UC Santa Barbara

The eigenstate thermalization hypothesis (ETH) has been widely accepted as the mechanism by which isolated non-integrable quantum systems thermalize and has cemented itself as a cornerstone of quantum many-body physics. With advancements in technology enabling the creation of such systems, further exploration and theoretical development of ETH are necessary.

First, we expand ETH to the regime of isolated non-integrable quantum systems with non- Abelian conserved charges. We show that our extension, the non-Abelian eigenstate thermalization hypothesis, indeed predicts thermal expectation values of local observables, filling a crucial gap toward a more general framework.

We further investigate ETH’s validity by examining the structure of observable matrix elements in the energy eigenstate basis. ETH-predicted matrix elements cannot be completely random and independent due to unrealistic consequences such as nonsensical results for the n-point correlation function of an observable. Nevertheless, assuming ETH, we discover a centered Jacobi ensemble distribution for the eigenvalue spectrum of a truncated observable operator in the energy basis. This analytical solution, converging to the Wigner semi-circle for small truncations, reinforces the intuitive notion that ETH applies within a limited energy window. Additionally, it serves as a benchmark for comparing numerical results, enabling the study of the correlations between energy eigenstates of a system.

Next, we delve into a critical inquiry: how can we differentiate between an energy eigenstate conforming to ETH and a genuinely thermal density matrix? Quantum Fisher information (QFI) offers a theoretical tool that distinguishes between these two states. However, the choice of state preparation protocol significantly influences QFI. To address this, we systematically examine the resulting QFI for both an energy eigenstate and a thermal density matrix across diverse experimental protocols.

Lastly, we explore entanglement negativity in the context of chaotic eigenstates. We study phase transitions in a simplified model of entanglement negativity to facilitate analytical tractability. This allows us to establish conditions on the volume fractions for a tripartite system where predictions for the entanglement negativity based on ETH align or deviate from thermal predictions.