  We study the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on an
infinity-by-$N$ square lattice for even $N$'s up to $14$. Previously, the
nonlinear sigma model perturbatively predicts that its spin rotational symmetry
asymptotically breaks when $N\rightarrow \infty$, i.e., when it is
two-dimensional (2D). However, we identified a critical width $N_c = 10$ for
which this symmetry breaks spontaneously. It defines a dimensional transition
from one-dimension (1D) including quasi-1D to 2D. The finite-size effect
differs from that of the $N$-by-$N$ lattice. The ground state (GS) energy per
site approaches the thermodynamic limit value, in agreement with the previously
accepted value, by one order of $1/N$ faster than when using $N$-by-$N$
lattices in the literature. We build and variationally solve a matrix product
state (MPS) on a chain, converting the $N$ sites in the rung into an effective
site. We show that the area law of entanglement entropy does not apply when $N$
increases in our method, and show that the reduced density matrix of each
effective site will have a saturating number of dominant diagonal elements with
increasing $N$. These two characteristics make the MPS rank needed to obtain a
demanded energy accuracy quickly saturate when $N$ is large, making our
algorithm efficient for large $N$'s. And, the latter enables space reduction in
MPS. Within the framework of MPS, we prove a theorem that the spin-spin
correlation at infinite separation is the square of staggered magnetization and
demonstrate that the eigenvalue structure of a building MPS unit of $\langle
g\mid g\rangle$, $\mid g\rangle$ being the GS, is responsible for order,
disorder and quasi-long-range order.
