Stoner transition in a two-dimensional electronic system at low-density: a diagrammatic Monte Carlo study

Stoner instabilities for fermions with repulsive interaction have been well studied within

the mean-field (ladder) approximation. In this study, we consider two-dimensional

electronic systems with one or two valleys and discuss a Stoner transition beyond the

ladder approximation. At weak coupling, the corrections to the ladder approximation

come predominantly from the renormalization of the particle-hole vertex in the particle-

particle channel, and the lowest-order corrections are logarithmically singular in the low-

density limit. To investigate the problem beyond the lowest order, we apply

diagrammatic Monte Carlo algorithm and treat ladder and non-ladder renormalizations

on equal footing. We find that in a one-valley system, a Stoner transition to a

ferromagnetism occurs at low density only when the interaction is sufficiently long-

ranged (more specifically, if there is a cutoff on the momentum transfer carried by the

interaction). In a two-valley system, the restriction is less severe. Here we find either a

direct Stoner transition into a quarter-metal state, in which spin and valley symmetries

are simultaneously broken, or a set of two Stoner transitions via an intermediate half-

metal state. The pathway is controlled by the anisotropy of the fermionic dispersion.