Scaling at the Chaos Threshold for Interacting Electrons in a Quantum Dot

The chaotic mixing by random two-body interactions of many-electron Fock states in a confined geometry is investigated. Two regimes are distinguished in the dependence of the typical number of Fock states that are mixed into an eigenstate on the interaction strength $V$, the excitation energy $\varepsilon$, and the level spacing $\Delta$. In both regimes the number is large (indicating delocalization in Fock space). However, only the large- $V$ regime is described by the golden rule (indicating chaotic mixing). The crossover region is characterized by a maximum in a scaling function that becomes more pronounced with increasing excitation energy. The scaling parameter that governs the transition is $\left(\varepsilon V/{\Delta }^2\right)\ln \left(\Delta /V\right)$.