Excited-state entanglement and thermal mutual information in random spin chains

Entanglement properties of excited eigenstates (or of thermal mixed states) are difficult to study with conventional analytical methods. We approach this problem for random spin chains using a recently developed real-space renormalization group technique for excited states (“RSRG-X”). For the random XX and quantum Ising chains, which have logarithmic divergences in the entanglement entropy of their (infinite-randomness) critical ground states, we show that the entanglement entropy of excited eigenstates retains a logarithmic divergence while the mutual information of thermal mixed states does not. However, in the XX case the coefficient of the logarithmic divergence extends from the universal ground-state value to a universal interval due to the degeneracy of excited eigenstates. These models are noninteracting in the sense of having free-fermion representations, allowing strong numerical checks of our analytical predictions.

Figure 1

Scaling of $\langle S_L^T\rangle$ in random XX chain, which is apparently different from the entanglement scaling of some excited states in homogeneous spin chains [53]. Here $J_i$'s are i.i.d. uniform random variables on the interval $[0,1]$. Example 1 $\left(\alpha =1/2\right)$. The blue, red, and green dots are data (averaged over 5000 samples) for $T=0,{10}^{-3}$, and $\infty$, respectively, in chains of 1000 spins. The lines are fits based on (2) and (4): $(ln2)(lnL)/3+0.86$ (blue), $(ln2)(lnL)/6+1.18$ (red), and $(ln2)(lnL)/6+2.09$ (green). Example 2 $\left(\alpha =1\right)$. The cyan dots are data (averaged over 3000 samples) for $T=\infty ,\delta ={10}^{-9}$ in chains of 500 spins. The cyan line is a fit based on (4): $(ln2)(lnL)/3+2.13$.

Figure 2

Scaling of $\langle I_L^T\rangle$ in random XX chain. The dots (from top to bottom) are data (averaged over 2500 samples) for $T=0,{10}^{-5},3\times {10}^{-5},{10}^{-4}$, and $3\times {10}^{-4}$ in chains of 200 spins. The line is a fit based on (6): $2(ln2)(lnL)/3+1.66$. $\langle I_L^T\rangle$ behaves as if $T=0$ for $L\ll L_c$ and saturates for $L\gg L_c$. Inset: Saturation value $\langle I_{L\gg L_c}^T\rangle$ vs temperature. The line is a fit based on (6): $4(ln2)[lnln(1/T\left)\right]/3+0.77$.

Figure 3

Scaling of $\langle S_L^T\rangle$ in critical random quantum Ising chain $\left(\delta =0\right)$. Here $J_i,h_i$'s are i.i.d. uniform random variables on the interval $[0,1]$. The blue, red, and green dots are data (averaged over 2500 samples) for $T=0,0.2$, and $\infty$, respectively, in chains of 400 spins. The lines are fits based on (10): $(ln2)(lnL)/6+0.51$ (blue), $(ln2)(lnL)/6+0.73$ (red), and $(ln2)(lnL)/6+1.09$ (green).