Expressivity of determinantal ansatzes for neural network wave functions

TL;DR

This paper investigates the expressivity of determinantal neural network wave functions in quantum many-body problems, establishing bounds and limitations for spin-dependent and spin-independent Hamiltonians, supported by numerical experiments.

Contribution

It introduces a theoretical framework for understanding the expressivity bounds of determinantal neural wave functions, especially regarding spin dependencies, and validates these bounds numerically.

Findings

Bounds between different wave function representations are established.

Full determinant wave functions are not suitable for spin-dependent Hamiltonians.

Numerical experiments confirm the theoretical bounds.

Abstract

Neural network wave functions have shown promise as a way to achieve high accuracy on the many-body quantum problem. These wave functions most commonly use a determinant or sum of determinants to antisymmetrize many-body orbitals which are described by a neural network. In many cases, the wave function is projected onto a fixed-spin state. Such a treatment is allowed for spin-independent operators; however, it cannot be applied to spin-dependent problems, such as Hamiltonians containing spin-orbit interactions. We show that for spin-independent Hamiltonians, a strict upper bound property is obeyed between a traditional Hartree-Fock like determinant, full spinor wave function, the full determinant wave function, and a generalized spinor wave function. The relationship between a spinor wave function and the full determinant arises because the full determinant wave function is the spinor wave function projected onto a fixed-spin, after which antisymmetry is implicitly restored in the spin-independent case. For spin-dependent Hamiltonians, the full determinant wave function is not applicable, because it is not antisymmetric. Numerical experiments on the H$_3$ molecule and two-dimensional homogeneous electron gas confirm the bounds.