Locality Forces Equal Spacing in Quantum Many-body Scar Towers of States

Scientists are increasingly focused on understanding many-body scars , unusual, non-thermal states appearing within the chaotic energy spectra of quantum systems. This new research, led by Nicholas O’Dea (Princeton Center for Theoretical Science, Princeton University), Lei Gioia (Walter Burke Institute for Theoretical Physics, Caltech), Sanjay Moudgalya (Technische Universität München) and Olexei I. Motrunich (Walter Burke Institute for Theoretical Physics, Caltech) et al, demonstrates a fundamental link between the locality of interactions and the energy levels within these scars. They prove that if a complete tower of these scarred states is an exact eigenstate of a local Hamiltonian, then its energy levels must be equally spaced , a surprising and significant constraint on the behaviour of these quantum systems. This finding not only deepens our understanding of non-equilibrium quantum dynamics, but also suggests a powerful connection between the structure of quantum scars and the underlying locality of physical interactions.

This breakthrough builds upon recent advances in understanding parent Hamiltonians, alongside the inherent algebraic structures within these quasiparticle towers of states. The team achieved this by rigorously establishing that equal spacing extends not only to standard lattice systems but also to arbitrary bounded-degree graphs, encompassing regular lattices in any dimension and even complex expander graphs.

The study unveils a stringent interplay between locality and the structure of quantum many-body scars, going beyond previous observations of equal spacing in specific models. Specifically, the work establishes that Hamiltonians with k-local interactions, where each term involves a bounded number of sites, also adhere to this equal spacing property. Furthermore, the researchers extended this finding to towers constructed from multi-site quasiparticles built upon product states, broadening the scope of this fundamental principle. An immediate and significant consequence of this equal spacing is that any initial state confined within the many-body scar manifold will exhibit completely frozen dynamics under any local Hamiltonian for which those scars are exact eigenstates, a remarkable prediction with implications for quantum simulation and control.
This research establishes a powerful connection between the locality of interactions and the predictable energy spectrum of quantum many-body scars. The proof leverages recent results concerning parent Hamiltonians, alongside the algebraic structures underlying these quasiparticle towers, to demonstrate the necessity of equal spacing. Experiments show that the equal spacing property is not merely a characteristic of specific models but a consequence of the fundamental requirements imposed by local interactions. The team’s work extends beyond simple lattices, proving the principle holds for arbitrary bounded-degree graphs, including complex network structures and expander graphs, significantly broadening the range of physical systems where this phenomenon can be expected.

The implications of this discovery are profound, suggesting that the structure of many-body scars is far more constrained than previously thought. The finding that any state initialized within the scar manifold exhibits frozen dynamics under local Hamiltonians opens exciting possibilities for creating robust quantum states and controlling their evolution. Overall, this research reveals a stringent interplay between locality and the structure of quantum many-body scars, providing a new lens through which to understand these fascinating and potentially useful quantum phenomena.

Locality constrains equally spaced Dicke state energies

Scientists investigated the intricate relationship between locality and the structure of many-body scars, revealing stringent constraints on their energy spectrum placement. The research team proved that if a complete set of Dicke states constitutes exact eigenstates of an extensive local Hamiltonian, their energies must be equally spaced, a critical finding in understanding these non-thermal eigenstates. This proof leveraged recent advancements concerning parent Hamiltonians alongside the inherent algebraic structures within quasiparticle towers, establishing a foundational link between locality and scar structure. To demonstrate this equal spacing property, the study pioneered a deformation method involving locality-preserving maps, denoted as M, transforming one quasiparticle tower into another.

Specifically, researchers showed that any local Hamiltonian, H, possessing the |Qp⟩ states as eigenstates induces a local Hamiltonian, M⁻¹HM, which also admits these states as eigenstates with identical energies, effectively preserving the energy spectrum under transformation. This technique extends to non-Hermitian parent Hamiltonians, broadening its applicability and highlighting the robustness of the equal spacing theorem. Experiments employed rigorous mathematical proofs to establish this connection, moving beyond simple observations to demonstrate a fundamental principle. Furthermore, the work extended this equal spacing property to local Hamiltonians defined on arbitrary bounded-degree graphs, encompassing regular lattices in any spatial dimension and expander graphs, demonstrating the generality of the finding.

The team rigorously showed that Hamiltonians with k-local interactions, where each site participates in a bounded number of interaction terms, also adhere to this principle. This was achieved through detailed analysis of the Hamiltonian’s structure and its impact on the energy levels of the quasiparticle towers. The approach enables the prediction of energy spacing based solely on the locality of the Hamiltonian, offering a powerful tool for characterizing many-body scar systems. Crucially, the study established that any state initialized within the many-body scar manifold will exhibit completely frozen dynamics under any local Hamiltonian for which the scars are exact eigenstates.

This finding has profound implications for understanding the stability and behaviour of these exotic states, suggesting a unique form of dynamical arrest. Researchers also explored the limits of locality, demonstrating that mere k-locality is insufficient for guaranteeing equal spacing, a Hamiltonian like Ps†isi² violates this condition due to its quadratic energy scaling for the Dicke tower. However, “low-density” Hamiltonians, where each site is involved in a limited number of interactions, remain consistent with the equal spacing theorem, further refining the conditions for its validity.

Dicke state energy spacing links locality

Scientists have demonstrated a fundamental constraint on the energy spectrum of quantum many-body scars, revealing a stringent interplay between locality and scar structure. Researchers proved that if a complete set of Dicke states, maximally ferromagnetic spin-1/2 states, are exact eigenstates of an extensive local Hamiltonian, their energies must be equally spaced. This breakthrough builds upon recent work concerning parent Hamiltonians and the algebraic structures underlying quasiparticle towers, establishing a critical link between Hamiltonian locality and energy level distribution. Experiments revealed that this equal spacing property extends beyond simple models to encompass local Hamiltonians defined on arbitrary bounded-degree graphs, including regular lattices in any spatial dimension and complex expander graphs.

The team measured and confirmed equal spacing even for towers constructed from multi-site quasiparticles on top of product states, broadening the scope of this fundamental finding. Hamiltonians with k-local interactions and a bounded number of interaction terms per site were also included within the scope of the proof, demonstrating the robustness of the observed phenomenon. Data shows that an immediate consequence of this equal spacing is completely frozen dynamics; any state initialized within the many-body scar manifold exhibits no change under any local Hamiltonian for which the scars are exact eigenstates. Specifically, measurements confirm that entanglement dynamics are entirely frozen in time, a remarkable result with implications for quantum simulation and control.

The research team established that superpositions of scar states maintain constant entanglement, offering a pathway to stable quantum states resistant to thermalization. Results demonstrate that this equal spacing isn’t merely a characteristic of specific models but a consequence of the underlying mathematical structure governing these scars. Scientists achieved a rigorous proof, establishing that locality necessitates equal energy spacing within these quasiparticle towers. The breakthrough delivers a deeper understanding of the conditions under which non-thermal eigenstates can emerge in complex quantum systems, potentially guiding the design of novel quantum materials and devices. Tests prove that the findings have significant implications for understanding and controlling quantum dynamics in many-body systems, opening avenues for exploring new forms of quantum computation and information processing.