Non-diagonal boundary conditions for gl(1|1) super spin chains

We study a one-dimensional model of free fermions with $\mathfrak{gl}(1|1)$ supersymmetry and demonstrate how non-diagonal boundary conditions can be incorporated into the framework of the graded Quantum Inverse Scattering Method (gQISM) by means of \emph{super matrices} with entries from a superalgebra. For super hermitian twists and open boundary conditions subject to a certain constraint, we solve the eigenvalue problem for the super transfermatrix by means of the graded algebraic Bethe ansatz technique (gABA) starting from a fermionic coherent state. For generic boundary conditions the algebraic Bethe ansatz can not be applied. In this case the spectrum of the super transfer matrix is obtained from a functional relation.