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

Wildeboer, Julia; Seidel, Alexander

doi:10.1103/physrevb.83.184430

Cited by 19 publications

(22 citation statements)

References 42 publications

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“…In particular, the results can be generalized to other lattices: Firstly, the PEPS description of RVB states applies to arbitrary graphs. All lattices with the "linear independence property" 8,27,28 are G-injective, and whenever the linearly independent blocks allow cover the lattice up to disconnected patches (such as in Fig. 9), this allows to interpolate between the RVB and the corresponding dimer state.…”

Section: Discussionmentioning

confidence: 99%

Resonating valence bond states in the PEPS formalism

et al. 2012

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We study resonating valence bond (RVB) states in the Projected Entangled Pair States (PEPS) formalism. Based on symmetries in the PEPS description, we establish relations between the toric code state, the orthogonal dimer state, and the SU(2) singlet RVB state on the kagome lattice: We prove the equivalence of toric code and dimer state, and devise an interpolation between the dimer state and the RVB state. This interpolation corresponds to a continuous path in Hamiltonian space, proving that the RVB state is the four-fold degenerate ground state of a local Hamiltonian on the (finite) kagome lattice. We investigate this interpolation using numerical PEPS methods, studying the decay of correlation functions, the change of overlap, and the entanglement spectrum, none of which exhibits signs of a phase transition.

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Section: Discussionmentioning

confidence: 99%

Resonating valence bond states in the PEPS formalism

et al. 2012

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“…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.…”

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Correlation Functions in SU(2)-Invariant Resonating-Valence-Bond Spin Liquids on Nonbipartite Lattices

Wildeboer¹,

Seidel²

2012

Phys. Rev. Lett.

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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.

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“…The proof of extensivity lies in the exact mappings between dimer coverings, six-vertex models, and the ice rules, and is known exactly in 2D [11] and by virtue of Pauling's exponential lower bound in 3D [16]. Although the dimer coverings are not orthogonal, detailed analyses [13,14,26] provide strong and growing evidence that the number of linearly independent states in such a system also scales exponentially with N , and may even equal the number of dimer coverings. The most important consequence of extensive degeneracy is complete dimensional reduction [15], meaning that all states in the Klein-point manifold are connected by local (zero-dimensional) processes of dimer rearrangement.…”

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Hubbard Model on the Pyrochlore Lattice: A 3D Quantum Spin Liquid

Normand

Nussinov

2014

Phys. Rev. Lett.

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We demonstrate that the insulating one-band Hubbard model on the pyrochlore lattice contains, for realistic parameters, an extended quantum spin-liquid phase. This is a three-dimensional spin liquid formed from a highly degenerate manifold of dimer-based states, which is a subset of the classical dimer coverings obeying the ice rules. It possesses spinon excitations, which are both massive and deconfined, and on doping it exhibits spin-charge separation. We discuss the realization of this state in effective S = 1/2 pyrochlore materials.PACS numbers: 75.10. Jm, 75.10.Kt, 75.40.Gb The quantum spin liquid [1] has become the focal point for our understanding of many of the most fundamental issues in strongly correlated systems. These include exotic quantum phases, quantum critical physics, the relevance of broken symmetries, topological order, entanglement, and the possibly fractional nature of elementary excitations in both gapped and gapless states [2]. The search for theoretical realizations of these ideas has led to numerous proposed models, which while highly informative have generally been too simple or abstract to apply to real materials [3]. The search for materials realizations is a very active field where much current attention is focused on kagome systems [4], triangular organics [5], and other frustrated S = 1/2 and S = 1 quantum magnets. However, materials complexities such as impurities, Dzyaloshinskii-Moriya interactions, spin-orbit coupling, and other anisotropies in real and spin space have to date caused strong departures from theoretical ideals.Frustrated quantum magnets offer one of the most promising routes to spin-liquid behavior [1,2]. Frustration presents a formidable barricade to theoretical understanding, because the ground manifold is quite generally a set of highly degenerate basis states, with little or no separation emerging in an exact treatment of the interactions [6]. Numerical calculations converge very slowly due to this proliferation of near-ground states [7]. Fluctuations in such a manifold may lead to a range of exotic phenomena [8][9][10], and the departures mentioned above are strong because any perturbation is strongly relevant in a highly degenerate system. Few exact results are available, although these afford essential insight [11][12][13][14].In this Letter, we discuss the one-band Hubbard model, showing that on a half-filled pyrochlore lattice it gives a highly frustrated intratetrahedral spin model with only weak perturbations. This model contains an exactly solvable Klein point, about which there is an extended region of parameter space where the ground state is a three-dimensional (3D) quantum spin liquid. This state hosts massive spinon excitations, which are deconfined and move in all three dimensions within the lattice. The parameter range for the spin-liquid phase lies exactly in the regime of many magnetic materials.The pyrochlore lattice, shown in Fig. 1, is a 3D array of corner-sharing tetrahedra, has cubic symmetry, and is a geometry widespread in transitio...

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

Cited by 19 publications

References 42 publications

Resonating valence bond states in the PEPS formalism

Resonating valence bond states in the PEPS formalism

Correlation Functions in SU(2)-Invariant Resonating-Valence-Bond Spin Liquids on Nonbipartite Lattices

Hubbard Model on the Pyrochlore Lattice: A 3D Quantum Spin Liquid

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