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Topological Signatures of the Optical Bound on Maximal Berry Curvature: Application to Two-Dimensional Time-Reversal Symmetric Insulators

Pok Man Chiu

Abstract

Unlike broken time-reversal symmetric (TRS) systems with a defined Chern number, directly measuring the bulk $Z_{2}$ invariant and Berry curvature (if nonzero) in topological insulators and their higher-order topological families remains an unsolved problem. Here, based on the refined trace-determinant inequality (TDI) involving the trace and determinant of the quantum metric and maximal Berry curvature (MBC), we establish an optical bound on the MBC for two-dimensional TRS insulators. By utilizing experimental data on the optical conductivity within a certain energy range, the topological signatures can be identified through the frequency integration of the optical bound. This is supported by the momentum integration of the refined TDI and its $f$-sum rule and topological extension, which provide a topological lower bound. Meanwhile, the decay of optical weight in the topologically trivial region can be controlled by the optical gap and the inverse mass tensor. We illustrate our approach using three representative topological models: the Kane-Mele model, mirror-protected insulator, and quadrupole insulator. Remarkably, we find that the quantized quantum volume (QV) in the mirror-protected insulator results from the Gauss-Bonnet theorem. Since QV has a topological lower bound, this suggests that double QV can be interpreted as an upper bound on the number of boundary states. Our findings offer a method for extracting the topological signatures of TRS insulators using optical conductivity data.

Topological Signatures of the Optical Bound on Maximal Berry Curvature: Application to Two-Dimensional Time-Reversal Symmetric Insulators

Abstract

Unlike broken time-reversal symmetric (TRS) systems with a defined Chern number, directly measuring the bulk invariant and Berry curvature (if nonzero) in topological insulators and their higher-order topological families remains an unsolved problem. Here, based on the refined trace-determinant inequality (TDI) involving the trace and determinant of the quantum metric and maximal Berry curvature (MBC), we establish an optical bound on the MBC for two-dimensional TRS insulators. By utilizing experimental data on the optical conductivity within a certain energy range, the topological signatures can be identified through the frequency integration of the optical bound. This is supported by the momentum integration of the refined TDI and its -sum rule and topological extension, which provide a topological lower bound. Meanwhile, the decay of optical weight in the topologically trivial region can be controlled by the optical gap and the inverse mass tensor. We illustrate our approach using three representative topological models: the Kane-Mele model, mirror-protected insulator, and quadrupole insulator. Remarkably, we find that the quantized quantum volume (QV) in the mirror-protected insulator results from the Gauss-Bonnet theorem. Since QV has a topological lower bound, this suggests that double QV can be interpreted as an upper bound on the number of boundary states. Our findings offer a method for extracting the topological signatures of TRS insulators using optical conductivity data.

Paper Structure

This paper contains 8 sections, 64 equations, 6 figures.

Table of Contents

  1. Factors Influencing the Decay Behavior of Quantum Weight: Crystalline Symmetry, Optical Gap, and Inverse Mass Tensor
  2. Euler Number and Its Relation to Chern Number
  3. 1: Quantum geometry of occupied bands
  4. 2. Trace-determinant inequality and its refinement for TIs, TCIs, and HOTIs with a topological lower bound
  5. 3: Optical bound
  6. 4: Souza-Wilkens-Martin sum rule and the relation between optical bound and quantum weight
  7. 5: Upper bound of the quantum weight and the optical gap
  8. 6: Numerical results on band structure and quantum weight during the topological phase transition for the three models

Figures (6)

  • Figure 1: Optical bound, optical weight, and topological signatures of the Kane-Mele model and TCI model of mirror-protected insulator. (a1, a2, b1, b2) The optical bound as a function of frequency with $\lambda_{\upsilon}=0.5t,2t$ in (a1,b1) and $\upsilon_{m}=1,5$ in (a2,b2). The red solid (green dashed) line represents the optical bound (generalized optical Hall conductivity) in the nontrivial (a1, a2) and trivial phases (b1, b2). (c1, c2) The optical weight $\mathcal{K}^{\omega_c}_{OP}$ as a function of frequency cutoff for four parameters. The red (black) lines correspond to the nontrivial (trivial) phase. (d1, d2) The topological signatures $\mathcal{K}$, $2\mathcal{V}$ and $\overline{C}$ as a function of a range of parameters. The back dashed line represents the number of boundary states. We calculate $\Re\sigma_{aa}$ and $\Im\overline{\sigma}_{ab}$ in units of $e^2/h$. Here, we set $(t,\lambda_{SO},\lambda_{R},\lambda_{\upsilon})=(1,0.2t,0,m)$ and $(t_1,t_2,t_{PH},\upsilon_s,\upsilon_{m})=(2,1.5,0.1,1.3,m)$ for the Kane-Mele model and TCI Wieder20, respectively.
  • Figure 2: Optical bounds, optical weight, and topological signatures of HOTI model of quadrupole insulator. (a1, a2, b1, b2) The optical bound as a function of frequency. The red solid (green dashed) line represents the optical bound (generalized optical Hall conductivity). In (a1, b1) and (a2, b2), we set $(\gamma_x, \gamma_y) = (0.25, 0.25)$, $(1.25, 1.25)$ and $(\gamma_x, \gamma_y) = (0.25, 0.5)$, $(1.5, 0.5)$, respectively. (c1, c2) The optical weight $\mathcal{K}^{\omega_c}_{OP}$ as a function of frequency cutoff for four parameters. The red (black) line corresponds to the nontrivial (trivial) phase. (d1, d2) The topological signatures $\mathcal{K}$, $2\mathcal{V}$, and $\mathcal{\overline{C}}$ as a function of a range of parameters. The black dashed and dotted lines represent the number of boundary states in the phases $\mathbf{p}^{\nu} = \left(\frac{1}{2}, \frac{1}{2}\right)$ and $\mathbf{p}^{\nu} = \left(0, \frac{1}{2}\right)$, respectively. We calculate $\Re\sigma_{aa}$ and $\Im\overline{\sigma}_{ab}$ in units of $e^2/h$. We take $(\gamma_x,\gamma_y) = (m,m)$ for panels (c1,d1) and $(\gamma_x,\gamma_y) = (m,0.5)$ for panels (c2,d2), respectively. Here, we set $(\lambda_x,\lambda_y)=(1,1)$.
  • Figure 3: Band structure and topological signatures of the Kane-Mele model with three sets of $(\lambda_{SO},\lambda_{R})$ parameters. (a1-a3) Band structure with (a1) $(\lambda_{SO},\lambda_{R})=(0.06t,0.05t)$, (a2) $(\lambda_{SO},\lambda_{R})=(0.06t,0)$, and (a3) $(\lambda_{SO},\lambda_{R})=(0.6t,0)$, in which we set $\lambda_{\upsilon}=0.1t$. (b1-b3) The topological signatures $\mathcal{K}$, $2\mathcal{V}$ and $\overline{C}$ with (b1) $(\lambda_{SO},\lambda_{R})=(0.06t,0.05t)$, (b2) $(\lambda_{SO},\lambda_{R})=(0.06t,0)$, and (b3) $(\lambda_{SO},\lambda_{R})=(0.6t,0)$ as a function of $m$. The black dashed line represents the number of boundary states. Here we set $t=1$. The presence of Rashba spin-orbit coupling, which breaks the mirror symmetry ($M_z$), will lower the value of MBC.
  • Figure 4: Euler numbers of the Kane-Mele model, the TCI model of mirror-protected insulator, and the HOTI model of quadrupole insulator as functions of $m$. (a) Euler number of the Kane-Mele model with $(t,\lambda_{SO},\lambda_{R},\lambda_{\upsilon})=(1,0.6t,0,m)$. (b) Euler number of the TCI model of mirror-protected insulator with $(t_1,t_2,t_{PH},\upsilon_s,\upsilon_{m})=(2,1.5,0.1,1.3,m)$. (c) Euler number of the HOTI model of quadrupole insulator with $(\lambda_x,\lambda_y,\gamma_x,\gamma_y) = (1,1,m,m)$.
  • Figure S1: Band structure of the three models as a function of $m$ along the selected Brillouin zone path near the topological phase transition. (a) $m\in(0,0.5)$. (b) $m\in(2.5,5)$. (c,d) $m\in(0.5,1.5)$.
  • ...and 1 more figures