Entanglement negativity via the replica trick: A quantum Monte Carlo approach

Motivated by recent developments in conformal field theory (CFT), we devise a quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. These quantities can serve as witnesses of criticality. In particular, we study several scale invariant combinations of the moments for the one-dimensional (1D) hard-core boson model. For two adjacent intervals unusual finite-size corrections are present, showing parity effects that oscillate with a filling dependent period. These are more pronounced in the presence of boundaries. For large chains we find perfect agreement with CFT. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. Remarkably, the finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderately large sizes.

Figure 1

Geometrical setup (chain partitions) used in this work: Two adjacent intervals ($A_1,A_2$) of equal length $\ell$ embedded in a chain (of length $L$) with periodic (a) and open (b) boundary conditions. (c) Two disjoint intervals.

Figure 2

Replica trick scheme for calculating the $n$th moment (here $n=3$) of ${\rho }_A^{T_2}$ in quantum Monte Carlo (QMC) simulations. Left: Three disconnected replicas (of area $L\times \beta$). On each replica periodic boundary conditions are used along the imaginary time direction $\widehat{\tau }$. Right: Topology of ${\mathcal{K}}_3$ for two adjacent intervals. Colored links now connect points on different replicas. Regions corresponding to different intervals $A_1,A_2$ (see Fig. 1) are shaded with different colors.

Figure 3

Two adjacent intervals: We show QMC data for $r_3$ versus $z\equiv \ell /L$ for a periodic chain of length $L=150$ of hard-core bosons at half filling for different temperatures. The dashed-dotted line is the zero temperature CFT result. Note that the crossing at $z=1/4$ is merely due to the definition of $r_n$. Inset: ${\ell }^{-1}{S}_3^{T_2}$ versus $z$ (QMC data) showing that the transposed Renyi entropy exhibits a volume law already at $T\sim 1$.

Figure 4

Scaling corrections of $r_3$ and $r_4$: Parity and boundary effects. $r_3$ (top) and $r_4$ (bottom) from QMC as functions of $z\equiv \ell /L$ at fixed $L=48$ and $T=0.01$ for a periodic chain of hard-core bosons at half (circles) and quarter (squares) filling. The dashed-dotted line is the zero temperature CFT result (same as in Fig. 3). Inset: Same as in the main figure but for open boundary conditions. The amplitude of the corrections is enhanced.

Figure 5

1D hard-core bosons: The ratio $R_3\left(y\right)$. QMC data at half filling, chain lengths $L=24,48,96$ (periodic boundary conditions), and temperatures $T=0.24/L$. (a) $R_3\left(y\right)$ vs cross ratio $y$. The dotted line highlights the oscillating behavior. The dashed-dotted line is the zero temperature CFT result. Inset: $d_3\equiv R_3^{\mathrm{CFT}}-R_3$ (at fixed $y$) versus ${\ell }^{-2/3}$, $\ell$ being the interval size. Dashed lines are one parameter fits to $\sim {\ell }^{-2/3}$. (b) Amplitude $d_3{\ell }^{2/3}$ of the corrections plotted versus $y$. Note the data collapse with the functions $g_3^{\left(q\right)}\left(y\right)$, $q$ being the parity of $\ell$.

Figure 6

Bose-Hubbard chain in the superfluid phase: Scale invariant ratio $R_3\left(y\right)$ versus the cross ratio $y$. QMC data are for a periodic chain at $U=2$ (corresponding to Luttinger parameter $K_L\approx 3.125$), and temperatures $T=0.96/L$. The dashed-dotted line is the asymptotic zero temperature CFT result.