Linear independence of nearest-neighbor valence bond states in several two-dimensional lattices

Abstract

We show for several two-dimensional lattices that the nearest neighbor valence bond states are linearly independent. To do so, we utilize and generalize a method that was recently introduced and applied to the kagome lattice by one of the authors. This method relies on the choice of an appropriate cell for the respective lattice, for which a certain local linear independence property can be demonstrated. Whenever this is achieved, linear independence follows for arbitrarily large lattices that can be covered by such cells, for {\em both} open and periodic boundary conditions. We report that this method is applicable to the kagome, honeycomb, square, squagome, two types of pentagonal, square-octagon, the star lattice, two types of archimedean lattices, three types of "martini" lattices, and to fullerene-type lattices, e.g., the well known "Buckyball". Applications of the linear independence property, such as the derivation of effective quantum dimer models, or the constructions of new solvable spin-1/2 models, are discussed.

9 pages, 10 figures, published version, reference corrected

References in corpus (11)

• Spin Dynamics of the Spin-1/2 Kagome Lattice Antiferromagnet ZnCu_3(OH)_6Cl_2
• Spin-Liquid State in the S = 1/2 Hyperkagome Antiferromagnet Na4Ir3O8
• Quantum magnetism in the paratacamite family: towards an ideal kagome lattice
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• Gapless spin liquids on the three dimensional hyper-kagome lattice of Na${}_4$Ir${}_3$O${}_8$
• Topological spin liquid on the hyper-kagome lattice of Na_4Ir_3O_8
• Na4Ir3O8 as a 3D spin liquid with fermionic spinons
• Exact bond percolation thresholds in two dimensions
• Generalized Hardcore Dimer Models approach to low-energy Heisenberg frustrated antiferromagnets: general properties and application to the kagome antiferromagnet
• Symmetry Breaking on the Three-Dimensional Hyperkagome Lattice of Na_4Ir_3O_8
• Variational RVB wave function for the spin-1/2 Heisenberg Model on honeycomb lattice

Cited by in corpus (14)

• Phase Diagram of a Frustrated Quantum Antiferromagnet on the Honeycomb Lattice: Magnetic Order versus Valence-Bond Crystal formation
• Resonating valence bond states in the PEPS formalism
• Exact Spin Liquid Ground States of the Quantum Dimer Model on the Square and Honeycomb Lattices
• Flat band, spin-1 Dirac cone, and Hofstadter diagram in the fermionic square kagome model
• Quantum dimer model for the spin-1/2 kagome Z2 spin liquid
• Correlation functions in SU(2)-invariant RVB spin liquids on nonbipartite lattices
• Nature of the phases in the frustrated XY model on the honeycomb lattice
• Ground state uniqueness of the twelve site RVB spin-liquid parent Hamiltonian on the kagome lattice
• Correlated valence-bond states
• Magnetic properties of a capped kagome molecule with 60 quantum spins
• Exact Solution and Correlations of a Quantum Dimer Model on the Checkerboard Lattice
• Flat-band full localization and symmetry-protected topological phase on bilayer lattice systems
• Entanglement studies of resonating valence bonds on the frustrated square lattice
• Quantum dimer model on fullerenes: resonance, scarring and confinement