Logarithmic Terms in Entanglement Entropies of 2D Quantum Critical Points and Shannon Entropies of Spin Chains

Universal logarithmic terms in the entanglement entropy appear at quantum critical points (QCPs) in one dimension (1D) and have been predicted in 2D at QCPs described by 2D conformal field theories. The entanglement entropy in a strip geometry at such QCPs can be obtained via the “Shannon entropy” of a 1D spin chain with open boundary conditions. The Shannon entropy of the $XXZ$ chain is found to have a logarithmic term that implies, for the QCP of the square-lattice quantum dimer model, a logarithm with universal coefficient $\pm 0.25$. However, the logarithm in the Shannon entropy of the transverse-field Ising model, which corresponds to entanglement in the 2D Ising conformal QCP, is found to have a singular dependence on the replica or Rényi index resulting from flows to different boundary conditions at the entanglement cut.

Figure 1

(a) The entanglement spectrum of the CQCP, $p_{\sigma }$, in a strip geometry is in one-to-one correspondence with field configurations $\{\sigma \}$ on the entanglement cut. These probabilities can be alternately understood as (b) partition functions for a constrained CFT in the strip geometry or (c) ground state amplitudes of a quantum chain.

Figure 2

(a) The logarithmic part of $F(n)$ at $\Delta =-1/2$. Data are included both without ($h=0$) and with ($h>0$) an edge magnetic field. $F(n,L)$ was computed for $L=2,4,\dots ,30$. Windows of 6 data points centered at $\ell$, $L=\{\ell -5,\dots ,\ell +5\}$, were fitted to the form $f_{1}L+\gamma \log (L)+f_0+f_{-1}{L}^{-1}$. The data displayed include $\gamma$ for $\ell =\{9,23\}$. The approach of $\gamma (n)$ to the dashed lines suggests convergence to $\gamma (n)=\mp \frac{1}{4}(n-1)$ as $\ell \to \infty$, for $h=0$ and $h>0$, respectively. The inclusion of a $L^{-1}$ term accelerates convergence, but the value of $\gamma (n)$ remains insensitive to within a few percent to the choice of fitting form for terms that vanish as $L\to \infty$, as discussed in Ref. 16. (b) The logarithmic contribution to the von Neumann entropy $S(L)$, as a function of the fitting window center $\ell$. As $\ell$ increases, the logarithm’s coefficient converges to $\mp 1/4$, depending on the boundary conditions. Successive lines are for $\Delta =\{-1/2,0,1/2\}$.

Figure 3

The logarithmic part of $F(n)$ for the transverse-field Ising model. $F(n,L)$ was computed for $L=2,4,\dots ,40$, and $\gamma (n)$ extracted by the same procedure as for the $XXZ$ model. The dashed lines are $\gamma (n)=-\frac{1}{8}n$ and $\gamma (n)=\frac{3}{8}n$. In the inset, the data are collapsed by plotting $\gamma (n)/n$ as a function of $(n-1){\gamma }^{\prime }(1)$.