Negativity spectrum in the random singlet phase

Entanglement features of the ground state of disordered quantum matter are often captured by an infinite-randomness fixed point that, for a variety of models, is the random singlet phase. Although a copious number of studies covers bipartite entanglement in pure states, at present, less is known for mixed states and tripartite settings. Our goal is to gain insights in this direction by studying the negativity spectrum in the random singlet phase. Through the strong disorder renormalization group technique, we derive analytic formulas for the universal scaling of the disorder-averaged moments of the partially transposed reduced density matrix. Our analytic predictions are checked against a numerical implementation of the strong disorder renormalization group and against exact computations for the XX spin chain (a model in which free fermion techniques apply). Importantly, our results show that the negativity and logarithmic negativity are not trivially related after the average over the disorder.

Figure 1

Cartoon of the random singlet phase in a tripartite setting $A_1\cup A_2\cup B$. The special case of two adjacent intervals (in red) embedded in a larger system is depicted.

Figure 2

Negativity moments ${\mathcal{E}}_{\alpha }$ and ${\widehat{\mathcal{E}}}_{\alpha }$ for the random XX chain: results of the ab initio computations for two adjacent intervals of equal length $\ell$. We report the absolute values, since for the considered values of $\alpha$, they are all negative quantities (i.e., the moments $M_{\alpha }^{T_2}$ are smaller than 1). The symbols correspond to the numerical data while the continuous lines are the analytic SDRG predictions Eqs. (55, 56, 57, 58), with a best fit for the unknown nonuniversal additive constants. The agreement between the simulation and SDRG predictions is excellent already for moderate values of $\ell$.

Figure 3

SDRG simulations for the negativity moments. Here we plot, as a function of $\alpha$, the prefactor of the logarithm ${\varepsilon }_{\alpha }^{e/o}$ [cf. (81)], as obtained by a fit of the numerical data. The symbols are the corresponding numerical data while the continuous lines are the analytic predictions in Eqs. (55, 56, 57, 58). The plots show an extremely good agreement between numerical data and analytic predictions.

Figure 4

Comparison between the SDRG simulations using the Ma-Dasgupta rule respectively for the XX chain Eq. (15) and the XXX chains Eq. (11). Similarly to Fig. 3, we plot, as a function of $\alpha$, the prefactor of the logarithm ${\varepsilon }_{\alpha }^{e/o}$ [cf. (81)], as obtained by a fit of the numerical data. The continuous lines are the analytic predictions in Eqs. (55, 56, 57, 58). The plots show an extremely good agreement between numerical data and analytic predictions.

Figure 5

Numerical comparison between the SDRG simulation for the XXX model using the initial coupling distribution Eq. (A1) with $\delta =1$ and $\delta =2$. Similarly to Fig. 3 and Fig. 4, we plot, as a function of $\alpha$, the prefactor of the logarithm ${\varepsilon }_{\alpha }^{e/o}$ [cf. (81)], as obtained by a fit of the numerical data. The continuous lines are the analytic predictions in Eqs. (55, 56, 57, 58). Analogous results hold for other values of $\delta$, and suggest the independence of the SDRG procedure with respect to the initial distribution of the couplings.