated with the density-matrix renormalization method (DMRG). We find a logarithmic behavior with an effective central charge $c^{\prime }$ varying with the length of the system. It flows to one in the ferromagnetic region and to zero in the antiferromagnetic region of the model. In the $XX$ case it has a nonuniversal limit and we recover previous results.

Figure 1

Comparison between DMRG and correlation function results for the $XX$ case and $J_{\mathit{imp}}=0.2$, 0.5 and $1.0∕1.1=0.9091$. Upper panel: Difference in the entanglement entropy $S$. Lower panel: Difference in the effective central charge $c^{\prime }$.

Figure 2

Spectrum of the reduced density matrix for $XXZ$ chains with various anisotropies. Upper figure: pure chains. Lower figure: defect strength $J_{\mathit{imp}}=0.5$. Please note the logarithmic scale. The insets show the first three eigenvalues.

Figure 3

Spectrum of the reduced density matrix for $J_{\mathit{imp}}=0.5$ and various sizes $L$. Upper figure: $\Delta =-0.9$. Lower figure: $\Delta =+0.9$. The insets show the first three eigenvalues.

Figure 4

(Color online) Entanglement entropy as a function of the system size for $J_{\mathit{imp}}=0.5$ and seven values of the anisotropy. The data correspond to $\Delta =-0.9;-0.5;-0.1;0;0.1;0.5;0.9$, from top to bottom.

Figure 5

(Color online) Effective central charge for $J_{\mathit{imp}}=0.5$. Upper figure: as a function of the system size for $\Delta =-0.5;-0.9;-0.1;0;0.1;0.5;0.9$, from top to bottom. Lower figure: as a function of the anisotropy. The step function is expected at $L=\infty$ and included as a guide for the eye.

Figure 6

(Color online) The same as Fig. 5, but for $J_{\mathit{imp}}=0.2$.

Figure 7

(Color online) The same as Fig. 5, but for $J_{\mathit{imp}}=1∕1.1=0.9091$. Please note the different vertical scale in the upper figure.

Figure 8

(Color online) Defect contribution $\delta S$ to the entanglement entropy for $J_{\mathit{imp}}=1∕1.1$. Upper figure: $\delta S$ as a function of $L$ for $\Delta =-0.9;-0.5;-0.1;0;0.1;0.5;0.9$, from top to bottom. Lower figure: Fit of the data (circles) for $\Delta =0.5$ according to Eq. (7), see text.