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arXiv:quant-ph/0606240

Analysis of entropy of XY Spin Chain

Fabio Franchini, Alexander Rudol'fovich Its, Bai-Qi Jin, V. E. Korepin

Published Jun 29, 2006• 5 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0606240

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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 compute the entropy of entanglement of two blocks of LLL spins at a distance ddd in the ground state of an Ising chain in an external transverse magnetic field. We numerically study the von Neumann entropy for different values of the transverse field. At the critical point we obtain analytical results for blocks of size L=1L=1L=1 and 2. In the general case, the critical entropy is shown to be additive when $d\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally, based on simple arguments, we derive an expression for the entropy at the critical point as a function of both LLL and ddd. This formula is in excellent agreement with numerical results.