2012/06/30 by Pasquale Calabrese, John Cardy, Erik Tonni · 3 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Central charge #Conformal map #Disjoint sets #Harmonic oscillator #Logarithm #Mathematical analysis #Mathematical physics #Mathematics #Path integral formulation #Physics #Physics of Superconductivity and Magnetism #Quantum #Quantum entanglement #Quantum field theory #Quantum many-body systems #Quantum mechanics #cond-mat.stat-mech #hep-th #quant-ph
paper · pdf · doi:10.1103/physrevlett.109.130502
published as Phys. Rev. Lett. 109, 130502 (2012) · 4 pages, 5 figures
openalex publication_date 2012/09/28 · arxiv created 2012/10/19 · arxiv updated 2012/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 \ensuremathρA^T2 of the reduced density matrix of a subsystem A=A1\ensuremath∪A2, and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity E=ln\ensuremath\Vert\ensuremathρA^T2\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 E\ensuremath∼(c/4)ln[\ensuremathℓ1\ensuremathℓ2/(\ensuremathℓ1+\ensuremathℓ2)] for the case of two adjacent intervals of lengths \ensuremathℓ1, \ensuremathℓ2 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.