Spin-polarized scanning tunneling microscopy measurement scheme for determining the quantum geometric tensor

Abstract

The quantum geometric tensor (QGT) embodies the geometry of the eigenstates of a system's Hamiltonian, and its full characterization across diverse quantum systems is essential. However, it is challenging to characterize the QGT of solid-state systems. Here we present an electric scheme to measure the complete QGT of two-dimensional solid-state systems by using spin-polarized scanning tunneling microscopy (STM), in which the spin texture is extracted from geometric amplitudes of Friedel oscillations induced by the intentionally introduced magnetic impurity, and then the QGT is derived from the momentum differential of spin texture. As a canonical spin model, the surface states of a topological insulator offer a promising way to demonstrate the scheme. In a slab of topological insulator, the gapped surface states host complete QGT, i.e., nonvanishing quantum metric and Berry curvature as its symmetric real part and the antisymmetric imaginary part. Thus, a detailed derivation guides the use of the developed scheme to measure the QGT of gapped surface states, even with an external magnetic field. This study opens a new avenue to directly measure the complete QGT of two-dimensional solid-state systems by using spin-polarized STM.

Figures (5)

• Figure 1: Schematic STM measurement scheme for the quantum geometric tensor. (a) In a topological insulator slab, spin-polarized Friedel oscillations are induced by the intentionally introduced magnetic impurity, which can be mapped out by the spin-polarized STM measurement. (b) Along any direction $\mathbf{R}$ in real space, the long-range Friedel oscillations are mainly contributed by the coupled backscattering states (red dots) on the constant-energy contour with group velocities (red arrows) parallel to ${\pm\mathbf{R}}$. For coupled backscattering states at $\mathbf{k}_{\pm}$, the spin vectors characterized by ($r_{\pm}, \Theta_{\pm}$) can be extracted from the geometric amplitudes of Friedel oscillations, which form the spin texture on the constant-energy contour. Based the energy-resolved STM measurement, the spin textures on different constant-energy (solid and dashed) contours should be provided. (c) If the STM measurement has sufficiently high momentum resolution for spin texture, i.e., sufficiently small $\delta k$ in (b) depending on the energy resolution of STM measurement, the quantum geometric tensor can be given through the momentum differential of spin texture, i.e., Eq. (\ref{['NQGT']}).
• Figure 2: Spin-polarized Friedel oscillations induced by a single magnetic impurity for isotropic gapped TISS. (a) $\delta\rho_{\alpha\beta}$ calculated by the $T$-matrix approach. (b) Comparison of $\delta\rho_{\alpha\beta}$ from the $T$-matrix approach (dotted lines) and the Born approximation (solid lines). (c) For the TISS with isotropic gapped Dirac cone (inset), the constant-energy contour is a circle. The amplitudes of real-space Friedel oscillations along $\mathbf{R}$ in (b) determine the momentum-space spin vectors ($r_{\pm}, \Theta_{\pm}$) of coupled backscattering states (red dots) with group velocities (red arrows) parallel to ${\pm\mathbf{R}}$. Here, $V_{0}=0.1$, $\varepsilon=0.2$, and $\delta=0.05$.
• Figure 3: Spin vector and quantum geometric tensor for gapped TISS. (a) The spin vectors characterized by ($r_{\pm}, \Theta_{\pm}$) for the coupled backscattering states. To consider $V_{0}=0.1$, the results are from the analytical scheme (solid lines) and the numerical scheme (circles). (b) The quantum geometric tensor including four independent components such as $\Omega_{z}$, $g_{xx}$, $g_{yy}$, $g_{xy}$ from the numerical scheme (light grey and black circles) and the analytical scheme (red lines), and $V_{0}=0.1$ is for red and light grey lines while $V_{0}=10^{-3}$ for black ones. Here, $\varepsilon=0.2$, $\delta=0.05$, and $\delta k=10^{-5}$.
• Figure 4: Spin-polarized Friedel oscillations induced by a single magnetic impurity for tilted gapped TISS. (a) $\delta\rho_{\alpha\beta}$ calculated by the $T$-matrix approach. (b) Comparison of $\delta\rho_{\alpha\beta}$ from the $T$-matrix approach (dotted lines) and the Born approximation (solid lines). (c) For the TISS with tilted Dirac cone (inset), the constant-energy contour is an ellipse. The amplitudes of real-space Friedel oscillations along $\mathbf{R}$ in (b) determine the momentum-space spin vectors ($r_{\pm}, \Theta_{\pm}$) of the coupled backscattering states (red dots) with group velocities (red arrows) parallel to ${\pm\mathbf{R}}$. Here, $\mathbf{t}=(0.3,0.3)$, $V_{0}=0.1$, $\varepsilon=0.2$, and $\delta=0.05$.
• Figure 5: Spin vector (a) and quantum geometric tensor (b) from the analytical scheme (solid lines) and exact expressions (dashed lines) for tilted gapped TISS. Here, $\mathbf{t}=(0.3,0.3)$, $V_{0}=0.1$, $\varepsilon=0.2$, $\delta=0.05$, and $\delta k=10^{-5}$.