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arXiv:1307.3485 · Physical Review B

Three- and four-body interactions from two-body interactions in spin models: A route to Abelian and non-Abelian fractional Chern insulators

Eliot Kapit, Steven H. Simon

Published Nov 13, 2013• 19 citations
arXivHTMLPDF

Abstract

We describe a method for engineering local $k+1$-body interactions ($k=1,2,3$) from two-body couplings in spin-$\frac{1}{2}$ systems. When implemented in certain systems with a flat single-particle band with a unit Chern number, the resulting many-body ground states are fractional Chern insulators which exhibit Abelian and non-Abelian anyon excitations. The most complex of these, with $k=3$, has Fibonacci anyon excitations; our system is thus capable of universal topological quantum computation. We then demonstrate that an appropriately tuned circuit of qubits could faithfully replicate this model up to small corrections, and further, we describe the process by which one might create and manipulate non-Abelian vortices in these circuits, allowing for direct control of the system's quantum information content.

DOI
10.1103/physrevb.88.184409

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We describe a method for engineering local k+1k+1k+1-body interactions (k=1,2,3k=1,2,3k=1,2,3) from two-body couplings in spin-12\frac{1}{2}21​ systems. When implemented in certain systems with a flat single-particle band with a unit Chern number, the resulting many-body ground states are fractional Chern insulators which exhibit Abelian and non-Abelian anyon excitations. The most complex of these, with k=3k=3k=3, has Fibonacci anyon excitations; our system is thus capable of universal topological quantum computation. We then demonstrate that an appropriately tuned circuit of qubits could faithfully replicate this model up to small corrections, and further, we describe the process by which one might create and manipulate non-Abelian vortices in these circuits, allowing for direct control of the system's quantum information content.
The recent theoretical discovery of fractional Chern insulators (FCIs) has provided an important new way to realize topologically ordered states in lattice models. In earlier works, on-site and nearest-neighbor Hubbard-like interactions have been used extensively to stabilize Abelian FCIs in systems with nearly flat, topologically nontrivial bands. However, attempts to use two-body interactions to stabilize non-Abelian FCIs, where the ground state in the presence of impurities can be massively degenerate and manipulated through anyon braiding, have proven very difficult in uniform lattice systems. Here, we study the remarkable effect of long-range interactions in a lattice model that possesses an exactly flat lowest band with a unit Chern number. When spinless bosons with two-body long-range interactions partially fill the lowest Chern band, we find convincing evidence of gapped, bosonic Read-Rezayi (RR) phases with non-Abelian anyon statistics. We characterize these states through studying topological degeneracies, the overlap between the ground states of two-body interactions and the exact RR ground states of three- and four-body interactions, and state counting in the particle-cut entanglement spectrum. Moreover, we demonstrate how an approximate lattice form of Haldane's pseudopotentials, analogous to that in the continuum, can be used as an efficient guiding principle in the search for lattice models with stable non-Abelian phases.
We study fermions on a triangular lattice model that exhibits topological flatbands characterized by nonzero Chern numbers. Our scheme stems from the well-known Hofstadter model but the next-nearest-neighbor hopping is introduced, which is crucial for tuning the lowest band to be nearly flat. Differing from previous proposals with the necessity of multiparticle interactions, we consider the more realistic long-range dipolar interaction combined with two-body short-range attractions between fermions. We show the realization of the non-Abelian $\ensuremath{\nu}=1/2$ Moore-Read fractional Chern insulators, and strong evidence for the existence of the more exotic $\ensuremath{\nu}=3/5$ Read-Rezayi fractional Chern insulators. Our results provide insights for the experimental realization of these exotic states by realistic two-body interactions and thus facilitates the implementation of the universal topological quantum computation.
We devise local lattice models whose ground states are model fractional Chern insulators-Abelian and non-Abelian topologically ordered states characterized by exact ground state degeneracies at any finite size and infinite entanglement gaps. Most saliently, we construct exact parent Hamiltonians for two distinct families of bosonic lattice generalizations of the Z_{k} parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling fractions ν=k/(C+1) and (ii) nematic states at ν=k/2, where C is the Chern number of the lowest band. In spite of a fluctuating Berry curvature, our construction is partially frustration free: the ground states reside entirely within the lowest band and exactly minimize a local (k+1) body repulsion term by term. In addition to providing the first known models hosting intriguing states such as higher Chern number generalizations of the Fibonacci anyon quantum Hall states, the remarkable stability and finite-size properties make our models particularly well suited for the study of novel phenomena involving, e.g., twist defects and proximity induced superconductivity, as well as being a guide for designing experiments.
We devise local lattice models whose ground states are \emph{ideal fractional Chern insulators}---Abelian and non-Abelian topologically ordered states characterized by exact ground state degeneracies at any finite size and infinite entanglement gaps. Most saliently, we construct exact parent Hamiltonians for two distinct families of bosonic lattice generalizations of the Zk\mathcal{Z}_kZk​ parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling fractions ν=k/(C+1)\nu=k/(\mathcal{C}+1)ν=k/(C+1) and (ii) nematic states at ν=k/2\nu=k/2ν=k/2, where C\mathcal{C}C is the Chern number of the lowest band in our models. In spite of a fluctuating Berry curvature, our construction is partially frustration free: the ground states reside entirely within the lowest band and exactly minimize a local (k+1)(k+1)(k+1)-body repulsion term by term. In addition to providing the first known models hosting intriguing states such as higher Chern number generalizations of the Fibonacci anyon quantum Hall states, the remarkable stability and finite-size properties make our models particularly well-suited for the study of novel phenomena involving e.g. twist defects and proximity induced superconductivity, as well as being a guide for designing experiments.
While the internal structure of Abelian topological order is well understood, how to characterize the non-Abelian topological order is an outstanding issue. We propose a distinctive scheme based on the many-body Chern number matrix to characterize non-Abelian multicomponent fractional quantum Hall states. As a concrete example, we study the many-body ground state of two-component bosons at the filling faction $\ensuremath{\nu}=4/3$ in topological flat band models. Utilizing density-matrix renormalization group and exact diagonalization calculations, we demonstrate the emergence of non-Abelian spin-singlet fractional quantum Hall effect under three-body interaction, whose topological nature is classified by sixfold degenerate ground states and a fractionally quantized Chern-number matrix.
We construct a class of lattice Hamiltonians whose single-particle spectrum consists of an arbitrary number of exactly degenerate flat bands that reproduce the analytic structure of the first ppp Landau levels restricted to the lattice. When combined with local bosonic contact interactions, these models become exactly solvable frustration-free parent Hamiltonians for FCIs that realize both Abelian and non-Abelian parton quantum Hall states. Using exact diagonalization, we confirm the expected zero-mode counting for variants of the model stabilizing the bosonic Jain-21 state as well as the non-Abelian 22- and 33-states, which are expected to support Ising- and Fibonacci-type anyons, respectively. Our construction provides an exactly solvable lattice realization of multi Landau-level physics and offers a new framework for studying FCIs with Chern number C>1C > 1C>1. More broadly, it supplies a family of idealized lattice models that capture the analytic structure of continuum Landau levels while remaining compatible with exponentially local hopping.