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Stability of fractional Chern insulators with a non-Landau level continuum limit

Bartholomew Andrews, Mathi Raja, Nimit Mishra, Michael P. Zaletel, and Rahul Roy
Phys. Rev. B 109, 245111 – Published 6 June 2024
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  • INTRODUCTION
  • MODEL
  • METHOD
  • RESULTS
  • DISCUSSION AND CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The stability of fractional Chern insulators is widely believed to be predicted by the resemblance of their single-particle spectra to Landau levels. We investigate the scope of this geometric stability hypothesis by analyzing the stability of a set of fractional Chern insulators that explicitly do not have a Landau level continuum limit. By computing the many-body spectra of Laughlin states in a generalized Hofstadter model, we analyze the relationship between single-particle metrics, such as trace inequality saturation, and many-body metrics, such as the magnitude of the many-body and entanglement gaps. We show numerically that the geometric stability hypothesis holds for Chern bands that are not continuously connected to Landau levels, as well as conventional Chern bands, albeit often requiring larger system sizes to converge for these configurations.

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    • Received 9 October 2023
    • Revised 24 May 2024
    • Accepted 28 May 2024

    DOI:https://doi.org/10.1103/PhysRevB.109.245111

    ©2024 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Fractional quantum Hall effectGeometric & topological phasesLandau levels
    1. Physical Systems
    2-dimensional systemsStrongly correlated systems
    1. Techniques
    Exact diagonalizationLattice models in condensed matter
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Bartholomew Andrews1,2, Mathi Raja1, Nimit Mishra2, Michael P. Zaletel1, and Rahul Roy2

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    Vol. 109, Iss. 24 — 15 June 2024

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    • Figure 1
      Figure 1

      (3,6,9) Hofstadter model. Sketch of the hopping terms in the (3,6,9) Hofstadter Hamiltonian (conjugates not drawn). The radii for the first 9 sets of nearest neighbors on a square lattice are 1, 2, 2, 5, 22, 3, 10, 13, 4.

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    • Figure 2
      Figure 2

      Single- and many-body spectra on the hexic line. (a) Single-particle band structure of the (3,6,9) Hofstadter model, defined as Eq. (1) with t1=1, n={1,3,6,9}, at the point (t3,t6,t9)=(−0.11,−0.15,0.05) on the hexic line, with nϕ=1/9. The Chern number C of each band is labeled and the bands are colored according to its sign. Note that bands 5 and 6 are not touching. The fluctuations of the lowest band about its mean are shown inset. (b) Butterfly plot showing the fractal single-particle energy spectrum as a function of nϕ. The bands are colored from blue to red according to their Chern number, in the range C∈[−10,10]. (c) Many-body energy spectrum for the 8-particle fermionic Laughlin state filling ν=1/3 of the lowest band at nϕ=1/24, stabilized using nearest-neighbor interactions Vij=δ〈ij〉. The many-body gap Δmb and quasidegeneracy spread δ are labeled, and the gap scaling with magnetic unit cell area q is shown inset. (d) Spectral flow of the 3 quasidegenerate grounds states shown in (c), as a flux Φ is threaded through the handle of the torus. (e) Particle entanglement spectrum corresponding to (c) obtained by tracing over half of the particles. The principal entanglement gap Δξ is labeled and the number of states below the gap obeys the (1,3) counting rule [4]. (f) Particle entanglement spectra corresponding to (c), in the k=0 momentum sector, obtained by tracing over NA particles. The entanglement gap is shaded gray.

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    • Figure 3
      Figure 3

      Stability of the bosonic Laughlin state on the quartic plane. (a) BZ-averaged TISM 〈T〉, scaled by the MUC area q, plotted in the quartic plane 1+4t3+9t6+16t9=0, for nϕ=1/16,1/49,1/81. The hexic line t9=(1−15t6)/64 and octic point t9=−1/56 are overlaid in green. The points corresponding to Fig. 4 are marked by blue crosses and the cross section corresponding to Fig. 7 is marked by a red dashed line. (b) Many-body gap Δmb, scaled by the MUC area q, with parameters corresponding to (a). The results are shown for the 8-particle bosonic Laughlin state stabilized by the contact interaction Vij=δij. (c) Quasidegeneracy spread δ, scaled by the MUC area q, corresponding to (b). (d) Principal entanglement gap Δξ, corresponding to (b) and (c).

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    • Figure 4
      Figure 4

      Many-body spectra for candidate bosonic Laughlin states with small and large TISM. Comparison of many-body energy and entanglement spectra for candidate 10-particle bosonic Laughlin FCIs at nϕ=1/81, on the quartic plane, with (a) (t6,t9)=(−0.25,−0.25), and (b) (t6,t9)=(0.25,0.25), corresponding to the blue crosses in Fig. 3. The ground-state energies are (a) Emb,0=−72.0 and (b) Emb,0=−74.6. The entanglement energies included in the (1,2) counting are colored red. The bottom panels show the corresponding finite-size scaling of the many-body and entanglement gaps. The many-body gaps are marked by blue dots and the entanglement gaps are marked by red crosses. For reference, the scaled many-body gap of the Hofstadter model in the thermodynamic continuum limit is shown as a blue dashed line [28, 39]. The lattice geometries are selected so that the total system is approximately square, with |1−Nx/Ny|≤50% in all cases. The PESs are computed with NA=⌊N/2⌋.

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    • Figure 5
      Figure 5

      Stability of the fermionic Laughlin state on the quartic plane. (a) BZ-averaged TISM 〈T〉, scaled by the MUC area q, plotted in the quartic plane 1+4t3+9t6+16t9=0, for nϕ=1/24,1/54,1/96. The hexic line t9=(1−15t6)/64 and octic point t9=−1/56 are overlaid in green. The point corresponding to Fig. 2 is marked by a red cross, the points corresponding to Fig. 6 are marked by blue crosses, and the cross section corresponding to Fig. 7 is marked by a red dashed line. (b) Many-body gap Δmb, scaled by the MUC area q, with parameters corresponding to (a). The results are shown for the 8-particle fermionic Laughlin state stabilized by the nearest-neighbor interaction Vij=δ〈ij〉. (c) Quasidegeneracy spread δ, scaled by the MUC area q, corresponding to (b). (d) Principal entanglement gap Δξ, corresponding to (b) and (c).

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    • Figure 6
      Figure 6

      Many-body spectra for candidate fermionic Laughlin states with small and large TISM. Comparison of many-body energy and entanglement spectra for candidate 10-particle fermionic Laughlin FCIs at nϕ=1/96, on the quartic plane, with (a) (t6,t9)=(−0.25,−0.25), and (b) (t6,t9)=(0.25,0.25), corresponding to the blue crosses in Fig. 5. The ground-state energies are (a) Emb,0=−72.1 and (b) Emb,0=−75.8. The entanglement energies included in the (1,3) counting are colored red. The bottom panels show the corresponding finite-size scaling of the many-body and entanglement gaps. The many-body gaps are marked by blue dots and the entanglement gaps are marked by red crosses. For reference, the scaled many-body gap of the Hofstadter model in the thermodynamic continuum limit is shown as a blue dashed line [28, 39]. The lattice geometries are selected so that the total system is approximately square, with |1−Nx/Ny|≤27% in all cases. The PESs are computed with NA=⌊N/2⌋.

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    • Figure 7
      Figure 7

      Stability of Laughlin states on cross sections of the quartic plane. Comparison of cross sections at (a) t6=−0.05 for the bosonic Laughlin state at nϕ=1/81, and (b) t6=0.25 for the fermionic Laughlin state at nϕ=1/96, corresponding to the red dashed lines in Figs. 3 and 5. The first panels show the scaled many-body gap, q(2)Δmb for bosons (fermions), and the principal entanglement gap Δξ, as a function of t9. The many-body gaps are marked by blue dots and the entanglement gaps are marked by red crosses. The maximum values of the gaps are marked with dashed lines, and the hexic line is marked with a solid green line, as in Figs. 3 and 5. The subsequent panels show the corresponding single-particle metrics in black. The second, third, and fourth panels show the scaling of the BZ-averaged TISM 〈T〉, the BZ-averaged DISM 〈D〉, and fluctuations of the Fubini-Study metric σg, respectively. The minima of the TISM and DISM are marked with a dashed line. Finally, the fifth and sixth panels show the scaling of the normalized Berry curvature fluctuations σ̂B and the gap-to-width ratio Δ/W. The logarithmic scale on the y axes is base 10.

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    • Figure 8
      Figure 8

      Additional results for the stability of the bosonic Laughlin state on the quartic plane. (a) BZ-averaged TISM 〈T〉, scaled by the MUC area q, plotted in the quartic plane 1+4t3+9t6+16t9=0, for nϕ=1/100,1/121,1/144,1/169,1/196,1/225,1/256. The hexic line t9=(1−15t6)/64 and octic point t9=−1/56 are overlaid in green. (b) Many-body gap Δmb, scaled by the MUC area q, with parameters corresponding to (a). The results are shown for the 8-particle bosonic Laughlin state stabilized by the contact interaction Vij=δij. (c) Quasidegeneracy spread δ, scaled by the MUC area q, corresponding to (b). (d) Principal entanglement gap Δξ, corresponding to (b) and (c). These results follow directly from Fig. 3.

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    • Figure 9
      Figure 9

      Additional results for the stability of the fermionic Laughlin state on the quartic plane. (a) BZ-averaged TISM 〈T〉, scaled by the MUC area q, plotted in the quartic plane 1+4t3+9t6+16t9=0, for nϕ=1/126,1/140,1/150,1/160,1/168,1/176,1/187. We select flux densities that admit almost square configurations (ε≤7%) and are approximately evenly spaced with respect to q. The hexic line t9=(1−15t6)/64 and octic point t9=−1/56 are overlaid in green. (b) Many-body gap Δmb, scaled by the MUC area q, with parameters corresponding to (a). The results are shown for the 8-particle fermionic Laughlin state stabilized by the nearest-neighbor interaction Vij=δ〈ij〉. (c) Quasidegeneracy spread δ, scaled by the MUC area q, corresponding to (b). (d) Principal entanglement gap Δξ, corresponding to (b) and (c). These results follow directly from Fig. 5.

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