  We discuss how to analytically obtain an -- essentially infinite -- Matrix
Product State (MPS) representation of the ground state of the XY model. On the
one hand this allows to illustrate how the Ornstein-Zernike form of the
correlation function emerges in the exact case using standard MPS language. On
the other hand we study the consequences of truncating the bond dimension of
the exact MPS, which is also part of many tensor network algorithms, and
analyze how the truncated MPS transfer matrix is representing the dominant part
of the exact quantum transfer matrix. In the gapped phase we observe that the
correlation length obtained from a truncated MPS approaches the exact value
following a power law in effective bond dimension. In the gapless phase we find
a good match between a state obtained numerically from standard MPS techniques
with finite bond dimension, and a state obtained by effective finite imaginary
time evolution in our framework. This provides a direct hint for a geometric
interpretation of Finite Entanglement Scaling at the critical point in this
case. Finally, by analyzing the spectra of transfer matrices, we support the
interpretation put forward by [V. Zauner at. al., New J. Phys. 17, 053002
(2015)] that the MPS transfer matrix emerges from the quantum transfer matrix
though the application of Wilson's Numerical Renormalisation Group along the
imaginary-time direction.
