๐ Understanding the Eigenstate Thermalization Hypothesis (ETH)
The Eigenstate Thermalization Hypothesis (ETH) is a central concept in quantum chaos and quantum statistical mechanics. It attempts to explain how closed quantum systems, when starting in a pure state far from equilibrium, can evolve to a state that is indistinguishable from thermal equilibrium. Instead of relying on external heat baths or ensemble averaging, ETH postulates that the thermal behavior is encoded within the eigenstates of the system's Hamiltonian itself.
๐ History and Background
The groundwork for ETH was laid in the latter half of the 20th century, with significant contributions from Marcos Rigol, Mark Srednicki, and others. It addresses the fundamental question of how quantum mechanics, which is time-reversal symmetric, can give rise to the irreversible behavior described by thermodynamics. ETH offers a resolution by suggesting that the observed thermalization arises from the structure of the Hamiltonian eigenstates, not from external influences or ensemble averaging.
โจ Key Principles of ETH
- โ๏ธ Eigenstate Structure: ETH posits that the expectation value of a physical observable $\hat{A}$ in an energy eigenstate $|E_i\rangle$ is a smooth function of the energy $E_i$: $\langle E_i | \hat{A} | E_i \rangle = A(E_i) + f(E_i)$, where $A(E)$ is a smooth function of energy and $f(E_i)$ is a small, system-specific fluctuation. This means that eigenstates within a narrow energy window have similar expectation values for physical observables.
- ๐งฎ Diagonal Ensemble: The diagonal ensemble is formed by the diagonal elements of the density matrix in the energy eigenbasis. ETH suggests that this ensemble is equivalent to the microcanonical ensemble at the corresponding energy. In other words, the long-time average of an observable is determined by the expectation values of that observable in the energy eigenstates.
- ๐ Off-Diagonal Elements: ETH predicts that off-diagonal matrix elements of observables, $\langle E_i | \hat{A} | E_j \rangle$, are exponentially small in the system size (e.g., number of particles) when $i \neq j$. They fluctuate randomly with zero mean and decay rapidly with increasing energy difference $|E_i - E_j|$. This ensures that there is no coherent superposition of eigenstates that would prevent thermalization.
๐ Real-World Examples and Applications
- ๐ง Quantum Quenches in Cold Atomic Gases: Experiments with ultra-cold atomic gases, where systems are rapidly driven out of equilibrium, have provided evidence supporting ETH. By observing the long-time behavior of observables after a quantum quench, researchers have found that the system thermalizes to a state predicted by ETH.
- ๐ป Numerical Studies of Many-Body Systems: Numerical simulations of interacting spin chains and fermionic systems have demonstrated the validity of ETH in various contexts. These studies involve calculating the energy eigenstates and matrix elements of observables to verify the predictions of ETH.
- ๐ Black Hole Information Paradox: ETH has implications for the black hole information paradox. It suggests that the information about the initial state of matter falling into a black hole is not lost but is scrambled and encoded in the fine-grained correlations of the Hawking radiation. The thermal appearance of the radiation is consistent with ETH.
๐งช Experimental Verification
Direct experimental verification of ETH is challenging due to the difficulty of measuring individual energy eigenstates. However, indirect evidence is often obtained by studying the dynamics of observables after a quantum quench or by comparing the properties of the system to the predictions of statistical mechanics. Experiments often involve observing the relaxation of local observables to their thermal equilibrium values.
๐ก Conclusion
The Eigenstate Thermalization Hypothesis provides a powerful framework for understanding thermalization in isolated quantum systems. It bridges the gap between quantum mechanics and statistical mechanics by attributing thermal behavior to the structure of the Hamiltonian eigenstates. While challenges remain in directly verifying ETH in complex systems, it continues to be a vital area of research in quantum chaos and many-body physics.