Form of wide-band limit for the broadening

Question

When considering the Green's function (GF) of a system coupled to a reservoir, the wide-band limit (WBL) is often assumed to simplify the discussion.

For instance, the reservoir retarded GF reads $g_k(\omega)=\frac{1}{\omega-\epsilon_k+i\eta}$ for its eigenstates labelled by $k$. It is coupled to some of the states of the central system labelled by $\alpha$ via a potential $v_{\alpha k}$ and its Hermitian conjugate $v_{k\alpha}$. Then the system retarded self-energy matrix is $$\Sigma_{\alpha\beta}(\omega)=\sum_k v_{\alpha k}\, g_k(\omega)\, v_{k\beta}. \qquad(*)$$

The WBL is an approximation that says one can set $$\Sigma_{\alpha\beta}(\omega)=-\frac{i}{2}\Gamma_{\alpha\beta}$$ with a constant broadening $\Gamma_{\alpha\beta}$. However, it is not very clear what matrix form that $\Gamma_{\alpha\beta}$ should take for practical calculations. Since it is an approximation, explicit calculation does not seem to be able to specify the matrix form. Instead, we probably need some physical argument or intuition. If there is only one relevant system state $\alpha$, it reduces merely to a number. This is the case for a single-level problem like in textbooks and some papers (e.g., p20 of this one).

But when the system is more than 1D and coupled to leads, even one interface of the system consists certainly of multiple sites/states labelled by $\alpha$. If we have transport calculations in mind, e.g., a system coupled to electrodes, let's say $\alpha,\beta$ belong to the sites of the central system's interface with one particular electrode and forget about other electrodes. In such a case, should we set $$\Gamma_{\alpha\beta}=\gamma\delta_{\alpha\beta}$$ for a constant $\gamma$? From (*) above, it seems reasonable to not assume $\Gamma_{\alpha\beta}=0$ when $\alpha\neq\beta$ for generic couplings. Then shall we use something like $$\Gamma_{\alpha\beta}=\gamma \quad \forall \alpha,\beta$$ or else? Unfortunately, I don't find any explicit discussion about this.

Answer 1

Typically tunneling matrix elements are characterized not only by the index of state within the "interacting" region $\alpha$, but also the index of the lead from/to which the tunneling occurs $\nu$, that is $$ \Sigma_{\alpha\beta}^{(\nu,\nu')}(\omega)=\sum_k v_{\alpha k\nu}\, g_k(\omega)\, v_{k\beta\nu'} $$ (In the OP this index could be implied in $k$, but it is worth writing explicitly.)

In the broad-band limit this degenerates into $$ \Sigma_{\alpha\beta}^{(\nu,\nu')}(\omega)=-\frac{i}{2}\Gamma_{\alpha\beta}^{(\nu)}\delta_{\nu,\nu'} $$ Actually, the self-energy entering the Green's function of the "interacting/central" region is summed over $\nu$, but the explicit lead index appears in the matrix factors describing the coupling to the leads, which enter expressions for transport coefficients.

Where do coefficients $\Gamma$ come from?
In general, $\Gamma_{\alpha\beta}^{(\nu)}$, just like level energies are determined by solving the appropriate Schrödinger equation prior to dealing with the transport problem. Moreover, the transfer Hamiltonian, describing uncoupled central region and leads as separate quantum regions, between which electrons hop, is itself an approximation or even a phenomenological ansatz. Thus, the broadenings $Gamma$ are phenomenological parameters, with the orders of magnitide best inferred experimentally, and relative magnitudes and phases determined by the physical problem itself.

In other words - our focus here is not on exactly determining the magnitudes of the parameters, but on modeling/describing transport phenomena. This is a completely general approach, and deserves a special remark: although physics prides itself in being quantitative science and making quantitative predictions, it should not be understood as carrying out exact first principles calculations when engineering devices - in fact, such calculations are unwieldy and often unrealistic, because measuring all the underlying parameters is technically very difficult, even if theoretically possible.

General approaches to $\Gamma$s

• The original paper by Meir, Wingreen, and Lee and many publications that followed assumed "proportionate" coupling to leads, i.e., broadenings with different indices $\Gamma$ differed only by a scalar coefficient, but otherwise were identical matrices. This allowed reducing the description of transport to calculating only the retarded Green's function $G^r$, which allowed producing a number of spectacular results. Generally calculations are more difficult and require full-fledged Green's function calculations to determine the "lesser" Keldysh Green's function $G^<$.
• A notable place where the "proportionate" coupling approach is impossible is when describing Aharonov-Bohm interferometers - since the broadenings need to have complex phases, modeling the magnetic flux. Example of such a calculation, from the same group, are here and here.
• Some problems describing Coulomb charging effects in quantum dots may require assigning big $\Gamma$ to some levels and small ones to other. In principle, this does not preclude the use of proportionate coupling. But since such problems are often studied using behavior of QDs inserted in AB rings, the previous comment applies.