Negativity spectrum of one-dimensional conformal field theories

The partial transpose ${\rho }_A^{T_2}$ of the reduced density matrix ${\rho }_A$ is the key object to quantify the entanglement in mixed states, in particular through the presence of negative eigenvalues in its spectrum. Here we derive analytically the distribution of the eigenvalues of ${\rho }_A^{T_2}$, which we dub negativity spectrum, in the ground state of gapless one-dimensional systems described by a conformal field theory (CFT), focusing on the case of two adjacent intervals. We show that the negativity spectrum is universal and depends only on the central charge of the CFT, similarly to the entanglement spectrum. The precise form of the negativity spectrum depends on whether the two intervals are in a pure or mixed state, and in both cases, a dependence on the sign of the eigenvalues is found. This dependence is weak for bulk eigenvalues, whereas it is strong at the spectrum edges. We also investigate the scaling of the smallest (negative) and largest (positive) eigenvalues of ${\rho }_A^{T_2}$. We check our results against DMRG simulations for the critical Ising and Heisenberg chains, and against exact results for the harmonic chain, finding good agreement for the spectrum, but showing that the smallest eigenvalue is affected by very large scaling corrections.

Figure 1

Partitions of the 1D pure systems considered in this work. Periodic boundary conditions are always implied. (a) The bipartition into two intervals $A_1$ and $A_2={\overline{A}}_1$. (b) The tripartition of the chain into two adjacent intervals $A_1$ and $A_2$ with $A\equiv A_1\cup A_2$ plus the remainder $B=\overline{A}$. In both (a) and (b), the partial transposition is performed with respect to the degrees of freedom in $A_2$.

Figure 2

Negativity spectrum of two adjacent intervals: survey of the CFT results for the number distribution function $n\left(\lambda \right)$ plotted versus the scaling variable $\xi \equiv [bln({\lambda }_M/{\left|\lambda \right|\left)\right]}^{1/2}$, with $b\equiv -ln{\lambda }_M$ and ${\lambda }_M$ the largest (positive) eigenvalue of ${\rho }_A^{T_2}$. The continuous line shows for comparison the CFT result for the entanglement spectrum. The dash-dotted and dotted lines denote $n\left(\lambda \right)$ for $\lambda >0$ and $\lambda <0$, respectively. The different colors correspond to the case with the two intervals in a pure state and a mixed state.

Figure 3

Negativity spectrum of two intervals in a pure state. DMRG results for the critical Ising chain (of sizes $L$) are compared with the CFT prediction for the tail distribution $n\left(\lambda \right)$ plotted as a function of $\xi \equiv [bln({\lambda }_M/{\left|\lambda \right|\left)\right]}^{1/2}$ [cf. Eq. (23)]. The subsystem size is always $\ell =L/2$. (a) and (b) report $n\left(\lambda \right)$ for $\lambda >0$ and $\lambda <0$, respectively. In both panels, the continuous line is the parameter-free CFT prediction.

Figure 4

Largest and smallest negativity eigenvalues for two adjacent equal intervals of length $\ell$ in a mixed state for a critical Ising chain of length $L$ as function of $z\equiv \ell /L$. (a) The largest positive eigenvalue ${\lambda }_M$ of ${\rho }_A^{T_2}$. The continuous line is the CFT prediction. (b) The smallest (negative) eigenvalue ${\lambda }_m$ of ${\rho }_A^{T_2}$.

Figure 5

Largest and smallest negativity eigenvalues for two adjacent intervals in a mixed state: same as in Fig. 4 for the $XXX$ chain. Note both in (a) and (b) the presence of oscillating scaling corrections.

Figure 6

Largest and smallest negativity eigenvalues for two adjacent intervals in a mixed state: same as in Fig. 4 for the harmonic chain. Note in (b) the very large scaling corrections for ${\lambda }_m$. The dashed line is the same curve for $-ln{\lambda }_M$ as in (a).

Figure 7

Negativity spectrum of two adjacent equal-length intervals in a mixed state: the number distribution function $n\left(\lambda \right)$ plotted as a function of $\xi \equiv [bln({\lambda }_M/{\left|\lambda \right|\left)\right]}^{1/2}$, with $b\equiv -ln{\lambda }_M$, and ${\lambda }_M$ the largest positive eigenvalue of ${\rho }_A^{T_2}$. The symbols are DMRG results for the critical Ising chain for several chain sizes $L$. The subsystem size is always $\ell =L/4$. (a) and (b) plot $n\left(\lambda \right)$ for positive and negative values of $\lambda$, respectively. In both panels the continuous line is the CFT prediction.

Figure 8

Negativity spectrum of two adjacent intervals in a mixed state: Same as in Fig. 7, for the $XXX$ chain. Note the degeneracy patterns at large $\left|\lambda \right|$.

Figure 9

The smallest single particle negativity level ${\nu }_{\min }$ plotted as a function of the chain size $L$. The circles are exact results for two adjacent equal-length intervals with $z\equiv \ell /L=1/4$, and $\ell$ being the size of one interval. The dash-dotted line is a fit to $a/(b+ln(L\left)\right)$, with $a,b$ the fitting parameters. The fit gives $a\sim 1.26$ and $b\sim 1.9$.