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

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