Effective Hamiltonian with position dependent mass and ordering problem

TL;DR

This paper derives an effective low-energy Hamiltonian for a tight-binding model with position-dependent hopping, showing that the ordering ambiguity in kinetic energy can be resolved by a more general form without changing the Hamiltonian.

Contribution

It introduces a generalized kinetic energy form for position-dependent mass systems, clarifying the ordering ambiguity in the Hamiltonian.

Findings

Effective Hamiltonian with position-dependent mass derived.

Ordering ambiguity in kinetic energy does not alter the Hamiltonian.

Proposes a more general kinetic energy form consistent with the Hamiltonian.

Abstract

We derive the effective low energy Hamiltonian for the tight-binding model with the hopping integral slowly varying along the chain. The effective Hamiltonian contains the kinetic energy with position dependent mass, which is inverse to the hopping integral, and effective potential energy. Changing of ordering in the kinetic energy leads to change of the effective potential energy and leaves the Hamiltonian the same one. Therefore, we can choose arbitrary von Roos ordering parameters in the kinetic energy without changing the Hamiltonian. Moreover, we propose a more general form for the kinetic energy than that of von Roos, which nevertheless together with the effective potential energy represent the same Hamiltonian.