arXiv:1110.3770 · Europhysics Letters (EPL)

New insights into the entanglement of disjoint blocks

Maurizio Fagotti

Published Jan 1, 2012• 33 citations

Abstract

We study the entanglement of two disjoint blocks in spin- chains obtained by merging solvable models, such as XX and quantum Ising models. We focus on the universal quantities that can be extracted from the Rényi entropies S α . The most important information is encoded in some functions denoted by F α . We compute F 2 and we show that F α − 1 and F v . N . , corresponding to the von Neumann entropy, can be negative, in contrast to what observed in all models examined so far. An exact relation between the entanglement of disjoint subsystems in the XX model and that in a chain embodying two quantum Ising models is a by-product of our investigations.

DOI: 10.1209/0295-5075/97/17007

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arXiv:1110.3770 · Europhysics Letters (EPL)

New insights into the entanglement of disjoint blocks

Maurizio Fagotti

Published Jan 1, 2012• 33 citations

arXiv HTML PDF

Abstract

DOI: 10.1209/0295-5075/97/17007

Related papers

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Maurizio Fagotti, Pasquale Calabrese

We study the Rényi entanglement entropies of two disjoint intervals in XY chains. We exploit the exact solution of the model in terms of free Majorana fermions and we show how to construct the reduced density matrix in the spin variables by taking the Jordan–Wigner string between the two blocks properly into account. From this we can evaluate any Rényi entropy of finite integer order. We study in detail critical XX and Ising chains and we show that the asymptotic results for large blocks agree with recent conformal field theory predictions if corrections to the scaling are included in the analysis correctly. We also report results for the gapped phase and after a quantum quench.We study the Rényi entanglement entropies of two disjoint intervals in XY chains. We exploit the exact solution of the model in terms of free Majorana fermions and we show how to construct the reduced density matrix in the spin variables by taking the Jordan–Wigner string between the two blocks properly into account. From this we can evaluate any Rényi entropy of finite integer order. We study in detail critical XX and Ising chains and we show that the asymptotic results for large blocks agree with recent conformal field theory predictions if corrections to the scaling are included in the analysis correctly. We also report results for the gapped phase and after a quantum quench.

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Quantifying entanglement of multiple subsystems is a challenging open problem in interacting quantum systems. Here, we focus on two subsystems of length $\ensuremath{\ell}$ separated by a distance $r=\ensuremath{\alpha}\ensuremath{\ell}$ and quantify their entanglement negativity ( $\mathcal{E}$ ) and mutual information ( $\mathcal{I}$ ) in critical random Ising chains. We find universal constant $\mathcal{E}(\ensuremath{\alpha})$ and $\mathcal{I}(\ensuremath{\alpha})$ over any distances, using the asymptotically exact strong disorder renormalization group method. Our results are qualitatively different from both those in the clean Ising model and random spin chains of a singlet ground state, like the spin- $\frac{1}{2}$ random Heisenberg chain and the random XX chain. While for random singlet states $\mathcal{I}(\ensuremath{\alpha})/\mathcal{E}(\ensuremath{\alpha})=2$, in the random Ising chain this universal ratio is strongly $\ensuremath{\alpha}$ dependent. This deviation between systems contrasts with the behavior of the entanglement entropy of a single subsystem, for which the various random critical chains and clean models give the same qualitative behavior. The reason is that $\mathcal{E}$ and $\mathcal{I}$ are sensitive to higher order correlations in the ground-state structure. Therefore, studying multipartite entanglement provides additional universal information in random quantum systems, beyond what we can learn from a single subsystem.Quantifying entanglement of multiple subsystems is a challenging open problem in interacting quantum systems. Here, we focus on two subsystems of length $\ensuremath{\ell}$ separated by a distance $r=\ensuremath{\alpha}\ensuremath{\ell}$ and quantify their entanglement negativity ($\mathcal{E}$) and mutual information ($\mathcal{I}$) in critical random Ising chains. We find universal constant $\mathcal{E}(\ensuremath{\alpha})$ and $\mathcal{I}(\ensuremath{\alpha})$ over any distances, using the asymptotically exact strong disorder renormalization group method. Our results are qualitatively different from both those in the clean Ising model and random spin chains of a singlet ground state, like the spin-$\frac{1}{2}$ random Heisenberg chain and the random XX chain. While for random singlet states $\mathcal{I}(\ensuremath{\alpha})/\mathcal{E}(\ensuremath{\alpha})=2$, in the random Ising chain this universal ratio is strongly $\ensuremath{\alpha}$ dependent. This deviation between systems contrasts with the behavior of the entanglement entropy of a single subsystem, for which the various random critical chains and clean models give the same qualitative behavior. The reason is that $\mathcal{E}$ and $\mathcal{I}$ are sensitive to higher order correlations in the ground-state structure. Therefore, studying multipartite entanglement provides additional universal information in random quantum systems, beyond what we can learn from a single subsystem.

arXiv:2205.14306Physical review. B./Physical review. BAug 2, 20223 citations

A bstract We take the paradigm of interacting spin chains, the Heisenberg spin-

\frac{1}{2}

1 2 XXZ model, as a reference system and consider interacting models that are related to it by Jordan-Wigner transformations and restrictions to sub-chains. An example is the fermionic analogue of the gapless XXZ Hamiltonian, which, in a continuum scaling limit, is described by the massless Thirring model. We work out the Rényi- α entropies of disjoint blocks in the ground state and extract the universal scaling functions describing the Rényi- α tripartite information in the limit of infinite lengths. We consider also the von Neumann entropy, but only in the limit of large distance. We show how to use the entropies of spin blocks to unveil the spin structures of the underlying massless Thirring model. Finally, we speculate about the tripartite information after global quenches and conjecture its asymptotic behaviour in the limit of infinite time and small quench. The resulting conjecture for the “residual tripartite information”, which corresponds to the limit in which the intervals’ lengths are infinitely larger than their (large) distance, supports the claim of universality recently made studying noninteracting spin chains. Our mild assumptions imply that the residual tripartite information after a small quench of the anisotropy in the gapless phase of XXZ is equal to − log 2.

XX

model.

\mathcal{E}

) and mutual information (

\mathcal{I}

) in critical random Ising chains. We find universal constant $\mathcal{E}(\ensuremath{\alpha})$ and $\mathcal{I}(\ensuremath{\alpha})$ over any distances, using the asymptotically exact strong disorder renormalization group method. Our results are qualitatively different from both those in the clean Ising model and random spin chains of a singlet ground state, like the spin-

\frac{1}{2}

random Heisenberg chain and the random XX chain. While for random singlet states $\mathcal{I}(\ensuremath{\alpha})/\mathcal{E}(\ensuremath{\alpha})=2$, in the random Ising chain this universal ratio is strongly $\ensuremath{\alpha}$ dependent. This deviation between systems contrasts with the behavior of the entanglement entropy of a single subsystem, for which the various random critical chains and clean models give the same qualitative behavior. The reason is that

\mathcal{E}

and

\mathcal{I}

are sensitive to higher order correlations in the ground-state structure. Therefore, studying multipartite entanglement provides additional universal information in random quantum systems, beyond what we can learn from a single subsystem.