Microscopic Foundation Of The Eigenstate Thermalization Hypothesis
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Microscopic Foundation of the Eigenstate Thermalization Hypothesis
The Eigenstate Thermalization Hypothesis (ETH) explains how isolated quantum many-body systems thermalize by proposing that each energy eigenstate is already thermal. The hypothesis is believed to be essential to the understanding of quantum chaos and implies various important thermodynamic relations. While by now there are many numerical validations of the ETH, only few analytical arguments supporting the ETH have been found (M. Rigol et al., Nature 452, 2008). One of them is based on a semiclassical limit (M. Srednicki, Phys. Rev. E 50, 1994). Another important reasoning was given by Deut...
Equilibrium and Nonequilibrium Statistical Mechanics: Principles and Concepts
Equilibrium and Non-equilibrium Statistical Mechanics is a source-book of great value to college and university students embarking upon a serious reading of Statistical Mechanics, and is likely to be of interest to teachers of the subject as well. Written in a lucid style, the book builds up the subject from basics, and goes on to quite advanced and modern developments, giving an overview of the entire framework of statistical mechanics. The equilibrium ensembles of quantum and classical statistical mechanics are introduced at length, indicating their relation to equilibrium states of thermodynamic systems, and the applications of these ensembles in the case of the ideal gas are worked out, pointing out the relevance of the ideal gas in respect of a number of real-life systems. The application to interacting systems is then taken up by way of explaining the virial expansion of a dilute gas. The book then deals with a number of foundational questions relating to the existence of the thermodynamic limit and to the equivalence of the various equilibrium ensembles. The relevance of the thermodynamic limit in explaining phase transitions is indicated with reference to the Yang-Lee theory and the Kirkwood-Salsburg equations for correlation functions. The statistical mechanics of interacting systems is then taken up again, with reference to the 1D and 2D Ising model and to the spin glass model of disordered systems. Applications of the Mean field theory are worked out, explaining the Landau-Ginzburg theory, which is then followed by the renormalization group approach to phase transitions. Interacting systems in the quantum context are referred to, addressing separately the cases of interacting bosons and fermions. The case of the weakly interacting bosons is explained in details, while the Landau theory for fermi liquids is also explained in outline. The book then goes on to a modern but readable account of non-equilibrium statistical mechanics, explaining the link with irreversible thermodynamcs. After an exposition of the Boltzmann equations and the linear response theory illustrated with reference to the hydrodynamic model, it explains the statistical mechanics of reduced systems, in the context of a number of reduction schemes. This is followed by an account of the relevance of dynamical chaos in laying down the foundations of classical statistical mechanics, where the SRB distributon is introduced in the context of non-equilibrium steady states, with reference to which the principle of minimum entropy production is explaned. A number of basic fluctuation relations are then worked out, pointing out their relation to irreversible thermodynamics. Finally, the book explains the relevance of quantum chaos in addressing foundational issues in quantum statistical mechanics, beginning with Berry’s conjecture and then going on to an exposition of the eigenstate thermalization (ETH) hypothesis, indicating how the latter is relevant in explaining the processes of equilibriation and thermalization in thermodynamic systems and their sub-systems.
Higher-Form Symmetry and Eigenstate Thermalization Hypothesis
The eigenstate thermalization hypothesis (ETH) is a successful framework providing criteria for thermalization in isolated quantum systems. Although numerical and theoretical analyses support the ETH as a fundamental mechanism for explaining thermalization in diverse systems, it remains a challenge to analytically identify whether particular systems satisfy the ETH. In quantum many-body systems and quantum field theories, phenomena that violate the ETH are expected to imply nontrivial thermalization processes, and are gathering increasing attention. This book elucidates how the existence of higher-form symmetries influences the dynamics of thermalization in isolated quantum systems. Under reasonable assumptions, it is analytically shown that a p-form symmetry in a (d+1)-dimensional quantum field theory leads to the breakdown of the ETH for many nontrivial (d−p)-dimensional observables. In the case of discrete higher-form (i.e., p ≥ 1) symmetry, this indicates the absence of thermalization for observables that are non-local but much smaller than the entire system size even though the system do have no local conserved quantities. The author provides numerical evidence for this argument for the (2+1)-dimensional Z2 lattice gauge theory. While local observables such as a plaquette operator thermalize even for mixed symmetry sectors, the non-local observable such as the one exciting a magnetic dipole instead relaxes to the generalized Gibbs ensemble that takes account of the Z2 1-form symmetry. The assumptions of the ETH-violation include the mixing of symmetry sectors within a given energy shell. This condition is rather challenging to verify because it requires information on the eigenstates in the middle of the spectrum. In the subsequent chapter, we further reconsider this assumption from the viewpoint of a projective phase to alleviate this difficulty. In the case of ZN symmetries, the difficulty can be circumvented considering ZN×ZN-symmetric theories with a projective phase, and then perturbing the Hamiltonian while preserving one of the ZN symmetries of interest. Additionally, the book also presents numerical analyses for (1+1)-dimensional spin chains and the (2+1)-dimensional Z2 lattice gauge theory to demonstrate this scenario.