We introduce a Monte Carlo scheme based on sampling of Pfaffians to investigate Anderson's resonating-valence-bond (RVB) spin liquid wave function on the kagome and the triangular lattice. This eliminates a sign problem that prevents utilization of the valence bond basis in Monte Carlo studies for non-bipartite lattices. Studying lattice sizes of up to 600 sites, we calculate singletsinglet and spin-spin correlations, and demonstrate how the lattice symmetry is restored within each topological sector as the system size is increased. Our findings are consistent with the expectation that the nearest neighbor RVB states describe a topological spin liquid on these non-bipartite lattices.Introduction. It has been almost four decades since Anderson proposed[1] the quantum spin liquid state. Its undiminished appeal stems from a variety of applications from high temperature superconductivity[2] to quantum computing [3,4]. The nature of the short ranged variant of Anderson's "resonating valence bond" (RVB) spin liquid as a topological phase became understood through a series of papers [5][6][7]. In particular, the invention of quantum dimer models [7] as an approximation to spin models finally lead to a lattice model exhibiting a topological RVB liquid phase [8,24]. This however, did not immediately address the (original) question whether this exotic phase could be stabilized within the phase diagram of SU (2)-invariant local spin-1/2 Hamiltonians. This was subsequently established for highly decorated lattices [9] and certain bipartite lattices [10], by finding a parent Hamiltonian for the simplest, i.e., nearest neighbor version of the prototypical RVB spin liquid wave function on such lattices. Work on quantum dimer models [8,[11][12][13], however, strongly suggests that nearest neighbor RVB states should be critical on bipartite lattices, as demonstrated recently [16,17]. They should describe a Z 2 -spin liquid with exponentially decaying correlations only in the nonbipartite case. While rigorously proven in the quantum dimer case, it is highly non-trivial establish this statement for the spin-1/2 RVB wave functions, due to orthogonality issues (cf, e.g., [14]). In the nonbipartite case, the nature of the correlation functions of the local spin and valence bond operator has not yet been studied systematically. This is largely due to a sign problem that will be addressed in this work. We finally mention that for the kagome case, the short-ranged RVB state studied here has been proven to be the ground state of a local parent Hamiltonian[18] (cf. also [19,20]). The present work will provide essential evidence from correlations demonstrating that the kagome lattice RVB ground state of the Hamiltonian given in [18] is a topological (Z 2 ) spin liquid.
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. arXiv:1310.8000v1 [cond-mat.str-el]
A small value of the spin gap in quantum antiferromagnets with strong frustration makes them susceptible to nominally small deviations from the ideal Heisenberg model. One of such perturbations, the anisotropic Dzyaloshinskii-Moriya interaction, is an important perturbation for the S = 1/2 kagome antiferromagnet, one of the current candidates for a quantum-disordered ground state. We study the influence of the DM term in a related one-dimensional system, the sawtooth chain that has valence-bond order in its ground state. Through a combination of analytical and numerical methods, we show that a relatively weak DM coupling, 0.115J, is sufficient to destroy the valence-bond order, close the spin gap, and turn the system into a Luttinger liquid with algebraic spin correlations. A similar mechanism may be at work in the kagome antiferromagnet.
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