Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1710.09350 · Physical review. B./Physical review. B

Stability of fractional Chern insulators in the effective continuum limit of Harper-Hofstadter bands with Chern number |C|>1

Bartholomew Andrews, Gunnar Möller

Published Jan 29, 2018• 51 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.97.035159

Related papers

Stability of Fractional Chern Insulators in the Effective Continuum Limit of $|C|>1$ Harper-Hofstadter Bands

Bartholomew Andrews, Gunnar Möller

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 $n_\phi\to 1/|C|$. 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 $\nu=2$ 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 $\nu=1$ in $|C|=2$ bands, we present additional data confirming the existence of a BIQHE state.

•Oct 25, 2017•0 citations

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

Gunnar Möller, Nigel R. Cooper

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.

arXiv:1504.06623•Physical Review Letters•Sep 16, 2015•107 citations

Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model

Bartholomew Andrews, Titus Neupert, Gunnar Möller

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.

arXiv:2107.06047•Physical review. B./Physical review. B•Sep 7, 2021•38 citations

Quantum-geometry stabilization of dilute fractional Chern insulators

Ying-Xing Ding, Limin Zhang, LI Wen-tong +2 more

Fractional Chern insulators have attracted broad interest as lattice analogs of fractional quantum Hall states without Landau levels. However, low-filling fractional Chern insulators are fragile because charge-ordered phases can compete strongly with the fractional topological liquid. Here, we propose a center-decorated kagome model, motivated by geometry-tunable artificial lattices, in which the center-site hopping $t_2$ provides a direct knob for the quantum geometry of an isolated $C=1$ flat band. Here quantum geometry refers to the Berry curvature and Fubini--Study metric, which determine the form factors of interactions projected into the Chern band. Exact diagonalization shows that tuning $t_2$ away from the flatness-optimized kagome limit reduces the trace-condition deviation, suppresses competing charge order, and enhances the many-body stability at both $ν=1/3$ and the more fragile $ν=1/5$ filling. At $ν=1/5$, this stability-enhanced window persists under nearby interaction profiles, including variations of the dominant third-neighbor repulsion and weak nearest-neighbor admixtures. Low-energy spectra, spectral flow, quasihole and entanglement counting, static structure factors, and the quantized total many-body Chern number $C_{\mathrm{tot}}=1$ consistently support Laughlin-like fractional Chern insulators. These results identify quantum-geometry engineering as a route to stabilizing dilute fractional Chern insulators beyond band-flatness optimization alone.

arXiv:2608.24013•Aug 25, 2026•0 citations

Ideal Fractional Chern Insulators

Jörg Behrmann, Zhao Liu, Emil J. Bergholtz

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 $\mathcal{Z}_k$ parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling fractions $\nu=k/(\mathcal{C}+1)$ and (ii) nematic states at $\nu=k/2$, where $\mathcal{C}$ 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 $(k+1)$-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.

•Dec 14, 2015•1 citation

Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model

David Bauer, Thomas Jackson, Rahul Roy

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.

arXiv:1504.07185•Physical review. B./Physical review. B•Jun 17, 2016•35 citations
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 \ensuremathν=r/(kr∣C∣+1),r\ensuremath∈Z\ensuremath{\nu}=r/(kr|C|+1),r\ensuremath{\in}\mathbb{Z}\ensuremathν=r/(kr∣C∣+1),r\ensuremath∈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\ensuremathϕ=1/∣C∣{n}_{\ensuremath{\phi}}=1/|C|n\ensuremathϕ​=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 \ensuremathν=1/(k∣C∣+1)\ensuremath{\nu}=1/(k|C|+1)\ensuremathν=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 \ensuremathν=2\ensuremath{\nu}=2\ensuremathν=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 \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=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 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.
Fractional Chern insulators have attracted broad interest as lattice analogs of fractional quantum Hall states without Landau levels. However, low-filling fractional Chern insulators are fragile because charge-ordered phases can compete strongly with the fractional topological liquid. Here, we propose a center-decorated kagome model, motivated by geometry-tunable artificial lattices, in which the center-site hopping t2t_2t2​ provides a direct knob for the quantum geometry of an isolated C=1C=1C=1 flat band. Here quantum geometry refers to the Berry curvature and Fubini--Study metric, which determine the form factors of interactions projected into the Chern band. Exact diagonalization shows that tuning t2t_2t2​ away from the flatness-optimized kagome limit reduces the trace-condition deviation, suppresses competing charge order, and enhances the many-body stability at both ν=1/3ν=1/3ν=1/3 and the more fragile ν=1/5ν=1/5ν=1/5 filling. At ν=1/5ν=1/5ν=1/5, this stability-enhanced window persists under nearby interaction profiles, including variations of the dominant third-neighbor repulsion and weak nearest-neighbor admixtures. Low-energy spectra, spectral flow, quasihole and entanglement counting, static structure factors, and the quantized total many-body Chern number Ctot=1C_{\mathrm{tot}}=1Ctot​=1 consistently support Laughlin-like fractional Chern insulators. These results identify quantum-geometry engineering as a route to stabilizing dilute fractional Chern insulators beyond band-flatness optimization alone.
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.
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.