Ground-state magnetization for interacting fermions in a disordered potential: Kinetic energy, exchange interaction, and off-diagonal fluctuations

We study a model of n interacting fermions in a disordered potential, which is assumed to generate uniformly fluctuating interaction matrix elements. We show that the ground-state magnetization is systematically decreased by off-diagonal fluctuations of the interaction matrix elements. This effect is neglected in the Stoner picture of itinerant ferromagnetism in which the ground-state magnetization is simply determined by the balance between ferromagnetic exchange and kinetic energy, and increasing the interaction strength always favors ferromagnetism. The physical origin of the demagnetizing effect of interaction fluctuations is the larger number K of final states available for interaction-induced scattering in the lower-spin sectors of the Hilbert space. We analyze the energetic role played by these fluctuations in the limits of small and large interactions U. In the small-$U$ limit we use second-order perturbation theory and identify explicitly transitions which are allowed for minimal spin and forbidden for higher spin. These transitions then on average lower the energy of the minimal spin ground state with respect to higher spin; we analytically evaluate the size of this reduction and find it to give a contribution ${\Delta }^s\propto {\mathrm{nU}}^2/\Delta$ to the spin gap between the two lowest-spin ground states. In terms of an average effective Hamiltonian, these contributions induce a ${\mathrm{nU}}^2{S}^2/\Delta$ term which decreases the strength of the ferromagnetic exchange, thereby delaying the onset of Stoner ferromagnetism, and generate a second, larger S term $\propto S^3,$ which results in a saturation of the ground-state spin before full polarization is achieved, in contrast to the Stoner scenario. For large interactions U we amplify on our earlier work [Ph. Jacquod and A. D. Stone, Phys. Rev. Lett. 84, 3938 (2000)] which showed that the broadening of the many-body density of states is proportional to $\sqrt{K}U$ and hence favors minimal spin. Numerical results are presented in both limits. After evaluating the effect of fluctuations, we discuss the competition between fluctuations plus kinetic energy and the exchange energy. We finally present numerical results for specific microscopic models and relate them to our generic model of fluctuations. We discuss the different physical situations to which such models may correspond, the importance of interaction fluctuations, and hence the relevance of our results to these situations and recall an experimental setup which we proposed in an earlier work to measure the importance of interaction fluctuations on the ground-state spin of lateral quantum dots in the Coulomb blockade regime.