We focus on the definition of 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 toward the determination of such “Froelich-transformed” Hamiltonain, in fact does not: we explicitly show how the 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.

Revisiting the Froehlich-type transformation when degenerate states are present

Acquarone M;
2007

Abstract

We focus on the definition of 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 toward the determination of such “Froelich-transformed” Hamiltonain, in fact does not: we explicitly show how the 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.
2007
Istituto dei Materiali per l'Elettronica ed il Magnetismo - IMEM
Froelich transformation
effective Hamiltonian
degenerate states
Su–Schrieffer–Heeger Hamiltonian
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.14243/40923
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