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arXiv:0906.0357 · Physical Review B

Linear independence of nearest-neighbor valence-bond states on the kagome lattice and construction of S U ( 2 ) -invariant spin- 1 2 Hamiltonian with a Sutherland-Rokhsar-Kivelson quantum liquid ground state

Alexander Seidel

Published Oct 29, 2009• 35 citations
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Abstract

A class of local $SU(2)$-invariant spin-$\frac{1}{2}$ 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 ${\mathbb{Z}}_{2}$ 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.

DOI
10.1103/physrevb.80.165131

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•Jan 1, 2005•1 citation
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.
The spin-12\frac{1}{2}21​ Heisenberg model on the kagome lattice, which is closely realized in layered Mott insulators such as ZnCu3(OH)6Cl2{\mathrm{ZnCu}}_{3}{(\mathrm{OH})}_{6}{\mathrm{Cl}}_{2}ZnCu3​(OH)6​Cl2​, is one of the oldest and most enigmatic spin-12\frac{1}{2}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{Z}_{2}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-12\frac{1}{2}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{Z}_{2}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 ddd, there is a transition between the Z2{Z}_{2}Z2​ spin liquid at small ddd and the diamond valence bond crystal at large ddd, the prediction of the present microscopic description for the two-dimensional lattice. These results show that, if the ground state of the spin-12\frac{1}{2}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.
Anderson's idea of a (short-ranged) resonating valence bond (RVB) spin liquid has been the first ever proposal of what we now call a topologically ordered phase. Since then, a wealth of exactly solvable lattice models have been constructed that have topologically ordered ground states. For a long time, however, it has been difficult to realize Anderson's original vision in such solvable models, according to which the ground state has an unbroken SU(2) spin rotational symmetry and is dominated by fluctuation of singlet bonds. The kagome lattice is the simplest lattice geometry for which parent Hamiltonians stabilizing a prototypical spin-1$/$2 short-ranged RVB wave function has been constructed and strong evidence has been given that this state belongs to a topological phase. The uniqueness of the desired RVB-type ground states has, however, not been rigorously proven for the simplest possible such Hamiltonian, which acts on 12 spins at a time. Rather, this uniqueness has been demonstrated for a longer ranged (19-site) variant of this Hamiltonian by Schuch et al., via making contact with powerful results for projected entangled-pair states. In this paper, we extend this result to the 12-site Hamiltonian. Our result is based on numerical studies on finite clusters, for which we demonstrate a ``ground state intersection property'' with implications for arbitrary system size. We also review the relations between various constructions schemes for RVB parent Hamiltonians found in the literature.
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.
Gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids have been proposed as candidates for the ground state of the S=1/2S=1/2S=1/2 quantum antiferromagnet on the kagome lattice. We extend the use of projected entangled pair states to construct (on the cylinder) resonating valence bond (RVB) states including both nearest-neighbor and next-nearest-neighbor singlet bonds. Our ansatz---dubbed ``simplex spin liquid''---allows for an asymmetry between the two types of triangles (of order 22%--3%2 in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological Z2{\mathbb{Z}}_{2}Z2​ structure is still preserved and, by considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB minimally entangled states. The spinon and vison coherence lengths are extracted from a finite size scaling with regard to the cylinder perimeter of the energy splittings of the four sectors and are found to be of the order of the lattice spacing. The entanglement spectrum of a partitioned (infinite) cylinder is found to be gapless, suggesting the occurrence, on a cylinder with real open boundaries, of gapless edge modes formally similar to Luttinger liquid (nonchiral) spin and charge modes. When inversion symmetry is spontaneously broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might have been observed in recent exact diagonalization studies.
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.