Entanglement Negativity in Quantum Field Theory

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 $A=A_1\cup A_2$, and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity $\mathcal{E}=\ln \Vert {\rho }_A^{T_2}\Vert$. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result $\mathcal{E}\sim (c/4)\ln [{\ell }_1{\ell }_2/({\ell }_1+{\ell }_2)]$ for the case of two adjacent intervals of lengths ${\ell }_1$, ${\ell }_2$ 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.

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 $\mathrm{Tr}{\rho }_A^n$ (top) and $\mathrm{Tr}({\rho }_A^{T_2}{)}^n$ (bottom) for $n=3$.

Figure 4

For two adjacent intervals of equal length $\ell <L/2$, we plot $r_n=\ln [\mathrm{Tr}({\rho }_A^{T_{A_2=\ell }}{)}^n/\mathrm{Tr}({\rho }_A^{T_{A_2=L/4}}{)}^n]$ as a function of $z=\ell /L$. The subtraction is chosen to cancel nonuniversal factors. The bottom-most panel shows ${\mathcal{E}}_1=\mathcal{E}-(\ln L)/4$, in which nonuniversal terms are absent. The continuous lines are the parameter-free CFT predictions.

Figure 5

Top: The ratio $R_n(y)$ in Eq. (17) 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 $d_n\equiv R_n^{\mathrm{CFT}}(y)-R_n(y)$ for $n=3$ displaying the unusual correction $L^{-2/n}$ [18]. The same is true for higher $n$ [16]. Bottom: The negativity $\mathcal{E}(y)$ is a universal scale-invariant function with an essential singularity at $y=0$.