Non-Abelian quantum geometric tensor in degenerate topological semimetals

The quantum geometric tensor characterizes the complete geometric properties of quantum states, with the symmetric part being the quantum metric and the antisymmetric part being the Berry curvature. We propose a generic Hamiltonian with globally degenerate ground states and give a general relation between the corresponding non-Abelian quantum metric and unit Bloch vector. This enables us to construct the relation between the non-Abelian quantum metric and Berry or Euler curvature. To be concrete, we present and study two topological semimetal models with globally degenerate bands under $CP$ and $C_{2}T$ symmetries. The topological invariants of these two degenerate topological semimetals are the Chern number and Euler class, respectively, which are calculated from the non-Abelian quantum metric with our constructed relations. Based on the adiabatic perturbation theory, we further obtain the relation between the non-Abelian quantum metric and the energy fluctuation. Such a nonadiabatic effect can be used to extract the non-Abelian quantum metric, which is numerically demonstrated for the two models of degenerate topological semimetals. Finally, we discuss the quantum simulation of the model Hamiltonians with cold atoms.

Figure 1

(a) The multi-Weyl monopoles of Hamiltonian $H_{CP}^{\left(n\right)}$ in the FBZ for $M_z=2$; the ellipses stand for monopoles, and $\pm Q$ indicate their topological charges. $Q=2$ for $H_{CP}^{\left(1\right)}$, and $Q=4$ for $H_{CP}^{\left(2\right)}$. (b) The first Chern number $\mathcal{C}$ for Hamiltonians $H_{CP}^{\left(1\right)}$ and $H_{CP}^{\left(2\right)}$ against the parameter $M_z$ for $k_z=0$. The bulk state and surface state for Hamiltonians $H_{CP}^{\left(1\right)}$ and $H_{CP}^{\left(2\right)}$. The energy spectra (c) for $H_{CP}^{\left(1\right)}$ for $M_z=2$ and $k_x=0$ and (d) for $H_{CP}^{\left(1\right)}$ for $M_z=2$ and $k_z=0$. The bulk state and surface state (e) for Hamiltonian $H_{CP}^{\left(2\right)}$ for $M_z=2$ and $k_x=0$ and (f) for Hamiltonian $H_{CP}^{\left(2\right)}$ for $M_z=2$ and $k_z=0$. (g) $\theta \left(k_x\right)$, the phases of the eigenvalues of the Wilson loops. ${\theta }_1$ and ${\theta }_2$ are for Hamiltonian $H_{CP}^{\left(1\right)}$ in Eq. (19) with $M_z=2$ (topological phase) and $M_z=4$ (trivial phase); $k_z=0$. (h) ${\theta }_3$ and ${\theta }_4$ are for Hamiltonian $H_{CP}^{\left(2\right)}$ in Eq. (24) with $M_z=2$ (topological phase) and $M_z=4$ (trivial phase); $k_z=0$. ${\alpha }_x={\alpha }_y={\alpha }_z=1$ for all panels.

Figure 2

(a) The monopoles of Hamiltonian $H_{C_{2}T}^{\left(n\right)}$ in the FBZ for $m_z=-2$. The ellipses stand for monopoles, and $\pm q$ indicate their topological charges; here $q=n$ for $H_{C_{2}T}^{\left(n\right)}$. (b) The Euler class for Hamiltonians $H_{C_{2}T}^{\left(1\right)}$ and $H_{C_{2}T}^{\left(2\right)}$ for $k_z=0$. (c) The energy spectra for Hamiltonian $H_{C_{2}T}^{\left(1\right)}$ with $m_z=-2$ and $k_x=\pi$; this is a topological Euler semimetal. (d) The energy spectra for Hamiltonian $H_{C_{2}T}^{\left(1\right)}$ with $m_z=-2$ and $k_z=\pi$; this is a Euler insulator. The bulk state and surface state (e) for Hamiltonian $H_{C_{2}T}^{\left(2\right)}$ with $m_z=-2$ and $k_x=\pi$ and (f) for Hamiltonian $H_{C_{2}T}^{\left(2\right)}$ with $m_z=-2$ and $k_z=\pi$. (g) ${\theta }_5$ and ${\theta }_6$ are transition functions for Hamiltonian $H_{C_{2}T}^{\left(1\right)}$ in Eq. (39) with $m_z=2$ (topological phase) and $m_z=4$ (trivial phase); $k_z=0$. (h) ${\theta }_7$ and ${\theta }_8$ are for Hamiltonian $H_{C_{2}T}^{\left(2\right)}$ in Eq. (39) with $m_z=2$ (topological phase) and $m_z=4$ (trivial phase); $k_z=0$. ${\alpha }_x={\alpha }_y={\alpha }_z=1$ for all panels.

Figure 3

(a) The analytical result of $F_{xy}^{11}$ for Hamiltonian $H_{CP}^{\left(1\right)}$, with $\text{Im}\left(F_{xy}^{11}\right)=0$. (b) The numerical result of $F_{xy}^{11}$ for Hamiltonian $H_{CP}^{\left(1\right)}$. (c) The analytical result of $\text{Im}\left(F_{xy}^{12}\right)$ for Hamiltonian $H_{C_{2}T}^{\left(1\right)}$. (d) The numerical result of $\text{Im}\left(F_{xy}^{12}\right)$ for Hamiltonian $H_{C_{2}T}^{\left(1\right)}$. The numerical results are obtained from full-time-dynamics simulations; $M_z=m_z=2$, and $k_z=0$ for all panels.

Figure 4

(a) The analytical result of $g_{xy}^{11}$ for Hamiltonian $H_{CP}^{\left(1\right)}$. (b) The numerical result of $g_{xy}^{11}$ for Hamiltonian $H_{CP}^{\left(1\right)}$. (c) The analytical result of $g_{xy}^{12}$ for Hamiltonian $H_{C_{2}T}^{\left(1\right)}$; it approaches zero over the entire BZ. (d) The numerical result of $g_{xy}^{12}$ for Hamiltonian $H_{C_{2}T}^{\left(1\right)}$. The numerical results are obtained from full-time-dynamics simulations, with $M_z=m_z=2$ and $k_z=0$.

Figure 5

Diagrammatic sketch of a four-level atomic system for simulating the Hamiltonians $H_{CP}^{\left(1\right)}$ and $H_{C_{2}T}^{\left(1\right)}$.