A simplified BBGKY hierarchy for correlated fermionic systems from a Stochastic Mean-Field approach

TL;DR

This paper introduces a simplified BBGKY hierarchy derived from the stochastic mean-field approach, providing a new analytical tool for studying correlations in fermionic systems, especially useful in weak coupling regimes and for short-time dynamics.

Contribution

It establishes an equivalence between the SMF approach and a simplified BBGKY hierarchy, offering analytical insights and practical approximations for correlated fermionic systems.

Findings

Simplified BBGKY captures correlations beyond mean-field.

Useful for weak coupling and short-time dynamics.

Full SMF remains necessary for long-time, strong coupling regimes.

Abstract

The stochastic mean-field (SMF) approach allows to treat correlations beyond mean-field using a set of independent mean-field trajectories with appropriate choice of fluctuating initial conditions. We show here, that this approach is equivalent to a simplified version of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy between one-, two-, ..., N-body degrees of freedom. In this simplified version, one-body degrees of freedom are coupled to fluctuations to all orders while retaining only specific terms of the general BBGKY hierarchy. The use of the simplified BBGKY is illustrated with the Lipkin-Meshkov-Glick (LMG) model. We show that a truncated version of this hierarchy can be useful, as an alternative to the SMF, especially in the weak coupling regime to get physical insight in the effect beyond mean-field. In particular, it leads to approximate analytical expressions for the quantum fluctuations both in the weak and strong coupling regime. In the strong coupling regime, it can only be used for short time evolution. In that case, it gives information on the evolution time-scale close to a saddle point associated to a quantum phase-transition. For long time evolution and strong coupling, we observed that the simplified BBGKY hierarchy cannot be truncated and only the full SMF with initial sampling leads to reasonable results.