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arXiv:1305.6948 · Physical Review Letters

Combined Topological and Landau Order from Strong Correlations in Chern Bands

Stefanos Kourtis, Maria Daghofer

Published Nov 21, 2014• 43 citations
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Abstract

We present a class of states with both topological and conventional Landau order that arise out of strongly interacting spinless fermions in fractionally filled and topologically nontrivial bands with Chern number C=±1. These quantum states show the features of fractional Chern insulators, such as fractional Hall conductivity and interchange of ground-state levels upon insertion of a magnetic flux. In addition, they exhibit charge order and a related additional trivial ground-state degeneracy. Band mixing and geometric frustration of the charge pattern place these lattice states markedly beyond a single-band description.

DOI
10.1103/physrevlett.113.216404

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We present a class of states with both topological and conventional Landau order that arise out of strongly interacting spinless fermions in fractionally filled and topologically nontrivial bands with Chern number C=±1. These quantum states show the features of fractional Chern insulators, such as fractional Hall conductivity and interchange of ground-state levels upon insertion of a magnetic flux. In addition, they exhibit charge order and a related additional trivial ground-state degeneracy. Band mixing and geometric frustration of the charge pattern place these lattice states markedly beyond a single-band description.
We study two models for spinless fermions featuring topologically nontrivial bands characterized by Chern numbers C=±1 at fractional filling. Using exact diagonalization, we show that, even for infinitely strong nearest-neighbor repulsion, the ground states of these models belong to the recently discovered class of quantum liquids called fractional Chern insulators (FCI). Thus, we establish that FCI states can arise even if interaction strengths are arbitrarily larger than the noninteracting band gap, going beyond the limits in which FCI states have been previously studied. The strong-coupling FCI states, therefore, depart from the usual isolated-band picture that parallels the fractional quantum Hall effect in Landau levels and demonstrate how a topologically ordered state can arise in a truly multiband system.
Lattice models forming bands with higher Chern number offer an intriguing possibility for new phases of matter with no analogue in continuum Landau levels. Here, we establish the existence of a number of new bulk insulating states at fractional filling in flat bands with a Chern number C = N > 1, forming in a recently proposed pyrochlore model with strong spin-orbit coupling. In particular, we find compelling evidence for a series of stable states at ν = 1/(2N + 1) for fermions as well as bosonic states at ν = 1/(N + 1). By examining the topological ground state degeneracies and the excitation structure as well as the entanglement spectrum, we conclude that these states are Abelian. We also explicitly demonstrate that these states are nevertheless qualitatively different from conventional quantum Hall (multilayer) states due to the novel properties of the underlying band structure.
Chern insulators are band insulators exhibiting a nonzero Hall conductance but preserving the lattice translational symmetry. We conclusively show that a partially filled Chern insulator at 1/3 filling exhibits a fractional quantum Hall effect and rule out charge-density wave states that have not been ruled out by previous studies. By diagonalizing the Hubbard interaction in the flat-band limit of these insulators, we show the following: The system is incompressible and has a 3-fold degenerate ground state whose momenta can be computed by postulating an generalized Pauli principle with no more than 1 particle in 3 consecutive orbitals. The ground state density is constant, and equal to 1/3 in momentum space. Excitations of the system are fractional statistics particles whose total counting matches that of quasiholes in the Laughlin state based on the same generalized Pauli principle. The entanglement spectrum of the state has a clear entanglement gap which seems to remain finite in the thermodynamic limit. The levels below the gap exhibit counting identical to that of Laughlin 1/3 quasiholes. Both the 3 ground states and excited states exhibit spectral flow upon flux insertion. All the properties above disappear in the trivial state of the insulator - both the many-body energy gap and the entanglement gap close at the phase transition when the single-particle Hamiltonian goes from topologically nontrivial to topologically trivial. These facts clearly show that fractional many-body states are possible in topological insulators.
The integer quantum Hall state occurs when Landau levels are fully occupied by fermions, while the fractional quantum Hall state usually emerges when the Landau level is partially filled by strongly correlated fermions or bosons. Here, we report two fractional Chern insulating states of hard-core bosons in a multiband lattice model hosting topological flat bands with a high Chern number. The previously proposed $\ensuremath{\nu}=1/3$ fractional Chern insulating state inherited from the high Chern number C=2C=2C=2 of the lowest topological flat band is revisited by the infinite density matrix renormalization group algorithm. In particular, we numerically identify a bosonic 1/21/21/2-Laughlin-like fractional Chern insulating state at the integer fillings. We show two lower topological flat bands jointly generate an effective C=1C=1C=1 Chern band with half filling. Furthermore, we find a strictly particle-hole-like symmetry between the $\ensuremath{\nu}$ and $3\ensuremath{-}\ensuremath{\nu}$ filling in our model. These findings extend our understanding of quantum Hall states and offer a route to realize fractional states in the system with multibands and high Chern numbers.
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.
Recent theoretical works have demonstrated various robust Abelian and non-Abelian fractional topological phases in lattice models with topological flat bands carrying Chern number C=1C=1C=1. Here we study hard-core bosons and interacting fermions in a three-band triangular-lattice model with the lowest topological flat band of Chern number C=2C=2C=2. We find convincing numerical evidence of bosonic fractional quantum Hall effect at the $\ensuremath{\nu}=1/3$ filling characterized by threefold quasidegeneracy of ground states on a torus, a fractional Chern number for each ground state, a robust spectrum gap, and a gap in quasihole excitation spectrum. We also observe numerical evidence of a robust fermionic fractional quantum Hall effect for spinless fermions at the $\ensuremath{\nu}=1/5$ filling with short-range interactions.