Multivariable functions of Grassmann numbers

Question

I'm trying to derive the closed form of the fermionic coherent state defined by the relation:

$$ f_i|\vec{\eta}\rangle = \eta_i |\vec{\eta}\rangle \tag{4.10} $$

My book (Atland and Simons, Condensed Matter Field Theory, pages 160-163 suggests that I use the state

$$ |\vec{\eta}\rangle = \exp\left[ -\sum_i \eta_i f_i^{\dagger} \right]|0 \rangle\tag{4.17} $$

where the $\eta_i$ are Grassmann numbers: $\{\eta_i,\eta_j\}=0$. I've noticed that Atland and Simons regularly use the relation:

$$ \exp\left[ -\sum_i \eta_i f_i^{\dagger} \right] = \left(1-\sum_i \eta_i f_i^{\dagger}\right) $$

(see, e.g. footnote 3 on pg 163, or the solutions to problems on pg 181). Why is this equality correct? I would think that if we have, for example, $i$ ranging from $1$ to $2$ then:

$$ \exp\left[-\eta_1 f_1^{\dagger}-\eta_2f_2^{\dagger}\right] = 1 + \partial_{\eta_1}\exp\left[...\right]\eta_1 + \partial_{\eta_2}\exp\left[...\right]\eta_2 \\ + \frac{1}{2}\partial_{\eta_1}\partial_{\eta_2}\exp\left[...\right]\eta_2\eta_1 + \frac{1}{2}\partial_{\eta_2}\partial_{\eta_1}\exp\left[...\right]\eta_1\eta_2 $$

Noticing that the last two terms are the same by anticommutation relations we find that:

$$ \exp\left[-\eta_1 f_1^{\dagger}-\eta_2f_2^{\dagger}\right] = 1 - \eta_1 f_1^{\dagger} - \eta_2 f_2^{\dagger} + \eta_1\eta_2 f_1^{\dagger}f_2^{\dagger}$$

In general, the Taylor expansion of an exponential of a linear combination of $N$ Grassmann variables should have nonzero terms up to order $N$ in the Grassmann variables? Is there a reason that Altland and Simons completely ignore the terms of order $>1$?

Answer 1

The formula that Ref. 1 uses is

$$\tag{*} \exp\left(-\sum_j \eta_j a_j^{\dagger} \right) ~=~ \prod_j\exp\left( - \eta_j a_j^{\dagger} \right) ~=~\prod_j \left(1- \eta_j a_j^{\dagger}\right). $$

1. Ref. 1 correctly applies [the Hermitian conjugate of] eq. (*) to the bra in answer (a) on p. 181. There is no mistake on p. 181.

2. Ref. 1 does not write a sum/product over $j$ in footnote 3 on p. 163. Instead Ref. 1 is only considering a single element $i$. In particular there is no sum implied over repeated $i$ indices! Footnote 3 is meant as a proof that the definition (4.17) for a coherent state of Fermionic operators has the property (4.10). To finish the proof, the reader is afterwards supposed to multiply on both sides with the missing $\prod_{j\neq i}$ factor, which is anyway effectively a passive spectator in the calculation. The full calculation of the ket is by the way a Hermitian conjugate version of answer (a) on p. 181.

References:

1. Altland and Simons (A&S), Condensed Matter Field Theory, 2nd edition, 2010.

Answer 2

For a univariable grassmann number $\eta_1$ it holds $\eta_1^2 = 0$ because of $ \{ \eta_1, \eta_1 \} = 0$. Hence all higher powers of this Grassmann number vanish. However, if there are multiple Grassmann numbers, linear combinations of these Grassmann numbers can be computed. It still holds $\eta_i^2 = 0$ for every $i$, but $\eta_i \eta_j \neq 0$ for $i \neq j$. You can show that if you take the $N+k$-th power of the linear combination $\sum_{i=1}^N a_i \eta_i$ for ordinary commutative numbers $a_i$ vanishes for $k > 0$ because in every term there are occuring Grassmann number squares.