Renyi entropy of the XY spin chain

DOI10.1088/1751-8113/41/2/025302zbMATH Open1189.82024arXiv0707.2534OpenAlexW3100848448WikidataQ59408945 ScholiaQ59408945MaRDI QIDQ5440341

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Publication date: 4 February 2008

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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 $alpha$ of the density matrix of the block. We calculate the asymptotic for $Loinfty$ analytically in terms of Klein's elliptic $lambda$ - function. We study the limiting entropy as a function of its parameter $alpha$. We show that up to the trivial addition terms and multiplicative factors, and after a proper re-scaling, 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 $alphaoalph{a}^{-1}$ and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when $alpha$ is a integer power of 2, this includes the purity $trh{o}^2$. We also analyze the behavior of the entropy as $alphao0$ and $infty$ and at the critical magnetic field and in the isotropic limit [XX model].

Full work available at URL: https://arxiv.org/abs/0707.2534

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