Exact Coupling Coefficient Distribution in the Doorway Mechanism

TL;DR

This paper derives exact formulas for the distribution of coupling coefficients in the doorway mechanism using random matrix theory, applicable to regular and chaotic systems with different symmetry properties, validated by numerical simulations.

Contribution

It provides the first exact analytical calculation of the maximum coupling coefficient distribution in the doorway mechanism for various background state regimes.

Findings

Exact distribution formulas match numerical simulations

Applicable to systems with preserved and broken time-reversal symmetry

Enhances understanding of doorway states in many-body systems

Abstract

In many--body and other systems, the physics situation often allows one to interpret certain, distinct states by means of a simple picture. In this interpretation, the distinct states are not eigenstates of the full Hamiltonian. Hence, there is an interaction which makes the distinct states act as doorways into background states which are modeled statistically. The crucial quantities are the overlaps between the eigenstates of the full Hamiltonian and the doorway states, that is, the coupling coefficients occuring in the expansion of true eigenstates in the simple model basis. Recently, the distribution of the maximum coupling coefficients was introduced as a new, highly sensitive statistical observable. In the particularly important regime of weak interactions, this distribution is very well approximated by the fidelity distribution, defined as the distribution of the overlap between the doorway states with interaction and without interaction. Using a random matrix model, we calculate the latter distribution exactly for regular and chaotic background states in the cases of preserved and fully broken time--reversal invariance. We also perform numerical simulations and find excellent agreement with our analytical results.