We derive exact results for close-packed dimers on the triangular kagome lattice (TKL), formed by inserting triangles into the triangles of the kagome lattice. Because the TKL is a nonbipartite lattice, dimer-dimer correlations are short ranged so that the ground state at the Rokhsar-Kivelson (RK) point of the corresponding quantum dimer model on the same lattice is a short-ranged spin liquid. Using the Pfaffian method, we derive an exact form for the free energy, and we find that the entropy is
31ln 2 per site, regardless of the weights of the bonds. The occupation probability of every bond is
41 in the case of equal weights on every bond. Similar to the case of lattices formed by corner-sharing triangles (such as the kagome and squagome lattices), we find that the dimer-dimer correlation function is identically zero beyond a certain (short) distance. We find in addition that monomers are deconfined on the TKL, indicating that there is a short-ranged spin liquid phase at the RK point. We also find exact results for the ground-state energy of the classical Heisenberg model. The ground state can be ferromagnetic, ferrimagnetic, locally coplanar, or locally canted, depending on the couplings. From the dimer model and the classical spin model, we derive upper bounds on the ground-state energy of the quantum Heisenberg model on the TKL.
In analogy to the quantum dimer model on the square lattice introduced by Rokhsar and Kivelson, we identify the low-energy Hilbert space of the spin-1/2 kagom\'e Heisenberg antiferromagnet with the set of dimerized states (nearest-neighbor singlets) and derive an effective Hamiltonian which describes resonance and includes fluctuations into next-nearest-neighbor singlets. Complete diagonalization of this effective Hamiltonian for systems as large as 36 spins makes possible the study of the low-energy thermodynamics of this quantum antiferromagnet and strongly supports the double peak feature in the heat capacity.
We consider a family of generalized Rokhsar-Kivelson (RK) Hamiltonians, which are reverse-engineered to have an arbitrary edge-weighted superposition of dimer coverings as their exact ground state at the RK point. We focus on a quantum dimer model on the triangular lattice, with doubly periodic edge weights. For simplicity we consider a
2×1 periodic model in which all weights are set to one except for a tunable horizontal edge weight labeled
α. We analytically show that the model exhibits a continuous quantum phase transition at
α=3, changing from a topological
Z2 quantum spin liquid (
α<3) to a columnar ordered state (
α>3). The dimer-dimer correlator decays exponentially on both sides of the transition with the correlation length
ξ∝1/∣α−3∣ and as a power-law at criticality. The vison correlator exhibits an exponential decay in the spin liquid phase, but becomes a constant in the ordered phase, which we explain in terms of loop statistics of the double-dimer model. Using finite-size scaling of the vison correlator, we extract critical exponents consistent with the 2D Ising universality class. Additionally, we analytically show that the topological Rényi entropy of order
∞ (topological min-entropy) changes from
log2 for the quantum spin liquid phase
α<3, to
0 for the ordered phase
α>3, thereby analytically confirming the topological nature of the phase transition.
We construct a generalized quantum dimer model on two-dimensional nonbipartite lattices, including the triangular lattice, the star lattice, and the kagome lattice. At the Rokhsar-Kivelson (RK) point, we obtain its exact ground states that are shown to be a fully gapped quantum spin liquid with the double-semion topological order. The ground-state wave function of such a model at the RK point is a superposition of dimer configurations with a nonlocal sign structure determined by counting the number of loops in the transition graph. We explicitly demonstrate the double-semion topological order in the ground states by showing the semionic statistics of monomer excitations. We also discuss possible implications of such double-semion resonating valence bond states to candidate quantum spin-liquid systems discovered experimentally and numerically in the past few years.
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.
Using series expansions around the dimer limit, we find that the ground state of the spin-1/2 Heisenberg antiferromagnet on the kagome lattice appears to be a valence bond crystal (VBC) with a 36 site unit cell, and ground-state energy per site $E=\ensuremath{-}0.433\ifmmode\pm\else\textpm\fi{}0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}$. It consists of a honeycomb lattice of ``perfect hexagons.'' The energy difference between the ground state and other ordered states with the maximum number of perfect hexagons, such as a stripe-ordered state, is of order
0.001J. The expansion is also done for the 36 site system with periodic boundary conditions; its energy per site is $0.005\ifmmode\pm\else\textpm\fi{}0.001\phantom{\rule{0.3em}{0ex}}\mathrm{J}$ lower than the infinite system, consistent with exact diagonalization results. Every unit cell of the VBC has two singlet states whose degeneracy is not lifted to sixth order in the expansion. We estimate this energy difference to be less than
0.001J. The dimerization order parameter is found to be robust. Two leading orders of perturbation theory give lowest triplet excitations to be dispersionless and confined to the perfect hexagons.