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arXiv:1612.06899 · Physical review. B./Physical review. B

Resonating valence bond states with trimer motifs

Hyun‐Yong Lee, Yun-Tak Oh, Jung Hoon Han, Hosho Katsura

Published Feb 21, 2017• 12 citations
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Abstract

A trimer resonating valence bond (tRVB) state consisting of an equal-weight superposition of trimer coverings on a square lattice is proposed. A model Hamiltonian of the Rokhsar-Kivelson type for which the tRVB becomes the exact ground state is written. The state is shown to have ${9}^{g}$ topological degeneracy on a genus $g$ surface and support ${\mathbb{Z}}_{3}$ vortex excitations. Correlation functions show exponential behavior with a very short correlation length consistent with the gapped spectrum. The classical problem of the degeneracy of trimer configurations is investigated by the transfer matrix method.

DOI
10.1103/physrevb.95.060413

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A trimer resonating valence bond (tRVB) state consisting of an equal-weight superposition of trimer coverings on a square lattice is proposed. A model Hamiltonian of the Rokhsar-Kivelson type for which the tRVB becomes the exact ground state is written. The state is shown to have 9g{9}^{g}9g topological degeneracy on a genus ggg surface and support Z3{\mathbb{Z}}_{3}Z3​ vortex excitations. Correlation functions show exponential behavior with a very short correlation length consistent with the gapped spectrum. The classical problem of the degeneracy of trimer configurations is investigated by the transfer matrix method.
Trimers are defined as two adjacent edges on a graph. We study the quantum states obtained as equal-weight superpositions of all trimer coverings of a lattice, with the constraint of having a trimer on each vertex: the so-called trimer resonating-valence-bond (tRVB) states. Exploiting their tensor network representation, we show that these states can host Z3{\mathbb{Z}}_{3}Z3​ topological order or can be gapless liquids with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ local symmetry. We prove that this continuous symmetry emerges whenever the lattice can be tripartite such that each trimer covers all the three sublattices. In the gapped case, we demonstrate the stability of topological order against dilution of maximal trimer coverings, which is relevant for realistic models where the density of trimers can fluctuate. Furthermore, we clarify the connection between gapped tRVB states and Z3{\mathbb{Z}}_{3}Z3​ lattice gauge theories by smoothly connecting the former to the Z3{\mathbb{Z}}_{3}Z3​ toric code, and discuss the nonlocal excitations on top of tRVB states. Finally, we analyze via exact diagonalization the zero-temperature phase diagram of a diluted trimer model on the square lattice and demonstrate that the ground state exhibits topological properties in a narrow region in parameter space. We show that a similar model can be implemented in Rydberg atom arrays exploiting the blockade effect. We investigate dynamical preparation schemes in this setup and provide a viable route for probing experimentally Z3{\mathbb{Z}}_{3}Z3​ quantum spin liquids.
We propose and study an SU(3) trimer resonating-valence-bond (tRVB) state with C4v{C}_{4v}C4v​ point-group symmetry on the square lattice. By devising a projected entangled-pair state representation, we show that all (connected) correlation functions between local operators in this SU(3) tRVB state decay exponentially, indicating its gapped nature. We further calculate the modular SSS and TTT matrices by constructing all nine topological sectors on a torus and establish the existence of Z3{\mathbb{Z}}_{3}Z3​ topological order in this SU(3) tRVB state.
We introduce the quantum dimer-pentamer model (QDPM) on the square lattice. This model is a generalization of the square-lattice quantum dimer model as its configuration space comprises fully packed hard-core dimer coverings as well as dimer configurations containing pentamers, where four dimers touch a vertex. Thus in the QDPM, the fully packed, hard-core constraint of the quantum dimer model is relaxed such that the local dimer number at each vertex is fixed modulo 3, resulting in an exact local Z3{Z}_{3}Z3​ gauge symmetry. We construct a local Hamiltonian for which the Rokhsar-Kivelson (RK) equal superposition state is the exact ground state and has a ninefold topological degeneracy on the torus. Using Monte Carlo calculations, we find no spontaneous symmetry breaking in the RK wave function and that its dimer-dimer correlation function decays exponentially. By doping the QDPM RK state with a pair of monomers, we demonstrate that Z3{Z}_{3}Z3​ electric charges are deconfined. Additionally, we introduce a Z3{Z}_{3}Z3​ magnetic string operator that we find decays exponentially and shows no signatures of magnetic vortex condensation. These numerical results suggest that the ground state of the QDPM is a dimer liquid with Z3{Z}_{3}Z3​ topological order.
We construct and study quantum trimer models and resonating SU(3)-singlet models on the kagome lattice, which generalize quantum dimer models and the resonating valence bond wave functions to a trimer and SU(3) setting. We demonstrate that these models carry a Z(3) symmetry which originates in the structure of trimers and the SU(3) representation theory, and which becomes the only symmetry under renormalization. Based on this, we construct simple and exact parent Hamiltonians for the model which exhibit a topological ninefold degenerate ground space. A combination of analytical reasoning and numerical analysis reveals that the quantum order ultimately displayed by the model depends on the relative weight assigned to different types of trimers-it can display either Z(3) topological order or form a symmetry-broken trimer crystal, and in addition possesses a point with an enhanced U(1) symmetry and critical behavior. Our results accordingly hold for the SU(3) model, where the two natural choices for trimer weights give rise to either a topological spin liquid or a system with symmetry-broken order, respectively. Our work thus demonstrates the suitability of resonating trimer and SU(3)-singlet ansatzes to model SU(3) topological spin liquids on the kagome lattice.
We introduce several exact models for SU(3) spin chains: (1) a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation 3‾\overline{\mathbf{3}}3 of SU(3) if the original spins of the model transform under representation 3. (2) a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement, and a Haldane gap in the excitation spectrum.
We construct and study quantum trimer models and resonating SU(3)-singlet models on the kagome lattice, which generalize quantum dimer models and the resonating valence bond wave functions to a trimer and SU(3) setting.We demonstrate that these models carry a Z 3 symmetry which originates in the structure of trimers and the SU(3) representation theory, and which becomes the only symmetry under renormalization.Based on this, we construct simple and exact parent Hamiltonians for the model which exhibit a topological ninefold degenerate ground space.A combination of analytical reasoning and numerical analysis reveals that the quantum order ultimately displayed by the model depends on the relative weight assigned to different types of trimers-it can display either Z 3 topological order or form a symmetry-broken trimer crystal, and in addition possesses a point with an enhanced U(1) symmetry and critical behavior.Our results accordingly hold for the SU(3) model, where the two natural choices for trimer weights give rise to either a topological spin liquid or a system with symmetry-broken order, respectively.Our work thus demonstrates the suitability of resonating trimer and SU(3)-singlet ansatzes to model SU(3) topological spin liquids on the kagome lattice.