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arXiv:2004.06602 · Physical review. B./Physical review. B

Fractional quantum Hall states for moiré superstructures in the Hofstadter regime

Bartholomew Andrews, Alexey A. Soluyanov

Published Jun 17, 2020• 55 citations
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Abstract

We study the transition of $\ensuremath{\nu}=1/3$ and $2/5$ fractional quantum Hall states of the honeycomb Hofstadter model as we tune to a two-orbital moir\'e superlattice Hamiltonian, motivated by the flat bands of twisted bilayer graphene in a perpendicular magnetic field. In doing so, we address the extent to which these states survive in moir\'e systems and analyze the nature of the transition. Through the use of a Peierls substitution, we determine the Landau-level splitting for the moir\'e Hamiltonian, and study the structure of the Chern bands for a range of magnetic flux per plaquette. We identify topological flat bands in the spectrum at low energies, with numerically tractable lattice geometries that can support the fractional quantum Hall effect. As we tune the model, we find that the orbital-polarized $\ensuremath{\nu}=1/3$ and $2/5$ states corresponding to the honeycomb Hofstadter model survive up to $\ensuremath{\approx}30%$ of typical moir\'e superlattice parameters, beyond which they transition into an insulating phase. We present evidence for this through density matrix renormalization group calculations on an infinite cylinder, by verifying the charge pumping, spectral flow, entanglement scaling, and conformal field theory edge counting. We conclude that fractional quantum Hall states from the Hofstadter model can persist up to hopping amplitudes of the same order as those typical for moir\'e superlattice Hamiltonians, which implies generally that fractional states for moir\'e superstructures can be discerned simply by analyzing the dominant terms in their effective Hamiltonians.

DOI
10.1103/physrevb.101.235312

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We study the transition of \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 and 2/52/52/5 fractional quantum Hall states of the honeycomb Hofstadter model as we tune to a two-orbital moir'e superlattice Hamiltonian, motivated by the flat bands of twisted bilayer graphene in a perpendicular magnetic field. In doing so, we address the extent to which these states survive in moir'e systems and analyze the nature of the transition. Through the use of a Peierls substitution, we determine the Landau-level splitting for the moir'e Hamiltonian, and study the structure of the Chern bands for a range of magnetic flux per plaquette. We identify topological flat bands in the spectrum at low energies, with numerically tractable lattice geometries that can support the fractional quantum Hall effect. As we tune the model, we find that the orbital-polarized \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 and 2/52/52/5 states corresponding to the honeycomb Hofstadter model survive up to \ensuremath≈30\ensuremath{\approx}30%\ensuremath≈30 of typical moir'e superlattice parameters, beyond which they transition into an insulating phase. We present evidence for this through density matrix renormalization group calculations on an infinite cylinder, by verifying the charge pumping, spectral flow, entanglement scaling, and conformal field theory edge counting. We conclude that fractional quantum Hall states from the Hofstadter model can persist up to hopping amplitudes of the same order as those typical for moir'e superlattice Hamiltonians, which implies generally that fractional states for moir'e superstructures can be discerned simply by analyzing the dominant terms in their effective Hamiltonians.
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.
We study how the stability of the fractional quantum Hall effect is influenced by the geometry of band structure in lattice Chern insulators. We consider the Hofstadter model, which converges to continuum Landau levels in the limit of small flux per plaquette. This gives us a degree of analytic control not possible in generic lattice models, and we are able to obtain analytic expressions for the relevant geometric criteria. These may be differentiated by whether they converge exponentially or polynomially to the continuum limit. We demonstrate that the latter criteria play a dominant role in predicting the physics of interacting particles in Hofstadter bands in this low flux density regime. In particular, we show that the many-body gap depends monotonically on a band-geometric criterion related to the trace of the Fubini-Study metric.
We study how the stability of the fractional quantum Hall effect is influenced by the geometry of band structure in lattice Chern insulators. We consider the Hofstadter model, which converges to continuum Landau levels in the limit of small flux per plaquette. This gives us a degree of analytic control not possible in generic lattice models, and we are able to obtain analytic expressions for the relevant geometric criteria. These may be differentiated by whether they converge exponentially or polynomially to the continuum limit. We demonstrate that the latter criteria play a dominant role in predicting the physics of interacting particles in Hofstadter bands in this low flux density regime. In particular, we show that the many-body gap depends monotonically on a band-geometric criterion related to the trace of the Fubini-Study metric.
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​.
We study the toplogical checkerboard lattice model around the $\ensuremath{\nu}=\frac{1}{3}$ fractional quantum Hall phase using numerical exact diagonalization without Landau level projections. We add local perturbations, modified hoppings, and on-site potentials, and observe phase transitions from the fractional quantum Hall phase to metallic and insulating phases when the strength and number of impurities is increased. In addition to evaluating the energy spectrum, we identify the phase diagrams by computing the topological Chern number of the many-body ground state manifold, and we show how the ground states lose their correlations due to the impurities by evaluating the spectrum of the one-body reduced density matrix. Our results show that the phase transition from the fractional quantum Hall phase to the metallic phase occurs for both impurity hoppings and potentials. Strong impurity hoppings cause a further transition into the insulating state, regardless of the sign of the hopping, when their density is high enough. In contrast, the same happens only for attractive potentials. Furthermore, the mixing to the higher band in a two-band model, generally denoted as Landau level mixing, is measured concluding that the lowest Landau level projection works well even with remarkably strong interactions and in the presence of impurities.
Twisted bilayer materials provide a versatile platform for realizing correlated topological states. Motivated by the recent experimental realization of atomic Bose-Einstein condensates in twisted bilayer optical lattices, we investigate the emergence of fractional topological phases in such systems. For experimentally realistic parameters, the single-particle spectrum hosts a pair of quasi-degenerate nearly flat moiré bands. We show using self-consistent Hartree-Fock calculations that atomic interactions spontaneously break time-reversal symmetry, lift the quasi-degeneracy, and generate an isolated nearly flat band with a nonzero Chern number. At fractional filling, exact diagonalization of the Hamiltonian projected to this band reveals a threefold degenerate ground-state manifold and a characteristic particle entanglement spectrum, providing strong evidence for a fractional Chern insulator. The resulting many-body gap is of order 10−3ER10^{-3}E_R10−3ER​, corresponding to a nanokelvin energy scale accessible to state-of-the-art cold-atom experiments. Our scheme requires neither spin-orbit coupling, as in transition metal dichalcogenides, nor a finite magnetic field, as in existing twisted-bilayer-graphene experiments, providing a highly tunable route to strongly correlated topological phases in twisted bilayer optical lattices.