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arXiv:0707.2534 · Journal of Physics A Mathematical and Theoretical

Renyi entropy of the XY spin chain

Fabio Franchini, Alexander Rudol'fovich Its, V. E. Korepin

Published Dec 19, 2007• 153 citations
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Abstract

We consider the one-dimensional XY quantum spin chain in a transverse magnetic field. We are interested in the Renyi entropy of a block of L neighboring spins at zero temperature on an infinite lattice. The Renyi entropy is essentially the trace of some power α of the density matrix of the block. We calculate the asymptotic for L → ∞ analytically in terms of Klein's elliptic λ-function. We study the limiting entropy as a function of its parameter α. We show that up to the trivial addition terms and multiplicative factors, and after a proper rescaling, the Renyi entropy is an automorphic function with respect to a certain subgroup of the modular group; moreover, the subgroup depends on whether the magnetic field is above or below its critical value. Using this fact, we derive the transformation properties of the Renyi entropy under the map α → α −1 and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when α is an integer power of 2; this includes the purity tr ρ 2 . We also analyze the behavior of the entropy as α → 0 and ∞ and at the critical magnetic field and in the isotropic limit (XX model).

DOI
10.1088/1751-8113/41/2/025302

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We consider the one-dimensional XY quantum spin chain in a transverse magnetic field. We are interested in the Renyi entropy of a block of L neighboring spins at zero temperature on an infinite lattice. The Renyi entropy is essentially the trace of some power α of the density matrix of the block. We calculate the asymptotic for L → ∞ analytically in terms of Klein's elliptic λ-function. We study the limiting entropy as a function of its parameter α. We show that up to the trivial addition terms and multiplicative factors, and after a proper rescaling, the Renyi entropy is an automorphic function with respect to a certain subgroup of the modular group; moreover, the subgroup depends on whether the magnetic field is above or below its critical value. Using this fact, we derive the transformation properties of the Renyi entropy under the map α → α −1 and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when α is an integer power of 2; this includes the purity tr ρ 2 . We also analyze the behavior of the entropy as α → 0 and ∞ and at the critical magnetic field and in the isotropic limit (XX model).
We consider the ground state of the XY model on an infinite chain at zero temperature. Following Bennett, Bernstein, Popescu, and Schumacher we use entropy of a sub-system as a measure of entanglement. Vidal, Latorre, Rico and Kitaev conjectured that von Neumann entropy of a large block of neighboring spins approaches a constant as the size of the block increases. We evaluated this limiting entropy as a function of anisotropy and transverse magnetic field. We used the methods based on integrable Fredholm operators and Riemann-Hilbert problem. The entropy is singular at phase transitions.
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.
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 ${p}_{\ensuremath{\alpha}}^{(p)}$ 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 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 ${p}_{\ensuremath{\alpha}}^{(p)}$ 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.
Entanglement in the ground state of the XY model on the infinite chain can be measured by the von Neumann entropy of a block of neighboring spins. We study a double scaling limit: the size of the block is much larger then 1 but much smaller then the length of the whole chain. In this limit, the entropy of the block approaches a constant. The limiting entropy is a function of the anisotropy and of the magnetic field. The entropy reaches minima at product states and increases boundlessly at phase transitions.
The spin-1/2 quantum anisotropic XY spin chain in a transverse random magnetic field parallel to the z axis is numerically studied by means of the density-matrix renormalization group. The dependence of the spontaneous magnetization and the energy gap on both the strength of the random magnetic field and the factor of anisotropy is determined. The critical line for the order-disorder phase transition is obtained and the resulting phase diagram is drawn. Our results are compatible with the fact that models with different factors of anisotropy fall within the same universality class as the quantum Ising model in a transverse random field.
We consider the Rényi entropies in the one-dimensional spin-1/2 Heisenberg XX chain in a magnetic field. The case n = 1 corresponds to the von Neumann ‘entanglement’ entropy. Using a combination of methods based on the generalized Fisher–Hartwig conjecture and a recurrence relation connected to the Painlevé VI differential equation we obtain the asymptotic behaviour, accurate to order , of the Rényi entropies for large block lengths . For n = 1, 2, 3, 10 this constitutes the 3, 6, 10, 48 leading terms respectively. The o(1) contributions are found to exhibit a rich structure of oscillatory behaviour, which we analyse in some detail both for finite n and in the limit .