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

Guanghua Liu, Guang-Shan Tian

The matrix product state (MPS) is utilized to investigate the ground state properties and quantum phase transitions (QPTs) of the dimerized antiferromagnetic Heisenberg (DAH) model. The ground state MPS wavefunctions determined by the infinite time-evolving block decimation (iTEBD) algorithm are shown to be very efficient descriptions of DAH model. In the thermodynamic limit, the quantum entanglement, the bond energy, and the nearest-neighbor correlations are calculated. It is revealed that the singular behavior of the bipartite entanglement can detect the QPTs directly. The critical point J c 2 = 1.0 is determined evidently, and the quantum phase transition is argued to belong to the second-order category. At the critical point, logarithmic divergent character of the block entanglement is observed, and the system can be described by a free bosonic field theory.

L.S. Lima

Christian Rüegg, Philip Merchant, MCMORROW Desmond Francis +3 more

Cheng Peng, Shi-Ju Ran, Maciej Lewenstein +1 more

Characterizing criticality in quantum many-body systems of dimension $\ge 2$ is one of the most important challenges of the contemporary physics. In principle, there is no generally valid theoretical method that could solve this problem. In this work, we propose an efficient approach to identify the criticality of quantum systems in higher dimensions. Departing from the analysis of the numerical renormalization group flows, we build a general equivalence between the higher-dimensional ground state and a one-dimensional (1D) quantum state defined in the imaginary time direction in terms of the so-called time matrix product state (tMPS). We show that the criticality of the targeted model can be faithfully identified by the tMPS, using the mature scaling schemes of correlation length and entanglement entropy in 1D quantum theories. We benchmark our proposal with the results obtained for the Heisenberg anti-ferromagnet on honeycomb lattice. We demonstrate critical scaling relation of the tMPS for the gapless case, and a trivial scaling for the gapped case with spatial anisotropy. The critical scaling behaviors are insensitive to the system size, suggesting the criticality can be identified in small systems. Our tMPS scheme for critical scaling shows clearly that the spin-1/2 kagomé Heisenberg antiferromagnet has a gapless ground state. More generally, the present study indicates that the 1D conformal field theories in imaginary time provide a very useful tool to characterize the criticality of higher dimensional quantum systems.

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.

Guifré Vidal