I am reading a paper and have no background in quantum mechanics. Any help/references would be appreciated.
The ground state for a system of $N$ non-interacting fermions on the unit torus $\mathbb T^d$ has the following simple form. In frequency space, $$\Big(\psi_N,\sum_j(-\Delta_{x_j})\psi_N\Big)=\sum_{p_i\in\mathbb Z^d,\,i=1,...,N}\Big(\sum_{j=1}^N p_j^2\Big)|\widehat\psi_N(p_1,...,p_N)|^2.$$ (This follows from Plancherel on $\mathbb T^d$.)
My question is about the following conclusion from this result. The paper I am reading says, Therefore, it is easy to see that the lowest energy configuration is given by $\widehat\psi_N={\bigwedge}_{i\in F_n}\delta_i$, where $F_N\subset\mathbb Z^d$ is the subset of $N$ distinct lattice points closest to the origin, and $\delta_i$ is the Kronecker delta at $i\in\mathbb Z^d$. Clearly, the set $F_N$ is an approximate ball $B_{R_N}(0)\cap\mathbb Z^d$ of radius $R_N\sim N^{1/d}$, and it is referred to as the Fermi sea.
My questions are:
How do I deduce from the equation that the lowest energy configuration is given by $\widehat\psi_N={\bigwedge}_{i\in F_n}\delta_i$, where $F_N\subset\mathbb Z^d$ is the subset of $N$ distinct lattice points closest to the origin?
Why is $F_N$ is an approximate ball $B_{R_N}(0)\cap\mathbb Z^d$ of radius $R_N\sim N^{1/d}$?