arXiv:1407.0324 · Journal of High Energy Physics

Conformal blocks and negativity at large central charge

Manuela Kulaxizi, Andrei Parnachev, Giuseppe Policastro

Published Sep 1, 2014• 87 citations

Abstract

We consider entanglement negativity for two disjoint intervals in 1+1 dimensional CFT in the limit of large central charge. As the two intervals get close, the leading behavior of negativity is given by the logarithm of the conformal block where a set of approximately null descendants appears in the intermediate channel. We compute this quantity numerically and compare with existing analytic methods which provide perturbative expansion in powers of the cross-ratio.

DOI: 10.1007/jhep09(2014)010

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arXiv:1407.0324 · Journal of High Energy Physics

Conformal blocks and negativity at large central charge

Manuela Kulaxizi, Andrei Parnachev, Giuseppe Policastro

Published Sep 1, 2014• 87 citations

arXiv HTML PDF

Abstract

DOI: 10.1007/jhep09(2014)010

Related papers

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Xueda Wen, Po-Yao Chang, Shinsei Ryu

We study the time evolution of the entanglement negativity after a local quantum quench in (1 + 1)-dimensional conformal field theories (CFTs), which we introduce by suddenly joining two initially decoupled CFTs at their end points. We calculate the negativity evolution for both adjacent intervals and disjoint intervals explicitly. For two adjacent intervals, the entanglement negativity grows logarithmically in time right after the quench. After developing a plateau-like feature, the entanglement negativity drops to the ground-state value. For the case of two spatially separated intervals, a light-cone behavior is observed in the negativity evolution; in addition, a long-range entanglement, which is independent of the distance between two intervals, can be created. Our results agree with the heuristic picture that quasiparticles, which carry entanglement, are emitted from the joining point and propagate freely through the system. Our analytical results are confirmed by numerical calculations based on a critical harmonic chain.We study the time evolution of the entanglement negativity after a local quantum quench in (1 + 1)-dimensional conformal field theories (CFTs), which we introduce by suddenly joining two initially decoupled CFTs at their end points. We calculate the negativity evolution for both adjacent intervals and disjoint intervals explicitly. For two adjacent intervals, the entanglement negativity grows logarithmically in time right after the quench. After developing a plateau-like feature, the entanglement negativity drops to the ground-state value. For the case of two spatially separated intervals, a light-cone behavior is observed in the negativity evolution; in addition, a long-range entanglement, which is independent of the distance between two intervals, can be created. Our results agree with the heuristic picture that quasiparticles, which carry entanglement, are emitted from the joining point and propagate freely through the system. Our analytical results are confirmed by numerical calculations based on a critical harmonic chain.

arXiv:1501.00568Physical Review BAug 6, 201592 citations

Holographic entanglement negativity for disjoint intervals in

AdS_3/CFT_2

A d S 3 / C F T 2

We advance a holographic construction for the entanglement negativity of bipartite mixed state configurations of two disjoint intervals in

(1+1)

dimensional conformal field theories (

CFT_{1+1}

) through the

AdS_3/CFT_2

correspondence. Our construction constitutes the large central charge analysis of the entanglement negativity for mixed states under consideration and involves a specific algebraic sum of bulk space like geodesics anchored on appropriate intervals in the dual

CFT_{1+1}

. The construction is utilized to compute the holographic entanglement negativity for such mixed states in

CFT_{1+1}

s dual to bulk pure

AdS_3

geometries and BTZ black holes respectively. Our analysis exactly reproduces the universal features of corresponding replica technique results in the large central charge limit which serves as a consistency check.

We develop a systematic method to extract the negativity in the ground state of a

1+1

dimensional relativistic quantum field theory, using a path integral formalism to construct the partial transpose ${\ensuremath{\rho}}_{A}^{{T}_{2}}$ of the reduced density matrix of a subsystem $A={A}_{1}\ensuremath{\cup}{A}_{2}$, and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity $\mathcal{E}=\mathrm{ln}\ensuremath{\Vert}{\ensuremath{\rho}}_{A}^{{T}_{2}}\ensuremath{\Vert}$. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result $\mathcal{E}\ensuremath{\sim}(c/4)\mathrm{ln}[{\ensuremath{\ell}}_{1}{\ensuremath{\ell}}_{2}/({\ensuremath{\ell}}_{1}+{\ensuremath{\ell}}_{2})]$ for the case of two adjacent intervals of lengths ${\ensuremath{\ell}}_{1}$, ${\ensuremath{\ell}}_{2}$ in an infinite system, where

c

is the central charge. For two disjoint intervals it depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We check our findings against exact numerical results in the harmonic chain.