Symmetry-protected Topological Order Research Articles

Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement

Calderbank-Shor-Steane (CSS) codes are a class of quantum error correction codes that contains the toric code and fracton models. A procedure called foliation defines a cluster state for a given CSS code. We use the CSS chain complex and its tensor product with other chain complexes to describe the topological structure in the foliated cluster state, and argue that it has a symmetry-protected topological order protected by generalized global symmetries supported on cycles in the foliated CSS chain complex. We demonstrate the so-called anomaly inflow between CSS codes and corresponding foliated cluster states by explicitly showing the equality of the gauge transformations of the bulk and boundary partition functions defined as functionals of defect world-volumes. We show that the bulk and boundary defects are related via measurement of the bulk system. Further, we provide a procedure to obtain statistical models associated with general CSS codes via the foliated cluster state, and derive a generalization of the Kramers-Wannier-Wegner duality for such statistical models with insertion of twist defects. We also study the measurement-assisted gauging method with cluster-state entanglers for CSS/fracton models based on recent proposals in the literature, and demonstrate a non-invertible fusion of duality operators. Using the cluster-state entanglers, we construct the so-called strange correlator for general CSS/fracton models. Finally, we introduce a new family of subsystem-symmetric quantum models each of which is self-dual under the generalized Kramers-Wannier-Wegner duality transformation, which becomes a non-invertible symmetry.

Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation

Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and provide a detailed discussion of its non-invertible symmetries. We demonstrate that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model with one rough boundary and one smooth boundary. We also discuss the generalization of the Hopf cluster state model to the Hopf ladder model through symmetry topological field theory. Furthermore, we introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra, which can be used to solve the Hopf cluster state model.

Higher-Order Cellular Automata Generated Symmetry-Protected Topological Phases and Detection Through Multi Point Strange Correlators

In computer and system sciences, higher-order cellular automata (HOCA) are a type of cellular automata that evolve over multiple time steps and generate complex patterns, which have various applications, such as secret-sharing schemes, data compression, and image encryption. In this paper, we introduce HOCA to quantum many-body physics and construct a series of symmetry-protected topological (SPT) phases of matter, in which symmetries are supported on a great variety of subsystems embbeded in the SPT bulk. We call these phases HOCA-generated SPT (HGSPT) phases. Specifically, we show that HOCA can generate not only well-understood SPTs with symmetries supported on either regular (e.g., linelike subsystems in the two-dimensional cluster model) or fractal subsystems, but also a large class of unexplored SPTs with symmetries supported on more choices of subsystems. One example is that has either fractal and linelike subsystem symmetries simultaneously or two distinct types of fractal symmetries simultaneously. Another example is in which chaotic-looking symmetries are significantly different from and thus cannot reduce to fractal or regular subsystem symmetries. We also introduce a new notation system to characterize HGSPTs. We prove that all possible subsystem symmetries in a square lattice can be locally simulated by an HOCA-generated symmetry. As the usual two-point strange correlators are trivial in most HGSPTs, we find that the nontrivial SPT orders can be detected by what we call . We propose a universal procedure to design the spatial configuration of the multi point strange correlators for a given HGSPT phase. Specifically, we find deep connections between multi point strange correlators and the spurious topological entanglement entropy (STEE), both exhibiting long-range behavior in a short-range entangled state. Our HOCA approaches and multi point strange correlators pave the way for a unified paradigm to design, classify, and detect phases of matter with symmetries supported on a great variety of subsystems, and also provide potential useful perspective in surpassing the computational irreducibility of HOCA in a quantum mechanical way. Published by the American Physical Society 2024

Symmetry-Enforced Many-Body Separability Transitions

We study quantum many-body mixed states with a symmetry from the perspective of , i.e., whether a mixed state can be expressed as an ensemble of short-range-entangled symmetric pure states. We provide evidence for “symmetry-enforced separability transitions” in a variety of states, where in one regime the mixed state is expressible as a convex sum of symmetric short-range-entangled pure states, while in the other regime, such a representation is not feasible. We first discuss the Gibbs state of Hamiltonians that exhibit spontaneous breaking of a discrete symmetry, and argue that the associated thermal phase transition can be thought of as a symmetry-enforced separability transition. Next we study cluster states in various dimensions subjected to local decoherence, and identify several distinct mixed-state phases and associated separability phase transitions, which also provides an alternative perspective on recently discussed “average symmetry-protected topological order.” We also study decohered p+ip superconductors, and find that if the decoherence breaks the fermion parity explicitly, then the resulting mixed state can be expressed as a convex sum of nonchiral states, while a fermion parity–preserving decoherence results in a phase transition at a nonzero threshold that corresponds to spontaneous breaking of fermion parity. Finally, we briefly discuss systems that satisfy the no low-energy trivial state property, such as the recently discovered good low-density parity-check codes, and argue that the Gibbs state of such systems exhibits a temperature-tuned separability transition. Published by the American Physical Society 2024

Edge Modes and Symmetry-Protected Topological States in Open Quantum Systems

Topological order offers possibilities for processing quantum information that can be immune to imperfections. However, the question of its stability out of equilibrium is relevant for experiments, where coupling to an environment is unavoidable. In this work, we demonstrate the robustness of certain aspects of Z2×Z2 symmetry-protected topological (SPT) order against a wide class of dissipation channels in the Lindblad and quantum trajectory formalisms of an open quantum system. This is illustrated using the one-dimensional ZXZ cluster Hamiltonian along with Pauli-string jump operators. We show that certain choices of dissipation retaining strong symmetries support a steady-state manifold consisting of two nonlocal qubits and for Hamiltonian perturbations preserving the global symmetry, states in this manifold remain metastable. In contrast, this metastability is destroyed upon breaking the above-mentioned symmetry. While the localized qubits of the cluster Hamiltonian are not conserved by the Lindbladian evolution, they do correspond to weak symmetries and thus retain a memory of their initial state at all times in the quantum trajectories. We utilize this feature to construct protocols to retrieve the quantum information either by monitoring jumps or error mitigation. Our work thus proposes a novel framework to study the dynamics of dissipative SPT phases and opens up the possibility of engineering entangled states relevant to quantum information processing. Published by the American Physical Society 2024

Symmetry-protected topological order as a requirement for measurement-based quantum gate teleportation

Symmetry-protected topological order as a requirement for measurement-based quantum gate teleportation

Universal quantum computing models:a resource-theoretic study

Quantum computing has been proven to be powerful, however, there are still great challenges for building real quantum computers due to the requirements of both fault-tolerance and universality. People still lack a systematic way to design fast quantum algorithms and identify the key quantum resources. In this work, we develop a resource-theoretic approach to characterize universal quantum computing models and the universal resources for quantum computing.<br>Our theory combines the framework of universal quantum computing model (UQCM) and the quantum resource theory (QRT). The former has played major roles in quantum computing, while the later was developed mainly for quantum information theory. Putting them together proves to be 'win-win': on one hand, using QRT can provide a resource-theoretic characterization of a UQCM, the relation among models and inspire new ones, and on the other hand, using UQCM offers a framework to apply resources, study relation among resources and classify them.<br>In quantum theory, we mainly study states, evolution, observable, and probability from measurements, and this motivates the introduction of different families of UQCMs. A family also includes generations depending on a hierarchical structure of resource theories. We introduce a table of UQCMs by first classifying two categories of models: one referring to the format of information, and one referring to the logical evolution of information requiring quantum error-correction codes. Each category contains a few families of models, leading to more than one hundred of them in total. Such a rich spectrum of models include some well-known ones that people use, such as the circuit model, the adiabatic model, but many of them are relatively new and worthy of more study in the future. Among them are the models of quantum von Neumann architectures established recently. This type of architecture or model circumvents the no-go theorems on both the quantum program storage and quantum control unit, enabling the construction of more complete quantum computer systems and high-level programming.<br>Correspondingly, each model is captured by a unique quantum resource. For instance, in the state family, the universal resource for the circuit model is coherence, for the local quantum Turing machine is bipartite entanglement, and for the cluster-state based, also known as measurement-based model is a specific type of entanglement relevant to symmetry-protected topological order. As program-storage is a central feature of the quantum von Neumann architecture, we find the quantum resources for it are quantum memories, which are dynamical resources closely related to entanglement. In other words, our classification of UQCMs also serves as a computational classification of quantum resources. This can be used to resolve the dispute over the computing power of resources, such as interference, entanglement, or contextuality. In all, we believe our theory lays down a solid framework to study computing models, resources, and design algorithms.

Dissipative symmetry-protected topological order

In this work, we investigate the interplay between dissipation and symmetry-protected topological order. We considered the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki model interacting with an environment where the dissipative dynamics are described by the Lindladian master equation. The Markovian dynamics is solved by the implementation of a tensor network algorithm for mixed states in the thermodynamic limit. We observe that, for time-reversal symmetric dissipation, the resulting steady state has topological signatures even if being a mixed state. This is seen in finite string-order parameters as well as in the degeneracy pattern of singular values in the tensor network decomposition of the reduced density matrix. We also show that such features do not appear for nonsymmetric dissipation. Our work opens the way toward a generalized and more practical definition of symmetry-protected topological order for mixed states induced by dissipation.

Model-Independent Learning of Quantum Phases of Matter with Quantum Convolutional Neural Networks.

Quantum convolutional neural networks (QCNNs) have been introduced as classifiers for gapped quantum phases of matter. Here, we propose a model-independent protocol for training QCNNs to discover order parameters that are unchanged under phase-preserving perturbations. We initiate the training sequence with the fixed-point wave functions of the quantum phase and add translation-invariant noise that respects the symmetries of the system to mask the fixed-point structure on short length scales. We illustrate this approach by training the QCNN on phases protected by time-reversal symmetry in one dimension, and test it on several time-reversal symmetric models exhibiting trivial, symmetry-breaking, and symmetry-protected topological order. The QCNN discovers a set of order parameters that identifies all three phases and accurately predicts the location of the phase boundary. The proposed protocol paves the way toward hardware-efficient training of quantum phase classifiers on a programmable quantum processor.

Construction of Non-Hermitian Parent Hamiltonian from Matrix Product States.

There are various research strategies used for non-Hermitian systems, which typically involve introducing non-Hermitian terms to preexisting Hermitian Hamiltonians. It can be challenging to directly design non-Hermitian many-body models that exhibit unique features not found in Hermitian systems. In this Letter, we propose a new method for constructing non-Hermitian many-body systems by generalizing the parent Hamiltonian method into non-Hermitian regimes. This allows us to build a local Hamiltonian using given matrix product states as its left and right ground states. We demonstrate this method by constructing a non-Hermitian spin-1 model from the asymmetric Affleck-Kennedy-Lieb-Tasaki state, which preserves both chiral order and symmetry-protected topological order. Our approach opens up a new paradigm for systematically constructing and studying non-Hermitian many-body systems, providing guiding principles for exploring new properties and phenomena in non-Hermitian physics.

Measurement-based quantum simulation of Abelian lattice gauge theories

Numerical simulation of lattice gauge theories is an indispensable tool in high energy physics, and their quantum simulation is expected to become a major application of quantum computers in the future. In this work, for an Abelian lattice gauge theory in dd spacetime dimensions, we define an entangled resource state (generalized cluster state) that reflects the spacetime structure of the gauge theory. We show that sequential single-qubit measurements with the bases adapted according to the former measurement outcomes induce a deterministic Hamiltonian quantum simulation of the gauge theory on the boundary. Our construction includes the (2+1)(2+1)-dimensional Abelian lattice gauge theory simulated on three-dimensional cluster state as an example, and generalizes to the simulation of Wegner’s lattice models M_{(d,n)}M(d,n) that involve higher-form Abelian gauge fields. We demonstrate that the generalized cluster state has a symmetry-protected topological order with respect to generalized global symmetries that are related to the symmetries of the simulated gauge theories on the boundary.Our procedure can be generalized to the simulation of Kitaev’s Majorana chain on a fermionic resource state. We also study the imaginary-time quantum simulation with two-qubit measurements and post-selections, and a classical-quantum correspondence, where the statistical partition function of the model M_{(d,n)}M(d,n) is written as the overlap between the product of two-qubit measurement bases and the wave function of the generalized cluster state.

Symmetries and field tensor network states

We study the interplay between symmetry representations of the physical and virtual space on the class of tensor network states for critical spins systems known as field tensor network states (fTNSs). These are by construction infinite-dimensional tensor networks whose virtual space is described by a conformal field theory (CFT). We can represent a symmetry on the physical index as a commutator with the corresponding CFT current on the virtual space. By then studying this virtual space representation we can learn about the critical symmetry-protected topological properties of the state, akin to the classification of symmetry-protected topological order for matrix product states. We use this to analytically derive the critical symmetry-protected topological properties of the two ground states of the Majumdar-Ghosh point with respect to the previously defined symmetries.

Learning of error statistics for the detection of quantum phases

We present a binary classifier based on neural networks to detect gapped quantum phases. By considering the errors on top of a suitable reference state describing the gapped phase, we show that a neural network trained on the errors can capture the correlation between the errors and can be used to detect the phase boundaries of the gapped quantum phase. We demonstrate the application of the method for matrix product state calculations for different quantum phases exhibiting local symmetry-breaking order, symmetry-protected topological order, and intrinsic topological order.

Emergent higher-symmetry protected topological orders in the confined phase of U(1) gauge theory

We consider compact ${U}^{\ensuremath{\kappa}}(1)$ gauge theory in $3+1$ dimensions with a general $2\ensuremath{\pi}$-quantized topological term ${\ensuremath{\sum}}_{I,J=1}^{\ensuremath{\kappa}}\frac{{K}_{IJ}}{4\ensuremath{\pi}}{\ensuremath{\int}}_{{M}^{4}}{F}^{I}\ensuremath{\wedge}{F}^{J}$, where $K$ is an integer symmetric matrix with even diagonal elements and ${F}^{I}=d{A}^{I}$. At energies below the gauge charges' gaps but above the monopoles' gaps, this field theory has an emergent ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry, where ${k}_{i}$ are the diagonal elements of the Smith normal form of $K$ and ${\mathbb{Z}}_{0}^{(1)}$ is regarded as a $U(1)$ 1-symmetry. In the ${U}^{\ensuremath{\kappa}}(1)$ confined phase, the boundary can have a phase whose infrared (IR) properties are described by Chern-Simons field theory. Such a phase has a ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry that can be anomalous. To show these results, we develop a bosonic lattice model whose IR properties are described by this continuum field theory, thus acting as its ultraviolet completion. The lattice model in the aforementioned limit has an exact ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry. We find that the short-range entangled gapped phase of the lattice model, corresponding to the confined phase of the ${U}^{\ensuremath{\kappa}}(1)$ gauge theory, is a symmetry protected topological (SPT) phase for the ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry, whose SPT invariant is $\phantom{\rule{1.0pt}{0ex}}{e}^{\phantom{\rule{1.0pt}{0ex}}i\phantom{\rule{1.0pt}{0ex}}\ensuremath{\pi}{\ensuremath{\sum}}_{I,J}{K}_{IJ}{\ensuremath{\int}}_{{\mathcal{M}}^{4}}{B}_{I}\ensuremath{\smile}{B}_{J}+{B}_{I}\underset{1}{\ensuremath{\smile}}\phantom{\rule{1.0pt}{0ex}}d{B}_{J}}\phantom{\rule{1.0pt}{0ex}}{e}^{\phantom{\rule{1.0pt}{0ex}}i\phantom{\rule{1.0pt}{0ex}}\ensuremath{\pi}{\ensuremath{\sum}}_{I<J}{K}_{IJ}{\ensuremath{\int}}_{{\mathcal{M}}^{4}}\phantom{\rule{1.0pt}{0ex}}d{B}_{I}\underset{2}{\ensuremath{\smile}}\phantom{\rule{1.0pt}{0ex}}d{B}_{J}}$. Here, the background $\mathbb{R}/\mathbb{Z}$-valued 2-cochains ${B}_{I}$ satisfy $\phantom{\rule{1.0pt}{0ex}}d{B}_{I}={\ensuremath{\sum}}_{I}{B}_{I}{K}_{IJ}=0\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.28em}{0ex}}1$ and describe the symmetry twist of the ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry. We apply this general result to a few examples with simple $K$ matrices. We find the nontrivial SPT order in the confined phases of these models and discuss its classifications using the fourth cohomology group of the corresponding 2-group.

Symmetry Protected Topological Order in Open Quantum Systems

We systematically investigate the robustness of symmetry protected topological (SPT) order in open quantum systems by studying the evolution of string order parameters and other probes under noisy channels. We find that one-dimensional SPT order is robust against noisy couplings to the environment that satisfy a strong symmetry condition, while it is destabilized by noise that satisfies only a weak symmetry condition, which generalizes the notion of symmetry for closed systems. We also discuss "transmutation" of SPT phases into other SPT phases of equal or lesser complexity, under noisy channels that satisfy twisted versions of the strong symmetry condition.

Topological order and entanglement dynamics in the measurement-only XZZX quantum code

We examine the dynamics of a $(1+1)$-dimensional measurement-only circuit defined by the stabilizers of the [[5,1,3]] quantum error correcting code interrupted by single-qubit Pauli measurements. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled state with a ${D}_{2}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a twofold bulk degeneracy. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase. Allowing more than one type of Pauli measurement increases the measurement-induced frustration and the SPT and SB order can be broken either simultaneously or separately at nonzero measurement rate. This yields a rich phase diagram and unanticipated critical behavior at the phase transitions. Although the correlation length exponent $\ensuremath{\nu}=\frac{4}{3}$ and the dynamical critical exponent $z=1$ are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take noninteger multiples of the percolation value. Remarkably, we identify a robust transient scaling regime for the purification dynamics of $L$ qubits. It reveals a modified dynamical critical exponent ${z}^{*}\ensuremath{\ne}z$, which is observable up to times $t\ensuremath{\sim}{L}^{{z}^{*}}$ and is reminiscent of the relaxation of critical systems into a prethermal state.

Topological edge modes without symmetry in quasiperiodically driven spin chains

We construct an example of a 1d quasiperiodically driven spin chain whose edge states can coherently store quantum information, protected by a combination of localization, dynamics, and topology. In a sharp departure from topological phases in static and periodically driven (Floquet) spin chains, this model does not rely upon microscopic symmetry protection: Instead, the edge states are protected purely by emergent dynamical symmetries. We explore the dynamical signatures of this emergent dynamical symmetry-protected topological (EDSPT) order through exact numerics, time evolving block decimation, and analytic high-frequency expansion, finding evidence that the EDSPT is a stable dynamical phase protected by bulk many-body localization up to (at least) stretched-exponentially long timescales, and possibly beyond. We argue that EDSPTs are special to the quasiperiodically driven setting, and cannot arise in Floquet systems. Moreover, we find evidence of a type of boundary critical with no known static or Floquet analogue, in which the edge spin dynamics transition from quasiperiodic to chaotic, leading to bulk thermalization.

Observation of fractional edge excitations in nanographene spin chains.

Fractionalization is a phenomenon in which strong interactions in a quantum system drive the emergence of excitations with quantum numbers that are absent in the building blocks. Outstanding examples are excitations with charge e/3 in the fractional quantum Hall effect1,2, solitons in one-dimensional conducting polymers3,4 and Majorana states in topological superconductors5. Fractionalization is also predicted to manifest itself in low-dimensional quantum magnets, such as one-dimensional antiferromagnetic S = 1 chains. The fundamental features of this system are gapped excitations in the bulk6 and, remarkably, S = 1/2 edge states at the chain termini7-9, leading to a four-fold degenerate ground state that reflects the underlying symmetry-protected topological order10,11. Here, we use on-surface synthesis12 to fabricate one-dimensional spin chains that contain the S = 1 polycyclic aromatic hydrocarbon triangulene as the building block. Using scanning tunnelling microscopy and spectroscopy at 4.5 K, we probe length-dependent magnetic excitations at the atomic scale in both open-ended and cyclic spin chains, and directly observe gapped spin excitations and fractional edge states therein. Exact diagonalization calculations provide conclusive evidence that the spin chains are described by the S = 1 bilinear-biquadratic Hamiltonian in the Haldane symmetry-protected topological phase. Our results open a bottom-up approach to study strongly correlated phases in purely organic materials, with the potential for the realization of measurement-based quantum computation13.

Symmetry-protected dissipative preparation of matrix product states

We propose and analyze a method for efficient dissipative preparation of matrix product states that exploits their symmetry properties. Specifically, we construct an explicit protocol that makes use of driven-dissipative dynamics to prepare a many-body quantum state that features symmetry-protected topological order and nontrivial edge excitations. The preparation protocol is protected from errors that respect the symmetry, allowing for robust experimental implementation without fine-tuned control. Numerical simulations show that the preparation time scales polynomially in system size $n$. Furthermore, we demonstrate that this scaling can be improved to $O({log}^{2}n)$ by using parallel preparation of individual segments and fusing them via quantum feedback. A concrete scheme using excitation of trapped neutral atoms into the Rydberg state via electromagnetically induced transparency is proposed and generalizations to a broader class of matrix product states are discussed.

Quantum scars and bulk coherence in a symmetry-protected topological phase

Formation of quantum scars in many-body systems provides a novel mechanism for enhancing coherence of weakly entangled states. At the same time, coherence of edge modes in certain symmetry protected topological (SPT) phases can persist away from the ground state. In this work we show the existence of many-body scars and their implications on bulk coherence in such an SPT phase. To this end, we study the eigenstate properties and the dynamics of an interacting spin-$1/2$ chain with three-site "cluster" terms hosting a $\mathbb{Z}_2 \times \mathbb{Z}_2$ SPT phase. Focusing on the weakly interacting regime, we find that eigenstates with volume-law entanglement coexist with area-law entangled eigenstates throughout the spectrum. We show that a subset of the latter can be constructed by virtue of repeated cluster excitations on the even or odd sublattice of the chain, resulting in an equidistant "tower" of states, analogous to the phenomenology of quantum many-body scars. We further demonstrate that these scarred eigenstates support nonthermal expectation values of local cluster operators in the bulk and exhibit signatures of topological order even at finite energy densities. Studying the dynamics for out-of-equilibrium states drawn from the noninteracting "cluster basis", we unveil that nonthermalizing bulk dynamics can be observed on long time scales if clusters on odd and even sites are energetically detuned. In this case, cluster excitations remain essentially confined to one of the two sublattices such that inhomogeneous cluster configurations cannot equilibrate and thermalization of the full system is impeded. Our work sheds light on the role of quantum many-body scars in preserving SPT order at finite temperature and the possibility of coherent bulk dynamics in models with SPT order beyond the existence of long-lived edge modes.