  The Kibble Zurek mechanism in a relativistic $\phi^{4}$ scalar field theory
in $D = (1 + 1)$ is studied using uniform matrix product states. The equal time
two point function in momentum space $G_{2}(k)$ is approximated as the system
is driven through a quantum phase transition at a variety of different quench
rates $\tau_{Q}$. We focus on looking for signatures of topological defect
formation in the system and demonstrate the consistency of the picture that the
two point function $G_{2}(k)$ displays two characteristic scales, the defect
density $n$ and the kink width $d_{K}$. Consequently, $G_{2}(k)$ provides a
clear signature for the formation of defects and a well defined measure of the
defect density in the system. These results provide a benchmark for the use of
tensor networks as powerful non-perturbative non-equilibrium methods for
relativistic quantum field theory, providing a promising technique for the
future study of high energy physics and cosmology.
