Multicomponent Skyrmion lattices and their excitations

Topological electrostatics

We present a theory of optimal topological textures in nonlinear sigma-models with degrees of freedom living in the GrassmannianGr(M,N)manifold. These textures describe skyrmion lattices ofN-component fermions in a quantising magnetic field, relevant to the physics of graphene, bilayer and other multicomponent quantum Hall systems near integer filling factorsν > 1. We derive analytically the optimality condition, minimizing topological charge density fluctuations, for a general Grassmannian sigma modelGr(M,N)on a sphere and a torus, together with counting arguments which show that for any filling factor and number of components there is a critical value of topological chargedcabove which there are no optimal textures. Belowdca solution of the optimality condition on a torus is unique, while in the case of a sphere one has, in general, a continuum of solutions corresponding to new non-Goldstone zero modes, whose degeneracy is not lifted (via a order from disorder mechanism) by any fermion interactions depending only on the distance on a sphere. We supplement our general theoretical considerations with the exact analytical results for the case ofGr(2,4), appropriate for recent experiments in graphene.

SU(4) Skyrmions in the ν=±1 Quantum Hall State of Graphene

We explore different Skyrmion types in the lowest Landau level of graphene at a filling factor ν=±1. In addition to the formation of spin and valley pseudospin Skyrmions, we show that another type of spin-valley entangled Skyrmions can be stabilized in graphene due to an approximate SU(4) spin-valley symmetry that is affected by sublattice symmetry-breaking terms. These Skyrmions have a clear signature in spin-resolved density measurements on the lattice scale, and we discuss the expected patterns for the different Skyrmion types.

Geometric stability of topological lattice phases

The fractional quantum Hall (FQH) effect illustrates the range of novel phenomena which can arise in a topologically ordered state in the presence of strong interactions. The possibility of realizing FQH-like phases in models with strong lattice effects has attracted intense interest as a more experimentally accessible venue for FQH phenomena which calls for more theoretical attention. Here we investigate the physical relevance of previously derived geometric conditions which quantify deviations from the Landau level physics of the FQHE. We conduct extensive numerical many-body simulations on several lattice models, obtaining new theoretical results in the process, and find remarkable correlation between these conditions and the many-body gap. These results indicate which physical factors are most relevant for the stability of FQH-like phases, a paradigm we refer to as the geometric stability hypothesis, and provide easily implementable guidelines for obtaining robust FQH-like phases in numerical or real-world experiments.

Fractional Chern insulators, topological insulators with partially filled bands, do not require large magnetic fields to form fractional quantum Hall states. Here, the authors investigate the correlation between the stability of such states and the bandgap to their creation on a variety of lattices.