  (Please refer to arXiv:1810.08050, which has completely different aims but
contains all the main contents of this paper) In this work, we propose to
access the information of criticality and excitations of one-dimensional
quantum systems by a matrix product state (MPS) defined in the (imaginary) time
direction. This state, dubbed as time MPS (tMPS), is a boundary state of tensor
network (TN) that represents the ground-state simulation after Trotter-Suzuki
decomposition. We show that the tMPS exhibits the structure of the continuous
MPS originally proposed for the field theories. The information of excitations,
e.g., dynamic correlation length and energy gap, can be accurately calculated
from the tMPS. The non-universal renormalization of velocity of the excitations
is given by a ratio between the correlations of the ground state and the tMPS.
When the system is at the quantum critical point, the tMPS is found to show the
logarithmic scaling law, where the scaling coefficient gives the central charge
that characterizes the criticality of the theory. Our work also implies that
the spectra of the transfer matrices defined by the ground state and tMPS
provide information of the low-lying masses of the theory in the continuous
limit. This could help to understand the role of the finiteness of the bond
dimension as an infra red regulator. Furthermore, we show from the perspective
of methodology that, the tMPS emerges from a generalized TN ab-initio
optimization principle scheme, which unifies the infinite density matrix
renormalization group and the infinite time-evolving block decimation
algorithms.

Strongly Correlated Electrons;Statistical Mechanics;Computational Physics