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arXiv:1908.00988

Interaction-driven plateau transition between integer and fractional Chern Insulators

Leon Schoonderwoerd, Frank Pollmann, Gunnar Möller

Published Aug 2, 2019• 6 citations
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Abstract

We present numerical evidence of an interaction-driven quantum Hall plateau transition between a $|C|>1$ Chern Insulator (CI) and a $ν= 1/3$ Laughlin state in the Harper-Hofstadter model. We study the model at flux densities $p/q$, where the lowest Landau level (LLL) manifold comprises $p$ 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 $ν=r/t$. These phases may compete at the same particle density when $p=t$. As a concrete example, we numerically explore the physics at flux density $n_ϕ = 3/11$, where we show evidence that a direct transition occurs between a CI and a $ν= 1/3$ Laughlin state, which we characterise in terms of its critical, topological and entanglement properties. We also show that strong interactions generically stabilise a $ν= 1/3$ 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 $n_ϕ$.

DOI
10.48550/arxiv.1908.00988

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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​.
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.
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 investigate topological phase transitions in Chern insulators within three-band models, focusing on the empty band and lowest band populated by spinless fermions. We consider Lieb and kagome lattices and notice phase transitions driven by the hopping integral between nearest-neighbors, which leads to the change of the lowest band Chern number from C=1C=1C=1 to C=−1C=-1C=−1. In the single-particle picture, different phases are examined by investigating corresponding entanglement spectra and the evolution of the entanglement entropy. Entanglement spectra reveal the spectral flow characteristic of topologically nontrivial systems before and after phase transitions. For the lowest band with 1/31/31/3 filling, Fractional Chern insulator (FCI) phases are identified by examining the ground state momenta, the spectral flow, and counting of the entanglement energy levels below the gap in the entanglement spectrum. A quasilinear dependence of the entanglement entropy α(nA)α(n_A)α(nA​) term is observed for FCI phase, similarly to linear behavior expected for Laughlin phase in the fractional quantum Hall effect. At the topological phase transition, for both empty and partially filled lowest bands, the energy gap closure and a discontinuity in the entanglement entropy are observed. This coincides with the divergence of a standard deviation of the Berry curvature. We notice also a phase transition driven by the nearest-neighbor interaction in the Lieb lattice, where the many-body energy gap closes. The phase transitions are shown to be stable for an arbitrary system size, thus predicted to be present in the thermodynamic limit. While our calculations are performed for an interaction energy far exceeding the gap between the two lowest energy bands, we note that the higher band does not affect the phase transitions, however destabilizes FCI phases.
Realizing strongly correlated topological phases of ultracold gases is a central goal for ongoing experiments. While fractional quantum Hall states could soon be implemented in small atomic ensembles, detecting their signatures in few-particle settings remains a fundamental challenge. In this work, we numerically analyze the center-of-mass Hall drift of a small ensemble of hardcore bosons, initially prepared in the ground state of the Harper-Hofstadter-Hubbard model in a box potential. By monitoring the Hall drift upon release, for a wide range of magnetic flux values, we identify an emergent Hall plateau compatible with a fractional Chern insulator state: The extracted Hall conductivity approaches a fractional value determined by the many-body Chern number, while the width of the plateau agrees with the spectral and topological properties of the prepared ground state. Besides, a direct application of Streda's formula indicates that such Hall plateaus can also be directly obtained from static density-profile measurements. Our calculations suggest that fractional Chern insulators can be detected in cold-atom experiments, using available detection methods.
We report on the numerically exact simulation of the dissipative dynamics governed by quantum master equations that feature fractional quantum Hall states as unique steady states. In particular, for the paradigmatic Hofstadter model, we show how Laughlin states can be to good approximation prepared in a dissipative fashion from arbitrary initial states by simply pumping strongly interacting bosons into the lowest Chern band of the corresponding single-particle spectrum. While pure (up to topological degeneracy) steady states are only reached in the low-flux limit or for extended hopping range, we observe a certain robustness regarding the overlap of the steady state with fractional quantum Hall states for experimentally well-controlled flux densities. This may be seen as an encouraging step towards addressing the long-standing challenge of preparing strongly correlated topological phases in quantum simulators.