Mass Partitions via Equivariant Sections of Stiefel Bundles

We consider a geometric combinatorial problem naturally associated to the geometric topology of certain spherical space forms. Given a collection of $m$ mass distributions on $\mathbb{R}^n$, the existence of $k$ affinely independent regular $q$-fans, each of which equipartitions each of the measures, can in many cases be deduced from the existence of a $\mathbb{Z}_q$-equivariant section of the Stiefel bundle $V_k(\mathbb{F}^n)$ over $S(\mathbb{F}^n)$, where $V_k(\mathbb{F}^n)$ is the Stiefel manifold of all orthonormal $k$-frames in $\mathbb{F}^n,\, \mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, and $S(\mathbb{F}^n)$ is the corresponding unit sphere. For example, the parallelizability of $\mathbb{R}P^n$ when $n = 2,4$, or $8$ implies that any two masses on $\mathbb{R}^n$ can be simultaneously bisected by each of $(n-1)$ pairwise-orthogonal hyperplanes, while when $q=3$ or 4, the triviality of the circle bundle $V_2(\mathbb{C}^2)/\mathbb{Z}_q$ over the standard Lens Spaces $L^3(q)$ yields that for any mass on $\mathbb{R}^4$, there exist a pair of complex orthogonal regular $q$-fans, each of which equipartitions the mass.Comment: 11 pages, final versio

Effect of Landau Level Mixing on Braiding Statistics

We examine the effect of Landau level mixing on the braiding statistics of quasiparticles of abelian and nonabelian quantum Hall states. While path dependent geometric phases can perturb the abelian part of the statistics, we find that the nonabelian properties remain unchanged to an accuracy that is exponentially small in the distance between quasiparticles.Comment: 4 page

Exact Solutions of Fractional Chern Insulators: Interacting Particles in the Hofstadter Model at Finite Size

We show that all the bands of the Hofstadter model on the torus have an exactly flat dispersion and Berry curvature when a special system size is chosen. This result holds for any hopping and Chern number. Our analysis therefore provides a simple rule for choosing a particularly advantageous system size when designing a Hofstadter system whose size is controllable, like a qubit lattice or an optical cavity array. The density operators projected onto the flat bands obey exactly the Girvin-MacDonald-Platzman algebra, like for Landau levels in the continuum in the case of $C=1$, or obey its straightforward generalization in the case of $C>1$. This allows a mapping between density-density interaction Hamiltonians for particles in the Hofstatder model and in a continuum Landau level. By using the well-known pseudopotential construction in the latter case, we obtain fractional Chern insulator phases, the lattice counterpart of fractional quantum Hall phases, that are exact zero-energy ground states of the Hofstadter model with certain interactions. Finally, the addition of a harmonic trapping potential is shown to lead to an appealingly symmetric description in which a new Hofstadter model appears in momentum space.Comment: 15 pages, 8 figures; Published versio

Exactly Solvable Lattice Models with Crossing Symmetry

We show how to compute the exact partition function for lattice statistical-mechanical models whose Boltzmann weights obey a special "crossing" symmetry. The crossing symmetry equates partition functions on different trivalent graphs, allowing a transformation to a graph where the partition function is easily computed. The simplest example is counting the number of nets without ends on the honeycomb lattice, including a weight per branching. Other examples include an Ising model on the Kagome' lattice with three-spin interactions, dimers on any graph of corner-sharing triangles, and non-crossing loops on the honeycomb lattice, where multiple loops on each edge are allowed. We give several methods for obtaining models with this crossing symmetry, one utilizing discrete groups and another anyon fusion rules. We also present results indicating that for models which deviate slightly from having crossing symmetry, a real-space decimation (renormalization-group-like) procedure restores the crossing symmetry