We present a study of a simple model antiferromagnet consisting of a sum of nearest-neighbor
SO(N) singlet projectors on the kagome lattice. Our model shares some features with the popular
S=1/2 kagome antiferromagnet but is specifically designed to be free of the sign problem of quantum Monte Carlo. In our numerical analysis, we find as a function of
N a quadrupolar magnetic state and a wide range of a quantum spin liquid. A solvable large-
N generalization suggests that the quantum spin liquid in our original model is a gapped
Z2 topological phase. Supporting this assertion, a numerical study of the entanglement entropy in the sign free model shows a quantized topological contribution.
We introduce a simple model of SO(N) spins with two-site interactions which is amenable to quantum Monte Carlo studies without a sign problem on nonbipartite lattices. We present numerical results for this model on the two-dimensional triangular lattice where we find evidence for a spin nematic at small N, a valence-bond solid at large N, and a quantum spin liquid at intermediate N. By the introduction of a sign-free four-site interaction, we uncover a rich phase diagram with evidence for both first-order and exotic continuous phase transitions.
In this paper, we do a complete classification of valence-bond crystals (VBCs) on the kagomé lattice based on general arguments of symmetry only and thus identify many new VBCs for different unit cell sizes. For the spin-1/2 Heisenberg antiferromagnet, we study the relative energetics of competing gapless spin liquids (SLs) and VBC phases within the class of Gutzwiller-projected fermionic wave functions using variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. By using a state-of-the-art optimization method, we conclusively show that the U(1) Dirac SL is remarkably stable towards dimerizing into all 6-, 12- and 36-site unit cell VBCs. This stability is also preserved on addition of a next-nearest-neighbor super-exchange coupling of both antiferromagnetic and ferromagnetic (FM) type. However, we find that a 36-site unit cell VBC is stabilized on addition of a very small next-nearest-neighbor FM super-exchange coupling, i.e. | J 2 | ≈ 0.045, and this VBC is the same in terms of space-group symmetry as that obtained in an effective quantum dimer model study. It breaks reflection symmetry, has a nontrivial flux pattern and is a strong dimerization of the uniform RVB SL.
A symmetry-based analysis (projective symmetry group) is used to study spin-liquid phases on the triangular and Kagom\'e lattices in the Schwinger boson framework. A maximum of eight distinct
Z2 spin-liquid states are found for each lattice, which preserve all symmetries. Out of these only a few have nonvanishing nearest-neighbor amplitudes, which are studied in greater detail. On the triangular lattice, only two such states are present---the first (zero-flux state) is the well-known state introduced by Sachdev, which on condensation of spinons leads to the 120\ifmmode^\circ\else\textdegree\fi{} ordered state. The other solution, which we call the $\ensuremath{\pi}$-flux state has not previously been discussed. Spinon condensation leads to an ordering wave vector at the Brillouin zone edge centers, in contrast to the 120\ifmmode^\circ\else\textdegree\fi{} state. While the zero-flux state is more stable with just nearest-neighbor exchange, we find that the introduction of either next-neighbor antiferromagnetic exchange or four-spin ring exchange (of the sign obtained from a Hubbard model) tends to favor the $\ensuremath{\pi}$-flux state. On the Kagom\'e lattice four solutions are obtained---two have been previously discussed by Sachdev, which on spinon condensation give rise to the
q=0 and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ spin-ordered states. In addition we find two states with significantly larger values of the quantum parameter at which magnetic ordering occurs. For one of them this even exceeds unity ${\ensuremath{\kappa}}_{c}\ensuremath{\approx}2.0$ in a nearest-neighbor model, indicating that if stabilized, could remain spin disordered for physical values of the spin. This state is also stabilized by ring-exchange interactions with signs as derived from the Hubbard model.
Using large scale quantum Monte Carlo simulations and dual vortex theory, we analyze the ground state phase diagram of hard-core bosons on the kagome lattice with nearest-neighbor repulsion. In contrast with the case of a triangular lattice, no supersolid emerges for strong interactions. While a uniform superfluid prevails at half filling, two novel solid phases emerge at densities rho=1/3 and rho=2/3. These solids exhibit an only partial ordering of the bosonic density, allowing for local resonances on a subset of hexagons of the kagome lattice. We provide evidence for a weakly first-order phase transition at the quantum melting point between these solid phases and the superfluid.
The spin-
21 Heisenberg model on the kagome lattice, which is closely realized in layered Mott insulators such as
ZnCu3(OH)6Cl2, is one of the oldest and most enigmatic spin-
21 lattice models. While the numerical evidence has accumulated in favor of a quantum spin liquid, the debate is still open as to whether it is a
Z2 spin liquid with very short-range correlations (some kind of resonating valence bond spin liquid), or an algebraic spin liquid with power-law correlations. To address this issue, we have pushed the program started by Rokhsar and Kivelson in their derivation of the effective quantum dimer model description of Heisenberg models to unprecedented accuracy for the spin-
21 kagome, by including all the most important virtual singlet contributions on top of the orthogonalization of the nearest-neighbor valence bond singlet basis. Quite remarkably, the resulting picture is a competition between a
Z2 spin liquid and a diamond valence bond crystal with a 12-site unit cell, as in the density-matrix renormalization group simulations of Yan et al. Furthermore, we found that, on cylinders of finite diameter
d, there is a transition between the
Z2 spin liquid at small
d and the diamond valence bond crystal at large
d, the prediction of the present microscopic description for the two-dimensional lattice. These results show that, if the ground state of the spin-
21 kagome antiferromagnet can be described by nearest-neighbor singlet dimers, it is a diamond valence bond crystal, and, a contrario, that, if the system is a quantum spin liquid, it has to involve long-range singlets, consistent with the algebraic spin liquid scenario.