Natural generalization of the ground-state Slater determinant to more than one dimension

TL;DR

This paper introduces a natural generalization of the Slater determinant for many-fermion ground states in higher dimensions, using algebraic invariant theory and bosonization techniques to describe the structure of the fermionic Hilbert space.

Contribution

It provides a novel algebraic framework and an explicit algorithm to describe the multi-dimensional fermionic wave functions beyond one dimension.

Findings

Identifies a finite set of 'shapes' as fundamental wave functions in higher dimensions.

Develops a bosonization approach using Euler bosons to generate excitations.

Shows that shapes serve as vacua for bosonic excitations, generalizing the Slater determinant concept.

Abstract

The basic question is addressed, how the space dimension $d$ is encoded in the Hilbert space of $N$ identical fermions. There appears a finite number $N!^{d-1}$ of many-body wave functions, called shapes, which cannot be generated by trivial combinatorial extension of the one-dimensional ones. A general algorithm is given to list them all in terms of standard Slater determinants. Conversely, excitations which can be induced from the one-dimensional case are bosonised into a system of distinguishable bosons, called Euler bosons, much like the electromagnetic field is quantized in terms of photons distinguishable by their wave numbers. Their wave functions are given explicitly in terms of elementary symmetric functions, reflecting the fact that the fermion sign problem is trivial in one dimension. The shapes act as vacua for the Euler bosons. They are the natural generalization of the single-Slater-determinant form for the ground state to more than one dimension. In terms of algebraic invariant theory, the shapes are antisymmetric invariants which finitely generate the $N$-fermion Hilbert space as a graded algebra over the ring of symmetric polynomials. Analogous results hold for identical bosons.