Signature of a quantum dimensional transition in the spin- 12 antiferromagnetic Heisenberg model on a square lattice and space reduction in the matrix product state

Lihua Wang, Kwang S. Kim

arXiv:1803.03779Physical review. B./Physical review. BApr 29, 20195 citations

Bulletin of the American Physical Society

Variational approach for 1D antiferromagnetic Heisenberg chain with matrix-product states

Ying-Jer Kao, Ling Wang, Anders W. Sandvik

Published Mar 14, 2008• 0 citations

OpenAlex

Related papers

Effects of edges in spin- 12 bond-alternating Heisenberg chains: matrix-product variational approach

Kei-ichi Funase, Shoji Yamamoto

arXiv:cond-mat/0412390Physics Letters ADec 15, 20044 citations

Matrix Product State, Quantum Entanglement, and Criticality in the One-Dimensional Dimerized Antiferromagnetic Heisenberg Model

刘光华, 田光善

理论物理通讯：英文版Jan 1, 20120 citations

Incommensurate matrix product state for quantum spin systems

Hiroshi Ueda, Isao Maruyama

We introduce a matrix product state (MPS) with an incommensurate periodicity by applying the spin-rotation operator of each site to a uniform MPS in the thermodynamic limit. The relation between the improvement of the variational method due to the spin-rotation operators and the spontaneous symmetry breaking of the MPS is studied. The optimized pitch of the rotational operator reflects the commensurate/incommensurate properties of spin-spin correlation functions in the $S=1/2$ Heisenberg chain and the $S=1/2$ ferromagnetic-antiferromagnetic zigzag chain.We introduce a matrix product state (MPS) with an incommensurate periodicity by applying the spin-rotation operator of each site to a uniform MPS in the thermodynamic limit. The relation between the improvement of the variational method due to the spin-rotation operators and the spontaneous symmetry breaking of the MPS is studied. The optimized pitch of the rotational operator reflects the commensurate/incommensurate properties of spin-spin correlation functions in the $S=1/2$ Heisenberg chain and the $S=1/2$ ferromagnetic-antiferromagnetic zigzag chain.

arXiv:1112.2075Physical Review BAug 27, 20122 citations

Variational Study of a Finite Heisenberg Chain

Eric Ashendorf, Joe Weiner, Jay D. Mancini +2 more

Bulletin of the American Physical SocietyMar 22, 20110 citations

Numerical Calculations of S=1 Heisenberg Antiferromagnetic Chain

M. Takahashi

Springer proceedings in physicsJan 1, 19920 citations

Signature of a quantum dimensional transition in the spin- 12 antiferromagnetic Heisenberg model on a square lattice and space reduction in the matrix product state

Lihua Wang, Kwang S. Kim

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, the nonlinear sigma model 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 ${N}_{c}=10$ 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 $N\ifmmode\times\else\texttimes\fi{}N$ 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 $N\ifmmode\times\else\texttimes\fi{}N$ 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 $\ensuremath{\langle}g|g\ensuremath{\rangle},|g\ensuremath{\rangle}$ being the GS is responsible for order, disorder, and quasi-long-range order.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, the nonlinear sigma model 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 ${N}_{c}=10$ 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 $N\ifmmode\times\else\texttimes\fi{}N$ 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 $N\ifmmode\times\else\texttimes\fi{}N$ 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 $\ensuremath{\langle}g|g\ensuremath{\rangle},|g\ensuremath{\rangle}$ being the GS is responsible for order, disorder, and quasi-long-range order.

arXiv:1803.03779Physical review. B./Physical review. BApr 29, 20195 citations

S=1/2

Heisenberg chain and the

S=1/2

ferromagnetic-antiferromagnetic zigzag chain.

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, the nonlinear sigma model 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

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 $N\ifmmode\times\else\texttimes\fi{}N$ 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 $N\ifmmode\times\else\texttimes\fi{}N$ 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 $\ensuremath{\langle}g|g\ensuremath{\rangle},|g\ensuremath{\rangle}$ being the GS is responsible for order, disorder, and quasi-long-range order.