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arXiv:1504.06623 · Physical Review Letters

Fractional Chern Insulators in Harper-Hofstadter Bands with Higher Chern Number

Gunnar Möller, Nigel R. Cooper

Published Sep 16, 2015• 107 citations
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Abstract

The Harper-Hofstadter model provides a fractal spectrum containing topological bands of any integer Chern number C. We study the many-body physics that is realized by interacting particles occupying Harper-Hofstadter bands with |C|>1. We formulate the predictions of Chern-Simons or composite fermion theory in terms of the filling factor ν, defined as the ratio of particle density to the number of single-particle states per unit area. We show that this theory predicts a series of fractional quantum Hall states with filling factors ν=r/(r|C|+1) for bosons, or ν=r/(2r|C|+1) for fermions. This series includes a bosonic integer quantum Hall state in |C|=2 bands. We construct specific cases where a single band of the Harper-Hofstadter model is occupied. For these cases, we provide numerical evidence that several states in this series are realized as incompressible quantum liquids for bosons with contact interactions.

DOI
10.1103/physrevlett.115.126401

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The Harper-Hofstadter model provides a fractal spectrum containing topological bands of any integer Chern number C. We study the many-body physics that is realized by interacting particles occupying Harper-Hofstadter bands with |C|>1. We formulate the predictions of Chern-Simons or composite fermion theory in terms of the filling factor ν, defined as the ratio of particle density to the number of single-particle states per unit area. We show that this theory predicts a series of fractional quantum Hall states with filling factors ν=r/(r|C|+1) for bosons, or ν=r/(2r|C|+1) for fermions. This series includes a bosonic integer quantum Hall state in |C|=2 bands. We construct specific cases where a single band of the Harper-Hofstadter model is occupied. For these cases, we provide numerical evidence that several states in this series are realized as incompressible quantum liquids for bosons with contact interactions.
We study the stability of composite fermion fractional quantum Hall states in Harper-Hofstadter bands with Chern number ∣C∣>1|C|>1∣C∣>1. From composite fermion theory, states are predicted to be found at filling factors $\ensuremath{\nu}=r/(kr|C|+1),r\ensuremath{\in}\mathbb{Z}$, with k=1k=1k=1 for bosons and k=2k=2k=2 for fermions. Here, we closely analyze these series in both cases, with contact interactions for bosons and nearest-neighbor interactions for (spinless) fermions. In particular, we analyze how the many-body gap scales as the bands are tuned to the effective continuum limit of Chern number ∣C∣|C|∣C∣ bands, realized near flux density ${n}_{\ensuremath{\phi}}=1/|C|$. Near these points, the Hofstadter model requires large magnetic unit cells that yield bands with perfectly flat dispersion and Berry curvature. We exploit the known scaling of energies in the effective continuum limit in order to maintain a fixed square aspect ratio in finite-size calculations. Based on exact diagonalization calculations of the band-projected Hamiltonian for these lattice geometries, we show that for both bosons and fermions, the vast majority of finite-size spectra yield the ground-state degeneracy predicted by composite fermion theory. For the chosen interactions, we confirm that states with filling factor $\ensuremath{\nu}=1/(k|C|+1)$ are the most robust and yield a clear gap in the thermodynamic limit. For bosons with contact interactions in ∣C∣=2|C|=2∣C∣=2 and ∣C∣=3|C|=3∣C∣=3 bands, our data for the composite fermion states are compatible with a finite gap in the thermodynamic limit. We also report new evidence for gapped incompressible states stabilized for fermions with nearest-neighbor interactions in ∣C∣>1|C|>1∣C∣>1 bands. For cases with a clear gap, we confirm that the thermodynamic limit commutes with the effective continuum limit within finite-size error bounds. We analyze the nature of the correlation functions for the Abelian composite fermion states and find that the correlation functions for ∣C∣>1|C|>1∣C∣>1 states are smooth functions for positions separated by ∣C∣|C|∣C∣ sites along both axes, giving rise to ∣C∣2{|C|}^{2}∣C∣2 sheets; some of which can be related by inversion symmetry. We also comment on two cases which are associated with a bosonic integer quantum Hall effect (BIQHE): For $\ensuremath{\nu}=2$ in ∣C∣=1|C|=1∣C∣=1 bands, we find a strong competing state with a higher ground-state degeneracy, so no clear BIQHE is found in the band-projected Hofstadter model; for $\ensuremath{\nu}=1$ in ∣C∣=2|C|=2∣C∣=2 bands, we present additional data confirming the existence of a BIQHE state.
Stability of Fractional Chern Insulators in the Effective Continuum Limit of ∣C∣>1|C|>1∣C∣>1 Harper-Hofstadter Bands
We study the stability of composite fermion fractional quantum Hall states in Harper-Hofstadter bands with Chern number ∣C∣>1|C|>1∣C∣>1. We analyze the states of the composite fermion series for bosons with contact interactions and (spinless) fermions with nearest-neighbor interactions. We examine the scaling of the many-body gap as the bands are tuned to the effective continuum limit nϕ→1/∣C∣n_\phi\to 1/|C|nϕ​→1/∣C∣. Near these points, the Hofstadter model realises large magnetic unit cells that yield bands with perfectly flat dispersion and Berry curvature. We exploit the known scaling of energies in the effective continuum limit in order to maintain a fixed square aspect ratio in finite-size calculations. Based on exact diagonalization calculations of the band-projected Hamiltonian, we show that almost all finite-size spectra yield the ground state degeneracy predicted by composite fermion theory. We confirm that states at low ranks in the composite fermion hierarchy are the most robust, and yield a clear gap in the thermodynamic limit. For bosons in ∣C∣=2|C|=2∣C∣=2 and ∣C∣=3|C|=3∣C∣=3 bands, our data for the composite fermion states are compatible with a finite gap in the thermodynamic limit. We also report new evidence for gapped incompressible states of fermions in ∣C∣>1|C|>1∣C∣>1 bands, which have large entanglement gaps. For cases with a clear spectral gap, we confirm that the thermodynamic limit commutes with the effective continuum limit. We analyze the nature of the correlation functions for the Abelian composite fermion states and find that they feature ∣C∣2|C|^2∣C∣2 smooth sheets. We examine two cases associated with a bosonic integer quantum Hall effect (BIQHE): for ν=2\nu=2ν=2 in ∣C∣=1|C|=1∣C∣=1 bands, we find a strong competing state with a higher ground state degeneracy, so no clear BIQHE is found in the band-projected Hofstadter model; for ν=1\nu=1ν=1 in ∣C∣=2|C|=2∣C∣=2 bands, we present additional data confirming the existence of a BIQHE state.
The Hofstadter model is a popular choice for theorists investigating the fractional quantum Hall effect on lattices, due to its simplicity, infinite selection of topological flat bands, and increasing applicability to real materials. In particular, fractional Chern insulators in bands with Chern number ∣C∣>1|C|>1∣C∣>1 can demonstrate richer physical properties than continuum Landau level states and have recently been detected in experiments. Motivated by this, we examine the stability of fractional Chern insulators with higher Chern number in the Hofstadter model, using large-scale infinite density matrix renormalization group (iDMRG) simulations on a thin cylinder. We confirm the existence of fractional states in bands with Chern numbers C=1,2,3,4,5C=1,2,3,4,5C=1,2,3,4,5 at the filling fractions predicted by the generalized Jain series [Phys. Rev. Lett. 115, 126401 (2015)]. Moreover, we discuss their metal-to-insulator phase transitions, as well as the subtleties in distinguishing between physical and numerical stability. Finally, we comment on the relative suitability of fractional Chern insulators in higher Chern number bands for proposed modern applications.
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 present numerical evidence of an interaction-driven quantum Hall plateau transition between a ∣C∣>1|C|>1∣C∣>1 Chern Insulator (CI) and a ν=1/3ν= 1/3ν=1/3 Laughlin state in the Harper-Hofstadter model. We study the model at flux densities p/qp/qp/q, where the lowest Landau level (LLL) manifold comprises ppp magnetic sub-bands. For weak interactions, the model realises integer CIs corresponding to filled sub-bands, while strongly interacting candidate states include fractional quantum Hall (FQH) states at LLL filling fractions ν=r/tν=r/tν=r/t. These phases may compete at the same particle density when p=tp=tp=t. As a concrete example, we numerically explore the physics at flux density nϕ=3/11n_ϕ = 3/11nϕ​=3/11, where we show evidence that a direct transition occurs between a CI and a ν=1/3ν= 1/3ν=1/3 Laughlin state, which we characterise in terms of its critical, topological and entanglement properties. We also show that strong interactions generically stabilise a ν=1/3ν= 1/3ν=1/3 Laughlin state even when the LLL is split into multiple bands, and introduce a powerful methodology to extract its topological entanglement entropy by exploiting the scaling of magnetic length with nϕn_ϕnϕ​.
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.