Emergent topology in thin films of nodal line semimetals

TL;DR

This work addresses how finite-size geometry modifies topology in three-dimensional nodal line semimetals by examining both drumhead surface-state hybridization and bulk-state confinement. Using a minimal two-band lattice model, it shows that surface-state decay can be oscillatory or monotonic, leading to a quasi-two-dimensional nodal-loop semimetal or a fully gapped trivial phase, respectively, and derives explicit conditions for the gap behavior. When confinement is imposed along in-plane directions, bulk nodal states hybridize to produce either two-dimensional Weyl cones with a 1D winding invariant or a fully gapped quasi-one-dimensional insulator with a thickness-dependent winding number; the invariants track the nodal-loop area and film geometry. The findings establish tunable finite-size topological phases in nodal-line materials, with potential experimental realization in thin films of PbTaSe2 and the ZrSiS family, and point to bulk-boundary signatures such as edge states and Fermi arcs as diagnostic tools.

Abstract

We investigate finite-size topological phases in thin films of nodal line semimetals (co-dimension 2) in three dimensions. By analyzing the hybridization of drumhead surface states, we demonstrate that such systems can transition into either a lower-dimensional nodal line state (co-dimension 1) or a fully gapped trivial phase. Additionally, we explore the hybridization of bulk states along the nodal loop when the system is finite in directions parallel to the loop's plane. This generally results in a topologically nontrivial gap. In films finite along a single in-plane direction, a partial gap opens, giving rise to two-dimensional Weyl cones characterized by a one-dimensional $\mathbb{Z}$ invariant. When the system is finite along both in-plane directions, a fully gapped phase appears, distinguished by a $\mathbb{Z}$ invariant whose value increases with film thickness. We further discuss the bulk-boundary correspondence associated with these emergent topological phases.

Figures (5)

• Figure 1: Drumhead surface states and finite-size gap in a slab geometry finite along the z-direction. The top and bottom rows correspond to $v_z \geq v$ and $v_z < v$, respectively. (a) Low-energy spectrum for $L_z = 50$, showing drumhead surface states. (b) Surface state wavefunctions localized at $z = 0$ and $z =L_z=50$ exhibit monotonic decay into the bulk. (c) Hybridization of these surface states leads to a finite gap $\Delta$. The resulting insulating state is topologically trivial. (d) The gap decreases exponentially with increasing system size $L_z$. (f) For $v_z < v$, surface state wavefunction exhibits oscillatory decay. (g) Hybridization in this regime results in a partial gap, with residual nodal loops appearing in the quasi-two-dimensional system. (h) Heatmap of the hybridization gap across momentum space, indicating that the spectrum remains gapless along two closed loops (highlighted in red), consistent with the analytical prediction (see Eq. \ref{['Eq:SNLAna']})
• Figure 2: Drumhead surface states exist inside the projection of the bulk nodal loop on the $k_x$-$k_y$ surface zone (inside the blue loop in the figure). For $v_z/v=0.8<1$, only some of the surface modes (shaded region) decays in an oscillatory fashion. In a finite size slab $L_z=10$, drumhead states which are non oscillatory get fully gapped out. When the drumhead surface states are oscillatory, some of them (on red loops) does not hybridize and remains gapless.
• Figure 3: Hybridization energy $E(k_x, k_y)$ of drumhead surface states as a function of $k_x$ for a fixed $k_y=0$, and $L_z=6$, $k_0=1.0$ and $v_z/v=0.2$. Solid lines represent exact diagonalization result and the energy computed from Eq. \ref{['Eq:DetB']} are shown by asterisk.
• Figure 4: (a) For a system finite along the y-direction, hybridization of bulk nodal loop states leads to partial gap opening and the emergence of Weyl cones in the low-energy spectrum. Parameters are $k_0=1.0$, $L_y=10$ and $v_z/v=1.0$. (b) Momentum-space locations of the emergent Weyl nodes along with their associated winding numbers. (c) Edge states (highlighted in blue) corresponding to the Weyl nodes (marked in red), exponentially localized near the $z = 1$ and $z = L_z = 50$ boundaries ($L_z \gg L_y$). In the thermodynamic limit along the $x$ and $z$ directions, the degeneracy of these edge states (indicated just below edge states) across different $k_x$ values is consistent with the $\mathbb{Z}$ classification of the Weyl nodes.
• Figure 5: When the system is finite along both in-plane directions ($x$ and $y$), a fully gapped phase emerges. (a) Gap $\Delta$ as a function of $k_0$ for a fixed system size $L_x=L_y=L=5$. (b) Log of the energy gap as a function of system size $L$ for a fixed $k_0=0.5$. The resulting insulating phase is characterized by a winding number $W$. Every gap closing-reopening transition is accompanied by a change in the winding number, reflecting the topological nature of the quasi-one-dimensional state.