Variational Monte Carlo Study of the 1/9-Magnetization Plateau in Kagome Antiferromagnets

Motivated by very recent experimental observations of the 1/9-magnetization plateaus in YCu_{3}(OH)_{6+x}Br_{3-x} and YCu_{3}(OD)_{6+x}Br_{3-x}, our study delves into the magnetic-field-induced phase transitions in the nearest-neighbor antiferromagnetic Heisenberg model on the kagome lattice using the variational Monte Carlo technique. We uncover a phase transition from a zero-field Dirac spin liquid to a field-induced magnetically disordered phase that exhibits the 1/9-magnetization plateau. Through a comprehensive analysis encompassing the magnetization distribution, spin correlations, chiral order parameter, topological entanglement entropy, ground-state degeneracy, Chern number, and excitation spectrum, we pinpoint the phase associated with this magnetization plateau as a chiral Z_{3} topological quantum spin liquid and elucidate its diverse physical properties.

Topological Classification of Correlations in 2D Electron Systems in Magnetic or Berry Fields

Recent topology classification of 2D electron states induced by different homotopy classes of mappings of the planar Brillouin zone into Bloch space can be supplemented by a homotopy classification of various phases of multi-electron homotopy patterns induced by Coulomb interaction between electrons. The general classification of such type is presented. It explains the topologically protected correlations responsible for integer and fractional Hall effects in 2D multi-electron systems in the presence of perpendicular quantizing magnetic field or Berry field, the latter in topological Chern insulators. The long-range quantum entanglement is essential for homotopy correlated phases in contrast to local binary entanglement for conventional phases with local order parameters. The classification of homotopy long-range correlated phases induced by the Coulomb interaction of electrons has been derived in terms of homotopy invariants and illustrated by experimental observations in GaAs 2DES, graphene monolayer, and bilayer and in Chern topological insulators. The homotopy phases are demonstrated to be topologically protected and immune to the local crystal field, local disorder, and variation of the electron interaction strength. The nonzero interaction between electrons is shown, however, to be essential for the definition of the homotopy invariants, which disappear in gaseous systems.

Homotopy Phases of FQHE with Long-Range Quantum Entanglement in Monolayer and Bilayer Hall Systems

Correlated phases in Hall systems have topological character. Multilayer configurations of planar electron systems create the opportunity to change topological phases on demand using macroscopic factors, such as vertical voltage. We present an analysis of such phenomena in close relation to recent experiments with multilayer Hall setups including GaAs and graphene multi-layers. The consequences of the blocking or not of the inter-layer electron tunneling in stacked Hall configurations are analyzed and presented in detail. Multilayer Hall systems are thus tunable topological composite nanomaterials, in the case of graphene-stacked systems by both intra- and inter-layer voltage.

Choreographed entanglement dances: Topological states of quantum matter

It has long been thought that all different phases of matter arise from symmetry breaking. Without symmetry breaking, there would be no pattern, and matter would be featureless. However, it is now clear that for quantum matter at zero temperature, even symmetric disordered liquids can have features, giving rise to topological phases of quantum matter. Some of the topological phases are highly entangled (that is, have topological order), whereas others are weakly entangled (that is, have symmetry-protected trivial order). This Review provides a brief summary of these zero-temperature states of matter and their emergent properties, as well as their importance in unifying some of the most basic concepts in nature.

Braiding with Borromean Rings in (3+1)-Dimensional Spacetime

While winding a particlelike excitation around a looplike excitation yields the celebrated Aharonov-Bohm phase, we find a distinctive braiding phase in the absence of such mutual winding. In this Letter, we propose an exotic particle-loop-loop braiding process, dubbed the Borromean rings braiding. In the process, a particle moves around two unlinked loops, such that its trajectory and the two loops form the Borromean rings or more general Brunnian links. As the particle trajectory does not wind with any of the loops, the resulting braiding phase is fundamentally different from the Aharonov-Bohm phase. We derive an explicit expression for the braiding phase in terms of the underlying Milnor's triple linking number. We also propose topological quantum field theories consisting of an AAB-type topological term which realize the braiding statistics.

Hierarchy Construction and Non-Abelian Families of Generic Topological Orders

We generalize the hierarchy construction to generic 2+1D topological orders (which can be non-Abelian) by condensing Abelian anyons in one topological order to construct a new one. We show that such construction is reversible and leads to a new equivalence relation between topological orders. We refer to the corresponding equivalence class (the orbit of the hierarchy construction) as "the non-Abelian family." Each non-Abelian family has one or a few root topological orders with the smallest number of anyon types. All the Abelian topological orders belong to the trivial non-Abelian family whose root is the trivial topological order. We show that Abelian anyons in root topological orders must be bosons or fermions with trivial mutual statistics between them. The classification of topological orders is then greatly simplified, by focusing on the roots of each family: those roots are given by non-Abelian modular extensions of representation categories of Abelian groups.