Rényi entropy and parity oscillations of anisotropic spin-$s$ Heisenberg chains in a magnetic field

Using the density matrix renormalization group, we investigate the Rényi entropy of the anisotropic spin-$s$ Heisenberg chains in a $z$-magnetic field. We considered the half-odd-integer spin-$s$ chains, with $s=1/2$, 3/2, and $5/2$, and periodic and open boundary conditions. In the case of the spin-1/2 chain we were able to obtain accurate estimates of the new parity exponents $p_{\alpha }^{(p)}$ and $p_{\alpha }^{(o)}$ that gives the power-law decay of the oscillations of the $\alpha$-Rényi entropy for periodic and open boundary conditions, respectively. We confirm the relations of these exponents with the Luttinger parameter $K$, as proposed by Calabrese et al. [Phys. Rev. Lett. 104, 095701 (2010)]. Moreover, the predicted periodicity of the oscillating term was also observed for some nonzero values of the magnetization $m$. We show that for $s>1/2$ the amplitudes of the oscillations are quite small and get accurate estimates of $p_{\alpha }^{(p)}$ and $p_{\alpha }^{(o)}$ become a challenge. Although our estimates of the new universal exponents $p_{\alpha }^{(p)}$ and $p_{\alpha }^{(o)}$ for the spin-3/2 chain are not so accurate, they are consistent with the theoretical predictions.

Figure 1

The Rényi entropy $S_{\alpha }(L,l)$ as a function of $l$ for the spin-1/2 XXZ chain with $\Delta =\sqrt{2}/2$. (a) $L=96$, PBC, $m=0$ and three values of $\alpha$ (see legend). (b) $L=180$, OBC, $m=0$, and two values of $\alpha$ (see legend). We added 0.3 in the values of the entropy for $\alpha =2$ in order to see both data in the same figure. (c) $L=96$ with $m=1/6$, and $L=80$ with $m=3/10$. In both bases $\alpha =3$ and PBC (d) $L=192$ with $m=1/6$, and $L=196$ with $m=3/10$. In both bases $\alpha =3$ and OBC.

Figure 2

The Rényi entropy $S_{\alpha }(L,l)$ as a function of $l$ for the spin-1/2 XXZ chain for $\Delta >1$. (a) $L=96$, $\Delta =2$, PBC, and three values of $\alpha$ (see legend). (b) $S_3(L,l)$ for chains with PBC, $\Delta =2$, and two values of magnetization $m$ (see legend). (c) $S_3(L,l)$ for chains with OBC, $\Delta =1.2$ and two values of $m$ (see legend).

Figure 3

(a) The Rényi entropy $S_3(L,l)$ vs $l$ for the spin-3/2 and spin-5/2 chains of size $L=96$, and two values of $\Delta$ (see legend). (b) $S_3(L,l)$ vs $l$ for the spin-3/2 chain of size $L=96$ and $\Delta =0.9980$. The solid circles are the DMRG data and the other symbols are fits (see legend). (c) Estimates of $p_{\alpha }^{(p)}$ as function of $1/$$l^{\mathrm{disc}}$ for the spin-1/2 and spin-3/2 chains with PBC. The symbols are the numerical data and the dashed lines connect the fitted data (see text). (d) $D_{10}(L,l)$ vs $l$ for the spin-3/2 chains at $\Delta =0.9980$ and two values of the magnetization (see the legend).