Entanglement negativity in quantum field theory

TL;DR

This paper introduces a systematic path integral and replica method to compute entanglement negativity in 1+1D quantum field theories, providing explicit formulas for conformal cases and validating with numerical results.

Contribution

It develops a novel systematic approach to calculate entanglement negativity in quantum field theories using path integrals and replica techniques, extending to conformal field theories.

Findings

Derived explicit negativity formulas for adjacent intervals in CFTs.

Showed negativity depends on harmonic ratio for disjoint intervals.

Validated theoretical results with numerical harmonic chain data.

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 rho_A^{T_2} of the reduced density matrix of a subsystem A=A1 U A2, and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity E=ln||\rho_A^{T_2}||. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result E\sim(c/4) ln(L1 L2/(L1+L2)) for the case of two adjacent intervals of lengths L1, L2 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.