Entanglement at a Two-Dimensional Quantum Critical Point: A Numerical Linked-Cluster Expansion Study

We develop a method to calculate the bipartite entanglement entropy of quantum models, in the thermodynamic limit, using a numerical linked-cluster expansion (NLCE) involving only rectangular clusters. It is based on exact diagonalization of all $n\times m$ rectangular clusters at the interface between entangled subsystems $A$ and $B$. We use it to obtain the Renyi entanglement entropy of the two-dimensional transverse field Ising model, for arbitrary real Renyi index $\alpha$. Extrapolating these results as a function of the order of the calculation, we obtain universal pieces of the entanglement entropy associated with lines and corners at the quantum critical point. They show NLCE to be one of the few methods capable of accurately calculating universal properties of arbitrary Renyi entropies at higher dimensional critical points.

Figure 1

(a) A sample of cluster shapes and sizes, as used in the rectangular NLCE. (b) To calculate the Renyi entropies of a given cluster, one considers all divisions into subregions $A$ and $B$ defined by translations perpendicular to the line (illustrated) or corner.

Figure 2

The corner term $c_2$ and line term $s_2$ as a function of $h/J$ for the second Renyi entanglement entropy using both high field and low field NLCE of orders 8, 12, 16, and 24. The dotted line denotes $h_c/J=3.044$ and the dashed line shows the series expansion results [20].

Figure 3

The coefficient $-a_{\alpha }$ from fitting $c_{\alpha }$ to the function $a_{\alpha }\log \sqrt{\mathcal{O}}+b_{\alpha }$ (purple dots) plotted along with other numerical estimates of $-a_{\alpha }$ from free scalar field theory [4] (green diamonds), tensor tree network [24] (red star), finite-T QMC [22] (yellow square), and series expansion [20, 21] (black circle). Standard error from the fit is shown for integer values of $\alpha$. Inset: $c_{\alpha }$ for $\alpha =1$, 2, 10 along with fits plotted vs $1/\sqrt{\mathcal{O}}$ on a logarithmic scale.

Figure 4

(a) Fits of the NLCE line term $s_{\alpha }$ to Eq. (5) for $\alpha =1$ from NLCE for $h=3.048$ (top), 3.075, and 3.1 (bottom). (b) The resulting linear term $A(h)$ from the fits in inset (a), fit to Eq. (6). (c) The normalized coefficient ${\overline{r}}_{\alpha }$ as a function of $\alpha$, with standard error from the fit shown for $\alpha$.