Kernels for non interacting fermions via a Green’s function approach with applications to step potentials

The quantum correlations of N non interacting spinless fermions in their ground state can be expressed in terms of a two-point function called the kernel. Here we develop a general and compact method for computing the kernel in a general trapping potential in terms of the Green’s function for the corresponding single particle Schrödinger equation. For smooth potentials in one dimension the method allows a simple alternative derivation of the local density approximation for the density and of the sine kernel in the bulk part of the trap in the large N limit. It also recovers the density and the kernel of the so-called Airy gas at the edge. This method allows the analysis of the quantum correlations in the ground state when the potential has a singular part with a fast variation in space. For the square step barrier of height V₀ in one dimension, we derive explicit expressions for the density and for the kernel. For large Fermi energy μ > V₀ it describes the interpolation between two regions of different densities in a Fermi gas, each described by a different sine kernel. Of particular interest is the critical point of the square well potential when μ = V₀. In this critical case, while there is a macroscopic number of fermions in the lower part of the step potential, there is only a finite O(1) number of fermions on the shoulder, and moreover this number is independent of μ. In particular, the density exhibits an algebraic decay ∼ 1/x², where x is the distance from the jump. Furthermore, we show that the critical behavior around μ = V₀ exhibits universality with respect to the shape of the barrier. This is established (i) by an exact solution for a smooth barrier (the Woods–Saxon potential) and (ii) by establishing a general relation between the large distance behavior of the kernel and the scattering amplitudes of the single-particle wave-function.