Summary of "Does a single eigenstate encode the full Hamiltonian?"
The paper "Does a single eigenstate encode the full Hamiltonian?" by Garrison and Grover investigates the implications of the Eigenstate Thermalization Hypothesis (ETH) on the information encoded within an individual eigenstate of a quantum many-body system. The central question of the paper is whether the properties of the entire Hamiltonian can be discerned from a single eigenstate within the system’s energy spectrum.
Background and Scope of the Study
The Eigenstate Thermalization Hypothesis posits that individual eigenstates at a finite energy density in non-integrable systems can yield equilibrium properties akin to those presented by thermal states. This paper extends ETH's implications by exploring which classes of operators (local or non-local) conform with ETH and, importantly, whether the full Hamiltonian's characteristics can be surmised from a single eigenstate.
Methodology
The authors formulate a strong form of ETH, which suggests that the reduced density matrix for any subsystem of an eigenstate should, in the thermodynamic limit, mimic a corresponding thermal reduced density matrix. This premise holds irrespective of the subsystem size relative to intrinsic system length scales, provided it is much smaller than the entire system. To validate their conjecture, the authors engage in exact diagonalization studies, focusing on a specific non-integrable 1D lattice quantum model with only energy conservation.
Additionally, the investigation extends to scenarios where the subsystem is a finite fraction of the total system. Here, the authors explore the validity of ETH for a diverse set of operators, including the von Neumann entanglement entropy, and demonstrate its equivalence with thermal entropy in numerous cases.
Results
The study provides numerical evidence supporting the conjecture that a single eigenstate at finite energy density contains comprehensive information about the system at different energy densities/temperatures. Among their key findings, the von Neumann entropy, calculated from a pure eigenstate, aligns with expected thermal entropy when the subsystem is less than half of the total system. Moreover, the authors demonstrate that, within appropriate constraints, these eigenstates can indeed allow for extrapolating thermodynamic properties of the system at various temperatures.
It’s noteworthy that their analyses identify certain operators, specifically those involving energy conservation, where ETH does not hold when the subsystem to system size ratio is finite, as indicated by discrepancies in subsystem energy variance.
Implications and Future Directions
The implications of this work, if the findings hold universally, provide a pathway to determine the full Hamiltonian properties using analytics on a single eigenstate. This result could simplify calculations in quantum statistical mechanics, requiring only a single eigenstate to understand multi-temperature system behaviors. Moreover, the exploration of non-local operator performance under ETH extends understanding of eigenstate complexity and thermalization processes.
For future work, the authors suggest further probing unequal-time correlators to see how they might similarly reflect the broader system properties from a single eigenstate. Moreover, the extension of results to time-evolved states poses an intriguing area for theoretical exploration, potentially impacting quantum computing and simulation efforts. Given the current experimental trajectory in cold atom and single-spin resolution imaging, these findings also open doors for practical advancements in atomic-scale measurement and control.
In conclusion, this paper provides a significant contribution to quantum statistical mechanics by extending the framework of ETH, emphasizing the potential comprehensive knowledge stored within single eigenstates, and detailing scenarios for future exploration and application.