Density of states in a system of interacting electrons

Question

When we are introduced to the density of states in typical band-theory problems we neglect interaction between electrons, and thus we define the density of states of a sigle particle as: $D(E)=2\int_{1st BZ}\delta (E-\epsilon_\mathbf{k})d\mathbf{k}$ since the possible states available to occupy for an electron lie within a band (I assume here there is just one band described by $\epsilon_\mathbf{k}$).

Now, when we switch on the interactions the concept of single-electron density of states seems ill-defined to me, since there is not such a thing as "possible energy of an electron". Take the 1D Hubbard model Hamiltonian within mean-field approximation:

$H=\sum_{k}(\epsilon_{k\uparrow}n_{k\uparrow}+ \epsilon_{k\downarrow}n_{k\downarrow})-U N\langle n_\uparrow \rangle \langle n_\downarrow \rangle$

where $\epsilon_{k\sigma}=-2t\cos k+\langle n_{-\sigma}\rangle U$. $U$ is the on-site Coulomb repulsion and $t$ the hopping term and $\sigma$ the spin (=$\pm$).

In this case the eigenstates are multiparticle states, and thus we can ask only for the energy of the system, or average of one particle. How can I then calculate the DOS in this context? How do I define my bands in first place if I don't know what's the dispersion relation?

Answer 1

You are right that in general it doesn't make sense to talk about single particle DOS in interacting systems. If the interaction is weak you can treat it as a perturbation, or as an effectively non-interacting system by rescaling some parameters (like in the Fermi liquid theory), and you can still talk about the DOS. In strongly interacting systems you can sometimes identify quasiparticle excitations which are weakly interacting and then talk about the DOS of those quasiparticles.

Answer 2

Meir-Wingreen formula provides a possible generalization of the density-of-states in the context of transport through an interacting region (see the references by Meir&Wingreen and Jauho&Haug): it is defined as a Fourier transform of a single particle Green's function and becomes identical with the one-particle DOS in the limit of no interactions.

This prescription of defining DOS as a Fourier transform of the single-particle Green's function works in a general case. However, its use is limited to the situations when one is interested in one-particle phenomena, such as electron transport through the interacting region or in bulk, when using the quantum kinetic equation (see the reviews by Rammer&Smith and Rammer). Once one is interested in multi-particle excitations, the basic concept of DOS is of limited applicability, although it has been applied to pairwise excitations, such as excitons or Copper pairs.

References:

1. Meir and Wingreen, "Landauer formula for the current through an interacting electron region"
2. Jauho and Haug, Quantum kinetics in transport and optics of semiconductors
3. Rammer & Smith, Quantum field-theoretical methods in transport theory of metals
4. Rammer, Quantum transport theory of electrons in solids: A single-particle approach

Answer 3

$\mu$ would be a proper generalization of a single partile's engery in many body systems. $$G=G(N)$$ $$\mu =\frac{\partial G}{\partial N} =G(N+1)-G(N) $$ $$\text{Density of states:} \quad \rho(\mu) := \frac{\partial N}{\partial \mu} $$