  We study the second-order quantum phase-transition of massive real scalar
field theory with a quartic interaction ($\phi^4$ theory) in (1+1) dimensions
on an infinite spatial lattice using matrix product states (MPS). We introduce
and apply a naive variational conjugate gradient method, based on the
time-dependent variational principle (TDVP) for imaginary time, to obtain
approximate ground states, using a related ansatz for excitations to calculate
the particle and soliton masses and to obtain the spectral density. We also
estimate the central charge using finite-entanglement scaling. Our value for
the critical parameter agrees well with recent Monte Carlo results, improving
on an earlier study which used the related DMRG method, verifying that these
techniques are well-suited to studying critical field systems. We also obtain
critical exponents that agree, as expected, with those of the transverse Ising
model. Additionally, we treat the special case of uniform product states (mean
field theory) separately, showing that they may be used to investigate
non-critical quantum field theories under certain conditions.
