Fermionic coherent state ($a|\theta ⟩=\theta |\theta ⟩$)

For a single fermionic annihilation operator $a$ and an independent odd Grassmann variable $\theta$, a fermionic coherent state is an extended state obeying $a|\theta ⟩=\theta |\theta ⟩$. With $\{a,\theta \}=0$ one may take $|\theta ⟩=\exp (-\theta a^{†})|0⟩$. Its overlap is exponential in the Grassmann labels. A resolution of identity with a Berezin integral gives the time-slice construction of a fermionic path integral. These formal eigenstates use Grassmann-valued coefficients.

The ordinary trace of an even fermionic operator requires $-\psi$ in the bra of its fermionic coherent state kernel. In one mode the diagonal matrix elements give $⟨-\psi |A|\psi ⟩=A_{00}-\bar{\psi }\psi A_{11}$, whose Gaussian Berezin integral is $A_{00}+A_{11}$. An untwisted bra gives $A_{00}-A_{11}$, the supertrace. This distinction produces the antiperiodic coherent-state thermal boundary conditions.