# Derivation of the Keldysh-Non-Linear-Sigma model

I have to give a presentation on the Keldysh Non Linear Sigma model in a proseminar at my university. I am new to the topic and I have had some problems with understanding the literature. My main source is the book Field Theory of Non-Equilibrium Systems by Kamenev.

There are some equations in the book that I am simply not able to reproduce. I would really like to understand the derivation of the NLSM especially because I also need to write a report on it and I would like to make a more pedagogical derivation than the one in the book. I would be really grateful if someone could help me out with this.

• The first equation I cannot reproduce is (11.23) on page 243 where the action for the Hubbard Stratonovic field $Q=R\Lambda R^{-1}$ in terms of the slow field $R_{t,t'}(\mathbf{r})$ is expressed. I don't see what I'm doing wrong here. The crucial step is to express $RG^{-1}R^{-1}=G^{-1} +\text{something}$. The given result is $$RG^{-1}R^{-1}=G^{-1} + i R\partial_t R^{-1}+iR\mathbf{v}_F \nabla_\mathbf{r} R^{-1}\, .$$ I was able to reproduce the part with the spatial derivative even though I'm a bit unsure about this linearization around the Fermi energy. Is there an angular average over the Fermi velocity implicit in this formula?
The part that confuses me more is the time derivative part. I tried to derive it by acting with $RG^{-1}R^{-1}$ on a test function and making partial integrations but did not arrive at the result. The result also seems counter intuitive to me because on the same page it is stated, that for stationary and uniform $R_{tt'}(\mathbf{r})=R_{t-t'}$, $G^{-1}$ and $R$ should commute. This would imply that $RG^{-1}R^{-1}=G^{-1}$. However, $\partial_t R_{t-t'}$ is not zero in general. Therefore, the formula given in (11.23) shouldn't work for the simplest case of the stationary and uniform field scenario. Does anybody see what I am missing here?
• The second equation I cannot reproduce is (11.28) $$iS(Q)=\frac{\pi\nu}{4}\text{Tr}\{D(\nabla_\mathbf{r}Q)^2-4\partial_t Q\}$$ on page 244. My problem here is related to the question in this (Derivation of the gradient expansion of the Keldysh nonlinear sigma model for disorder metals) post on the forum. However, this question was not completely resolved, and I am stuck at a slightly different place. I was able to reproduce the time part in (11.28). I was also able to bring the spatial part to the last expression given in the footnote (5). Namely, $$iS[Q]_{\text{spatial}}\propto \text{Tr}\{ [1+\Lambda](R\nabla_\mathbf{r} R^{-1})[1-\Lambda](R\nabla_\mathbf{r} R^{-1})\} \, ,$$ but I didn't manage to bring this to the final expression. Does anybody have an idea on how one treats such an expression?
• Lastly I was not able to derive the Usadel equation (11.29) on page 245. I arrived at the correct expressions for the terms linear in $W$ but I couldn't understand the last step $$\text{Tr}\{W_{t_1t_2}(\partial_{t_1} + \partial_{t_2})\underline{Q}_{t_1 t_2}\}=\text{Tr}(W\{\partial_t,\underline{Q}\})\, .$$ When I tried to derive this by acting on test functions and using partial integration in $t_2$ I got a commutator and not an anti commutator. Does anybody see my mistake?

If nobody can help me with these questions, does anybody know a source where this derivation is done in a more slow paced manner?