Finite-size corrections of the entanglement entropy of critical quantum chains

Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement $\alpha$-Rényi entropy of a single interval for several critical quantum chains. We considered models with $U\left(1\right)$ symmetry such as the spin-1/2 $XXZ$ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of $\alpha$, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the $\alpha$-Rényi entropies. We conjecture that the exponent of the leading finite-size correction of the $\alpha$-Rényi entropies is $p_{\alpha }=2X_{\epsilon }/\alpha$ for $\alpha >1$ and $p_1=\nu$, where $X_{\epsilon }$ denotes the dimensions of the energy operator of the model and $\nu =2$ for all the models.

Figure 1

Results of the Rényi entropy $S_{\alpha }(L,\ell )$ and of the function $D_{\alpha }(L,\ell )$ for the Ising and Blume-Capel models with PBC and size $L=96$. (a) $S_{\alpha }(L,\ell )$ vs $\ell$ for two values of $\alpha$ and some values of coupling (see legend). The symbols are the numerical data and the solid lines connect the fitted points using Eq. (3). (b) $D_{\alpha }(L,\ell )$ vs $\ell$ for $\alpha =3$. We added 0.03 in the values of the Ising entropy in order to see better both data in the same figure.

Figure 2

Rényi entropy $S_{\alpha }(L,\ell )$ and the difference $D_{\alpha }(L,\ell )$ for the 3SPM with lattice size $L=96$. The symbols are the numerical data and the solid lines connect the fitted points (see text). (a) $S_{\alpha }(L,\ell )$ vs $\ell$ for $\lambda =1$, and some values of $\alpha$ (see legend). (b) $D_{\alpha }(L,\ell )$ vs $\ell$ for $\lambda =1$, $\alpha =2$, and $\alpha =3$.

Figure 3

Results of the Rényi entropy $S_{\alpha }(L,\ell )$ and the difference $D_{\alpha }(L,\ell )$ for the spin-1 Fateev-Zamolodchikov quantum chain with PBC and size $L=72$. The symbols are the numerical data and the solid lines connect the fitted points (see text). (a) $S_{\alpha }(L,\ell )$ vs $\ell$ for $\epsilon =1$, $\delta =0.5$, and some values of $\alpha$ (see legend). (b) Same as (a) but for $\epsilon =-1$. (c) $D_{\alpha }(L,\ell )$ vs $\ell$ for $\epsilon =1$, $\alpha =2$, and $\alpha =3$ at $\delta =0.5$.