A contour for the entanglement negativity of bosonic Gaussian states

Abstract

We construct a contour function for the logarithmic negativity and the logarithm of the moments of the partial transpose of the reduced density matrix for multimode bosonic Gaussian states of a free lattice model. In one spatial dimension, numerical results are obtained for harmonic chains either in the ground state or at finite temperature, by considering, respectively, either a subsystem made by two adjacent or disjoint blocks on the line or a bipartition of the circle. The contour function of the logarithmic negativity diverges only at the entangling points, while the contour function for the logarithm of the moments of the partial transpose is divergent also at the boundary of the bipartite subsystem, as functions of the position. In a two-dimensional conformal field theory, analytic expressions that describe these divergencies are discussed. In one spatial dimension, we explore the partial derivative of the logarithmic negativity of two adjacent intervals with respect to the logarithm of the harmonic ratio of their lengths while their ratio and the other parameters are kept fixed. Considering the ground state of the harmonic chain on the line and in the massive regime, we report numerical results showing that this quantity displays a monotonically decreasing behaviour.

Figures (14)

• Figure 1: Mode participation function \ref{['tilde-pk-i-def']} for an infinite chain on the line and in its ground state, when $A$ is the union of two disjoint blocks made by $L_1 =100$ and $L_2 =50$ consecutive sites, separated by $d = 10$ consecutive sites. Here $\omega L_1 = 10^{-10}$. The green plane highlights the threshold mode at $k = k_\ast$, defined in (\ref{['threshold-mode-def']}).
• Figure 2: Contribution of the $k$-mode to the logarithmic negativity for $1 \leqslant k < k_\ast$ (see \ref{['neg-from-sigma-k']} and \ref{['threshold-mode-def']}), in the same setup of Fig. \ref{['fig-3D-MPF-example']}.
• Figure 3: Mode participation function \ref{['tilde-pk-i-def']} for two adjacent blocks with $L_1=2L_2 = 100$ in the infinite harmonic chain in its ground state. The parameter $\omega$ changes in the different panels as follows: $\omega L_1 = 10^{-10}$ (top left), $\omega L_1 = 40$ (top right), $\omega L_1 = 500$ (bottom left) and $\omega L_1 = 2000$ (bottom right). The green straight line corresponds to the threshold mode at $k=k_\ast$.
• Figure 4: Contour function $\mathsf{E}_A(i)$ (see (\ref{['contour-function-NEG-projector']})) for the logarithmic negativity of two adjacent blocks with $L_1 = 2L_2$ in the infinite harmonic chain with $\omega L_1 = 10^{-10}$ in its ground state. The black and magenta dashed lines correspond to the analytical proposals given by \ref{['neg-adj-proposal']} and \ref{['Ryu-contour-adj-on-A']} respectively. The inset in the top panel highlights the quality of the fit, while the one in the bottom panel zooms in on the region around the entangling point.
• Figure 5: Contour function $\mathsf{E}_A^{(n)}(i)$ for the logarithm of the moments of $\gamma_A^{\textrm{\tiny $\Gamma_2$}}$ in (\ref{['contour-function-NEG-projector']}) in the massless regime and for various values of $n$, in the same setup of Fig. \ref{['fig-adj']}. The black and green dashed lines correspond respectively to \ref{['neg-contour-adjacent-cft']} and to its modified version, obtained by replacing $\Delta^{(2)}_n$ with $\Delta_n$.