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arXiv:1704.02063 · Physical review. B./Physical review. B

Z3 topological order in the quantum dimer-pentamer model

Owen D. Myers, C. M. Herdman

Published Nov 27, 2017• 3 citations
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Abstract

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 ${Z}_{3}$ 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 ${Z}_{3}$ electric charges are deconfined. Additionally, we introduce a ${Z}_{3}$ 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 ${Z}_{3}$ topological order.

DOI
10.1103/physrevb.96.174434

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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 define a quantum monomer-dimer model in the space of maximal dimer coverings of quasicrystalline Penrose tilings. Since Penrose tilings do not admit perfect dimer coverings, as shown by F. Flicker et al., PRX 10, 011005 (2020), monomers are necessarily present in our model. The model features a frustration-free Rokhsar-Kivelson (RK) point where the ground state is a uniform superposition of all the exponentially many maximal dimer coverings, despite the presence of a finite density of monomers. We map our model to a Z2\mathbb{Z}_2Z2​ gauge theory with matter and calculate various correlators to characterize the phase of the system at the RK point using classical Monte Carlo calculations. Specifically, we compute the dimer-dimer and vison-vison correlators, as well as open Wilson lines and closed Wilson loops corresponding to the monomers and the visons. We find that both the dimer-dimer and vison-vison correlators decay exponentially with distance. The open Wilson lines and closed Wilson loops decay exponentially with the same correlation length, indicating that the gauge theory is in the confined phase, which implies that the system is likely in an ordered phase.
We construct a generalized quantum dimer model on two-dimensional nonbipartite lattices, including the triangular lattice, the star lattice, and the kagome lattice. At the Rokhsar-Kivelson (RK) point, we obtain its exact ground states that are shown to be a fully gapped quantum spin liquid with the double-semion topological order. The ground-state wave function of such a model at the RK point is a superposition of dimer configurations with a nonlocal sign structure determined by counting the number of loops in the transition graph. We explicitly demonstrate the double-semion topological order in the ground states by showing the semionic statistics of monomer excitations. We also discuss possible implications of such double-semion resonating valence bond states to candidate quantum spin-liquid systems discovered experimentally and numerically in the past few years.
We consider a family of generalized Rokhsar-Kivelson (RK) Hamiltonians, which are reverse-engineered to have an arbitrary edge-weighted superposition of dimer coverings as their exact ground state at the RK point. We focus on a quantum dimer model on the triangular lattice, with doubly periodic edge weights. For simplicity we consider a 2×12\times12×1 periodic model in which all weights are set to one except for a tunable horizontal edge weight labeled ααα. We analytically show that the model exhibits a continuous quantum phase transition at α=3α=3α=3, changing from a topological Z2\mathbb{Z}_2Z2​ quantum spin liquid (α<3α<3α<3) to a columnar ordered state (α>3α>3α>3). The dimer-dimer correlator decays exponentially on both sides of the transition with the correlation length ξ∝1/∣α−3∣ξ\propto1/|α-3|ξ∝1/∣α−3∣ and as a power-law at criticality. The vison correlator exhibits an exponential decay in the spin liquid phase, but becomes a constant in the ordered phase, which we explain in terms of loop statistics of the double-dimer model. Using finite-size scaling of the vison correlator, we extract critical exponents consistent with the 2D Ising universality class. Additionally, we analytically show that the topological Rényi entropy of order ∞\infty∞ (topological min-entropy) changes from log⁡2\log2log2 for the quantum spin liquid phase α<3α<3α<3, to 000 for the ordered phase α>3α>3α>3, thereby analytically confirming the topological nature of the phase transition.
Topologically ordered quantum systems have robust physical properties, such as quasiparticle statistics, and ground-state degeneracy, which do not depend on the microscopic details of the Hamiltonian. We consider topological phase transitions under a deformation such as an effective string tension on a Z3{\mathbb{Z}}_{3}Z3​ topological state. This is studied in terms of the gauge-symmetry-preserved quantum state renormalization group, first proposed by He, Moradi, and Wen [Phys. Rev. B 90, 205114 (2014)]. In this approach modular matrices SSS and TTT can be obtained and used as order parameters to characterize the topological properties of the phase and determine phase transitions. From a mapping to a classical two-dimensional Potts model on the square lattice, the critical string tension, at which the transition to a topologically trivial phase takes place, can be obtained analytically and agrees with the numerically determined value. Such a transition can be generalized to a ZN{\mathbb{Z}}_{N}ZN​ topological model under a string tension and determined in the same way. With different deformations, the Z3{\mathbb{Z}}_{3}Z3​ topological phase can also be driven to a critical phase, which contains, in the large deformation limit, the wave function analogous to the Rokhsar-Kivelson point in the quantum dimer model in one case and the fully packed loop model in another case.
We study a generalized quantum hard-core dimer model on the square and honeycomb lattices, allowing for first and second neighbor dimers. At generalized Rokhsar-Kivelson points, the exact ground states can be constructed, and ground-state correlation functions can be equated to those of interacting (1+1)-dimensional Grassmann fields. When the concentration of second neighbor dimers is small, the ground-state correlations are shown to be short ranged corresponding to a (gaped) spin liquid phase. On a 2-torus, the ground states exhibit fourfold topological degeneracy. On a finite cylinder we have found a dramatic even-odd effect depending on the circumference and propose that this can be used as a numerical diagnostic of gapped spin-liquid phases, more generally.
We examine the topological order in the resonating singlet valence plaquette (RSVP) phase of the hard-core quantum plaquette model (QPM) on the face centered cubic (FCC) lattice. To do this, we construct a Rohksar-Kivelson type Hamiltonian of local plaquette resonances. This model is shown to exhibit a Z3{\mathbb{Z}}_{3}Z3​ topological order, which we show by identifying a Z3{\mathbb{Z}}_{3}Z3​ topological constant (which leads to a 33{3}^{3}33-fold topological ground state degeneracy on the 3-torus) and topological pointlike charge and looplike magnetic excitations which obey Z3{\mathbb{Z}}_{3}Z3​ statistics. We also consider an exactly solvable generalization of this model, which makes the geometrical origin of the Z3{\mathbb{Z}}_{3}Z3​ order explicitly clear. For other models and lattices, such generalizations produce a wide variety of topological phases, some of which are novel fracton phases.