Quantum geometric bounds in spinful systems with trivial band topology

Abstract

We derive quantum geometric bounds in spinful systems with spin topology characterized by a single $\mathbb{Z}$ index protected by a spin gap. Our bounds provide geometric conditions on the spin topology, distinct from the known quantum geometric bounds associated with Wilson loops and nontrivial band topologies. As a result, we obtain broader bounds in time-reversal symmetric systems with a nontrivial $\mathbb{Z}_2$ index and also bounds in systems with a trivial $\mathbb{Z}_2$ index, where the quantum metric should be otherwise unbounded. We benchmark these findings with first-principles calculations in elemental bismuth realizing a nontrivial even spin-Chern number. Moreover, we connect these bounds to optical responses and show their robustness in the presence of disorder within a real space marker formulation, demonstrating that spin-resolved quantum geometry is observable in realistic experimental settings of impure materials. Finally, we connect spin bounds to quantum Cramér-Rao bounds that are central to quantum metrology, showing that elemental Bi and other spin-topological phases hold promises for topological free fermion quantum sensors.

Figures (4)

• Figure 1: Illustration of the main results of this work. (a) For a non-magnetic material such as ultrathin bismuth, the energy bands may not have a definite spin value, due to spin-orbit entanglement, as illustrated schematically in panel (b), but the spin structure of energetically isolated bands (green dashed box) with projector $P$ can still be defined via the projected spin operator ($PS_zP$). The electronic topology, captured by the Kane-Mele invariant $\nu_{\mathbb{Z}_2}$ is only protected for odd relative Wilson loop windings of the bands in $P$, $\mathcal{W}[P]$, while even relative Wilson loop windings of the bands can gap out (c). As long as the energy and spin gap (defined in Sec. I of the Supplemental Material SI) stay open, however, the spin topology, captured by the spin-Chern number $C_s$, is well-defined, and the spin-Wilson loop winding shown in panel (d) cannot gap. This has direct consequences for the lower bound of the integrated quantum metric, which is crucial for quantum metrology. Previously, only phases with $\nu_{\mathbb{Z}_2} = 1$ [green in panel (e)] had a known lower bound. We formulate a new bound for phases with even $C_s$ [orange in panel (e)], and extend the bound for phases with odd $C_s>1$.
• Figure 2: Quantum (spin-)geometry in ultrathin Bi. (a) Crystal structure, corresponding to the puckered $Pmn2_1$ phase UltrathinBiExperimental of ultrathin Bi. (b) Band structure. (c) Spin-$z$ band structure with manifestly large spin gap $\Delta S_P$. (d) Topological winding of projected spin operator $PS_zP$ eigenstates in a spin-Wilson loop. (e) Metrics, where we observe that the quantum metric ($g_{ij}$) pattern is supported by the spin-resolved quantum metrics $g^\uparrow_{ij}$, $g^\downarrow_{ij}$. The spin-topological geometric bound is satisfied at every point within the BZ, with spin-Berry curvatures bounding the spin-resolved quantum metric from below (f). We find as expected that $(2\pi)^{-1}\int_{\mathrm{BZ}}\Omega_{xy}^{\uparrow} = -(2\pi)^{-1}\int_{\mathrm{BZ}}\Omega_{xy}^{\downarrow} = 2$, whereas $(2\pi)^{-1}\int_{\mathrm{BZ}}\Omega_{xy}= 0.0$. By contrast, $(2\pi)^{-1}\int_{\mathrm{BZ}} \mathrm{Tr}[g^{\uparrow}]=(2\pi)^{-1}\int_{\mathrm{BZ}} \mathrm{Tr}[g^{\downarrow}] = 15.1$ whereas $(2\pi)^{-1}\int_{\mathrm{BZ}} \mathrm{Tr}[g] = 25.6$, demonstrating an enhanced quantum geometry in spin-topological Bi with $|C^\sigma_s| = 2$.
• Figure 3: Robustness of spin-topologically bounded quantum geometry in the presence of disorder. (a) Spin-resolved metric markers $\frac{1}{2\pi} \text{Tr}[g^{\sigma}({\bf r})]$ and (b) spin-Chern markers $C^\sigma({\bf r})$ in disordered ultrathin Bi flake system of $1746$ atoms, with random potential disorder of strength ${W = 1.5~\text{eV}}$. Averaging over 120 atoms in the center yields: ${\frac{1}{2\pi} \overline{\text{Tr}~g^{\uparrow}({\bf r})}=20.2}$ and ${\overline{C^{\uparrow}(\boldsymbol{r})} = 1.8}$. (c) Scaling of single-point metrics $g^{(\sigma)}_{\text{sp}} \equiv \frac{1}{2\pi} \text{Tr}[{g}^{(\sigma)}_{ij}]_\text{sp}$, (d) single-point spin-Chern numbers $C^\sigma_\text{sp}$, (e) spin gap $\Delta S_p$, and (f) energy gap $\Delta$E against disorder strength $W$ in ultrathin Bi system of 784 atoms, averaged over 20 disorder realizations.
• Figure 4: Scaling behavior of the quantum metric in ultrathin bismuth with SOC, where $\mathrm{SOC} = 1$ corresponds to the unperturbed Bi (for further information, see Sec. VI of the Supplemental Material SI). We observe that (a) the metric diverges close to the critical value SOC $=1.2$, where the spin-Chern numbers (b) become ill defined. In this transition, the spin gap $\Delta S_P$(c) remains well-defined up to the critical point, while the energy gap $\Delta E$(d) closes. The analytical argument for the metric scaling is detailed in Sec. VII of the Supplemental Material SI.