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arXiv:1101.1621 · Physical Review B

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

Julia Wildeboer, Alexander Seidel

Published May 26, 2011• 15 citations
arXivHTMLPDF

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 both open and periodic boundary conditions. We report that this method is applicable to the kagome, honeycomb, square, squagome, two types of pentagonal, square-octagonal, star, two types of Archimedean, three types of ``martini'', and 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 construction of new solvable spin-$1/2$ models, are discussed.

DOI
10.1103/physrevb.83.184430

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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 both open and periodic boundary conditions. We report that this method is applicable to the kagome, honeycomb, square, squagome, two types of pentagonal, square-octagonal, star, two types of Archimedean, three types of ``martini'', and 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 construction of new solvable spin-1/21/21/2 models, are discussed.
A class of local SU(2)SU(2)SU(2)-invariant spin-12\frac{1}{2}21​ Hamiltonians is studied that has ground states within the space of nearest-neighbor valence-bond states on the kagome lattice. Cases include ``generalized Klein'' models without obvious non-valence-bond ground states, as well as a ``resonating valence-bond'' Hamiltonian whose unique ground states within the nearest-neighbor valence-bond space are four topologically degenerate ``Sutherland-Rokhsar-Kivelson'' (SRK) -type wave functions, which are expected to describe a gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquid. The proof of this uniqueness is intimately related to the linear independence of the nearest-neighbor valence-bond states on quite general and arbitrarily large kagome lattices, which is also established in this work. It is argued that the SRK ground states are also unique within the entire Hilbert space depending on properties of the generalized Klein models. Applications of the strategies developed in this work to other lattice types are also discussed.
We consider dimers on the star lattice (aka the 3-12, Fisher, expanded kagome or triangle-honeycomb lattice). We show that dimer coverings on this lattice have Z_2 arrow and pseudo-spin representations analogous to those for the kagome lattice, and use these to construct an exactly solvable quantum dimer model (QDM) with a Rokhsar-Kivelson (RK) ground state. This QDM, first discussed by Moessner and Sondhi from a different point of view, is the star-lattice analogue of a kagome-lattice QDM analyzed by Misguich et al. We give a detailed analysis of various properties of the classical equal-weight dimer model on the star lattice, most of which are related to those of the RK state. Using both the arrow representation and the fermionic path integral formulation of the Pfaffian method, we calculate the number of dimer coverings, dimer occupation probabilities, and dimer, vison, and monomer correlation functions. We show that a dimer is uncorrelated with dimers further away than on neighboring triangles, that the vison-vison correlation function vanishes, and that the monomer-monomer correlation function equals 1/4 regardless of the monomer positions (with one exception). A key result in the fermionic approach is the vanishing of the two-point Green function beyond a short distance. These properties are very similar to those of dimers on the kagome lattice. We also discuss some generalizations involving arrow representations for dimer coverings on "general Fisher lattices" and their "reduced" lattices, with the kagome, squagome, and triangular kagome lattice being examples of the latter.
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.
We construct a family of rotationally invariant, local, S=1/2 Klein Hamiltonians on various lattices that exhibit ground state manifolds spanned by nearest-neighbor valence bond states. We show that with selected perturbations such models can be driven into phases modeled by well understood quantum dimer models on the corresponding lattices. Specifically, we show that the perturbation procedure is arbitrarily well controlled by a new parameter which is the extent of decoration of the reference lattice. This strategy leads to Hamiltonians that exhibit i) Z2Z_2Z2​ RVB phases in two dimensions, ii) U(1) RVB phases with a gapless ``photon'' in three dimensions, and iii) a Cantor deconfined region in two dimensions. We also construct two models on the pyrochlore lattice, one model exhibiting a Z2Z_2Z2​ RVB phase and the other a U(1) RVB phase.
We consider dimers on the star lattice (aka the 3-12, Fisher, expanded kagome or triangle-honeycomb lattice). We show that dimer coverings on this lattice have Z_2 arrow and pseudo-spin representations analogous to those for the kagome lattice, and use these to construct an exactly solvable quantum dimer model (QDM) with a Rokhsar-Kivelson (RK) ground state. This QDM, first discussed by Moessner and Sondhi from a different point of view, is the star-lattice analogue of a kagome-lattice QDM analyzed by Misguich et al. We give a detailed analysis of various properties of the classical equal-weight dimer model on the star lattice, most of which are related to those of the RK state. Using both the arrow representation and the fermionic path integral formulation of the Pfaffian method, we calculate the number of dimer coverings, dimer occupation probabilities, and dimer, vison, and monomer correlation functions. We show that a dimer is uncorrelated with dimers further away than on neighboring triangles, that the vison-vison correlation function vanishes, and that the monomer-monomer correlation function equals 1/4 regardless of the monomer positions (with one exception). A key result in the fermionic approach is the vanishing of the two-point Green function beyond a short distance. These properties are very similar to those of dimers on the kagome lattice. We also discuss some generalizations involving arrow representations for dimer coverings on general Fisher lattices and their reduced lattices, with the kagome, squagome, and triangular kagome lattice being examples of the latter.
We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome, and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb, and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension X_{t2}=4 predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent. The amplitude of the power-law correction associated with X_{t2}=4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.