Thermalization in the two-body random ensemble

Abstract

Using the ergodicity principle for the expectation values of several types of observables, weinvestigate the thermalization process in isolated fermionic systems. These are described bythe two-body random ensemble, which is a paradigmatic model to study quantum chaosand especially the dynamical transition from integrability to chaos. By means of exactdiagonalizations we analyze the relevance of the eigenstate thermalization hypothesis aswell as the influence of other factors, such as the energy and structure of the initial state,or the dimension of the Hilbert space. We also obtain analytical expressions linkingthe degree of thermalization for a given observable with the so-called number ofprincipal components for transition strengths originating at a given energy, with thedimensions of the whole Hilbert space and microcanonical energy shell, and with thecorrelations generated by the observable. As the strength of the residual interaction isincreased, an order-to-chaos transition takes place, and we show that the onset ofWigner spectral fluctuations, which is the standard signature of chaos, is notsufficient to guarantee thermalization in finite systems. When all the signatures ofchaos are fulfilled, including the quasicomplete delocalization of eigenfunctions,the eigenstate thermalization hypothesis is the mechanism responsible for thethermalization of certain types of observables, such as (linear combinations of)occupancies and strength function operators. Our results also suggest that fullychaotic systems will thermalize relative to most observables in the thermodynamiclimit.