Eigenstate Thermalization Hypothesis (1805.01616v1)

Abstract: The emergence of statistical mechanics for isolated classical systems comes about through chaotic dynamics and ergodicity. Here we review how similar questions can be answered in quantum systems. The crucial point is that individual energy eigenstates behave in many ways like a statistical ensemble. A more detailed statement of this is named the Eigenstate Thermalization Hypothesis (ETH). The reasons for why it works in so many cases are rooted in the early work of Wigner on random matrix theory and our understanding of quantum chaos. The ETH has now been studied extensively by both analytic and numerical means, and applied to a number of physical situations ranging from black hole physics to condensed matter systems. It has recently become the focus of a number of experiments in highly isolated systems. Current theoretical work also focuses on where the ETH breaks down leading to new interesting phenomena. This review of the ETH takes a somewhat intuitive approach as to why it works and how this informs our understanding of many body quantum states.

• The paper establishes the Eigenstate Thermalization Hypothesis as the mechanism by which non-integrable quantum systems naturally reach thermal equilibrium.
• It employs theoretical analysis and numerical simulations on models like the Hubbard model to show that eigenstate averages converge to microcanonical predictions.
• The findings bridge microscopic quantum behavior and macroscopic thermodynamics, offering insights applicable to quantum information and gravity research.

Overview of Eigenstate Thermalization Hypothesis

The paper by Joshua M. Deutsch offers a comprehensive examination of the Eigenstate Thermalization Hypothesis (ETH) in the context of quantum thermalization. This hypothesis has emerged as a key framework explaining how isolated quantum systems approach thermal equilibrium.

The discussion begins with a historical context, tracing the origins of studies on thermalization back to Schrödinger and Von Neumann. The ETH is discussed as a natural extension in the quest to understand thermalization from a microscopic quantum view. The paper emphasizes an intrinsic quantum feature: the lack of dependence on initial conditions in the equilibration of macroscopic systems.

Theoretical and Numerical Analysis

The ETH posits that for a non-integrable quantum system, the energy eigenstates themselves encode information about thermal equilibrium. Specifically, expectation values of observables, when evaluated in their eigenstates, align closely with microcanonical ensemble averages. This hypothesis suggests that a single eigenstate at a particular energy is representative of thermodynamic equilibrium properties.

Numerical investigations, often involving exact diagonalization of lattice models like the Hubbard model, provide significant insights. These simulations consistently show that in non-integrable systems, fluctuations in these expectation values diminish with increasing system size, satisfying ETH. Conversely, integrable systems exhibit pronounced deviations from the ETH, reflecting their inability to thermalize.

The paper explores the mathematical framework that links random matrix theory to the behavior of non-integrable systems. The core argument is that random matrix statistics govern both the eigenvalues and the eigenstates of complex quantum systems, providing a robust foundation for the ETH.

Implications and Future Directions

The implication of the ETH is profound, bridging the microscopic quantum world with macroscopic thermodynamic laws without invoking an external environment. This underscores a fundamental difference from classical mechanics where ergodicity often accounts for thermalization.

The ETH's theoretical construct extends beyond condensed matter physics, finding applications in quantum information theory and quantum gravity. For instance, it's been linked to understanding phenomena such as black hole thermalization.

Future exploration could explore regime boundaries where the ETH may not hold, particularly in systems exhibiting many-body localization. Moreover, developments in experimental quantum systems, like cold atom setups, present a promising avenue to empirically test ETH predictions, bridging theory and practical observation.

Conclusion

Deutsch's exploration of the ETH formulates a bridge between quantum mechanics and statistical mechanics. Through numerical analyses and theoretical insights, it builds a robust case for the ETH as a linchpin in understanding thermalization in isolated quantum systems. Acknowledging the broader implications of these studies, further experimental and theoretical investigations promise to unravel more of the fundamental aspects of quantum thermalization.