Frustrated quantum antiferromagnetism is an important physical phenomena leading to many of the key properties of quantum matter, such as quantum spin liquids and topological order. Here, the authors employ novel techniques based on tensor network methods for the triangle-honeycomb lattice in order to compute ground-state properties of a frustrated quantum antiferromagnet on the star lattice, with high accuracy, in the thermodynamic limit. Importantly, their findings contradict some of the existing results in the literature on this model. In particular, they find a new resonating valence-bond solid (VBS) phase that had not been observed before, as well as a different VBS structure at the isotropic point than the one previously conjectured by other methods. Moreover, by using tensor network techniques, they also determine the existence of a continuous quantum phase transition in the system between different VBS orders.
We propose that a valence-bond-solid (VBS) order can be stabilized in certain two-dimensional antiferromagnets due to spin-lattice coupling. In contrast to the VBS state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type, the spin-lattice coupling-induced VBS state can occur when
2S is not an integer multiple of the coordination number
z.
S=2 spins on the triangular lattice with
z=6 is discussed as an example. Within the Schwinger boson mean-field theory the ground state is derived as the direct product of states, one of which represents the conventional long-range ordered spins, and the other given by the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ modulation of the valence bond amplitudes. Excitation spectrum for the modulated valence bond state is worked out within the single-mode approximation. The spectrum offers a new collective mode, distinct from the spin wave excitations of the magnetically ordered ground state, and observable in neutron scattering.
Various lattice geometries and boundary conditions are used to investigate valence-bond-solid (VBS) ordering in the ground state of an
S=1/2 square-lattice quantum spin model---the
J-
Q model, in which four- or six-spin interactions
Q are added to the standard Heisenberg exchange
J. Ground-state results for finite systems (with up to thousands of spins) are obtained using an unbiased projector quantum Monte Carlo method. It is found that great care has to be taken when extrapolating the order parameter to infinite lattice size, in particular, in cylinder geometry. Even though strong VBS order exists in two dimensions, and is established clearly with increasing system size on $L\ifmmode\times\else\texttimes\fi{}L$ lattices (or ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ lattices with a fixed aspect ratio
Lx/Ly of order 1), only short-range VBS correlations are observed on long cylinders (when ${L}_{x}\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed
Ly). The correlation length increases with the cylinder width, until long-range order sets in at a ``critical'' width. This width is very large even when the 2D order is relatively strong. For example, for a system in which the order parameter is
70 of the largest possible value,
Ly=8 is required for ordering. Extrapolations of the VBS order parameter based on correlation functions (the square of the order parameter) for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices can also be misleading. For a
20-ordered system, results for
L up to $\ensuremath{\approx}$20 appear to extrapolate clearly to a vanishing order parameter, while for larger lattices the scaling behavior crosses over and extrapolates to a nonzero value (with exponentially small finite-size corrections). The VBS order parameter also exhibits interesting edge effects related the known emergent U(1) symmetry close to a ``deconfined'' critical point, which, if not considered properly, can lead to wrong conclusions for the thermodynamic limit. The observed finite-size behavior for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices and long cylinders is very similar to that predicted for a
Z2 spin liquid. The results therefore raise concerns about recent numerical work claiming
Z2 spin-liquid ground states in 2D frustrated quantum spin systems, in particular, the Heisenberg model with nearest and next-nearest-neighbor couplings. Based on the results presented so far, a VBS state in this system cannot be ruled out.
In this paper, we propose a method to understand the nature of the quantum disorder phase of the two-dimensional (2D) high-spin frustrated model. The ground state and excitation properties of a fully frustrated 2D spin-1 model are studied based on a model whose ground state can be found exactly. By analogy to the pseudopotential approach in the fractional quantum Hall effect, we conclude that the ground states of the fully frustrated spin-1 model are doubly degenerate valence-bond solid (VBS) states along the horizontal or vertical direction of the square lattice. We also find that the ground state could be characterized by a nonzero string order, which rarely happened in the 2D case. The method that we used reveals the connection between the fractional quantum Hall effect and the frustrated 2D antiferromagnetism system. The VBS states capture the main character of the disordered phase in the 2D spin-1 frustrated system and can be verified by a numerical method.
We calculate the bipartite von Neumann and second R\'enyi entanglement entropies of the ground states of spin-1/2 dimerized Heisenberg antiferromagnets on a square lattice. Two distinct dimerization patterns are considered: columnar and staggered. In both cases, we concentrate on the valence bond solid (VBS) phase and describe such a phase with the bond-operator representation. Within this formalism, the original spin Hamiltonian is mapped into an effective interacting boson model for the triplet excitations. We study the effective Hamiltonian at the harmonic approximation and determine the spectrum of the elementary triplet excitations. We then follow an analytical procedure, which is based on a modified spin-wave theory for finite systems and was originally employed to calculate the entanglement entropies of magnetic ordered phases, and calculate the entanglement entropies of the VBS ground states. In particular, we consider one-dimensional (line) subsystems within the square lattice, a choice that allows us to consider line subsystems with sizes up to ${L}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}1000$. We combine such a procedure with the results of the bond-operator formalism at the harmonic level and show that, for both dimerized Heisenberg models, the entanglement entropies of the corresponding VBS ground states obey an area law as expected for gapped phases. For both columnar-dimer and staggered-dimer models, we also show that the entanglement entropies increase but they seem to not diverge as the dimerization decreases and the system approaches the N\'eel-VBS quantum phase transition. Finally, the entanglement spectra associated with the VBS ground states are presented.