Revisiting the Froehlich-type transformation when degenerate states are present

TL;DR

This paper revisits the Froehlich-type transformation for degenerate states, demonstrating that degeneracies do not hinder the derivation of effective Hamiltonians and can be included without issues, as shown through the SSH model.

Contribution

It provides a detailed proof that degenerate states can be incorporated into the Froehlich transformation without complications, extending its applicability.

Findings

Degenerate states contribute with vanishing matrix elements.

Effective Hamiltonian accurately describes renormalized energies.

Method applies to few-body systems without thermodynamic limit.

Abstract

We focus on the definition of the unitary transformation leading to an effective second order Hamiltonian, inside degenerate eigensubspaces of the non-perturbed Hamiltonian. We shall prove, by working out in detail the Su-Schrieffer-Heeger Hamiltonian case, that the presence of degenerate states, including fermions and bosons, which might seemingly pose an obstacle towards the determination of such "Froehlich-transformed" Hamiltonian, in fact does not: we explicitly show how degenerate states may be harmlessly included in the treatment, as they contribute with vanishing matrix elements to the effective Hamiltonian matrix. In such a way, one can use without difficulty the eigenvalues of the effective Hamiltonian to describe the renormalized energies of the real excitations in the interacting system. Our argument applies also to few-body systems where one may not invoke the thermodynamic limit to get rid of the "dangerous" perturbation terms.