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arXiv:quant-ph/0412118 · Physical Review A

Entanglement in theXXspin chain with an energy current

Viktor Eisler, Zoltán Zimborás

Published Apr 11, 2005• 58 citations
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Abstract

We consider the ground state of an XX chain that is constrained to carry a current of energy. The von Neumann entropy of a block of $L$ neighboring spins, describing entanglement of the block with the rest of the chain, is computed. Recent calculations have revealed that the entropy in the XX model diverges logarithmically with the size of the subsystem. We show that the presence of the energy current increases the prefactor of the logarithmic growth. This result indicates that the emergence of the energy current gives rise to an increase of entanglement.

DOI
10.1103/physreva.71.042318

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We consider the ground state of an XX chain that is constrained to carry a current of energy. The von Neumann entropy of a block of LLL neighboring spins, describing entanglement of the block with the rest of the chain, is computed. Recent calculations have revealed that the entropy in the XX model diverges logarithmically with the size of the subsystem. We show that the presence of the energy current increases the prefactor of the logarithmic growth. This result indicates that the emergence of the energy current gives rise to an increase of entanglement.
It is known that the entropy of a block of spins of size L embedded in an infinite pure critical spin chain diverges as the logarithm of L with a prefactor fixed by the central charge of the correspondent conformal field theory. For a class of strongly random spin chains, it has been shown that the correspondent block entropy still remains universal and diverges logarithmically with an ‘effective’ central charge. By computing the entanglement entropy for a family of models which includes the N -states random Potts chain and the Z N clock model, we give some definitive answer to recent conjectures about the behaviour of this effective central charge. In particular, we show that the ratio between the entanglement entropy in the pure and in the disordered system is model dependent and we provide a series of critical models where the entanglement entropy grows from the pure to the random case.
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.
Recent studies have shown that logarithmic divergence of entanglement entropy as a function of the size of a subsystem is a signature of criticality in quantum models. We demonstrate that the ground-state entanglement entropy of nnn sites for the ferromagnetic Heisenberg spin-12\frac{1}{2}21​ chain of the length LLL in a sector with fixed magnetization yyy per site grows as $\frac{1}{2}{\mathrm{log}}_{2}[n(L\ensuremath{-}n)∕L]C(y)$, where $C(y)=2\mathrm{\ensuremath{\pi}}e(\frac{1}{4}\ensuremath{-}{y}^{2})$.
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.