Construction of size-consistent effective Hamiltonians

Abstract

For a system consisting of two interacting subsystems, effective Hamiltonians for one subsystem are usually constructed by integrating out the degrees of freedom of the second subsystem. In contrast to usual treatments, our method for deriving effective Hamiltonians is based on the introduction of cumulants. Cumulants guarantee size consistency, a property that is not always evident in other approaches. The formalism applies for any temperature. Our method is based on a matrix formulation in the framework of a recently introduced cumulantprojection method. This method allows the treatment of strongly correlated systems, where conventional diagrammatic techniques are difficult to apply. The relation with perturbation theory is also shown. As a simple application we discuss the small polaron problem. Evaluating the one-electron dispersion relation in the strong-coupling case shows good agreement with recent results from exact Lanczos diagonalization. Our method has substantial advantages over perturbation theory.

Spectral properties of the 2D Holstein polaron

“…As mentioned above such processes are of particular importance in the crossover region. Thus we expect an even better agreement including higher-order corrections [20]. The more important point, however, seems to be the use of the incomplete Lang-Firsov transformation in the derivation of (34).…”

“…After inserting (20)- (25) into (19), the τ -integration is easily accomplished. As a next step, the summation over the Matsubara frequencies iω ς has to be carried out.…”

“…On the other hand, the most interesting effects will be expected if the characteristic electronic and phononic energy scales are not well separated (λ ∼ 1; α ∼ 1). For example, very recent numerical investigations, performed in the crossover region for 1D chains with up to 20 sites, indicate that the polaronic band structure markedly deviates from a simple tight-binding dispersion [18][19][20].…”

Direct equivalence between quantum phase transition phenomena in radiation-matter and magnetic systems: Scaling of entanglement

“…In order to eliminate the cavity degrees of freedom and obtain a size-consistent H qb (i.e. a reduced qubit Hamiltonian which scales with the system's size) we follow the procedure sketched out by Polatsek and Becker [11]. The resulting effective qubit Hamiltonian can be written as…”

Construction of size-consistent effective Hamiltonians for systems with arbitrary Hilbert space

“…As an alternative to perturbation theory, the evaluation of the effective Hamiltonian (15) can also be done in the framework of the Mori-Zwanzig projection method 22,23 as was shown in Ref. 21. In the following we want to restrict ourselves to the perturbation result (18).…”