Parity Effects in the Scaling of Block Entanglement in Gapless Spin Chains

We consider the Rényi $\alpha$ entropies for Luttinger liquids (LL). For large block lengths $\ell$, these are known to grow like $\ln \ell$. We show that there are subleading terms that oscillate with frequency $2k_F$ (the Fermi wave number of the LL) and exhibit a universal power-law decay with $\ell$. The new critical exponent is equal to $K/(2\alpha )$, where $K$ is the LL parameter. We present numerical results for the anisotropic XXZ model and the full analytic solution for the free fermion (XX) point.

Figure 1

Parity effects in Rényi entropies in the XXZ model at $\Delta =-1/2$. $S_{\alpha }(\ell ,L)$ for several $L$ and $\alpha =1,2,\infty$. The straight lines indicate the asymptotic slopes $(1+{\alpha }^{-1})/6$.

Figure 2

Corrections to scaling $d_{\alpha }(\ell )=S_{\alpha }(\ell )-S_{\alpha }^{\mathrm{CFT}}(\ell )$ for the XX model and four different values of $\alpha$. Red circles and black squares correspond to even and odd $\ell$, respectively.

Figure 3

Corrections to scaling in the finite length XX model for odd $L$. $F_2(x)$ and $F_{\infty }(x)$ ($x=\ell /L$) obtained from 69 different systems with lengths in the range $17\le L\le 4623$ exhibit perfect data collapse.

Figure 4

Left: The additive constant in the Rényi entropies $c_{\alpha }^{\prime }$ (bottom) and the multiplicative constant $c_{\alpha }$ for the moments $\mathrm{Tr}{\rho }_A^{\alpha }$ (top in log scale) as function of $\alpha$ at fixed $\Delta$. The drawn lines are the exact values for $\Delta =0$. Right: $c_{\alpha }^{\prime }$ as function of $\Delta$ at fixed $\alpha$.

Figure 5

Universal scaling function $f_{\alpha }{F}_{\alpha }(x)$ (where $x=\ell /L$) for the XXZ chain for several values of $\alpha$ and $\Delta$. Left: Several $\alpha$ and $\Delta$ for odd $L$. Right: even $L$. In the latter case, $F_{\alpha }(x)$ is practically independent on $x$.