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Extending Topological Bound on Quantum Weight Beyond Symmetry-Protected Topological Phases

Yi-Chun Hung, Yugo Onishi, Hsin Lin, Liang Fu, Arun Bansil

Abstract

The quantum metric encodes the geometric structure of Bloch wave functions and governs a wide range of physical responses. Its Brillouin-zone integral, the quantum weight, appears in the structure factor and provides lower bounds on observables such as the optical gap and dielectric constant. In symmetry-protected topological (SPT) phases, the nontrivial band topology imposes a lower bound on the quantum weight and constraints on the observables. Here, we generalize the topological bound on quantum geometry to encompass systems beyond the SPT phases. We show that topological invariants defined via the projected spectrum lower-bound the quantum weight with a symmetry-breaking correction to the quantum metric. Our proposed bound holds even when the underlying symmetries are broken, and it would be amenable to experimental verification via the optical conductivity sum rule under external fields. We illustrate our theory by adding a nonzero spin-orbit coupling term to a spin Chern insulator model, where we show that our proposed bound applies even though the conventional topological bound does not hold.

Extending Topological Bound on Quantum Weight Beyond Symmetry-Protected Topological Phases

Abstract

The quantum metric encodes the geometric structure of Bloch wave functions and governs a wide range of physical responses. Its Brillouin-zone integral, the quantum weight, appears in the structure factor and provides lower bounds on observables such as the optical gap and dielectric constant. In symmetry-protected topological (SPT) phases, the nontrivial band topology imposes a lower bound on the quantum weight and constraints on the observables. Here, we generalize the topological bound on quantum geometry to encompass systems beyond the SPT phases. We show that topological invariants defined via the projected spectrum lower-bound the quantum weight with a symmetry-breaking correction to the quantum metric. Our proposed bound holds even when the underlying symmetries are broken, and it would be amenable to experimental verification via the optical conductivity sum rule under external fields. We illustrate our theory by adding a nonzero spin-orbit coupling term to a spin Chern insulator model, where we show that our proposed bound applies even though the conventional topological bound does not hold.
Paper Structure (8 sections, 62 equations, 4 figures)

This paper contains 8 sections, 62 equations, 4 figures.

Table of Contents

  1. S1. Remarks on the proof of the band-gap bound for SCIs with $C_s=2$
  2. S2. Positive semidefiniteness and real symmetry of the quantum geometric tensors
  3. S3. $K_c$ as a generic correction
  4. A. Necessary conditions for $K_c\neq0$: $K\neq0$ and broken $\hat{O}$-symmetry
  5. B. An example of $K_c=0$ with a broken $\hat{O}$-symmetry and $K\neq0$
  6. S4. Proof of $\text{tr}[G_{c,+}]=\text{tr}[G_{c,-}]$
  7. S5. Optical origin of the topological bound
  8. S6. Comments on experimental implication of the topological bound for the case of $\mathcal{\hat{O}}_{\mathcal{P}}$ with multiple sectors

Figures (4)

  • Figure 1: Schematics of (a) band structure of an insulator and (b) various sectors categorized by $\lambda_n^{(\pm)}$ in the spectrum of $\mathcal{\hat{O}_{P}}$ (projected spectrum). Here, $Q=1-P$.
  • Figure 2: Schematics of the band structure of a SCI with narrow occupied-state bandwidth $t$ and a large band gap $E_{\text{gap}}$ (a) before and (b) after applying a Zeeman field with energy scale $E_{h_z}$. $E_f$ denotes the Fermi level, which is tuned to half-filling.
  • Figure 3: Schematics of (a) the band structure and (b) the spectrum of $\mathcal{\hat{S}}^z_{\,\,\mathcal{P}}$ for the SCI model in Eq. \ref{['eq:model']} with $\Delta = m$, $\tilde{\mu} = 1.2$, $\tilde{\lambda} = 1.6$, and $E_0=m\Delta^2/2$. Colormap shows the Berry curvature distribution $\Omega_{xy}$ with $\Omega_0 = 10^{-4}$. (c) $K$, $K_c$, and $K+K_c$ as functions of $\tilde{\lambda}$ with $\tilde{\mu}=1$ and $\Delta=m$. Dashed line marks the topological bound set by $\sum_{\alpha=\pm}|C_\alpha|$.
  • Figure S1: Schematic band structure showing a narrow occupied-state bandwidth and a large band gap after applying a tunable physical field $F_{\hat{O}}$ with energy scale $E_{F_{\hat{O}}}$. Dashed lines mark the Fermi levels for two optical conductivity measurements.