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

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.

Generic example of algebraic bosonisation

“…[9] for the case N = 3, d = 2, which can be written in terms of the z i alone. The other three involvez i on one or both rows of the determinant, while two other shapes, with two and four nodes [9], similarly involve both z i andz i , for a total of N ! d−1 = 6 shapes.…”

Fundamental invariants of many-body Hilbert space

“…It remains to be seen whether they will be practical. At least, there is a chance to manage complexity, because one can imagine 11 truncating the list (18) to limit the calculation to one ideal, or band of excitations. Such an approach is similar in spirit to a fixed-node approximation 24,25 in usual simulations.…”

“…Proofs of the main results have appeared elsewhere. 11 The primary ambition, unfulfilled at present, is to obtain realistic wave functions of few-body systems, without necessarily improving on existing numerical methods either in terms of speed or accuracy in the calculation of binding energies. In this sense the philosophy is converse to that of DFT, which consciously sacrifices realism of the wave functions in favor of precise calculation of energies.…”

Entropy of pure states: not all wave functions are born equal

“…It appears now that these foundational developments did not go as far as mathematically possible [4]. Many-body Hilbert space has an additional finer structure, which is difficult to visualize in the standard formulations.…”