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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.