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In A strongly correlated metal built from Sachdev-Ye-Kitaev models by Song et al. they wish to calculate the generating function for a system with quenched disorder. In the Keldysh formalism, this corresponds to the averaged (over the random variables $U_{ijkl}$) partition function $$Z[U_{ijkl}] = \frac{\text{Tr}[\rho U]}{\text{Tr}[\rho]}, $$ with $\rho = e^{-\beta H}$ and $U = e^{- i H(0^-) (t_f-t_0)}e^{- i H(0^+) (t_0-t_f)}$. Note that $U$ describes the forward and then backward evolution.

It is at the next step that I get lost, however, because they seemingly jump to the aforementioned average (6) $$\bar{Z} = \text{Tr}\bar{U},$$ or something of that nature (note this jump exists in Sec. B in Methods as well). Here there are two main points, the average over the denominator is never mentioned again, and neither is the part of $\rho$ in the numerator.

This seems to be a typical jump that is made in SYK calculations, see for instance the jump from 2.9 to 2.11 in Universal Aspects of Quantum-Critical Dynamics in and Out of Equilibrium by Steinberg

So are these jumps something I am missing, or some unmentioned approximation, or a mistake?

Background on Keldysh formalism use in quenched averaging

When dealing with quenched disorder systems the standard generating function is the averaged free energy $\overline{\ln e^{-\beta H}}$, which is notoriously difficult to calculate. This is in contrast to the average over the exponential $\overline{e^{-\beta H}}$. This is one of the often stated advantages of the Keldysh formalism, as mentioned in 11.2 of Field Theory of Non-Equilibrium Systems by Kamenev. This is because we then focus on the average of $$ Z[U_{ijkl}] = \frac{\text{Tr}[\rho U]}{\text{Tr}[\rho]}.$$

However the precise unitary time evolution operator $U$ used in the literature changes. It seems to fall into two catergories:

  1. It is often assumed that interactions are adiabatically switched on, and that the initial state $\rho$ is not dependent on the disorder in the first place. My guess is that the assumption is that the system thermalizes over a long enough time such that the new generating function is still a generating function for the interacting system.
  2. In the SYK case they really seem to imply that interaction exist from the state, which, at least based on the explicit mathematics,leaves one with averaging a combination of $e^{-\beta H}$ and $\ln e^{-\beta H}$, which is glossed over. My guess is that the assumption is that of Bogoliubov weakening of initial conditions. I imagine that since the initial state is in the asymptotic past $t_0 \to -\infty$, that it is assumed that one may just ignore that part of the contour as long as the correct boundary conditions are specified.

So as a second request, I would ask for any thoughts that can (explicitly) explain what is being assumed in these cases.

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    $\begingroup$ I think your question is about a step that the paper (and many other papers on this subject) omitted: they applied the replica trick to handle the disorder average and go to the replica diagonal solution. A careful explanation of the omitted steps is in Appendix A of arxiv.org/abs/1711.08467 $\endgroup$
    – Meng Cheng
    Nov 21 '21 at 15:59
  • $\begingroup$ That appears to be the case. So another way of stating this would be: They consider initial states for which the system is self averaging, i.e., where the quenched average can be replaced by an annealed average? $\endgroup$
    – Jan
    Nov 21 '21 at 16:13
  • $\begingroup$ That is equivalent to the validity of the replica-diagonal solution. If not, then the ground state would be something like spin glasses. I do not know how this is justified in general, but at least there is some good numerical evidence. $\endgroup$
    – Meng Cheng
    Nov 21 '21 at 16:29

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