Year: 2012
Type: article
Abstract: 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.
Source: Physical Review Letters
Authors: Pasquale Calabrese, John Cardy, Erik Tonni
Language: English
Topic: Quantum many-body systems
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1969360334