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arXiv:1406.6364 · Physical Review B

Double-semion topological order from exactly solvable quantum dimer models

Yang Qi, Zheng‐Cheng Gu, Hong Yao

Published Oct 6, 2015• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.92.155105

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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.
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 derive exact results for close-packed dimers on the triangular kagome lattice (TKL), formed by inserting triangles into the triangles of the kagome lattice. Because the TKL is a nonbipartite lattice, dimer-dimer correlations are short ranged so that the ground state at the Rokhsar-Kivelson (RK) point of the corresponding quantum dimer model on the same lattice is a short-ranged spin liquid. Using the Pfaffian method, we derive an exact form for the free energy, and we find that the entropy is 13ln 2\frac{1}{3}\text{ln}\text{ }231​ln 2 per site, regardless of the weights of the bonds. The occupation probability of every bond is 14\frac{1}{4}41​ in the case of equal weights on every bond. Similar to the case of lattices formed by corner-sharing triangles (such as the kagome and squagome lattices), we find that the dimer-dimer correlation function is identically zero beyond a certain (short) distance. We find in addition that monomers are deconfined on the TKL, indicating that there is a short-ranged spin liquid phase at the RK point. We also find exact results for the ground-state energy of the classical Heisenberg model. The ground state can be ferromagnetic, ferrimagnetic, locally coplanar, or locally canted, depending on the couplings. From the dimer model and the classical spin model, we derive upper bounds on the ground-state energy of the quantum Heisenberg model on the TKL.
Quantum dimer models typically arise in various low energy theories like those of frustrated antiferromagnets. We introduce a quantum dimer model on the kagome lattice which stabilizes an alternative Z2{\mathbb{Z}}_{2}Z2​ topological order, namely the so-called ``double semion'' order. For a particular set of parameters, the model is exactly solvable, allowing us to access the ground state as well as the excited states. We show that the double semion phase is stable over a wide range of parameters using numerical exact diagonalization. Furthermore, we propose a simple microscopic spin Hamiltonian for which the low-energy physics is described by the derived quantum dimer model.
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 study a general class of easy-axis spin models on a lattice of corner sharing even-sided polygons with all-to-all interactions within a plaquette. The low-energy description corresponds to a quantum dimer model on a dual lattice of even coordination number with a multidimer constraint. At an appropriately constructed frustration-free Rokhsar-Kivelson (RK) point, the ground-state wavefunction can be exactly mapped onto a classical vertex model on the dual lattice. When the dual lattice is bipartite, the vertex models are bonded and are self-dual under Wegner's duality, with the self-dual point corresponding to the RK point of the original multidimer model. We argue that the self-dual point is a critical point based on known exact solutions to some of the vertex models. When the dual lattice is nonbipartite, the vertex model is arrowed, and we use numerical methods to argue that there is no phase transition as a function of the vertex weights. Motivated by these wavefunction dualities, we construct two other distinct families of frustration-free Hamiltonians whose ground states can be mapped onto these vertex models. Many of these RK Hamiltonians provably host Z2{\mathbb{Z}}_{2}Z2​ topologically ordered phases.