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.
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 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=1,2,3,4,5 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 the stability of composite fermion fractional quantum Hall states in Harper-Hofstadter bands with Chern number
∣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=1 for bosons and
k=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∣ 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 and
∣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 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 states are smooth functions for positions separated by
∣C∣ sites along both axes, giving rise to
∣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 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 bands, we present additional data confirming the existence of a BIQHE state.
We study the stability of composite fermion fractional quantum Hall states in Harper-Hofstadter bands with Chern number
∣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∣. 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 and
∣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 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 smooth sheets. We examine two cases associated with a bosonic integer quantum Hall effect (BIQHE): for
ν=2 in
∣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 in
∣C∣=2 bands, we present additional data confirming the existence of a BIQHE state.
We present a theory of fractional Chern insulator stabilization against charge-ordered states. We argue that the phase competition is captured by an effective interaction range, which depends on both the bare interaction range and quantum geometric properties. We claim that short effective interaction ranges stabilize fractional states while longer-range interactions favor charge-ordered states. To confirm our hypothesis, we conduct a numerical study of the generalized Hofstadter model using the density matrix renormalization group. Our theory offers an interpretation of the geometric stability hypothesis and generalizes it, providing a unifying framework for several approaches to fractional phase stabilization. Finally, we propose a route toward experimental verification of the theory and possible implications for fractional states in bands with higher Chern numbers.