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arXiv:2205.14306 · Physical review. B./Physical review. B

Multipartite entanglement in the random Ising chain

Jay S. Zou, Helen S. Ansell, István A. Kovács

Published Aug 2, 2022• 3 citations
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Abstract

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 ($\mathcal{E}$) and mutual information ($\mathcal{I}$) 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-$\frac{1}{2}$ 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 $\mathcal{E}$ and $\mathcal{I}$ 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.

DOI
10.1103/physrevb.106.054201

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We investigate the logarithmic negativity in strongly disordered spin chains in the random-singlet phase. We focus on the spin-12\frac{1}{2}21​ random Heisenberg chain and the random XXXXXX 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.
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.
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.
The entanglement entropy in clean, as well as in random quantum spin chains has a logarithmic size-dependence at the critical point. Here, we study the entanglement of composite systems that consist of a clean and a random part, both being critical. In the composite, antiferromagnetic XX-chain with a sharp interface, the entropy is found to grow in a double-logarithmic fashion S∼ln⁡ln⁡(L){\cal S}\sim \ln\ln(L)S∼lnln(L), where LLL is the length of the chain. We have also considered an extended defect at the interface, where the disorder penetrates into the homogeneous region in such a way that the strength of disorder decays with the distance lll from the contact point as ∼l−κ\sim l^{-\kappa}∼l−κ. For κ<1/2\kappa<1/2κ<1/2, the entropy scales as S(κ)≃(1−2κ)S(κ=0){\cal S}(\kappa) \simeq (1-2\kappa){\cal S}(\kappa=0)S(κ)≃(1−2κ)S(κ=0), while for κ≥1/2\kappa \ge 1/2κ≥1/2, when the extended interface defect is an irrelevant perturbation, we recover the double-logarithmic scaling. These results are explained through strong-disorder RG arguments.
Quantum entanglement at critical points is often marked by universal characteristics. Here, the entanglement entropy is calculated at the quantum multicritical point of the random transverse-field Ising model (RTIM). We use an efficient implementation of the strong disorder renormalization group method in two and three dimensions for two types of disorder. For cubic subsystems we find a universal logarithmic corner contribution to the area law $bln(\ensuremath{\ell})$ that is independent of the form of disorder. Our results agree qualitatively with those at the quantum critical points of the RTIM, but with new bbb prefactors due to having both geometric and quantum fluctuations at play. By studying the vicinity of the multicritical point, we demonstrate that the corner contribution serves as an ``entanglement susceptibility,'' a useful tool to locate the phase transition and to measure the correlation length critical exponents.
Abstract Disordered quantum magnets are not only experimentally relevant, but offer efficient computational methodologies to calculate the low energy states as well as various measures of quantum correlations. We present a systematic analysis of quantum entanglement in the paradigmatic random transverse-field Ising model in two dimensions, using an efficient implementation of the asymptotically exact strong disorder renormalization group method. The phase diagram is known to be governed by three distinct infinitely disordered fixed points (IDFPs) that we study here. For square subsystems, it has been recently established that quantum entanglement has a universal logarithmic correction due to the corners of the subsystem at all three IDFPs. This corner contribution has been proposed as an ‘entanglement susceptibility’, a useful tool to locate the phase transition and measure the correlation length critical exponent. Towards a deeper understanding, we quantify how the corner contribution depends on the shape of the subsystem. While the corner contribution remains universal, the shape-dependence is qualitatively different in each universality class, also confirmed by line segment subsystems, a special case of skeletal entanglement. Therefore, unlike in conformally invariant systems, in general different subsystem shapes are versatile probes to unveil new universal information on phase transitions in disordered quantum systems.