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arXiv:1407.8521 · Physical Review B

Double semion phase in an exactly solvable quantum dimer model on the kagome lattice

Oliver Buerschaper, Siddhardh C. Morampudi, Frank Pollmann

Published Nov 26, 2014• 24 citations
arXivHTMLPDF

Abstract

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 ${\mathbb{Z}}_{2}$ 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.

DOI
10.1103/physrevb.90.195148

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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 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.
Extend of the Z2\mathbb{Z}_2Z2​-spin liquid phase on the Kagomé-lattice
The Z2\mathbb{Z}_2Z2​ topological phase in the quantum dimer model on the Kagomé-lattice is a candidate for the description of the low-energy physics of the anti-ferromagnetic Heisenberg model on the same lattice. We study the extend of the topological phase by interpolating between the exactly solvable parent Hamiltonian of the topological phase and an effective low-energy description of the Heisenberg model in terms of a quantum-dimer Hamiltonian. Therefore, we perform a perturbative treatment of the low-energy excitations in the topological phase including free and interacting quasi-particles. We find a phase transition out of the topological phase far from the Heisenberg point. The resulting phase is characterized by a spontaneously broken rotational symmetry and a unit cell involving six sites.
We revisit the description of the low-energy singlet sector of the spin-12\frac{1}{2}21​ Heisenberg antiferromagnet on kagome in terms of an effective quantum dimer model. With the help of exact diagonalizations of appropriate finite-size clusters, we show that the embedding of a given process in its kagome environment leads to dramatic modifications of the amplitudes of the elementary loop processes, an effect not accessible to the standard approach based on the truncation of the Hamiltonian to the nearest-neighbor valence-bond basis. The resulting parameters are consistent with a Z2{Z}_{2}Z2​ spin liquid rather than with a valence-bond crystal, in agreement with the last density matrix renormalization group results.
Strongly correlated systems on geometrically frustrated lattices can stabilize a large number of interesting phases that includes a wide array of novel Mott insulators in both bosonic and electronic systems. Charge fluctuations in a Mott insulator are suppressed due to strong mutual interaction among the particles. The presence of frustration is of particular importance as the physics it offers is often rich, unexpectedly complicated, and continues to raise many open questions. The thesis elucidates some of these issues on a kagome lattice where strong interactions among the particles in the Mott phase impose non-trivial local constraints depending on the filling fraction on the lattice. These Mott insulators, in addition to featuring unusual magnetic and/or charge ordering, can also harbor topologically ordered states of quantum matter, e.g., resonating valence bond liquids realized in certain quantum dimer models on non-bipartite lattices. The dimer models can be regarded as low-energy effective theories for different types of bosonic models in the strong-coupling limit. Exploring this connection is a central theme of this thesis with the aim of realizing novel strongly correlated ground states. Past studies of these models have revealed the existence of various ordered and disordered phases with distinct signatures. Among these low-energy phases, the presence of a stable topological liquid at a particular point, known as Rokhsar-Kivelson point, in the phase diagram is notable. The classical versions of the dimer model are also known to have garnered a vast interest in various fields ranging from problems of pure mathematical origin to ones in physical chemistry as well as statistical physics. Pioneered by Kasteleyn, several analytical works came forward to exactly calculate the partition function of the problem from which other physical observables can be derived. Classical numerical methods are extensively applied to these models to verify the analytical predictions. We introduce a new classical algorithm here to compute the correlation functions of a classical dimer model on a square (bipartite) and a triangular (non-bipartite) lattice based on a tensor network construction. The method, called tensor network renormalization group, turns out to be a powerful tool for simulating short-ranged gapped systems as inferred from our results benchmarked against the classical Monte-Carlo technique and compared with past analytical studies. One should note that the quantum dimer model at the Rokhsar-Kivelson point can also be described as an infinite temperature canonical ensemble of classical dimers because of the particular structure of the ground state which is an equal weight superposition in the configuration manifold. The geometry of the lattice plays a pivotal role in deciding the nature of the phases that arise in the dimer models. Many physical properties of the dimer liquid phase can be extracted in the simple classical setting which certainly allows for a deep understanding of the classical models to be developed. The liquid phase is gapped on non-bipartite lattices and gapless on bipartite lattices, which is reflected in the decay of correlation functions with spatial distances. In general on non-bipartite lattices, the topological nature of the dimer liquid is characterized by a Z2 topological order which survives even when the model is perturbed away from the Rokhsar-Kivelson point. Stability of this liquid phase not only depends on the lattice geometries but notably on dimer concentrations also. In this context, we focus on a particular variant of the dimer model on a triangular lattice which is known as the quantum fully packed loop model. The model is composed of nonintersecting closed loops made of dimers and governed by the same Hamiltonian as the quantum dimer model. The loop model provides an effective low-energy description of a strongly correlated bosonic system at 1/3 filling on the kagome lattice. The corresponding Bose-Hubbard Hamiltonian consists of nearest-neighbor hopping and all possible repulsive interactions within a hexagonal plaquette. Conspicuous features of the zero-temperature phase diagram for this model include (i) presence of a stable Z2 liquid even without any Rokhsar-Kivelson potential term (in distinction to the standard quantum dimer model), and (ii) an unconventional phase transition from the liquid phase to a novel crystalline phase that has nematic order (dubbed lattice nematic). For a deeper understanding of the physics, a mapping to an Ising gauge theory is presented. The gauge theoretic description provides a useful way to predict the nature of the quantum phase transition to lie in the O(3) universality class. Finally a fermionic model at the same 1/3 filling is considered in which the ground state exhibits a number of exotic local orderings resulting from the spin-charge interplay of electrons. The Hamiltonian comprises nearest-neighbor hopping, strong on-site Coulomb interaction, and repulsive interaction terms only between nearest-neighbors. In the strong correlation limit, this fermionic problem maps to a two-color fully packed loop model – a model in which the loop segments carry an additional quantum number as color on a honeycomb lattice. The effective theory is governed by coherent three-particle ring exchanges and nearest-neighbor antiferromagnetic spin exchanges. The competition between these two leads to a phase diagram composed of a novel plaquette ordered state (known as the plaquette phase) that undergoes phase transition to a new kind of charge ordered state which we call a short loop phase. From our numerical analysis, we conclude that the plaquette phase features an unusual antiferromagnetic order with gapless spin excitations while the charge-ordered state is subjugated by spin fluctuations of localized electrons arranged in small hexagonal loops on the kagome lattice.
We investigate the quantum phases of a frustrated antiferromagnetic Heisenberg spin-1/2 model Hamiltonian on a kagom\'e-strip chain (KSC), a one-dimensional analog of the kagom\'e lattice, and construct its phase diagram in an extended exchange-parameter space. The isolated unit cell of this lattice comprises of five spin-1/2 particles, giving rise to several types of magnetic ground states in a unit cell: a spin-3/23/23/2 state as well as spin-1/21/21/2 states with and without additional degeneracies. We explore the ground-state properties of the fully connected thermodynamic system using exact diagonalization and density matrix renormalization group methods, identifying several distinct quantum phases. All but one of the phases exhibit gapless spin excitations. The exception is a dimerized spin-gapped phase that covers a large part of the phase diagram and includes the uniformly exchange coupled system. We argue that this phase can be understood by perturbing around the limit of decoupled unit cells where each unit cell has six degenerate ground states. We use degenerate perturbation theory to obtain an effective Hamiltonian, a Kugel-Khomskii model with an anisotropic spin-one orbital degree of freedom, which helps explain the origin of dimerization.
We present a study of a simple model antiferromagnet consisting of a sum of nearest-neighbor SO(N)\mathrm{SO}(N)SO(N) singlet projectors on the kagome lattice. Our model shares some features with the popular S=1/2S=1/2S=1/2 kagome antiferromagnet but is specifically designed to be free of the sign problem of quantum Monte Carlo. In our numerical analysis, we find as a function of NNN a quadrupolar magnetic state and a wide range of a quantum spin liquid. A solvable large-NNN generalization suggests that the quantum spin liquid in our original model is a gapped Z2{\mathbb{Z}}_{2}Z2​ topological phase. Supporting this assertion, a numerical study of the entanglement entropy in the sign free model shows a quantized topological contribution.