  Tensor network methods have progressed from variational techniques based on
matrix-product states able to compute properties of one-dimensional
condensed-matter lattice models into methods rooted in more elaborate states
such as projected entangled pair states aimed at simulating the physics of
two-dimensional models. In this work, we advocate the paradigm that for
two-dimensional fermionic models, matrix-product states are still applicable to
significantly higher accuracy levels than direct embeddings into
one-dimensional systems allow for. To do so, we exploit schemes of fermionic
mode transformations and overcome the prejudice that one-dimensional embeddings
need to be local. This approach takes the insight seriously that the suitable
exploitation of both the manifold of matrix-product states and the unitary
manifold of mode transformations can more accurately capture the natural
correlation structure. By demonstrating the residual low levels of entanglement
in emerging modes, we show that matrix-product states can describe ground
states strikingly well. The power of the approach is exemplified by
investigating a phase transition of spin-less fermions for lattice sizes up to
10x10.
