Signature of a quantum dimensional transition in the spin-1/2 antiferromagnetic Heisenberg model on a square lattice and space reduction in the matrix product state

We study the spin-1/2 antiferromagnetic Heisenberg model on an infinity x N square lattice for even N's up to 14. Previously, the nonlinear sigma model perturbatively predicted that its spin-rotational symmetry breaks asymptotically with N -> infinity, i.e., when it becomes two dimensional (2D). However, we identify a critical width N-c= 10 for which this symmetry breaks spontaneously. It shows the signature of a dimensional transition from one dimensional (1D) including quasi-1D to 2D. The finite-size effect differs from that of the N x 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 x N lattices in the literature. Methodwise, we build and variationally solve a matrix product state (MPS) on a chain, converting the N sites in each rung into an effective site. We show that the area law of entanglement entropy does not apply when N increases in our method and the reduced density matrix of each effective site has a saturating number of dominant diagonal elements with increasing N. These two characteristics make the MPS rank needed to obtain a desired energy accuracy quickly saturate when N is large, making our algorithm efficient for large N's. Furthermore, 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 < g vertical bar g >, vertical bar g > being the GS is responsible for order, disorder, and quasi-long-range order.