Quantum geometric bound and ideal condition for Euler band topology

Understanding the relationship between quantum geometry and topological invariants is a central problem in the study of topological states. In this work, we establish the relationship between the quantum metric and the Euler curvature in two-dimensional systems with space-time inversion $I_{ST}$ symmetry satisfying $I_{ST}^2=+1$. As $I_{ST}$ symmetry imposes the reality of the wave function with vanishing Berry curvature, the well-known inequality between the quantum metric and the Berry curvature is not meaningful in this class of systems. We find that the non-Abelian quantum geometric tensor of two real bands exhibits an intriguing inequality between the off-diagonal Berry curvature and the quantum metric, which in turn gives the inequality between the quantum volume and the Euler invariant. Moreover, we show that the saturation condition of the inequality is deeply related to the ideal condition for Euler bands, which provides a criterion for the stability of fractional topological phases in interacting Euler bands. Our findings demonstrate the potential of the quantum geometry as a powerful tool for characterizing symmetry-protected topological states and their interaction effect.

Figure 1

(a), (b) The change of the quantum volume and Euler invariant of the lower two flat bands for three-band models where a direct transition between a topological insulator (TI) with $e_2\ne 0$ and a normal insulator (NI) with $e_2=0$ occurs.

Figure 2

(a) The band structure of the four-band model $H_4^a\left(\mathbf{k}\right)$ when $m_3=0$. Both red and blue bands are doubly degenerate when $m_3=0$ while the degeneracy is lifted when $m_3\ne 0$. (b) The quantum volume and Euler invariant of the red band in (a).

Figure 3

(a) The lattice structure of TBG with the twist angle $\theta$. (b) The change of the quantum volume and Euler invariant as a function of $\alpha$ at the chiral limit with ${\omega }_0/{\omega }_1=0$. ${\mathrm{vol}}_g/2\pi =\left|e_2\right|$ holds only at magic angles. (c) Similar plot for different ${\omega }_0/{\omega }_1$. (d) Plot of ${\alpha }_{\text{min}}$ at which the Dirac velocity $v_D$, the flat-band bandwidth $W$, or ${\mathrm{vol}}_g$ becomes minimized at a given ${\omega }_0/{\omega }_1$ as a function of ${\omega }_0/{\omega }_1$.