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arXiv:quant-ph/0606178

Entropy of XY Spin Chain and Block Toeplitz Determinants

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

Published Jun 21, 2006• 23 citations
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Abstract

We consider entanglement in the ground state of the XY spin model on infinite chain. We use von Neumann entropy of a sub-system as a measure of entanglement. The entropy of a large block of neighboring spins approaches a constant as the size of the block increases. We prove rigorously expression for limiting entropy which was published before. We observe that the entropy reaches minimum at product states but increases boundlessly at phase transitions.

DOI
10.48550/arxiv.quant-ph/0606178

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We consider entanglement in the ground state of the XY spin model on infinite chain. We use von Neumann entropy of a sub-system as a measure of entanglement. The entropy of a large block of neighboring spins approaches a constant as the size of the block increases. We prove rigorously expression for limiting entropy which was published before. We observe that the entropy reaches minimum at product states but increases boundlessly at phase transitions.
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.
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.
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 neighbouring spins. We study a double scaling limit: the size of the block is much larger than 1 but much smaller than the length of the whole chain. The entropy of the block has an asymptotic limit in the gapped regimes. We study this limiting entropy as a function of the anisotropy and of the magnetic field. We identify its minima at product states and its divergencies at the quantum phase transitions. We find that the curves of constant entropy are ellipses and hyperbolas, and that they all meet at one point ( essential critical point ). Depending on the approach to the essential critical point, the entropy can take any value between 0 and ∞. In the vicinity of this point, small changes in the parameters cause large change of the entropy.
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.