# Topological invariant for interacting systems using single particle green functions?

Why Single particle green's function is (preferred) used to find topological for interacting systems?

$N_1 =\frac{\epsilon_{ijk}}{24 \Pi ^2} \int dw d^3k G \partial_i G^{-1}G\partial_jG^{-1}G\partial_kG^{-1}$

I have some points unclear

1.what is the motivation behind using single particle green funcion

1. How we can physically explain above invariant formula?

2. Does Single particle green function can give us only information about single particle edge states and will it give any information about many body edge states?

• So there is no reference ? Mayhap's arxiv.org/abs/1011.2273 or arxiv.org/abs/1104.1602 would help. Otherwise, a quick answer, Volovik has always been using the Green's function to describe homotopy/homology in condensed matter, as were using the pioneers in condensed matter, as e.g. old papers by Luttinger about the stability of the Fermi surface (please see references there : physics.stackexchange.com/q/69358/16689 ), so I guess it's for the same reason. The relevant question is: why it is not widely used ? – FraSchelle Mar 31 '15 at 5:31

I assume this is the formula for Chern number in a Chern insulator. The physical reason that such a formula exists is that this is exactly what Kubo formula gives you for the Hall conductance, which holds for interacting systems too.

The reason to use Green's function to define topological invariants is that it is straight forward to generalize Green's functions to interacting systems. In topological band theory, on the other hand, topological invariants are defined in terms of the single-particle (Bloch) eigenstates of the filled bands, e.g. the Berry phase

\begin{align} \gamma_\alpha &= \int d \vec{k} \cdot \vec{A}_\alpha(\vec{k})\\ \vec{A}_\alpha(\vec{k}) &= - i <\phi_\alpha(\vec{k})|\nabla_k|\phi_\alpha(\vec{k})> \end{align}

where $|\phi_\alpha(\vec{k})>$ is an occupied (single-particle!) Bloch state. The drawback with this formalism is: What do you do if your system is interacting?

There are three very very very interesting papers that proof that instead of using your formula, which needs a frequency integral, topological invariants for interacting systems can also be calculated from a so called topological Hamiltonian

\begin{align} H_{top}(\vec{k}) &= -G^{-1}(\vec{k},i\omega=0) = H_0(\vec{k}) + \Sigma(\vec{k},i\omega=0) \end{align}

where $H_0$ is noninteracting, $G$ the full Green's function and $\Sigma(\vec{k},i\omega=0)$ the self-energy evaluated at the Matsubara frequency $i\omega=0$, which is defined for $T=0$. Can you believe how easy this makes everything?? The papers are

• Equivalent topological invariants of topological insulators (Wang Z Qi X Zhang S)
• Simplified Topological Invariants for Interacting Insulators (Wang Z Zhang S)
• Topological Hamiltonian as an exact tool for topological invariants (Wang Z Yan B)

Edit: I think I misread your question. The answer is, the single-particle Green's function of an interacting system still contains all information about that system. The single-particle Green's function from an interacting system and that from a noninteracting system look completely different. The imaginary part of the latter are just delta functions, whereas the interacting one yields a generally broadened spectral function