Effective delocalization in the one-dimensional Anderson model with stealthy disorder

This paper demonstrates that introducing "stealthy" disorder with a vanishing power spectrum in a continuous band of wave numbers to the one-dimensional Anderson model induces effective delocalization, where the localization length scales with an arbitrarily high inverse power of the disorder strength, allowing it to exceed system sizes at fixed disorder levels.

The Gist

Imagine you are walking down a long, narrow hallway. The floor is covered in a random pattern of bumps and potholes. In the world of physics, this is like an electron trying to move through a "disordered" material. Usually, if the floor is messy enough, the electron gets stuck, bouncing back and forth until it can't go anywhere. This is called Anderson Localization, and it's like the electron getting trapped in a pit of mud.

For decades, scientists thought that if you made the floor messy (disordered), the electron would always get stuck eventually, no matter how long the hallway was. The only way to stop it was to make the floor perfectly smooth (no disorder) or perfectly repeating (like a tiled floor).

But this paper discovers a "magic trick" that breaks the rules.

The researchers found a way to design a floor that looks messy but has a hidden secret: it's "stealthy."

The "Stealthy" Floor Analogy

Imagine you are trying to trip a runner by throwing random obstacles in their path.

• Normal Disorder: You throw rocks of all sizes everywhere. The runner trips immediately.
• Stealthy Disorder: You decide to never throw obstacles that are the "right size" to trip the runner.

In physics terms, the "size" of the obstacle is related to its wavelength. The electron has a specific rhythm (wavelength) as it moves.

• In a normal messy system, there are obstacles of every size, so eventually, there's one that matches the electron's rhythm perfectly, causing it to bounce back (scatter) and get stuck.
• In this new "stealthy" system, the researchers engineered the disorder so that no obstacles exist within a specific range of sizes. It's like a "gap" in the noise.

The "Ghost" Effect

Because there are no obstacles of the "right size" to trip the electron, the electron doesn't just walk a little further; it walks much, much further.

The paper shows that by tuning this "stealthiness," you can make the electron travel so far that it effectively crosses the entire length of the hallway without ever getting stuck, even though the hallway is still full of random bumps.

• The Old Rule: If you make the floor messy, the electron stops after a short distance.
• The New Rule: If you make the floor "stealthy," the electron can walk across a city-sized hallway without tripping, even if the bumps are still there.

The "Volume Knob" of Chaos

The authors introduce a dial called $\chi$ (Chi). Think of this as a "Stealthiness Knob."

• Knob at 0: The floor is totally random and messy. The electron gets stuck quickly.
• Knob at 1/2: The floor is a perfect, repeating pattern (like a tiled floor). The electron zooms through perfectly.
• Knob in the Middle: This is the sweet spot. You have a messy floor, but you've tuned it so that the "bad" sizes of obstacles are missing.

The magic is that as you turn this knob, the distance the electron can travel doesn't just grow a little bit; it explodes. The paper proves that for certain settings, the electron can travel a distance that is mathematically "infinite" compared to the size of the system. It's as if the electron has found a secret tunnel that bypasses the mud entirely.

Why Does This Matter?

This isn't just about electrons in a lab. The math works the same for:

1. Light: Imagine a laser beam trying to pass through a foggy window. If the fog is "stealthy," the light might pass right through without scattering, making the window look clear even if it's full of particles.
2. Sound: You could design a wall that blocks sound in a normal way, but if you make it "stealthy," it might let specific sounds pass through while blocking others, or let sound travel much further than expected.
3. Quantum Computers: If you can stop particles from getting "stuck" in disorder, you might be able to build better, more stable quantum computers that don't lose their information as easily.

The Bottom Line

The paper reveals that disorder doesn't always mean "stop." If you arrange the disorder carefully—specifically, if you hide the "bad" sizes of the mess—you can create a material that looks chaotic but behaves like a super-highway for waves. It's a new way to control how energy, light, and information move through the world, turning a "messy" situation into a "transparent" one.

Technical Summary

1. Problem Statement

The paper investigates the phenomenon of Anderson localization in a one-dimensional (1D) quantum system subject to stealthy disorder.

• Context: In standard 1D Anderson models with uncorrelated random potentials, any amount of disorder leads to the exponential localization of wavefunctions. The localization length $\xi$ scales as $W^{-2}$ for small disorder strength $W$.
• The Gap: While correlated disorder (e.g., speckle potentials, random dimers) can suppress localization, the specific case of stealthy hyperuniform disorder—where the power spectrum of the disorder vanishes over a continuous band of wave numbers—had not been fully analytically characterized in the context of the 1D tight-binding Anderson model.
• Objective: To determine how the "stealthiness" of the disorder (the size of the spectral gap) affects the localization length and whether it can lead to effective delocalization (where $\xi$ exceeds the system size) at fixed disorder strength.

2. Methodology

The authors employ a dual approach combining analytical perturbation theory and large-scale numerical simulations.

A. Analytical Approach: Perturbation Theory

• Model: A 1D tight-binding Hamiltonian $H = H_0 + V$, where $H_0$ is the kinetic energy and $V$ is a random on-site potential $w_j$.
• Disorder Definition: The disorder is generated via Fourier filtering to satisfy a specific power spectrum $S(q) = |w_q|^2/W^2$. For stealthy hyperuniform disorder, $S(q) = 0$ for $|q| < k_0$, where $k_0$ defines the stealthiness parameter $\chi = k_0 / (2\pi)$.
• Self-Energy Expansion: The localization length $\xi$ is related to the imaginary part of the on-shell self-energy $\Sigma(k, \epsilon_k)$ via the mean survival time $\tau$: $1/\xi \propto -\text{Im}\Sigma$.
• Diagrammatic Expansion: The authors expand the self-energy in powers of the disorder strength $W$. They analyze Feynman diagrams up to high orders to determine the leading non-vanishing term contributing to the imaginary part of the self-energy.
  • Key Insight: In 1D, back-scattering (momentum transfer $q \approx 2k$) is the primary mechanism for localization. If the power spectrum $S(q)$ vanishes at $q=2k$, the first-order term vanishes.
  • Higher Orders: If $S(q)$ also prevents scattering combinations at second order (e.g., $q_1 + q_2 = 2k$), the leading contribution shifts to higher orders ($W^4, W^6$, etc.).

B. Numerical Approach

• System Sizes: Simulations were performed on systems with sizes up to $L \approx 10^6$ (specifically $8 \times 10^5$ in the main text, $5 \times 10^6$ in supplemental).
• Metric: The localization length was extracted from the fractal dimension ($D_\alpha$) of the eigenstates. The relationship used is $D_\alpha \approx 1 + \ln(\xi)/\ln(L)$.
• Validation: The numerical scaling of $\xi$ with $W$ was compared against the perturbative predictions for various values of $\chi$ and energy $E$.

3. Key Contributions and Results

A. Effective Delocalization via Spectral Gaps

The central finding is that for a fixed energy $E$ and fixed disorder strength $W$, there exists a range of the stealthiness parameter $\chi$ such that the localization length $\xi$ exceeds the system size $L$.

• Mechanism: The vanishing of the power spectrum over a continuous band suppresses specific scattering channels. As $\chi$ increases, the number of vanishing terms in the perturbation expansion of the self-energy increases.
• Scaling Law: Unlike uncorrelated disorder where $\xi \sim W^{-2}$, stealthy disorder yields a scaling of $\xi \sim W^{-2n}$, where $n$ is an integer that increases with $\chi$.
  • If the first non-vanishing term is of order $W^{2n}$, then $\xi \propto W^{-2n}$.
  • For sufficiently large $\chi$ (but $< 0.5$), $n$ can be arbitrarily large, making $\xi$ effectively infinite for any finite system size.

B. Phase Diagram of Localization

The authors constructed a phase diagram (Fig. 3 in the paper) mapping the scaling exponent $n$ as a function of momentum $k$ (energy) and the stealthiness parameter $\chi$ (or $k_0$).

• Boundaries: The boundaries between regions of different scaling ($W^{-2}, W^{-4}, W^{-6}, \dots$) are determined by the solvability of momentum conservation equations involving the stealthy gap.
  • Example: For $2k < k_0$, the first-order term vanishes. If $k_0 > \pi - k$, the second-order term also vanishes, pushing the leading order to $W^{-6}$ or higher.
• Separatrices: The transitions between these regimes are sharp and predicted analytically by the conditions where the momentum conservation equations for back-scattering have no solution within the allowed $q$-space (outside the gap).

C. Numerical Confirmation

• The numerical data confirms the perturbative predictions.
• Scaling Verification: For $\chi = 0.1$, $\xi \sim W^{-2}$. For $\chi = 0.2, 0.25, 0.3$, the data shows a clear transition to $\xi \sim W^{-4}$.
• Disorder Strength Dependence: As $W$ decreases, the numerical transitions in $\xi$ vs. $\chi$ become sharper and align perfectly with the theoretical vertical lines predicted by the perturbation theory.

D. Connection to Level Splitting

The paper also analyzes the splitting of degenerate energy levels in the presence of stealthy disorder.

• In uncorrelated disorder, level splitting $\Delta E \propto W$.
• In stealthy disorder, if the matrix element $\langle \psi_1 | V | \psi_2 \rangle$ vanishes due to the spectral gap, the splitting scales as $\Delta E \propto W^n$.
• This confirms that the suppression of localization is intrinsically linked to the suppression of scattering-induced level mixing.

4. Significance and Implications

• Fundamental Physics: The work challenges the intuition that 1D systems are always localized for any disorder. It demonstrates that correlated disorder with specific spectral properties can induce "effective delocalization" without requiring the system to be periodic or quasi-periodic.
• Universality: The mechanism relies solely on the spectral properties of the disorder (the vanishing power spectrum). Therefore, the results apply not only to electronic tight-binding models but also to photonic and phononic wave systems.
• Experimental Relevance:
  • Optical Transparency: The findings provide a theoretical basis for recent observations of high transparency in stealthy hyperuniform layered media, explaining why Anderson localization is suppressed.
  • Realization: The model can be realized in current experimental platforms, including ultracold atoms in optical lattices, waveguide arrays, and photonic crystals, where the disorder potential can be programmatically engineered to have specific spectral gaps.
• Future Directions: The paper suggests that this mechanism could be extended to interacting many-body systems (many-body localization), though this remains an open question.

Conclusion

Vanoni et al. demonstrate that stealthy disorder acts as a control knob to tune the localization length in 1D systems. By engineering a continuous spectral gap in the disorder potential, one can suppress the leading-order scattering terms, causing the localization length to scale as $W^{-2n}$ with arbitrarily large $n$. This results in a regime of effective delocalization where wavefunctions remain extended across macroscopic system sizes, offering a new pathway to control wave transport in disordered media.