⚛️ quantum physics

Coexistence of Anderson Localization and Quantum Scarring in Two Dimensions

This paper demonstrates that finite two-dimensional disordered systems with periodic confinement exhibit a coexistence of low-energy Anderson localization and high-energy quantum scarring, producing distinct, observable signatures in spatial intensity patterns and spectral statistics despite the theoretical prediction of universal localization in the thermodynamic limit.

Original authors: Fartash Chalangari, Anant Vijay Varma, Joonas Keski-Rahkonen, Esa Räsänen

Published 2026-04-08

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CC BY 4.0

Original authors: Fartash Chalangari, Anant Vijay Varma, Joonas Keski-Rahkonen, Esa Räsänen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 you are dropping a handful of marbles into a large, square room. The floor of this room isn't flat; it's covered in a grid of shallow, circular bowls (like a waffle iron). Scattered randomly across this floor are also some sticky, bumpy patches of glue (the "disorder").

Now, imagine these marbles are actually quantum waves (like ripples in a pond) rather than solid balls. The paper you shared investigates what happens to these ripples as they move around this tricky floor.

Here is the story of what they found, explained simply:

1. The Two Opposite Worlds

Usually, physicists expect quantum waves to behave in one of two ways:

The "Stuck" Mode (Anderson Localization): If the floor is very bumpy and the wave doesn't have much energy, it gets trapped. It's like a marble rolling into a deep bowl and getting stuck there. It can't move anywhere else. This is called Anderson Localization.
The "Free" Mode (Ergodicity): If the wave has a lot of energy and the floor is smooth, it spreads out evenly across the whole room, bouncing off walls and filling every corner. It forgets where it started. This is called being Ergodic.

2. The Surprise: A Third Option

The researchers discovered something strange happening in the middle. They found a "Goldilocks" zone where both of these behaviors happen at the same time, but in different parts of the room.

Even more surprisingly, they found a third type of behavior that looks like a ghostly highway.

The "Scar" Effect: Imagine that even though the floor is bumpy, the waves somehow decide to travel only in straight lines, like cars on a highway, ignoring the bumps on the sides. These lines are called "scars."
Why "Scars"? In classical physics, if you throw a ball in a chaotic room, it bounces everywhere randomly. But sometimes, the ball finds a specific path it keeps repeating. In quantum mechanics, the wave "remembers" this path and concentrates its energy along it, leaving a "scar" on the map of the room.

3. The Magic Ingredient: Energy and Size

Why do these three things happen together? It comes down to Energy and Size.

Low Energy (The Stuck Marbles): When the waves are low-energy, they are weak. The random bumps on the floor trap them immediately. They are stuck in small pockets.
Medium Energy (The Free Runners): As the waves get stronger (higher energy), they can jump over the bumps. They start to spread out and fill the room like a gas.
High Energy (The Highway Builders): This is the magic part. When the waves get very energetic, they become so fast and sharp that they start to "see" the grid of bowls on the floor again. Instead of just bouncing randomly, they lock onto specific straight paths (the "highways") created by the geometry of the bowls.

The Twist: The paper shows that in a room of a specific size, you can have:

Low-energy waves stuck in the corners.
Medium-energy waves spreading everywhere.
High-energy waves zooming in straight lines (scars).

All three exist at the same time in the same room!

4. The "Variational" Trick

You might ask, "But the floor is bumpy! How can the waves stay in a straight line?"

Usually, bumps destroy straight lines. But the researchers found that the waves are smart. They use a "variational" trick. Think of it like a hiker trying to cross a field of mud.

If the mud is random, the hiker usually gets stuck.
But if the hiker looks at the whole field, they might find a specific path where the mud is slightly less sticky, or where the bumps cancel each other out.
The quantum waves do this automatically. They "choose" the specific straight path that fits best with the random bumps, creating a stable "scar" that survives the disorder.

5. Why Does This Matter?

This isn't just a math game. It changes how we understand the future of technology.

Electronics: If we can control these "highway" scars, we might build computer chips that guide electricity in very specific directions without it getting lost or scattered.
Lasers and Light: We could design lasers that keep their beam tight and focused even if the material inside is a bit messy.
Cold Atoms: Scientists can use this to trap atoms in specific patterns to build new types of sensors.

The Big Picture

For a long time, scientists thought that if you added enough disorder (bumps) to a system, everything would eventually get stuck (localized). This paper says: "Not so fast!"

If the system is the right size and the energy is just right, the waves can find a way to organize themselves into beautiful, straight-line patterns (scars) even in a messy environment. It's like finding a perfectly straight line drawn in a pile of sand, just because the wind blew in a very specific way.

In short: Nature is messy, but sometimes, that messiness creates hidden highways that waves can travel on, defying the expectation that they should just get lost.

1. Problem Statement

The paper addresses a fundamental tension in mesoscopic physics regarding the behavior of quantum eigenstates in finite, two-dimensional (2D) disordered systems.

The Conflict: Standard scaling theory (Abrahams et al.) predicts that in the thermodynamic limit ( $L \to \infty$ ), all eigenstates in 2D non-interacting systems with time-reversal symmetry and uncorrelated disorder are Anderson localized (AL). Conversely, "Quantum Scarring" describes non-ergodic states where probability density concentrates along classical periodic orbits (POs), typically observed in clean chaotic systems or specific structured potentials.
The Gap: While AL dominates at low energies and scarring is known in clean systems, the mechanism by which these two distinct non-ergodic regimes (conventional localization and variational scarring) coexist within the same finite system across the energy spectrum remains elusive. Specifically, how finite-size effects and energy-dependent localization lengths allow for the simultaneous presence of localized, ergodic, and scarred states in realistic mesoscopic systems is not fully understood.

2. Methodology

The authors investigate a finite 2D continuum Hamiltonian using numerical simulations and scaling analysis.

Model System:
- Geometry: A square domain ( $L \times L$ ) with Dirichlet hard-wall boundaries.
- Potential: A periodic array of circular "Fermi wells" ( $V_{ext}$ ) acting as a confining lattice, combined with randomly distributed Gaussian impurities ( $V_{imp}$ ) representing disorder.
- Hamiltonian: $H = -\frac{1}{2}\nabla^2 + V_{ext}(\mathbf{r}) + V_{imp}(\mathbf{r})$ (in atomic units).
- Parameters: The system is tuned to be non-integrable and classically chaotic. Disorder strength is controlled by the ratio of the average impurity amplitude to the well depth ( $\langle A \rangle / V_0$ ).
Numerical Techniques:
- Eigenstate Calculation: The lowest $\sim 10^3 - 10^4$ eigenstates are computed using the imaginary-time propagation method.
- Classification Metrics:
  - Inverse Participation Ratio (IPR, $P_2$ ): To distinguish between localized ( $P_2 \sim L^0$ ), extended ( $P_2 \sim L^{-2}$ ), and multifractal states.
  - Scar Metric ( $S_n$ ): A novel metric quantifying the anisotropy of probability density along lattice rows/columns (POs). It combines probability weight contrast with spatial elongation factors.
  - Level Spacing Statistics: The ratio of consecutive level spacings ( $\tilde{\eta}$ ) is used to distinguish between Poisson statistics (localized) and Wigner-Dyson/GOE statistics (chaotic/ergodic).
- Scaling Analysis: Simulations are performed across varying system sizes ( $L$ ) to observe finite-size crossover regimes.

3. Key Contributions

Demonstration of Coexistence: The paper provides rigorous evidence that Anderson localization, fully ergodic delocalization, and variational quantum scarring can coexist within the same finite 2D disordered system, separated primarily by energy.
Mechanism of Variational Scarring in Disorder: The authors clarify that while disorder typically destroys conventional scars, it is the central ingredient for variational scarring. In a periodic potential, local disorder lifts the degeneracy of states associated with families of classical POs. Degenerate perturbation theory selects specific superpositions that extremize the overlap with the impurity landscape, resulting in anisotropic, scarred states.
Finite-Size Crossover Theory: The study elucidates how the interplay between the energy-dependent localization length $\xi(E)$ $ξ (E)$ and the system size $L$ $L$ creates distinct regimes:
- Low Energy ( $L \gg \xi(E)$ ): Dominated by Anderson localization.
- Intermediate Energy: Extended, ergodic states where the de Broglie wavelength is too large to resolve the lattice structure.
- High Energy ( $L \lesssim \xi(E)$ and $\lambda \sim$ lattice spacing): Variational scars emerge as anisotropic outliers within the delocalized background.

4. Key Results

Spectral Regimes:
- Low Energy: Eigenstates exhibit strong Anderson localization with exponential decay tails. The IPR is high and independent of system size ( $D_2 \approx 0$ ).
- Intermediate Energy: States are fully delocalized and ergodic, following Wigner-Dyson statistics. The IPR scales as $L^{-2}$ ( $D_2 \approx 2$ ).
- High Energy: A subset of states exhibits variational scarring. These states have high IPR (indicating localization) but are distinct from AL states because they are highly anisotropic (channel-like) and follow specific classical periodic orbits. They show intermediate scaling behavior ( $0 < D_2 < 2$ ) that drifts toward ergodic scaling as $L$ increases.
Scaling Behavior:
- As system size $L$ increases, the energy window required to reach the high-energy scarred regime shifts upward (following Weyl's law).
- In the thermodynamic limit, true metallic phases are excluded, but the mesoscopic window allows for the observation of these coexisting phases.
Spectral Statistics:
- Contrary to the expectation that strong disorder drives systems toward Poisson statistics, the authors find that in the high-energy scarred regime, the level spacing statistics can shift toward Wigner-Dyson (GOE) behavior. This is because the scarred states, while non-ergodic in spatial distribution, retain spectral correlations characteristic of the underlying chaotic dynamics, distinct from the clustering of localized states.
Kinetic vs. Potential Energy:
- Scarred states are characterized by anomalously high kinetic-to-potential energy ratios ( $\langle T \rangle / \langle V \rangle$ ), indicating they are "kinetic-dominated" modes that form quasi-one-dimensional channels, a feature absent in coarse-grained tight-binding models.

5. Significance and Implications

Beyond Tight-Binding Models: The results highlight limitations in standard tight-binding Anderson models, which often fail to capture the smooth, correlated disorder and continuum interference effects necessary for variational scarring. The continuum nature of the wavefunction is crucial for the formation of these anisotropic structures.
Experimental Relevance: The predicted signatures (spatial intensity patterns, specific IPR scaling, and level statistics) are directly observable in current mesoscopic platforms, including:
- Electronic systems: Semiconductor quantum wells and graphene quantum dots.
- Photonic systems: Optical media with controlled disorder.
- Cold atoms: Trapped atomic gases in optical lattices.
Theoretical Impact: The work bridges the gap between Anderson localization and quantum chaos, demonstrating that "weak ergodicity breaking" (scarring) is not mutually exclusive with localization phenomena but can coexist as a finite-size effect driven by the specific interplay of energy, disorder, and periodic confinement.

In summary, the paper establishes that in finite 2D systems, the energy dependence of the localization length creates a "crossover window" where Anderson localization, ergodicity, and variational scarring coexist, offering a new perspective on non-ergodic behavior in disordered quantum matter.