Influence of Hopping Integrals and Spin-Orbit Coupling on Quantum Oscillations in Kagome Lattices

Abstract

Motivated by recent experiments on CsTi$_3$Bi$_5$ and RbTi$_3$Bi$_5$~[Rehfuss \textit{et al.}, Phys.\ Rev.\ Mater.\ \textbf{8}, 024003 (2024)], we theoretically investigate the effects of hopping integrals and spin-orbit coupling (SOC) on quantum oscillations in kagome lattice models. Our tight-binding models successfully capture the distinct quantum oscillation features observed in experiments, when a relatively strong SOC is included. It is more important that, by discussing the effect of the next-nearest-neighbor term $t_2$, we provide a coherent explanation for their different topological responses. For the case of $t_2 = 0$, the small hybridization gap between adjacent bands with opposite Berry curvatures allows magnetic breakdown to occur under a strong magnetic field, enabling charge carriers to tunnel between the bands and thereby effectively masking the intrinsic topological character. In contrast, for $t_2 \neq 0$, the hybridization gap is significantly enlarged by $t_2$, which suppresses magnetic breakdown and confines electrons to individual orbits with opposite Berry curvatures, thereby revealing the nontrivial Berry phase ($φ_B \approx π$). Consequently, we identify the lattice-driven hopping $t_2$ as a critical control parameter that regulates the experimental observability of the topological phase in CsTi$_3$Bi$_5$ and RbTi$_3$Bi$_5$. These findings underscore the key role of the $t_2$ term and show that tuning lattice parameters can effectively control topological signatures in quantum transport.

Figures (10)

• Figure 1: (Color online) Schematic of the kagome lattice model with four sites (1, 2, 3, 4) per unit cell (pink shaded area). Nearest-neighbor vectors are $\bm{a}_1 = (1,0)a$ and $\bm{a}_2 = (1/2,\sqrt{3}/2)a$, where $a$ is the nearest-neighbor distance.
• Figure 2: (Color online) Band structures and Fermi surfaces for the two hopping mechanisms. The energy bands are indexed from bottom to top as Band 1, Band 2, Band 3, and Band 4. Panels (c) and (d) illustrate the Fermi surfaces taken at $E=0$ corresponding to the band structures in (a) and (b), respectively. (a, c) Results for the $t_2=0$ case (nearest-neighbor hopping only). The Fermi surfaces in (c) originate from the crossing of Band 2 and Band 3 at the Fermi level. (b, d) Results for the $t_2=-0.01$ case (including next-nearest-neighbor hopping). The SOC is set to $\lambda=0$ for all panels.
• Figure 3: (Color online) (a,e) DOS at $E_F=0$ versus inverse magnetic field $1/B$ for the $t_2=0$ case (top) and $t_2=-0.01$ case (bottom) without SOC. (b,f) Corresponding Fourier transform amplitudes versus frequency. (c,g) Landau fan diagrams plotting Landau level index $N$ versus $1/B$ for the $\gamma$-frequency peak. (d,h) The main DOS oscillations versus $1/B$, with filters set between 1850–1900 T, to isolate the $\gamma$frequency contribution for constructing the Landau fan diagrams of the $t_2=0$ and $t_2=-0.01$ cases, respectively.
• Figure 4: (Color online) Same as Fig. \ref{['fig:3']}, but with spin-orbit coupling strength $\lambda=0.05t$. The DOS, Fourier transform amplitudes, and Landau fan diagrams are shown for the $t_2=0$ (top) and $t_2=-0.01$ (bottom) cases, highlighting the effect of SOC on the Berry phase.
• Figure 5: (Color online) Calculated electronic structures and topological phase diagrams for $t_2 = 0$ and $t_2=-0.01$. (a, b) Calculated nanoribbon band structures for (a) $t_2=0$ and (b) $t_2=-0.01$ with fixed SOC strength $\lambda=0.05$. Both cases exhibit gapless topological edge states crossing the bulk gap, indicating an intrinsically nontrivial topological phase. (c, d) Topological phase diagrams showing the evolution of the Chern number $\mathcal{C}$ (blue circles) and the bulk energy gap $\Delta_E$ (red squares) as a function of $\lambda$ for (c) $t_2=0$ and (d) $t_2=-0.01$. The persistent non-zero Chern number ($\mathcal{C}=\pm 1$) confirms that both cases are topologically non-trivial in the bulk, despite differences in experimental quantum oscillation signatures.