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arXiv:2201.09220 · Physical Review Research

Quantum many-body scars in spin-1 Kitaev chains

Wen‐Long You, Zhuan Zhao, Jie Ren, Gaoyong Sun, Liangsheng Li, Andrzej M. Oleś

Published Feb 10, 2022• 29 citations
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Abstract

To provide a physical example of quantum scars, we study the many-body scars in the spin-1 Kitaev chain where the so-called PXP Hamiltonian is exactly embedded in the spectra. Regarding the conserved quantities, the Hilbert space is fragmented into disconnected subspaces and we explore the associated constrained dynamics. The continuous revivals of the fidelity and the entanglement entropy when the initial state is prepared in $|{\mathbb{Z}}_{k}\ensuremath{\rangle}$ ($k=2,3$) state illustrate the essential physics of the PXP model. We study the quantum phase transitions in the one-dimensional spin-1 Kitaev-Heisenberg model using the density-matrix renormalization group and Lanczos exact diagonalization methods, and determine the phase diagram. We parametrize the two terms in the Hamiltonian by the angle $\ensuremath{\phi}$, where the Kitaev term is $K\ensuremath{\equiv}sin(\ensuremath{\phi})$ and competes with the Heisenberg $J\ensuremath{\equiv}cos(\ensuremath{\phi})$ term. One finds a rich ground-state phase diagram as a function of the angle $\ensuremath{\phi}$. Depending on the ratio $K/J\ensuremath{\equiv}tan(\ensuremath{\phi})$, the system either breaks the symmetry to one of distinct symmetry broken phases, or preserves the symmetry in a quantum spin-liquid phase with frustrated interactions. We find that the scarred state is stable for perturbations which obey ${\mathbb{Z}}_{2}$-symmetry, while it becomes unstable against Heisenberg-type perturbations.

DOI
10.1103/physrevresearch.4.013103

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To provide a physical example of quantum scars, we study the many-body scars in the spin-1 Kitaev chain where the so-called PXP Hamiltonian is exactly embedded in the spectra. Regarding the conserved quantities, the Hilbert space is fragmented into disconnected subspaces and we explore the associated constrained dynamics. The continuous revivals of the fidelity and the entanglement entropy when the initial state is prepared in ∣Zk\ensuremath⟩|{\mathbb{Z}}_{k}\ensuremath{\rangle}∣Zk​\ensuremath⟩ (k=2,3k=2,3k=2,3) state illustrate the essential physics of the PXP model. We study the quantum phase transitions in the one-dimensional spin-1 Kitaev-Heisenberg model using the density-matrix renormalization group and Lanczos exact diagonalization methods, and determine the phase diagram. We parametrize the two terms in the Hamiltonian by the angle \ensuremathϕ\ensuremath{\phi}\ensuremathϕ, where the Kitaev term is K\ensuremath≡sin(\ensuremathϕ)K\ensuremath{\equiv}sin(\ensuremath{\phi})K\ensuremath≡sin(\ensuremathϕ) and competes with the Heisenberg J\ensuremath≡cos(\ensuremathϕ)J\ensuremath{\equiv}cos(\ensuremath{\phi})J\ensuremath≡cos(\ensuremathϕ) term. One finds a rich ground-state phase diagram as a function of the angle \ensuremathϕ\ensuremath{\phi}\ensuremathϕ. Depending on the ratio K/J\ensuremath≡tan(\ensuremathϕ)K/J\ensuremath{\equiv}tan(\ensuremath{\phi})K/J\ensuremath≡tan(\ensuremathϕ), the system either breaks the symmetry to one of distinct symmetry broken phases, or preserves the symmetry in a quantum spin-liquid phase with frustrated interactions. We find that the scarred state is stable for perturbations which obey Z2{\mathbb{Z}}_{2}Z2​-symmetry, while it becomes unstable against Heisenberg-type perturbations.
Experimental observation of coherent oscillations in a Rydberg atom chain [H. Bernien et al., Nature (London) 551, 579 (2017)] has led to the discovery of quantum many-body scars (QMBSs) which is a new paradigm for ergodicity breaking. The experimental findings in the Rydberg chain can be well captured by a kinetically constrained model called the ``PXP'' model, which has been shown to host the eigenstate thermalization hypothesis (ETH)-violating scar states in the middle of the spectrum. Much effort has been put into identifying similar kinetically restricted systems that show a violation of ETH. In this paper, we study the QMBSs that can arise in one such model, namely the spin-1 Kitaev chain, where owing to some conserved quantities, the Hilbert space gets fragmented into unequal disconnected subspaces. Recently, You et al. [Phys. Rev. Res. 4, 013103 (2022)] showed that the ground state sector of this chain can be mapped exactly onto the prototypical PXP model and thus hosts QMBSs. Here, we demonstrate that the phenomenon of scarring is also present in other sectors, and in particular, we identify a sector that exhibits substantially more scarring than the ground state one. We propose an initial state and numerically demonstrate that its fidelity revivals are pronounced and longer lived than those in the PXP model.
To establish a solid-state-based framework for the coexistence of quantum many-body scars and quantum criticality, we investigate the spin-1 Kitaev chain with uniaxial single-ion anisotropy (SIA). In the subspace with uniform Z2{\mathbb{Z}}_{2}Z2​ gauge fields, this model can be exactly mapped to the spin-1/2 effective detuned PXP Hamiltonian, where the SIA plays a role of the static detuning term. The quench dynamics starting from the product states is symmetric between positive and negative values of the SIA, while a quantum phase transition from the Kitaev spin liquid to the dimer phase only occurs at the critical point with a negative Dc{D}_{c}Dc​, implying the spontaneous breaking of the translational symmetry. We find that the coherent oscillations of quantum fidelity and certain local observables are sustained against small SIA perturbations in a quantum quench from special initial states. While the oscillation amplitudes of these observables decay with time as the SIA strength is increased, the system completely thermalizes upon approaching the critical point. In contrast, the initial polarized state, which shows an absence of revivals of quantum fidelity, will exhibit long revivals for D<DcD<{D}_{c}D<Dc​. Finally, we investigate the evolution of phase boundaries of the Kitaev spin liquid and dimer phase by introducing Heisenberg interactions, which spoil the Z2{\mathbb{Z}}_{2}Z2​ gauge fields. A complete phase diagram is given by the infinite time-evolving block decimation method and the ground-state properties of each phase are accurately captured by various spin correlations. Our paper opens the door to understanding exotic connections between many-body scars and quantum criticality in systems with higher spins.
This thesis investigates the anomalous dynamics observed in a Rydberg atom experiment in the broadest sense: a strongly interacting spin chain, without integrability or disorder, exhibits coherent many-body revivals ensuing only from specific initial states. This non-thermal behavior is attributed to the presence of quantum many-body scars (QMBS), a novel class of ergodicity breaking phenomena. Unlike other non-thermalizing systems such as integrable models and many-body localization, the mechanism of ergodicity breaking in QMBS is quite subtle, relying on a (vanishingly) small subset of anomalous eigenstates, known as scar states: one could observe ergodicity breaking only if initial states have appreciable overlap with these scar states. In other words, systems hosting QMBS are indistinguishable from typical chaotic, thermalizing systems, except for scar states. While the theoretical understanding of QMBS has advanced through the investigation of various toy quantum models, ranging from the Affleck-Kennedy-Lieb-Tasaki (AKLT) model to the Hubbard model, these insights do not fully resolve the mysterious oscillations in the Rydberg experiment: crucially the wavefunction corresponding to the experiment, and its relation to many other toy models in the literature remain open questions. Furthermore, beyond mathematical curiocity, the observability of non-thermal dynamics suggests that QMBS may yield distinct physical effects. the nature of which remains to be determined. This dissertation addresses these issues by solving two concrete problems: (i) the mathematical characterization of scar states in the PXP model, the effective model for the experiment, and (ii) the identification of physical consequences of QMBS in both short- and long-time dynamics. In the first part, we clarify the algebraic structure underlying the scar states of the PXP model and derive analytic ansatz for these wavefunctions. The proposed forms make explicit the hidden algebraic structure of the PXP scars and demonstrate how this structure is shared with many other QMBS Hamiltonians, including the AKLT model. The second part investigates dynamical signatures and robustness. Given the fragility of many-body eigenstates to perturbations, identifying which scar characteristics survive realistic perturbations is crucial. In closed systems, we show that out-of-time-ordered correlators (OTOCs) and the associated Lyapunov exponent can distinguish QMBS from generic chaotic states, at least within a semiclassical regime. Specifically, the OTOC growth associated with scar states is parametrically slower and remains slow under small ``scar-breaking'' perturbations. For open systems governed by non-Hermitian dynamics, we introduce a novel non-equilibrium phase, the QMBS phase, where the long-time wavefunction is localized in a small corner of the Hilbert space. We further identify a first-order transition between the QMBS phase and a thermal phase, where the wavefunction becomes delocalized.
Quantum scars are nonthermal eigenstates characterized by low entanglement entropy, initially detected in systems subject to nearest-neighbor Rydberg blockade, the so-called PXP model. While most of these special eigenstates elude an analytical description and seem to hybridize with nearby thermal eigenstates for large systems, some of them can be written as matrix product states with size-independent bond dimension. We study the response of these exact quantum scars to perturbations by analyzing the scaling of the fidelity susceptibility with system size. We find that some of them are anomalously stable at first order in perturbation theory, in sharp contrast to the eigenstate thermalization hypothesis. However, this stability seems to break down when all orders are taken into account. We further investigate models with larger blockade radius and find a set of exact quantum scars that we write down analytically and compare with the PXP exact eigenstates. We show that they exhibit the same robustness against perturbations at first order.
Recently, it has been proposed that higher-spin analogs of the Kitaev interactions K>0K>0K>0 may also occur in a number of materials with strong Hund's and spin-orbit coupling. In this work, we use Lanczos diagonalization and density matrix renormalization group methods to investigate numerically the S=1S=1S=1 Kitaev-Heisenberg model. The ground-state phase diagram and quantum phase transitions are investigated by employing local and nonlocal spin correlations. We identify two ordered phases at negative Heisenberg coupling J<0J<0J<0: a ferromagnetic phase with $\ensuremath{\langle}{S}_{i}^{z}{S}_{i+1}^{z}\ensuremath{\rangle}>0$ and an intermediate left-left-right-right phase with $\ensuremath{\langle}{S}_{i}^{x}{S}_{i+1}^{x}\ensuremath{\rangle}\ensuremath{\ne}0$. A quantum spin liquid is stable near the Kitaev limit, while a topological Haldane phase is found for J>0J>0J>0.
We investigate the stability of quantum many-body scars under perturbations, within the PXP model. We numerically compute the fidelity and average correlations to monitor the state evolution and to identify revivals. The results indicate that, on the one hand, the entanglement entropy of PXP scars exhibit great sensitivity, in the sense that their profile approaches the ones expected for thermal states already for very small perturbations. On the other hand, other scar signatures, such as the revivals of states having large overlap with scars, show remarkable robustness. Additionally, we examined the effects of minor disturbances on initial states that previously exhibited high overlap with scars and consistent revivals. Our analysis revealed that different types of disturbances can induce markedly different behaviors, such as partially "freezing" the chain, leading to sustained oscillations, or accelerating the process of thermalization.