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Localization renormalization and quantum Hall systems

Bartholomew Andrews, Dominic Reiss, Fenner Harper, and Rahul Roy
Phys. Rev. B 109, 125132 – Published 19 March 2024
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  • INTRODUCTION
  • LOCALIZATION RENORMALIZATION
  • QUANTUM HALL EXAMPLES
  • DISCUSSION
  • CONCLUSION
  • ACKNOWLEDGMENTS
    • Supplemental Material
    References

    Abstract

    The obstruction to constructing localized degrees of freedom is a signature of several interesting condensed matter phases. We introduce a localization renormalization procedure that harnesses this property and apply our method to distinguish between topological and trivial phases in quantum Hall and Chern insulators. By iteratively removing a fraction of maximally localized orthogonal basis states, we find that the localization length in the residual Hilbert space exhibits a power-law divergence as the fraction of remaining states approaches zero, with an exponent of ν=0.5. In sharp contrast, the localization length converges to a system-size-independent constant in the trivial phase. We verify this scaling using a variety of algorithms to truncate the Hilbert space and show that it corresponds to a statistically self-similar expansion of the real-space projector. This result is in accord with a renormalization group picture and motivates the use of localization renormalization as a versatile numerical diagnostic for quantum Hall systems.

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    • Received 21 October 2023
    • Revised 24 February 2024
    • Accepted 28 February 2024

    DOI:https://doi.org/10.1103/PhysRevB.109.125132

    ©2024 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Chern insulatorsInteger quantum Hall effectLandau levelsLocalization
    1. Physical Systems
    Two-dimensional electron system
    1. Techniques
    Finite-size scalingRenormalization
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Bartholomew Andrews1,2,*, Dominic Reiss1,*, Fenner Harper1, and Rahul Roy1,†

    • 1Department of Physics and Astronomy, University of California at Los Angeles, 475 Portola Plaza, Los Angeles, California 90095, USA
    • 2Department of Physics, University of California at Berkeley, 100 South Dr, Berkeley, California 94720, USA

    • *These authors contributed equally to this work.
    • †rroy@physics.ucla.edu

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    Vol. 109, Iss. 12 — 15 March 2024

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    Images

    • Figure 1
      Figure 1

      Removal lattice for Landau levels. Convergence of the removal lattice Lρ in Landau levels, sketched for three values of ρ. The area of the Lρ unit cell aρ2=2π/(1−ρ) is shaded pink and the disk radius R is marked with an arrow.

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    • Figure 2
      Figure 2

      Localization scaling in Landau levels. (Left) Localization length ξ(ρ) obtained by the simultaneous elimination of a lattice of localized states in the (a) n=0, (b) n=1, and (c) n=2 Landau levels. The system size R is depicted in different colors. At small ρ and large system sizes, the relationship is linear indicating power-law behavior in the thermodynamic limit. (Right) The quality function of data collapse d(ν) for the (a) n=0, (b) n=1, and (c) n=2 Landau levels; optimization of the quality function results in critical exponents of (a) ν0LL=0.500±0.005, (b) ν1LL=0.500±0.006, and (c) ν2LL=0.500±0.006.

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    • Figure 3
      Figure 3

      Projector expansion in Landau levels. (i) Sketch of the averaged real-space projector expansion as ρ→0 in the LLL. (ii) (Left) Real-space radial profile of the projector P(r,0)=〈r|PρLL|0〉. The rotationally and translationally averaged projector, normalized with respect to the initial value, P̂av, is shown in the main plot. The original unaveraged projector, normalized with respect to the maximum value, P̂, is shown inset. The projectors are presented for the (a) n=0, (b) n=1, and (c) n=2 Landau levels, at a system size of R=40 in the main plot and R=20 in the inset, with ρ0=1.5. The boundary of the system at r=20 in the inset is marked with a dashed line. Note that the projectors are plotted in reverse order, such that the κ=0 line is on top. (ii) (Right) Finite-size scaling of the second moment of the projector for the (a) n=0, (b) n=1, and (c) n=2 Landau levels. The lines of best fit, for the linear region of the largest system size, are overlaid in black. The corresponding higher moments, ξγ=〈rγ〉, are shown inset for R=20.

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    • Figure 4
      Figure 4

      Haldane model on a square lattice. (a) Mapping of the Haldane model onto a square lattice. The basis vectors {a1,a2} are given in units of the lattice constant and the unit cell is shaded gray. The t1 hoppings are colored green and the t2 hoppings are colored red (blue) according to their A (B) sublattice, with the direction of the arrows corresponding to a positive complex phase. (b) Phase diagram for the Haldane model, as a function of complex phase ϕ and chemical potential M, colored according to the sign of the Chern number for the lowest band C1. The three selected parameter sets are at ta={t2=0.1,M=0}, tb={t2=0.2,M=0.1}, and tc={t2=0.1,M=1}. For all parameter sets, ϕ=π/2, and in all cases we set t1=1.

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    • Figure 5
      Figure 5

      Removal lattice for the Haldane model. Convergence of the removal lattice Lρ for the square-lattice Haldane model, sketched for three values of ρ. The area of the L unit cell a2=1 is shaded gray, the area of the Lρ unit cell aρ2=1/(1−ρ) is shaded pink, and the linear system size L is marked with an arrow.

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    • Figure 6
      Figure 6

      Localization scaling in the Haldane model. (Left) Localization length ξ(ρ) obtained by the simultaneous elimination of a lattice of localized degrees of freedom, applied to the Haldane model with parameters (a) ta, (b) tb, and (c) tc. The system size L is depicted in different colors. (Right) The quality function of data collapse d(ν) for systems in the topological Haldane model with parameters (a) ta and (b) tb; optimization of the quality function results in critical exponents of (a) νta=0.503±0.011 and (b) νtb=0.499±0.012.

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    • Figure 7
      Figure 7

      Projector expansion in the Haldane model. (Left) Real-space radial profile of the projector P(r,0)=〈r|PρCI|0〉. The rotationally and translationally averaged projector, normalized with respect to the initial value, P̂av, is shown in the main plot. The original unaveraged projector, normalized with respect to the maximum value, P̂, is shown inset. The projectors are presented for the (a) ta, (b) tb, and (c) tc phases of the Haldane model, at a system size of L=28 in the main plot and L=20 in the inset, with ρ0=1.5. The boundary of the system at r=10 in the inset is marked with a dashed line. Note that the projectors are plotted in reverse order, such that the κ=0 line is on top. (Right) Finite-size scaling of the second moment of the projector for the (a) ta, (b) tb, and (c) tc phases of the Haldane model. The lines of best fit, for the linear region of the largest system size, are overlaid in black. The corresponding higher moments, ξγ=〈rγ〉, are shown inset for L=20.

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