Many-body position operator in lattice fermionic systems with periodic boundary conditions

TL;DR

This paper derives a total position operator for lattice fermionic systems with periodic boundaries, demonstrating its properties and applying it to analyze phase transitions and wavefunctions.

Contribution

It introduces a Hermitian total position operator for lattice fermions with periodic boundary conditions, enabling calculation of moments and analysis of physical phenomena.

Findings

Operator is Hermitian and generates translations in momentum space.

Time derivative of the operator corresponds to total current.

Finite size scaling applied to phase transitions and wavefunctions.

Abstract

A total position operator $X$ in the position representation is derived for lattice fermionic systems with periodic boundary conditions. The operator is shown to be Hermitian, the generator of translations in momentum space, and its time derivative is shown to correspond to the total current operator in a periodic system. The operator is such that its moments can be calculated up to any order. To demonstrate its utility finite size scaling is applied to the Brinkman-Rice transition as well as metallic and insulating Gutzwiller wavefunctions.