Implications of the two nodal domains conjecture for ground state fermionic wave functions

The nodes ofmany-bodywave functions aremathematical objects important in many different fields of physics.
They are at the heart of the quantum Monte Carlo methods but outside this field their properties are neither widely
known nor studied. In recent years a conjecture, already proven to be true in several important cases, has been put
forward related to the nodes of the fermionic ground state of a many-body system, namely that there is a single
nodal hypersurface that divides configuration space into only two connected domains. While this is obviously
relevant to the fixed node diffusion Monte Carlo method, its repercussions have ramifications in various fields
of physics as diverse as density functional theory or Feynman and Cohen’s backflow wave function formulation.
To illustrate this we explicitly show that, even if we knew the exact Kohn-Sham exchange correlation functional,
there are systems for which we would obtain the exact ground state energy and density but a wave function quite
different from the exact one. This paradox is only apparent since the Hohenberg-Kohn theorem relates the energy
directly to the density and the wave function is not guaranteed to be close to the exact one. The aim of this paper
is to stimulate the investigation of the properties of the nodes of many-body wave functions in different fields
of physics. Furthermore, we explicitly show that this conjecture is related to the phenomenon of avoided nodal
crossing but it is not necessarily caused by electron correlation, as sometimes has been suggested in the literature.
We explicitly build a many-body uncorrelated example whose nodal structure shows the same phenomenon.