Entanglement negativity in random spin chains

We investigate the logarithmic negativity in strongly disordered spin chains in the random-singlet phase. We focus on the spin-$\frac{1}{2}$ random Heisenberg chain and the random $XX$ chain. We find that for two arbitrary intervals, the disorder-averaged negativity and the mutual information are proportional to the number of singlets shared between the two intervals. Using the strong-disorder renormalization group (SDRG), we prove that the negativity of two adjacent intervals grows logarithmically with the intervals' length. In particular, the scaling behavior is the same as in conformal field theory, but with a different prefactor. For two disjoint intervals the negativity is given by a universal simple function of the cross ratio, reflecting scale invariance. As a function of the distance of the two intervals, the negativity decays algebraically in contrast with the exponential behavior in clean models. We confirm our predictions using a numerical implementation of the SDRG method. Finally, we also implement density matrix renormalization group simulations for the negativity in open spin chains. The chains accessible in the presence of strong disorder are not sufficiently long to provide a reliable confirmation of the SDRG results.

Figure 1

(a) Partition of an infinite spin chain corresponding to two disjoint intervals. The region of interest is $A_1\cup A_2$. Two adjacent intervals correspond to $r=0$. (b) Cartoon of the random-singlet phase. The lines connect pairs of spins forming $\text{SU}\left(2\right)$ singlets. The entanglement negativity ${\mathcal{E}}_{A_1:A_2}$ is proportional to the number of singlets $n_{A_1:A_2}$ shared between $A_1$ and $A_2$ (here $n_{A_1:A_2}=1$).

Figure 2

Numerical strong-disorder renormalization group: flow of the couplings probability distribution $P\left({\beta }^{{}_{\left(m\right)}}\right)$ as a function of the SDRG step $m$. Different histograms correspond to different $m$. Results are for the Heisenberg spin chain with $L={10}^6$ sites. The width of $P\left({\beta }_i^{{}_{\left(m\right)}}\right)$ increases upon increasing $m$. Inset: $P\left({\beta }^{{}_{\left(m\right)}}\right)$ for $m=99\times {10}^4$, compared with the infinite-randomness fixed point (IRFP) result $P^{*}\left(\beta \right)=e^{-\beta /\mathrm{\Gamma }}/\mathrm{\Gamma }$ (dashed-dotted line) with ${\mathrm{\Gamma }}^{\left(m\right)}$ given in (14).

Figure 3

The single-interval entanglement entropy for the disordered Heisenberg spin chain. The data correspond to the average over $73000$ disorder realizations. Notice that (here and in the following figures) we report also the statistical errors, although the error bars are smaller than the symbol sizes. Different symbols correspond to different chain sizes $L$ and dashed-dotted lines are one-parameter fit to the analytic prediction from the SDRG. (a) A single interval in a periodic chain: $S_A$ plotted versus the interval length $\ell$. (b) Same data as in (a) plotted against the modified chord length (44). (c) A single interval starting from the boundary of a chain with OBC versus the modified chord length (44). (d) A single interval in the bulk of an open chain.

Figure 4

Shifted logarithmic negativity ${\mathcal{E}}_{A_1:A_2}^s$ (48) between two adjacent intervals of equal length $\ell$ in the random Heisenberg chain. ${\mathcal{E}}_{A_1:A_2}^s$ is plotted against $\ell /L$. The symbols correspond to the average over $73000$ disorder realizations. Different symbols correspond to different system sizes. (a) For a periodic chain, the data are perfectly described by (49) with only one fitting parameter. (b) For an open chain, we can only use the prediction for the infinite chain which describes well the data as long as $\ell \ll L$.

Figure 5

Logarithmic negativity ${\mathcal{E}}_{A_1:A_2}$ between two disjoint intervals of equal length $\ell$ in the random Heisenberg spin chain plotted against the cross ratio $x$. The data are averaged over $73000$ disorder realizations. Different symbols correspond to different system sizes. (a) For a periodic chain, the cross ratio $x$ is given by (50) and the dashed line is the analytic SDRG prediction (51). (b) For an open chain, we consider the infinite-volume definition of the cross ratio (5) and the dashed-dotted line is the SDRG prediction (51) to which the data tend for large chains.

Figure 6

Reduced density matrix with matrix product states (MPS) for a chain with $L=8$ sites and open boundary conditions. (a) The MPS representation for a generic wave function $|\mathrm{\Psi }\rangle$. The basic tensor $T_{{\nu }_i,{\nu }_{i+1}}^{{\sigma }_i}$ at site $i$ of the chain is shown in the shaded box. The vertical leg denotes the physical index ${\sigma }_i$, while the horizontal ones are the virtual indices ${\nu }_i$ (summed over). Tensors in the bulk have rank 3, while at the edges have rank 2. (b) The tripartition of the chain as $A_1\cup B_1\cup A_2\cup B_2$. $A_1$ and $A_2$ are two equal blocks formed by two spins. The two blocks are at distance two (spins in $B_1$). (c) The reduced density matrix ${\rho }_{A_1\cup A_2}$. Only the tensors living in $A_1\cup B_1\cup A_2$ appear. (d) Definition of the transfer matrices $E_1, E_2, E_{12}$ corresponding to parts $A_1, A_2$, and $B_1$ of the chain.

Figure 7

The definition of the tensor $N_1$ [cf. (67)] from the singular value decomposition (SVD) of the transfer matrix $E_1$. The SVD is performed with respect to the index pairs $({\nu }_{1L},{\nu }_{1R})$ and $({\nu }_{1L}^{\prime },{\nu }_{1R}^{\prime })$. The rhombi denote the diagonal matrix containing the square root of the singular values. The same definition holds for $N_2$.

Figure 8

(a) The reduced density matrix of two intervals ${\rho }_{A_1\cup A_2}$ in the MPS framework. The tensor $N_1$ is defined in Fig. 7 (a similar definition holds for $N_2$), whereas $E_{12}$ is as in Fig. 6. The indices are as in (67). The index pairs $(m,m^{\prime })$ and $(n,n^{\prime })$ refer to intervals $A_1$ and $A_2$, respectively. (b) The partially transposed reduced density matrix ${\rho }_{A_1\cup A_2}^{T_2}$. Here, the partial transposition is performed with respect to subsystem $A_2$, and it corresponds to the exchange $n\leftrightarrow n^{\prime }$.

Figure 9

The logarithmic negativity ${\mathcal{E}}_{A_1:A_2}$ between the two (equal-length) adjacent intervals $A_1\cup A_2$ in the $XX$ chain. Here, the two intervals are at the center of the chain. ${\mathcal{E}}_{A_1:A_2}$ is plotted against the length of one interval $\ell$. The symbols are DMRG data for a chain with length up to $L=128$. The dashed-dotted line is the CFT prediction $1/4ln\ell$ in the thermodynamic limit.

Figure 10

Entanglement entropy in the random $XX$ spin chain. (a) Check of the DMRG convergence. The von Neumann entropy $S_A$ for a single interval at the center of an open chain plotted versus the interval length $\ell$. The empty symbols denote DMRG results for chains with $L=16,24$ and disorder strength $\delta =\frac{3}{2}$. The data are averaged over $\sim {10}^3$ disorder realizations. The full symbols are exact results obtained using free-fermion techniques. (b) Same geometry as in (a). The symbols are DMRG data for $L=16,24,28,32$ and $\frac{1}{4}\le \delta \le \frac{3}{2}$. Here, $\delta =0$ and $\delta \to \infty$ correspond to the clean case and the IRFP, respectively. For $\delta =1$ one has the box distribution. The dashed-dotted line is the CFT prediction for $\delta =0$. The dotted line is the strong-disorder renormalization group (SDRG) prediction in the thermodynamic limit.

Figure 11

Entanglement negativity in the random $XX$ spin chain. (a) Rényi entropy $S_A^{(1/2)}$ for a single interval at the center of an open chain plotted versus the interval length $\ell$. The data (circles) are obtained by means of free-fermionic techniques which allow us to reach large systems sizes ($L=200$). Note the crossover from the scaling of the clean model at short lengths (dotted line) to the SDRG one at large distances (dashed-dotted line). (b) The logarithmic negativity ${\mathcal{E}}_{A_1:A_2}$ for two adjacent intervals of equal length $\ell$ (in the center of an open chain) plotted as a function of $\ell$. The symbols denote DMRG results for chains with $L=16,24,28,32,63$ and disorder strength $\frac{1}{4}\le \delta \le \frac{3}{2}$. The data are averaged over $\sim {10}^3$ disorder realizations. The dashed-dotted and dotted lines are the CFT prediction for $\delta =0$ and the SDRG result, respectively.