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arXiv:1103.2103 · Physical Review B

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

J. C. Xavier, Francisco Castilho Alcaraz

Published Jun 29, 2011• 39 citations
arXivHTMLPDF

Abstract

Using the density matrix renormalization group, we investigate the R\'enyi 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 ${p}_{\ensuremath{\alpha}}^{(p)}$ and ${p}_{\ensuremath{\alpha}}^{(o)}$ that gives the power-law decay of the oscillations of the $\ensuremath{\alpha}$-R\'enyi 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 ${p}_{\ensuremath{\alpha}}^{(p)}$ and ${p}_{\ensuremath{\alpha}}^{(o)}$ become a challenge. Although our estimates of the new universal exponents ${p}_{\ensuremath{\alpha}}^{(p)}$ and ${p}_{\ensuremath{\alpha}}^{(o)}$ for the spin-3/2 chain are not so accurate, they are consistent with the theoretical predictions.

DOI
10.1103/physrevb.83.214425

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Using the density matrix renormalization group, we investigate the R'enyi entropy of the anisotropic spin-sss Heisenberg chains in a zzz-magnetic field. We considered the half-odd-integer spin-sss chains, with s=1/2s=1/2s=1/2, 3/2, and 5/25/25/2, 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\ensuremathα(p){p}_{\ensuremath{\alpha}}^{(p)}p\ensuremathα(p)​ and p\ensuremathα(o){p}_{\ensuremath{\alpha}}^{(o)}p\ensuremathα(o)​ that gives the power-law decay of the oscillations of the \ensuremathα\ensuremath{\alpha}\ensuremathα-R'enyi entropy for periodic and open boundary conditions, respectively. We confirm the relations of these exponents with the Luttinger parameter KKK, 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 mmm. We show that for s>1/2s>1/2s>1/2 the amplitudes of the oscillations are quite small and get accurate estimates of p\ensuremathα(p){p}_{\ensuremath{\alpha}}^{(p)}p\ensuremathα(p)​ and p\ensuremathα(o){p}_{\ensuremath{\alpha}}^{(o)}p\ensuremathα(o)​ become a challenge. Although our estimates of the new universal exponents p\ensuremathα(p){p}_{\ensuremath{\alpha}}^{(p)}p\ensuremathα(p)​ and p\ensuremathα(o){p}_{\ensuremath{\alpha}}^{(o)}p\ensuremathα(o)​ for the spin-3/2 chain are not so accurate, they are consistent with the theoretical predictions.
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Abstract We introduce a family of inhomogeneous XX spin chains whose squared couplings are a polynomial of degree at most four in the site index. We show how to obtain an asymptotic approximation for the Rényi entanglement entropy of all such chains in a constant magnetic field at half filling by exploiting their connection with the conformal field theory of a massless Dirac fermion in a suitably curved static background. We study the above approximation for three particular chains in the family, two of them related to well-known quasi-exactly solvable quantum models on the line and the third one to classical Krawtchouk polynomials, finding an excellent agreement with the exact value obtained numerically when the Rényi parameter α is less than one. When α ≥ 1 we find parity oscillations, as expected from the homogeneous case, and show that they are very accurately reproduced by a modification of the Fagotti-Calabrese formula. We have also analyzed the asymptotic behavior of the Rényi entanglement entropy in the non-standard situation of arbitrary filling and/or inhomogeneous magnetic field. Our numerical results show that in this case a block of spins at each end of the chain becomes disentangled from the rest. Moreover, the asymptotic approximation for the case of half filling and constant magnetic field, when suitably rescaled to the region of non-vanishing entropy, provides a rough approximation to the entanglement entropy also in this general case.
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