Wavefunction structure in quantum many-fermion systems with k-body interactions: conditional q-normal form of strength functions

Abstract

For finite quantum many-particle systems modeled with say m fermions in N single particle states and interacting with k-body interactions (k ⩽ m), the wavefunction structure is studied using random matrix theory. The Hamiltonian for the system is chosen to be H = H 0(t) + λV(k) with the unperturbed H 0(t) Hamiltonian being a t-body operator and V(k) a k-body operator with interaction strength λ. Representing H 0(t) and V(k) by independent Gaussian orthogonal ensembles of random matrices in t and k fermion spaces, respectively, the first four moments, in m-fermion spaces, of the strength functions F κ (E) are derived; the strength functions contain all of the information about wavefunction structure. With E denoting the H energies or eigenvalues and κ denoting unperturbed basis states with energy E κ , the F κ (E) give the spreading of the κ states over the eigenstates E. It is shown that the first four moments of F κ (E) are essentially the same as that of the conditional q-normal distribution given in (Szabowski 2010 Electron. J. Probab. 15 1296). This naturally gives asymmetry in F κ (E) with respect to E as E κ increases and also the peak value changes with E κ . Thus, the wavefunction structure in quantum many-fermion systems with k-body interactions in general follows the conditional q-normal distribution.