Laser-induced topological phases in monolayer amorphous carbon

This paper proposes that circularly polarized laser light can drive monolayer amorphous carbon into topological phases characterized by regular and anomalous edge modes, demonstrating that local atomic coordination and the spectral localizer marker are key to engineering topological states in disordered, non-crystalline materials.

The Gist

Imagine you have a messy pile of LEGO bricks. Unlike a perfect LEGO castle where every piece fits in a strict, repeating grid (like a crystal), this pile is random, jumbled, and chaotic. In the world of physics, this is called an amorphous material. For a long time, scientists believed that to create "magic" properties—specifically topological phases, which are special states of matter that conduct electricity perfectly along their edges without losing energy—you needed a perfect, orderly crystal.

This paper is like a magic trick that proves you don't need order to create magic. The researchers show that if you shine a specific kind of laser light on a messy sheet of carbon atoms (amorphous carbon), you can turn that chaos into a highly organized, "topological" highway for electrons.

Here is the breakdown of their discovery using simple analogies:

1. The Setup: The Messy Carbon Sheet

Think of graphene (the material in pencil lead) as a perfect honeycomb made of carbon atoms. It's like a pristine, tiled floor.
Now, imagine monolayer amorphous carbon. This is also made of carbon atoms, but instead of a perfect honeycomb, the atoms are arranged in random polygons (triangles, pentagons, heptagons). It's like a floor where the tiles are all different shapes and sizes, jumbled together. Usually, this messiness destroys the special "topological" properties that physicists love.

2. The Magic Wand: The Laser

The researchers didn't try to fix the mess. Instead, they used a circularly polarized laser as a "magic wand."

• The Analogy: Imagine the laser is a giant, spinning fan blowing on a crowded dance floor. The dancers (electrons) are moving randomly. When the fan spins in a circle, it forces the dancers to start moving in a coordinated, swirling pattern, even though the floor itself is still messy.
• The Result: This laser light shakes the atoms in a rhythmic, repeating way (called Floquet engineering). This rhythmic shaking creates a "force field" that organizes the electrons, opening up a "gap" where no electrons can exist in the middle of the material, but allowing them to flow freely along the edges.

3. The Discovery: Two Types of "Edge Highways"

When they turned on the laser, they found two types of special highways appearing on the edges of the material:

• The "Regular" Highway (0-gap): This is a standard topological path that appears at the normal energy level. It's like a dedicated bike lane that appears on the side of a chaotic road.
• The "Anomalous" Highway (π-gap): This is a stranger, more exotic path that appears at a different energy level. It's like a second, invisible bike lane that only exists because of the laser's rhythm.
• The Catch: The "Regular" highway was very stable, even in the messy carbon. The "Anomalous" highway was a bit wobbly and harder to keep open, but it was still there!

4. The Secret Ingredient: Local Coordination

The most surprising part of the paper is why this works.

• The Old Rule: Scientists thought you needed a perfect, repeating grid (long-range order) to get these highways.
• The New Rule: The researchers found that you only need local order.
• The Analogy: Imagine a neighborhood. Even if the houses are different sizes and the streets are crooked (amorphous), as long as every house has exactly three neighbors (three connections), the neighborhood functions as a topological system.
• The Test: They tried adding "defects" where a carbon atom had four neighbors instead of three. As soon as they added too many of these four-neighbor atoms, the magic highways disappeared.
• The Lesson: It's not about the big picture being perfect; it's about the immediate neighborhood of every atom being consistent. If the local rules are followed, the global magic happens.

5. Why This Matters

This is a big deal for two reasons:

1. Abundance: Crystals are hard to make perfectly. Amorphous materials (like glass or disordered carbon) are everywhere and easy to make. This means we might be able to build topological devices out of cheap, messy materials.
2. Control: We can turn these properties "on" and "off" just by switching a laser on and off. It's like having a light switch for a super-conducting highway.

Summary

The paper tells us that chaos doesn't have to mean disorder in function. By using a rhythmic laser, we can take a messy, jumbled sheet of carbon and force its electrons to behave like they are in a perfect crystal, creating special edge highways that could revolutionize future electronics. The key isn't a perfect grid; it's just making sure every atom has the right number of neighbors.

Technical Summary

Here is a detailed technical summary of the paper "Laser-induced topological phases in monolayer amorphous carbon" by Arnob Kumar Ghosh, Quentin Marsal, and Annica M. Black-Schaffer.

1. Problem Statement

Topological phases of matter have traditionally been studied in crystalline materials, where long-range translational symmetry and specific lattice symmetries (like bipartiteness in graphene) are essential for defining topological invariants (e.g., Chern numbers). While "Floquet engineering" (using time-periodic perturbations like circularly polarized lasers) has successfully induced topological phases in crystalline systems (e.g., the Haldane model in graphene), it remains unclear if these mechanisms apply to amorphous materials. Amorphous solids lack long-range order and translation symmetry, making standard momentum-space topological characterization impossible. The central problem is whether a non-topological amorphous material can be driven into a topological phase and how to characterize such a phase without translational symmetry.

2. Methodology

The authors employ a combination of theoretical modeling, numerical simulation, and advanced topological characterization tools:

• Material Model: They model monolayer amorphous carbon, an experimentally realizable material (via electron beam irradiation of graphene) that retains local $sp^2$ hybridization (threefold coordination) but lacks the hexagonal long-range order of crystalline graphene. The lattice is generated via Voronoi tessellation of random points.
• Hamiltonian: A spinless tight-binding Hamiltonian is used for the $p_z$ orbitals with nearest-neighbor hopping ($t_h$). Hopping amplitudes decay exponentially with bond length variations to account for geometric disorder.
• Floquet Engineering: The system is driven by a circularly polarized laser with vector potential $\mathbf{A}(t) = A(\cos \Omega t, \sin \Omega t)$. This breaks time-reversal symmetry. The interaction is introduced via the Peierls substitution.
• Numerical Approach:
  • The time-periodic Hamiltonian is treated using Floquet theory. The time-evolution operator $U(T,0)$ is computed via Trotterization (200 steps).
  • The effective time-independent Floquet Hamiltonian $H_F = -i \ln[U(T,0)]$ is derived to analyze the quasienergy spectrum (periodic in $\Omega$, mapped to $(-\Omega/2, \Omega/2]$).
• Topological Characterization:
  • Since momentum space is undefined, the authors utilize the Spectral Localizer method. This provides a real-space, energy-resolved topological invariant (Chern number).
  • The spectral localizer operator $L_{x,y,E}$ combines position operators ($X, Y$) and the Floquet Hamiltonian ($H_F$).
  • The local Chern number is calculated as $C_{x,y,E} = \frac{1}{2} \text{sig}[L_{x,y,E}]$, where "sig" is the matrix signature.
  • To ensure computational efficiency, they employ LDLT decomposition to avoid full diagonalization of the large spectral localizer matrix.
• Defect Analysis: The study introduces fourfold-coordinated defects (by merging neighboring sites) to test the robustness of the topology against changes in local atomic coordination.

3. Key Contributions

1. Extension of Floquet Topology to Amorphous Systems: The paper demonstrates that monolayer amorphous carbon, despite lacking long-range order and bipartite symmetry, can be driven into topological phases.
2. Real-Space Topological Invariant: It successfully applies the spectral localizer to characterize Floquet topological phases in disordered systems, overcoming the limitations of traditional $k$-space methods.
3. Role of Local Coordination: The study establishes that local atomic coordination (specifically threefold coordination) is the critical factor for topology in amorphous materials, rather than long-range crystalline order.
4. Experimental Feasibility: The authors provide realistic estimates for laser parameters (wavelength, intensity) required to observe these phases, confirming experimental viability.

4. Key Results

• Induced Edge Modes: Circularly polarized laser light induces topological edge states in monolayer amorphous carbon at two distinct quasienergies:
  • Regular modes at quasienergy 0: These are robust and clearly localized at the edges, similar to crystalline graphene.
  • Anomalous modes at quasienergy $\pm \pi$: These are generally less stable in the amorphous system due to disorder-induced bulk states closing the gap, though stable $\pi$-modes exist in specific parameter regimes (low frequency, specific amplitude).
• Phase Diagrams: The authors map the average Chern numbers ($\bar{C}_0$ and $\bar{C}_\pi$) as a function of laser amplitude ($A$) and frequency ($\Omega$).
  • The phase diagram for amorphous carbon closely resembles that of crystalline graphene, showing regions of non-trivial topology ($\bar{C} \neq 0$).
  • The $\pi$-gap phase in amorphous carbon exhibits higher variance (instability) compared to the 0-gap, attributed to a smaller spectral gap protecting the edge states.
• Impact of Defects (Local Coordination):
  • The topological phase is robust against a moderate density of fourfold-coordinated defects.
  • However, as the fraction of fourfold-coordinated sites ($n_4$) increases, the two energy bands merge (around $n_4 \approx 0.5$), the spectral gap closes, and the system becomes topologically trivial.
  • At $n_4 = 1$ (fully fourfold-coordinated), the system mimics a trivial metallic square lattice with no topological gap.
  • Conclusion: Topology in these amorphous systems relies on the existence of two bands emerging from the sublattice degree of freedom inherent to threefold coordination. Fourfold coordination destroys this sublattice structure and the resulting topology.
• Experimental Parameters: To reach the topological phase, the required photon energy is $1.35 - 27$eV (UV to IR), with laser intensities in the range of$10^{12} - 10^{16}$W/cm$^2$. The authors note that intensities near$10^{13}$W/cm$^2$ (achievable with short pulses) are likely experimentally viable without destroying the sample.

5. Significance

This work fundamentally shifts the paradigm of topological matter by proving that long-range crystalline order is not a prerequisite for topological phases. Instead, local atomic coordination is the governing principle.

• Versatility: It opens a vast new "playground" for topological physics using abundant amorphous materials (like carbon), which are easier to synthesize and more diverse than high-quality crystals.
• Control: It demonstrates that topology in disordered systems can be dynamically controlled via external fields (lasers), offering a new knob for engineering quantum states.
• Future Directions: The findings suggest that other amorphous systems with specific local coordination environments could host controllable topological states, potentially leading to new applications in robust quantum transport and optical devices that are insensitive to structural defects.