Emergent topology in thin films of nodal line semimetals

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This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a piece of material that isn't quite a solid metal and isn't quite an insulator (a material that blocks electricity). Instead, it's a Nodal Line Semimetal. Think of this material as a giant, 3D spiderweb of energy. Inside this web, there are specific "highways" where electrons can travel without any resistance. In a normal 3D block of this material, these highways form a complete, closed circle (a loop) floating in the middle of the energy landscape.

Now, imagine taking this giant 3D block and slicing it into a very thin sheet, like a piece of paper or a slice of bread. This is what the paper investigates: What happens to these energy highways when you squeeze the material into a thin film?

The author, Faruk Abdulla, discovers that squeezing the material doesn't just make it smaller; it fundamentally changes the rules of the road, creating new types of "traffic patterns" that didn't exist before. Here is the breakdown using simple analogies:

1. The "Drumhead" and the Two-Sided Sandwich

In a thick block of this material, the surface has special "drumhead" states. Imagine the material is a drum, and the surface is the drum skin. In a thick drum, the skin on the top and the skin on the bottom are far apart; they don't know each other exists.

But when you make the film very thin (like a sandwich with only a few layers of bread), the top skin and the bottom skin get so close that they start to "talk" to each other. This is called hybridization.

The paper finds that what happens next depends on how the "sound" (the electron wave) travels through the bread:

The Smooth Decay (The Trivial Phase): If the wave fades away smoothly and steadily as it moves from the top to the bottom, the two surfaces merge perfectly. The energy highway disappears completely, and the material becomes a boring, insulating block where electricity can't flow.
The Bumpy Decay (The New Semimetal): If the wave wiggles up and down (oscillates) as it fades, it's like a wave that hits a "dead zone" at the bottom. When the top and bottom surfaces meet, these wiggles sometimes cancel each other out perfectly at specific spots. Instead of closing the highway, this creates new, smaller loops of energy highways inside the thin film. It's like taking a giant circular racetrack and shrinking it down into a few smaller, concentric circles.

2. Squeezing from the Sides (The Wire Effect)

The paper also looks at what happens if you squeeze the material not just from top to bottom, but from the sides (making it a thin wire).

Squeezing One Side (The 2D Weyl Cones): If you squeeze the material from just one side (making it a thin slab), the big circular highway breaks apart. It shatters into isolated "islands" of energy. These islands act like 2D Weyl nodes.
- Analogy: Imagine a circular racetrack. If you pinch the track from one side, the circle breaks, and you are left with two separate, straight sections of track. These sections act like special portals (Weyl cones) that allow electrons to flow in very specific, protected ways.
Squeezing Both Sides (The Fully Gapped Wire): If you squeeze the material from both sides (making it a very thin wire), the circular highway is crushed completely. All the energy gaps close, and the material becomes a fully insulating wire.
- The Twist: However, this isn't just a boring wire. It's a topologically special wire. The number of "twists" in the energy structure (called a winding number) depends on how thick the wire is. You can actually tune the "topological charge" of the wire just by changing its thickness, like turning a dial.

3. Why Does This Matter?

Think of this like Lego building.

In the past, scientists knew how to build stable structures with big blocks (3D materials).
This paper shows that if you build with very thin, flat layers (thin films), the rules change. You can create new, stable structures (topological phases) that are impossible to build with big blocks.

The key takeaway is that geometry controls physics. By simply changing the shape and thickness of a material, you can force it to switch between being a conductor, an insulator, or a new type of exotic semimetal with its own unique "traffic laws" for electrons.

Summary in a Nutshell

The Setup: A 3D material with a circular energy highway.
The Action: Slicing it into a thin film.
The Result:
- If the film is thin and the waves wiggle, the big highway splits into smaller loops.
- If you squeeze the sides, the highway breaks into special islands (Weyl nodes).
- If you squeeze everything, the highway vanishes, but the material becomes a magic wire whose properties can be tuned by its thickness.

This research gives engineers a new "recipe book" for designing future electronics, where they can control how electricity flows just by cutting the material to the right size and shape.

1. Problem Statement

Topological insulators and semimetals are characterized by robust boundary states arising from non-trivial bulk topology. In finite-size systems, boundary states on opposite surfaces can overlap and hybridize, often opening an energy gap. While this phenomenon is well-studied in topological insulators and Weyl semimetals (where bulk nodes are isolated points), the behavior of Nodal Line Semimetals (NLSMs) under finite-size confinement remains less explored.

NLSMs possess one-dimensional closed nodal loops in momentum space and associated two-dimensional "drumhead" surface states. The central problem addressed is: How do finite-size effects (quantum confinement) alter the topological phases of 3D NLSMs when the system is thinned into films or wires? Specifically, the paper investigates whether hybridization of surface states and bulk nodal states leads to trivial gapped phases, lower-dimensional nodal states, or new topological phases, and how these depend on the confinement geometry and material parameters.

2. Methodology

The author employs a combination of analytical derivation and numerical diagonalization using a minimal two-band tight-binding model defined on a cubic lattice.

Model Hamiltonian: The study utilizes a chiral-symmetric Hamiltonian $H(\mathbf{k})$ defined by:
$H(\mathbf{k}) = \tilde{v} (M_k - \cos k_z a)\tau_x + v_z \sin k_z a \tau_z$
where $M_k = 2 + \cos k_0 a - \cos k_x a - \cos k_y a$ . This model supports a nodal loop in the $k_x$ - $k_y$ plane at $k_z=0$ .
Symmetry: The system possesses chiral symmetry ( $S H S^{-1} = -H$ ), which protects the nodal loop and allows for the definition of winding numbers.
Confinement Geometries:
1. Surface Hybridization: The system is finite along the $z$ -direction (perpendicular to the nodal loop plane) but infinite in $x$ and $y$ .
2. Bulk Hybridization (Slab): The system is finite along a single in-plane direction (e.g., $y$ ).
3. Bulk Hybridization (Wire): The system is finite along both in-plane directions ( $x$ and $y$ ).
Analytical Techniques:
- Derivation of surface state wavefunctions for semi-infinite slabs to determine decay characteristics (oscillatory vs. monotonic).
- Construction of exact wavefunctions for finite slabs satisfying open boundary conditions to calculate hybridization energies.
- Use of the "Particle in a Box" method (PiBM) to approximate bulk state hybridization in confined geometries.
Topological Invariants: The study characterizes phases using:
- $\mathbb{Z}_2$ invariant for 2D nodal loops.
- $\mathbb{Z}$ winding numbers for 2D Weyl nodes and 1D gapped phases.

3. Key Contributions and Results

A. Hybridization of Drumhead Surface States (Finite along $z$ )

The paper demonstrates that the fate of the system depends critically on the ratio of group velocities $v_z/v$ (perpendicular vs. parallel to the nodal loop).

Non-Oscillatory Decay ( $|v_z| \ge |v|$ ): Surface states decay monotonically into the bulk. When confined in a finite slab, states from opposite surfaces hybridize fully, opening a global energy gap. The system transitions into a topologically trivial insulator.
Oscillatory Decay ( $|v_z| < |v|$ ): Surface states decay with an oscillatory envelope, vanishing periodically in space.
- Partial Gap: Hybridization is suppressed at specific momenta where the wavefunction nodes align with the boundaries.
- Emergent Phase: The system transitions into a quasi-2D nodal line semimetal featuring multiple concentric nodal loops.
- Predictability: The positions of the residual gapless loops are analytically predictable based on the film thickness ( $L_z$ ) and the oscillation wavelength.
- Topology: This phase is characterized by a $\mathbb{Z}_2$ invariant, distinguishing gapped regions in momentum space.

B. Hybridization of Bulk Nodal States (Finite along in-plane directions)

When confinement is applied parallel to the nodal loop plane, the bulk nodal states themselves hybridize.

Single In-Plane Confinement (Slab Geometry, finite $y$ ):
- The continuous nodal loop breaks into discrete points in momentum space.
- Result: The system becomes a 2D Weyl semimetal with isolated Weyl nodes (conical crossings).
- Topology: These nodes carry an integer $\mathbb{Z}$ winding number. The bulk-boundary correspondence predicts the emergence of Fermi-arc edge states when the system is further confined in the $z$ -direction.
- Independence: Unlike surface hybridization, this gap-opening mechanism is independent of the $v_z/v$ ratio.
Double In-Plane Confinement (Wire Geometry, finite $x$ and $y$ ):
- All bulk nodal states hybridize, leading to a fully gapped quasi-1D phase.
- Topology: The phase is a topological insulator characterized by a 1D $\mathbb{Z}$ winding number.
- Tunability: The value of the winding number increases with the thickness of the sample ( $L_x, L_y$ ) and the size of the original nodal loop (controlled by parameter $k_0$ ). This provides a mechanism to engineer topological invariants via geometric control.

4. Significance and Implications

Richness of Finite-Size Topology (FST): The work reveals that NLSMs exhibit a much richer landscape of finite-size phases compared to Weyl semimetals. While Weyl semimetals typically transition to Chern insulators or 2D Weyl nodes, NLSMs can transition to lower-dimensional nodal loops, 2D Weyl semimetals, or 1D topological insulators depending on the confinement direction and velocity ratios.
Mechanism for Phase Engineering: The paper establishes that the topological phase of a thin film is not fixed but can be tuned by:
1. Film Thickness: Controlling the hybridization strength and the number of emergent nodal loops or winding numbers.
2. Material Parameters: Tuning the ratio of group velocities ( $v_z/v$ ) to switch between trivial and nodal phases.
3. Geometry: Choosing between slab, wire, or bulk-like confinement.
Experimental Relevance: The findings are directly applicable to experimental realizations of NLSM thin films and van der Waals heterostructures. The authors suggest materials like PbTaSe $_2$ and the ZrSiS family as promising candidates for observing these emergent phases.
Bulk-Boundary Correspondence: The study reinforces the bulk-boundary correspondence in reduced dimensions, showing how emergent topological invariants in finite systems (like the $\mathbb{Z}$ winding number in 1D wires) dictate the existence of robust edge states.

In conclusion, this paper provides a comprehensive theoretical framework for understanding how dimensionality reduction transforms the topology of nodal line semimetals, offering new pathways for designing topological devices through geometric confinement.