A direct basis approach to nonorthogonality in second quantization. Theory and application

TL;DR

This paper introduces a direct basis formalism for nonorthogonal basis sets in second quantization, simplifying operator formulation and manipulation without dual basis complications, demonstrated through Hubbard and Heisenberg models.

Contribution

It proposes a novel direct basis approach that preserves Hermiticity and simplifies quantum operator formulation in nonorthogonal basis sets, avoiding dual basis complexities.

Findings

Effective formulation of quantum operators in direct basis

Application to Hubbard Hamiltonian and Heisenberg dynamics

Simplified matrix manipulations for N-particle operators

Abstract

We present a direct basis formalism for using nonorthogonal basis sets in the second quantization framework. As an alternative to the dual basis formalism, a direct basis retains the Hermiticity relation between the creation and annihilation operators, with which the form of quantum operators -- e.g. the number operator and the Hamiltonian -- can be readily interpreted and manipulated. To tackle the difficulty of formulating quantum operators in the direct basis, we introduce the coefficient matrix and the generalized creation and annihilation operators, with which an arbitrary N-particle operator can be generated by simple matrix manipulations with the metric tensor of a general basis set. We illustrate the application of the direct basis formalism with the Hubbard Hamiltonian and a dynamical study with the Heisenberg equations of motion