Entanglement negativity in quantum field theory

TL;DR

A systematic method to extract the negativity in the ground state of a 1+1 dimensional relativistic quantum field theory is developed, using a path integral formalism to construct the partial transpose ρ(A)(T(2) of the reduced density matrix of a subsystem and applied to conformal field theories.

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||ρ_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.

Figures (5)

• Figure 1: We consider the entanglement between two blocks $A_1$ and $A_2$ embedded in the ground-state of a larger system.
• Figure 2: Top: The reduced density matrix $\rho_A$ of two disjoint intervals. Middle: Partial transpose with respect to the second interval $\rho_A^{T_{2}}$. Bottom: Reversed partial transpose $\rho_A^{C_{2}}$.
• Figure 3: Path integral representation of ${\rm Tr}\rho_A^n$ (top) and ${\rm Tr}(\rho_A^{T_{2}})^n$ (bottom) for $n=3$.
• Figure 4: For two adjacent intervals of equal length $\ell<L/2$, we plot $r_n=\ln [{\rm Tr}(\rho_A^{T_{A_2=\ell}})^n/{\rm Tr}(\rho_A^{T_{A_2=L/4}})^n]$ as function of $z=\ell/L$. The subtraction is chosen to cancel non-universal factors. The bottommost panel shows ${\cal E}_1={\cal E}-(\ln L)/4$ in which non-universal terms are absent. The continuous lines are the parameter free CFT predictions.
• Figure 5: Top: The ratio $R_n(y)$ in Eq. (\ref{['Rndef']}) as function of $y$ for several $L$ and for $n=3,4$. The continuous lines are the parameter free CFT predictions. The inset shows a finite size scaling analysis for $d_n\equiv R^{{\textrm{\tiny CFT}}}_n(y)-R_n(y)$ for $n=3$ displaying the unusual correction $L^{-2/n}$unusual. The same is true for higher $n$cct-prep. Bottom: The negativity ${\cal E}(y)$ is a universal scale invariant function with an essential singularity at $y=0$.