Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation

Area of Science:

• Quantum statistical mechanics
• Condensed matter theory
• Many-body physics

Background:

• Macroscopic thermal equilibrium (MATE) describes isolated quantum systems in pure states.
• Thermalization occurs if the Hamiltonian satisfies the eigenstate thermalization hypothesis (ETH), where every eigenvector is in MATE.
• Previous work proved ETH for a perturbed Hamiltonian of free fermions.

Purpose of the Study:

• To investigate thermalization in degenerate quantum systems.
• To prove the ETH for a specific non-perturbed Hamiltonian of free fermions.
• To establish a general condition for thermalization in systems with multiple eigenbases.

Main Methods:

• Mathematical proof of the ETH for the non-degenerate Hamiltonian of free fermions.
• Analysis of the relationship between the existence of one MATE eigenbasis and the prevalence of MATE across other eigenbases.
• Demonstration that a small generic perturbation typically leads to ETH satisfaction and thermalization.

Main Results:

• The ETH is proven to hold for the non-perturbed Hamiltonian of N free fermions, implying all initial states thermalize.
• The existence of a single eigenbasis in MATE generally implies that most eigenbases are also in MATE.
• Generic perturbations of the Hamiltonian lead to ETH satisfaction and universal thermalization of all states.

Conclusions:

• Degenerate Hamiltonians can exhibit thermalization if the ETH holds for all their eigenbases.
• The study provides a general framework for understanding thermalization in quantum systems, linking MATE properties across different eigenbases.
• The findings confirm that generic perturbations ensure ETH and thermalization in quantum many-body systems.