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Generating quantum multicriticality in topological insulators by periodic driving

Paolo Molignini, Wei Chen, and R. Chitra
Phys. Rev. B 101, 165106 – Published 7 April 2020
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Abstract
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Article Text
  • INTRODUCTION
  • MODEL AND TOPOLOGICAL DESCRIPTION
  • DETECTION OF TPTs
  • QUANTUM CRITICALITY OF THE TPT AND…
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We demonstrate that the prototypical two-dimensional Chern insulator hosts exotic quantum multicriticality in the presence of an appropriate periodic driving: a linear Dirac-like transition coexists with a quadratic nodal looplike transition. This nodal loop gap closure is characterized by an enhanced chiral-mirror symmetry that is induced by the driving procedure. The existence of multiple universality classes can be unambiguously captured by extracting critical exponents and scaling laws with a single renormalization group approach based on the curvature function of the stroboscopic Floquet Hamiltonian. This procedure is effective regardless of whether the topological phase transitions are associated with anomalous edge modes or not. We comment on possible experimental realizations of the model and detection schemes for the curvature function.

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    • Received 23 June 2019
    • Accepted 6 March 2020

    DOI:https://doi.org/10.1103/PhysRevB.101.165106

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Chern insulators
    1. Physical Systems
    Floquet systemsNode-line semimetals
    1. Techniques
    Renormalization group
    Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

    Authors & Affiliations

    Paolo Molignini1,*, Wei Chen1,2, and R. Chitra1

    • 1Institute for Theoretical Physics, ETH Zürich, 8093 Zurich, Switzerland
    • 2Department of Physics, PUC-Rio, 22451-900 Rio de Janeiro, Brazil

    • *Present address: Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, United Kingdom.

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    Vol. 101, Iss. 16 — 15 April 2020

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    Images

    • Figure 1
      Figure 1

      (a) Depiction of the periodically driven 2D tight-binding model discussed in the main text. We consider a periodic four-step driving protocol for which the first hopping is different from the next three, J≡J1≠J2=J3=J4≡J̃. (b) Phase diagram obtained from the time-integrated topological invariant, indicating the type of topological excitations in each phase. The arrows, also shown in Fig. 2, indicate the two classes of TPTs studied, while the red circles highlight multicritical points. (c), (d) CRG flow diagram evaluated at k0=(0,0) and k0=(0,π) along direction ki=kx. The color codes indicate the log of the numerator of the CRG equation, log∂ki2F(k0,M), with yellow being high values (critical lines) and blue low values (fixed lines). Choosing other HSPs [i.e., (π,0) and (π,π)] or the direction ki=ky leads to similar flow diagrams.

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    • Figure 2
      Figure 2

      Illustration of different TPTs (indicated by the arrows in Fig. 1) in the FCI as a function of tuning parameter M=J (b) or M=J̃ (d). (a) Quasienergy dispersion exhibiting Dirac-cones with linear gap closure at quasienergy π for J̃=13n−J. b) Behavior of the Berry curvature across the TPT with linear gap closure (Lorentzians). (c) Quasienergy dispersion with quadratic gap closure at quasienergy 0 for J̃=J. (d) Behavior of the Berry curvature across the TPT with quadratic gap closures, where non-Lorentzian pairs of peaks flip sign and change direction.

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    • Figure 3
      Figure 3

      Illustration of the gap closures of the quasienergy dispersion θ(k) at (a) J=0.1[4π],J̃=0.3[4π], (b) J=0.2[4π],J̃=0.6[4π], (c) J=0.3[4π],J̃=0.9[4π], (d) J=0.6[4π],J̃=0.6[4π]. The dashed red lines indicate the cuts shown in Fig. 4.

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    • Figure 4
      Figure 4

      Cuts of the quasienergy dispersion θ(kx) along the ky value of the HSPs, illustrating the kind of gap closure for each topological phase transition. The values of the energy parameters are as in Fig. 3.

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    • Figure 5
      Figure 5

      Power law fit (solid lines) of the diverging quantities in the nodal loop semimetal extracted from the normalized Berry curvature. The fitted data is indicated by the dots.

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    • Figure 6
      Figure 6

      (a) Two-dimensional parametric fit of the quasienergy dispersion (upper band) to Eq. (13) around the HSP k0=(0,−π) for the TPT at J̃=J=0.6 in the FCI. The color bar on the left indicates the absolute error of the fit shown as a contour plot. (b) Dependence of the fitting parameters A, p, and M on the position of the TPT J̃=J, showing that only the overall scaling A changes along the transition line.

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    • Figure 7
      Figure 7

      (a) The normalized absolute value of the Berry curvature across the topological phase transition for the Floquet-Chern insulator (J̃ in units of 4π). (b) The normalized absolute value of the Berry curvature across the topological phase transition for the nodal loop semimetal as a function of μ, with ν=β=1.0 and α=−1.0.

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