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arXiv:1405.1817 · Physical Review B

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

Yichen Huang, Joel E. Moore

Published Dec 15, 2014• 58 citations
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Abstract

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.

DOI
10.1103/physrevb.90.220202

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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.
Quantum information theoretical measures are useful tools for characterizing quantum dynamical phases. However, employing them to study excited states of random spin systems is a challenging problem. Here, we report results for the entanglement entropy (EE) scaling of excited eigenstates of random XX antiferromagnetic spin chains with long-range (LR) interactions decaying as a power law with distance with exponent $\ensuremath{\alpha}$. To this end, we extend the real-space renormalization group technique for excited states (RSRG-X) to solve this problem with LR interactions. For comparison, we perform numerical exact diagonalization (ED) calculations. From the distribution of energy-level spacings, as obtained by ED for up to $N\ensuremath{\sim}18$ spins, we find indications of a delocalization transition at ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1$ in the middle of the energy spectrum. With RSRG-X and ED, we show that for $\ensuremath{\alpha}>{\ensuremath{\alpha}}^{*}$ the EE of excited eigenstates retains a logarithmic divergence similar to the one observed for the ground state of the same model, while for $\ensuremath{\alpha}<{\ensuremath{\alpha}}^{*}$ EE displays an algebraic growth with the subsystem size $l, {S}_{l}\ensuremath{\sim}{l}^{\ensuremath{\beta}}$, with $0<\ensuremath{\beta}<1$. We find that ${\ensuremath{\alpha}}^{*}\ensuremath{\approx}1$ coincides with the delocalization transition ${\ensuremath{\alpha}}_{c}$ in the middle of the many-body spectrum. An interpretation of these results based on the structure of the RG rules is proposed, which is due to rainbow proliferation for very long-range interactions $\ensuremath{\alpha}\ensuremath{\ll}1$. We also investigate the effective temperature dependence of the EE, allowing us to study the half-chain EE of eigenstates at different energy densities, where we find that the crossover in EE occurs at ${\ensuremath{\alpha}}^{*}<1$.
We study the infinite-temperature properties of an infinite sequence of random quantum spin chains using a real-space renormalization group approach, and demonstrate that they exhibit nonergodic behavior at strong disorder. The analysis is conveniently implemented in terms of SU(2)_{k} anyon chains that include the Ising and Potts chains as notable examples. Highly excited eigenstates of these systems exhibit properties usually associated with quantum critical ground states, leading us to dub them "quantum critical glasses." We argue that random-bond Heisenberg chains self-thermalize and that the excited-state entanglement crosses over from volume-law to logarithmic scaling at a length scale that diverges in the Heisenberg limit k→∞. The excited state fixed points are generically distinct from their ground state counterparts, and represent novel nonequilibrium critical phases of matter.
Quantifying entanglement of multiple subsystems is a challenging open problem in interacting quantum systems. Here, we focus on two subsystems of length $\ensuremath{\ell}$ separated by a distance $r=\ensuremath{\alpha}\ensuremath{\ell}$ and quantify their entanglement negativity (E\mathcal{E}E) and mutual information (I\mathcal{I}I) in critical random Ising chains. We find universal constant $\mathcal{E}(\ensuremath{\alpha})$ and $\mathcal{I}(\ensuremath{\alpha})$ over any distances, using the asymptotically exact strong disorder renormalization group method. Our results are qualitatively different from both those in the clean Ising model and random spin chains of a singlet ground state, like the spin-12\frac{1}{2}21​ random Heisenberg chain and the random XX chain. While for random singlet states $\mathcal{I}(\ensuremath{\alpha})/\mathcal{E}(\ensuremath{\alpha})=2$, in the random Ising chain this universal ratio is strongly $\ensuremath{\alpha}$ dependent. This deviation between systems contrasts with the behavior of the entanglement entropy of a single subsystem, for which the various random critical chains and clean models give the same qualitative behavior. The reason is that E\mathcal{E}E and I\mathcal{I}I are sensitive to higher order correlations in the ground-state structure. Therefore, studying multipartite entanglement provides additional universal information in random quantum systems, beyond what we can learn from a single subsystem.
We examine the real space renormalization group method of finding the excited eigenstates (RSRG-X) of the XXXXXX spin-1/2 chain from entanglement perspectives. The eigenmodes of the entanglement Hamiltonian, especially the maximally entangled mode and the corresponding entanglement energies are studied and compared with the predictions of RSRG-X. Our numerical results demonstrate the accuracy of the RSRG-X method in the strong-disorder limit, and quantify its error when applied to the weak-disorder regime. Overall, our results validate the RSRG-X method qualitatively, but also show that its accuracy decreases with increasing temperature.
In this paper, we study the ground-state entanglement properties of finite XXXXXX spin-1/2 chains with random couplings using the Jordan-Wigner transformation. We divide the system into two parts and study the reduced density matrices (RDMs) of its subsystems. Due to the free-fermion nature of the problem, the RDMs take the form of that of a free-fermion thermal ensemble. Finding the spectrum of the corresponding entanglement Hamiltonian and corresponding eigenvectors and comparing them with the real-space renormalization-group (RSRG) treatment, we establish the validity of the RSRG approach for entanglement in the limit of strong disorder but also find its limitations when disorder is weak. In the latter case, our paper provides a way to visualize the ``effective spins'' that form long-distance singlet pairs.
We study the spreading of quantum correlations and information in a one-dimensional quantum spin chain with critical disorder encoded in an infinite randomness fixed point. Specifically, we focus on the dynamics after a quantum quench of the R\'enyi entropies, mutual information, and entanglement negativity in the prototypical XXZ spin chain with random bonds and anisotropy parameters. We provide analytic predictions in the scaling regime based on real-space renormalization-group methods. We support these findings through numerical simulations in the noninteracting limit, where we can access the scaling regime.