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arXiv:1807.03254 · Physical review. B./Physical review. B

SU(3) trimer resonating-valence-bond state on the square lattice

Xiaoyu Dong, Ji-Yao Chen, Hong-Hao Tu

Published Nov 8, 2018• 9 citations
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Abstract

We propose and study an SU(3) trimer resonating-valence-bond (tRVB) state with ${C}_{4v}$ 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 $S$ and $T$ matrices by constructing all nine topological sectors on a torus and establish the existence of ${\mathbb{Z}}_{3}$ topological order in this SU(3) tRVB state.

DOI
10.1103/physrevb.98.205117

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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.
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 generalize the construction of the spin-1/2 SU(2) resonating valence bond (RVB) state to the case of the self-conjugate 6\mathbf{6}6 representation of SU(4). As for the case of SU(2) [J.-Y. Chen and D. Poilblanc, Phys. Rev. B 97, 161107(R) (2018)], we use the projected entangled pair state (PEPS) formalism to derive a simple (two-dimensional) family of generalized SU(4) RVB states on the square lattice. We show that, when longer-range SU(4)-singlet bonds are included, a local gauge symmetry is broken down from U(1) to Z2{\mathbb{Z}}_{2}Z2​, leading to the emergence of a short-range spin liquid. Evidence for the topological nature of this spin liquid is provided by the investigation of the Renyi entanglement entropy of infinitely long cylinders and of the modular matrices. Relevance to microscopic models and experiments of ultracold atoms is discussed.
We construct a topological spin-liquid (TSL) model on the kagome lattice, with SU(3)\mathrm{SU}(3)SU(3) symmetry with the fundamental representation at each lattice site, based on projected entangled pair states (PEPS). Using the PEPS framework, we can adiabatically connect the model to a fixed-point model (analogous to the dimer model for resonating valence bond states) which we prove to be locally equivalent to a Z3{\mathbb{Z}}_{3}Z3​ quantum double model. Numerical study of the interpolation reveals no sign of a phase transition or long-range order, characterizing the model conclusively as a gapped TSL. We further study the entanglement spectrum of the model and find that while it is gapped, it exhibits branches with vastly different velocities, with the slow branch matching the counting of a chiral SU(3)1\mathrm{SU}{(3)}_{1}SU(3)1​ Conformal Field Theory, suggesting that it can be deformed to a model with chiral SU(3)1\mathrm{SU}{(3)}_{1}SU(3)1​ entanglement 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.
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.