Finite-Size Effects in Integrable Systems - Lecture 3

Explore finite-size effects in integrable systems through this comprehensive lecture by Fabian Essler from the International Centre for Theoretical Sciences. Delve into topics such as relaxation of time averages, the diagonal ensemble, traversals and revivals in finite systems, and macro states in free theories. Examine conservation laws in interacting integrable models, including the string hypothesis and composition principle. Learn about energy eigenstates, occupation numbers, and particle/hole densities in the context of integrable systems. Gain insights into the thermodynamic entropy and generalized micro-canonical ensemble. The lecture concludes with a Q&A session, providing an opportunity to deepen understanding of these advanced concepts in mathematical physics and condensed matter theory.

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Relaxation of time averages & the "Diagonal Ensemble"
It is believed that for local observables
diagonal ensemble
Finite-size effects: traversals & revivals
This "traversal" is a generic effect and is very different from a revival:
Revivals are related to regularities in the spectrum of H:
Example:
Revivals in split ID Bose gases
Exercise: [Requires Mathematica to plot]
VI. Macro states & conservation laws in free theories
Mode occ. #s are conserved & related to extensive cons. laws with local densities
Macro states in the thermodynamic limit
Expectation values of local operators in macro states
Generalized Micro-Canonical Ensemble
Counting the number of micro states in a large, finite L thhat correspond to the same macro state nk
Thermodynamic entropy
A. Energy eigenstates
Solve the Schrodinger eqn
B. Analog of Occupation numbers: "String Hypothesis"
Composition principle:
C. Analogs of particle/hole densities
Each set of positive functions
VIII. Conservation laws in interacting integrable models Example:
Q&A