Quantum Many-Body Scars and Hilbert Space Fragmentation: A Review of Exact Results (2109.00548v2)

Abstract: The discovery of Quantum Many-Body Scars (QMBS) both in Rydberg atom simulators and in the Affleck-Kennedy-Lieb-Tasaki (AKLT) spin-1 chain model, have shown that a weak violation of ergodicity can still lead to rich experimental and theoretical physics. In this review, we provide a pedagogical introduction to and an overview of the exact results on weak ergodicity breaking via QMBS in isolated quantum systems with the help of simple examples such as the fermionic Hubbard model. We also discuss various mechanisms and unifying formalisms that have been proposed to encompass the plethora of systems exhibiting QMBS. We cover examples of equally-spaced towers that lead to exact revivals for particular initial states, as well as isolated examples of QMBS. Finally, we review Hilbert Space Fragmentation, a related phenomenon where systems exhibit a richer variety of ergodic and non-ergodic behaviors, and discuss its connections to QMBS.

• The paper presents exact analytical models showing how a subset of eigenstates, or quantum many-body scars, defies conventional thermalization.
• The paper details how Hilbert space fragmentation partitions quantum systems into disconnected subspaces with distinctive ergodic and non-ergodic behaviors.
• The paper underscores experimental relevance by linking theoretical frameworks to observed long-lived oscillations in Rydberg atom simulators.

Quantum Many-Body Scars and Hilbert Space Fragmentation: A Review of Exact Results

The paper "Quantum Many-Body Scars and Hilbert Space Fragmentation: A Review of Exact Results," authored by Sanjay Moudgalya, B. Andrei Bernevig, and Nicolas Regnault, provides a comprehensive overview of recent advancements in the paper of weak ergodicity breaking in quantum systems. This is accomplished through the detailed exploration of quantum many-body scars (QMBS) and Hilbert space fragmentation, with a focus on exact results and theoretical frameworks.

Quantum Many-Body Scars

QMBS are a fascinating phenomenon wherein a small subset of a system's eigenstates avoids thermalization, persisting with non-ergodic behavior despite the surrounding chaos in the energy spectrum. Such states emerge prominently in models like the PXP model, relevant to Rydberg atom simulators, and have challenged the conventional understanding based on the Eigenstate Thermalization Hypothesis (ETH).

The paper reviews several exact results related to QMBS, starting with their origins in simple, analytically solvable models. Examples such as the AKLT model and $\eta$-pairing states in the Hubbard model serve as foundational cases demonstrating the mechanisms behind QMBS. These systems exhibit towers of equally spaced eigenstates that lead to observable phenomena, such as quantum revivals from specific initial states.

The authors categorize these phenomena under a unifying spectrum generating algebra framework, providing a broad lens to understand the role of weak symmetry-breaking terms that preserve non-thermal states, even within otherwise chaotic spectra.

Hilbert Space Fragmentation

Hilbert space fragmentation is a closely related concept that describes a more extensive partitioning of the Hilbert space into dynamically disconnected subspaces. This phenomenon can be seen as an extension of the QMBS phenomenon, occurring in systems with conserved quantities such as dipole moments or certain higher symmetries.

The paper discusses exact models where fragmentation arises, including examples like spin chains with dipole conservation. These models showcase a rich variety of ergodic and non-ergodic behaviors, elucidated through formalism and backed by exact analytical solutions.

Implications and Future Directions

The implications of this research are manifold. Practically, QMBS have been experimentally observed to produce long-lived oscillations in quantum simulators, as seen with Rydberg atom systems. Theoretically, they challenge the canonical view of quantum thermalization, providing new insights and stimulating further research into quantum dynamics beyond ETH.

The paper suggests several future directions for research. These include exploring higher-dimensional systems, understanding the stability of QMBS against perturbations, and expanding the unifying frameworks to accommodate more diverse quantum systems. The authors also highlight the need for extending the paper of fragmentation and examining its implications for quantum dynamics.

In conclusion, this review provides a detailed and critical examination of QMBS and Hilbert space fragmentation, revealing a landscape of complex and intriguing quantum behaviors. These phenomena open new avenues for both theoretical exploration and experimental investigation, promising to advance our understanding of quantum many-body systems significantly.