Please use this identifier to cite or link to this item:

`http://arks.princeton.edu/ark:/88435/dsp01n009w241h`

Title: | Quantum Geometry of the "Fuzzy-Lattice" Hubbard Model and the Fractional Chern Insulator |

Authors: | Vijay, Ksheerasagar |

Advisors: | Haldane, Frederick D. |

Contributors: | Sondhi, Shivaji |

Department: | Physics |

Class Year: | 2013 |

Abstract: | The fractional quantum Hall effect (FQHE), describing the quantization of the Hall conductance in terms of rational fractional multiples of a fundamental constant, has been experimentally observed in two-dimensional systems of charged fermions confined to Landau orbits in the presence of a strong magnetic field. The effect is particularly interesting from a theoretical standpoint as an example of an interacting topological phase of matter that cannot be described in terms of the Landau symmetry-breaking picture of phase transitions. Properties of FQH states such as quasi-particle excitations obeying fractional statistics and topological ground-state degeneracy have been explained, in part, by proposing trial many-body wavefunctions such as the Laughlin and Read-Rezayi series. Recent studies of interacting particles on tight-binding lattices with broken time-reversal symmetry reveal FQH phases at zero magnetic field (fractional Chern insulators, FCI). In a partially-filled Landau level, the non-commutative Landau-orbit guiding-centers are the residual degrees of freedom, requiring a "quantum geometry" Hilbert-space description, since a real-space SchrÃ¶dinger description can only apply in the "classical geometry" of unprojected coordinates. The continuum description does not apply on a lattice, where we describe the emergence of the FCI from a noncommutative quantum lattice geometry. We define a "fuzzy lattice" by projecting a one-particle bandstructure with more than one orbital per unit cell into a single band, and then renormalizing the orbital on each site to unit weight. The resulting overcomplete basis of local states is analogous to a basis of more than one coherent state per flux quantum in a Landau level. The overlap matrix characterizes "quantum geometry" on the "fuzzy lattice", defining a "quantum distance" measure and Berry fluxes through elementary lattice triangles. By studying quantum geometry at transitions between topologically-distinct instances of a fuzzy lattice, we numerically observe features of the FCI, including the emergence of topological edge states consistent with the chiral Luttinger liquid theory of the FQH edge, and power-law decay of the overlap between "fuzzy" states, suggesting the presence of strongly-correlated phase. We comment on current work to generalize this "fuzzy-orbital" construction for lattice models with higher Chern number. |

Extent: | 77 pages |

URI: | http://arks.princeton.edu/ark:/88435/dsp01n009w241h |

Access Restrictions: | Walk-in Access. This thesis can only be viewed on computer terminals at the Mudd Manuscript Library. |

Type of Material: | Princeton University Senior Theses |

Language: | en_US |

Appears in Collections: | Physics, 1936-2016 |

Files in This Item:

File | Size | Format | |
---|---|---|---|

Vijay_Sagar_Senior_Thesis.pdf | 59.42 MB | Adobe PDF | Request a copy |

Items in Dataspace are protected by copyright, with all rights reserved, unless otherwise indicated.