Surface states and finite size effects in triple-fold semimetals

TL;DR

This work develops a analytical continuum framework to study surface states and finite-size effects in 3D triple-fold (pseudospin-1) semimetals. Using a two-node model, it uncovers two Fermi arcs per triple-fold node (one linking the nodes and one radiating outward), with the arcs’ topology tied to the Chern-number structure $\mathcal{C}(k_z)=\text{sgn}(m-k_z^2)$. Finite-thickness films reveal rich arc hybridization patterns, including petal-like loops and central loops, controlled by thickness via critical lengths $L_{cr,1}$ and $L_{cr,2}$ and an energy scale $E_{\rm sim}\approx 0.43\gamma m$. To address bulk finite-size spectra, a doubly-degenerate triple-fold model is analyzed, yielding quantized bulk levels $k_y=\pi n/L$ and two surface branches, one gapped, one gapless, with explicit density-of-states expressions. The results provide analytic predictions for ARPES and STM probes and motivate extensions beyond the low-energy description to map the full Brillouin zone of multi-fold semimetals.

Abstract

Triple-fold or pseudospin-1 semimetals belong to a class of multi-fold materials in which linearly dispersive bands and flat bands intersect at the same point, forming triple-fold crossing points. We conduct an analytical investigation of topologically protected Fermi arc surface states and finite-size effects in three-dimensional (3D) triple-fold and doubly degenerate triple-fold semimetals in continuum low-energy models. Higher topological charge of the triple-fold crossing points leads to two Fermi arcs connecting the nodes. For a single triple-fold crossing point, we found that no term in the Hamiltonian with momentum-independent elements can open a gap, prompting us to consider doubly-degenerate triple-fold fermions, where the gap can be opened by mixing the degenerate copies. Thin films of triple-fold semimetals allow for mixing between the surface and bulk states in addition to the discretization of energy levels of the latter.

Figures (5)

• Figure 1: The energy dispersion given in Eq. \ref{['fa-bulk-spectrum']} in the model of 3D pseudospin-1 semimetal with two nodes as a function of $k_x$ and $k_z$ at $k_y=0$. We set $v_F=\gamma\sqrt{m}$.
• Figure 2: Fermi arcs at the semimetal-vacuum interface for $v_F=\gamma\sqrt{m}$. Green regions represent the projections of the bulk spectrum onto the surface Brillouin zone. Blue curves correspond to the bottom surface states, while red curves correspond to the top surface. Black dots and numbers $\pm2$ mark triple-fold crossing points and the corresponding topological charges.
• Figure 3: Fermi arcs for a thin film of a triple-fold semimetal are shown as colored curves. The color indicates the spatial localization of the surface states: $\rho_t$ and $\rho_b$ are probability densities at the top and bottom surfaces, respectively. Points, whose value of $\ln\left(\rho_t/\rho_b\right)$ is above $4.5$ (below $-4.5$), are still marked as red (blue). Black dots and numbers $\pm2$ mark triple-fold crossing points and the corresponding topological charges. The projections of bulk states of semimetal at $L\to \infty$ are shown by green regions. In all panels, we fix $v_F=\gamma \sqrt{m}$.
• Figure 4: Energy spectrum ($E\geq 0$) of a slab of doubly-degenerate triple-fold semimetal with finite thickness $L$ at $\mu \to \infty$ defined by Eqs. \ref{['reducedsixfsol-1']} and \ref{['reducedsixfsol-2']}. The states above the orange dotted line correspond to bulk states (blue lines), while below that line, only surface states are possible, which are plotted in magenta. The flat band is plotted in red. The green dotted line represents the surface Dirac cone, which appears in the $L\rightarrow\infty$ limit.
• Figure 5: Dispersion relation of surface states given in Eq. \ref{['Surfsemiinfinite']} for finite values of gap parameter $\mu$ (pink) and at $\mu\rightarrow\infty$ (green). We use the dimensionless variables, see the text after Eq. \ref{['S-def']} for the definition.