  Employing matrix product states as an ansatz, we study the non-thermal phase
structure of the (1+1)-dimensional massive Thirring model in the sector of
vanishing total fermion number with staggered regularization. In this paper,
details of the implementation for this project are described. To depict the
phase diagram of the model, we examine the entanglement entropy, the fermion
bilinear condensate and two types of correlation functions. Our investigation
shows the existence of two phases, with one of them being critical and the
other gapped. An interesting feature of the phase structure is that the theory
with non-zero fermion mass can be conformal. We also find clear numerical
evidence that these phases are separated by a transition of the
Berezinskii-Kosterlitz-Thouless type. Results presented in this paper establish
the possibility of using the matrix product states for probing this type of
phase transition in quantum field theories. They can provide information for
further exploration of scaling behaviour, and serve as an important ingredient
for controlling the continuum extrapolation of the model.
