How accurately can the microcanonical ensemble describe small isolated quantum systems?

Area of Science:

• Quantum physics
• Condensed matter theory
• Statistical mechanics

Background:

• The microcanonical ensemble is a fundamental concept in statistical mechanics, describing isolated quantum systems.
• Understanding the validity of the microcanonical ensemble in small, nonintegrable quantum systems is crucial for theoretical physics.
• The hard-core Bose-Hubbard model provides a tractable platform for studying quantum dynamics.

Purpose of the Study:

• To numerically investigate the accuracy of the microcanonical ensemble in small, isolated quantum systems undergoing quantum quenches.
• To determine how the accuracy of the microcanonical ensemble scales with system size and Hilbert space dimension.
• To identify the underlying physical mechanisms responsible for the observed scaling behaviors.

Main Methods:

• Numerical simulations of quantum quenches in a nonintegrable hard-core Bose-Hubbard model.
• Analysis of the system's behavior across a range of system sizes and Hilbert space dimensions.
• Comparison of simulation results with predictions from the microcanonical ensemble.

Main Results:

• The accuracy of the microcanonical ensemble increases with the dimension of the Hilbert space (D) as 1/D within a specific system size range.
• This rapid improvement is attributed to the absence of correlations between many-body energy eigenstates.
• Outside this range, the accuracy scales as 1/√D or algebraically with system size.

Conclusions:

• The microcanonical ensemble can accurately describe isolated quantum systems after quantum quenches, particularly in regimes where Hilbert space dimension is large.
• The scaling of accuracy provides insights into the role of correlations and eigenstate properties in thermalization.
• These findings have implications for understanding thermalization and statistical mechanics in finite quantum systems.