Ergodicity-breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians (1904.04266v2)

Abstract: We show that the combination of charge and dipole conservation---characteristic of fracton systems---leads to an extensive fragmentation of the Hilbert space, which in turn can lead to a breakdown of thermalization. As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total $S^z$) and its associated dipole moment. First, we consider a minimal model including only three-site terms and find that the infinite temperature auto-correlation saturates to a finite value---showcasing non-thermal behavior. The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local $S^z$ basis, arising from the interplay of dipole conservation and local interactions. Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: the system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states. More generally, for any finite range of interactions, the system still exhibits non-thermal eigenstates appearing throughout the entire spectrum. We compare our results to charge and dipole moment conserving random unitary circuit models for which we reach identical conclusions.

• The paper reveals that Hilbert space fragmentation from dipole conservation causes ergodicity breaking, with finite saturation in auto-correlation functions.
• The paper uses a minimal three-site interaction model and additional four-site terms to distinguish transitions between non-ergodic and thermalizing regimes.
• The paper highlights that the interplay between conserved quantities and interaction range critically determines the emergence of frozen states and non-thermal behavior.

Ergodicity-Breaking in Dipole-Conserving Hamiltonians

The paper presented in this paper explores the intricate dynamics of one-dimensional spin-1 models that conserve both charge and dipole moment, revealing significant insights into ergodicity-breaking mechanisms caused by Hilbert space fragmentation. The research primarily investigates the consequences of these conservation laws in quantum systems, particularly focusing on their impact on thermalization processes. The authors present a comprehensive analysis, starting with a minimal model consisting of three-site interactions and extending to systems with additional four-site terms, thus unveiling a spectrum of behaviors from non-ergodic to thermalizing regimes.

Theoretical Insights and Model Description

The crux of the paper lies in understanding how specific symmetries in Hamiltonians can lead to ergodicity-breaking. The authors consider spin-1 systems with Hamiltonians that conserve both charge $Q$ and dipole moment $P$. This conservation is a haLLMark of fracton systems, which exhibit constrained dynamics due to the localized nature of their excitations. The paper begins with a minimal Hamiltonian, $H_3$, composed solely of three-site terms. The model exhibits a striking fragmentation of the Hilbert space into an exponential number of invariant subspaces. This fragmentation serves as the primary mechanism for the observed breakdown in thermalization, manifested as saturation to finite values in infinite-temperature auto-correlation functions.

Numerical Results and Analysis

The numerical simulations reveal that the infinite-temperature auto-correlation function for the $H_3$ model saturates to finite values that do not diminish with increasing system size, which deviates from expectations based on the Eigenstate Thermalization Hypothesis (ETH). This non-ergodic behavior is attributed to a prolific fragmentation of the Hilbert space into many independent subspaces, including 'frozen' states that are product states in the local $S^z$-basis. These frozen states remain invariant under the Hamiltonian dynamics, thereby hindering thermalization.

Upon extending the model to include four-site interactions ($H_3 + H_4$), the results indicate a dilution of the fragmentation effect. The additional terms contribute to a 'weak fragmentation,' where although exponentially many invariant subspaces persist, they are no longer dominant enough to prevent thermalization at typical initial states. This transition highlights a critical point: the addition of longer-range interactions alters the connectivity of invariant subspaces within the Hilbert space, facilitating thermal behavior.

Implications and Future Directions

This research adds a significant dimension to understanding how conservation laws can affect the dynamics in quantum many-body systems. The distinction between strong and weak fragmentation offers a nuanced perspective on ergodicity-breaking, suggesting that the interplay between conservation laws and interaction ranges is central to dictating the thermalization properties of a system. The presence of non-thermal eigenstates even in systems with longer-ranged interactions could have implications for understanding quantum many-body scars, which relate to non-ergodic behavior amidst an otherwise thermalizing spectrum.

Looking forward, extending these findings to higher-dimensional systems or exploring their manifestation in experimental setups, such as those with strong electric fields or engineered defects, could provide further validation and insights. Moreover, the paper invites further exploration into the classification and characterization of non-local conservation laws and their systematic labeling, which could deepen the connection between ergodicity and symmetries in complex quantum systems.

Conclusion

This paper elucidates how dipole conservation in spin-1 chains leads to ergodicity-breaking through a severe Hilbert space fragmentation. It delineates the transition from strong to weak fragmentation upon introducing longer-range terms, thereby contributing significantly to the rich tapestry of non-ergodic phenomena in quantum mechanics. The work presents a sophisticated understanding that could influence both theoretical perspectives and practical approaches in managing and predicting the dynamic properties of quantum systems.