Confusion taking functional derivatives of the generating functional (Coleman)

I'm reading Piers Coleman's Introduction to Many Body Physics and I've run into a strange issue.

In section 8.4, he considers a free theory in the presence of a source $[\bar\eta, \eta]$:

\begin{align} H_0 &= \sum_\lambda \epsilon_\lambda \psi^\dagger_\lambda \psi^~_\lambda \\ V_0(\tau) &= \sum_\lambda \left[\bar\eta_\lambda(\tau)\psi^~_\lambda(\tau) + \psi^\dagger_\lambda(\tau) \eta_\lambda(\tau) \right] \end{align}

where $\eta$ and $\bar\eta$ are complex numbers if $\psi$ is bosonic or Grassmann numbers if $\psi$ is fermionic.

The generating functional is then

\begin{align} Z_0[\bar\eta, \eta] &= Z_0 \left\langle T \exp \left[\int_0^\beta d\tau \sum_\lambda\left( \bar\eta_\lambda(\tau)\psi^~_\lambda(\tau) + \psi^\dagger_\lambda(\tau) \eta_\lambda(\tau) \right) \right] \right\rangle \\ &= Z_0~\exp \left[ \sum_\lambda \int_0^\beta d\tau_1 d\tau_2 ~\bar\eta_\lambda(\tau_1) G_\lambda(\tau_1 - \tau_2)\eta_\lambda(\tau_2) \right] \end{align}

where $G_\lambda(\tau_1 - \tau_2) = - \langle T \psi_\lambda(\tau_1)\psi^\dagger_\lambda(\tau_2) \rangle$ and $Z_0$ is just the generating functional for the free Hamiltonian.

He then goes on to say that \begin{align} \frac{\delta \ln Z_0[\bar\eta, \eta]}{\delta ~\bar\eta(1)} &= \langle \psi(1) \rangle~~,\\ \frac{\delta \ln Z_0[\bar\eta, \eta]}{\delta ~\eta(2)} &= \zeta~\langle \psi^\dagger(2) \rangle \end{align}

where $\zeta$ is $-1$ if the field is Fermionic or $+1$ if it is Bosonic. This is all fine, but then he claims that

\begin{align} \frac{\delta^2 \ln Z_0[\bar\eta, \eta]}{\delta\eta(2)~\delta \bar\eta(1)} &= \langle T~ \psi(1)\psi^\dagger(2) \rangle - \langle \psi(1) \rangle \langle \psi^\dagger(2) \rangle~~. \end{align}

However, when I try to verify this I do not get the second term because $$\ln Z_0[\bar\eta, \eta] = \ln Z_0 - \sum_\lambda \int_0^\beta d\tau_1 d\tau_2 ~\bar\eta_\lambda(\tau_1) G_\lambda(\tau_1 - \tau_2)\eta_\lambda(\tau_2)$$

and two functional derivatives of this simply gives me

\begin{align} \frac{\delta^2 \ln Z_0[\bar\eta, \eta]}{\delta\eta(2)~\delta \bar\eta(1)} &= -G_\lambda(\tau_1 - \tau_2)\\ &= \langle T~ \psi(1)\psi^\dagger(2) \rangle \end{align}

but no second term! After much investigating in other textbooks I suspect that the second term that Coleman claims should appear only appear in theories where $$\frac{\delta ~\bar\eta(1)}{\delta ~\eta(2)} \neq 0$$

but in this case, even the complex or Grassmannian nature of $\bar\eta$ and $\eta$ should guarantee that the above functional derivative is zero. Hence I don't see why the second term should appear.

I have tried calculating this thing the long way without using that $\ln e^{~x} = x$ and it also gave me this selfsame result. Does anyone have any idea on what I may be doing wrong or if there is a mistake in the book?

Note: I lapsed partway into a condensed notation where $\psi_\lambda(\tau_1) \equiv \psi(1)$ for convenience.

• Not sure about his notation, but it's possible that $G_\lambda$ is supposed to be the connected correlation function. So in some sense it's defined to be two point time ordered correlator minus the product of the one-point functions. In QFT, we often set the sources to zero, and so don't have to worry about the one-point functions being non-vanishing. – user2309840 Nov 2 '17 at 3:12
• He explicitly defines $G_\lambda$ as shown above though. – Mason Nov 2 '17 at 4:02