Topological defects (kinks) in a relativistic $\phi^{4}$ scalar field theory
in $D=(1+1)$ are studied using the matrix product state tensor network. The one
kink state is approximated as a matrix product state and the kink mass is
calculated. The approach used is quite general and can be applied to a variety
of theories and tensor networks. Additionally, the contribution of
kink-antikink excitations to the ground state is examined and a general method
to estimate the scalar mass from equal time ground state observables is
provided. The scalar and kink mass are compared at strong coupling and behave
as expected from universality arguments. This suggests that the matrix product
state can adequately capture the physics of defect-antidefect excitations and
thus provides a promising technique to study challenging non-equilibrium
physics such as the Kibble-Zurek mechanism of defect formation.