🔬 condensed matter

Classical versus quantum Anderson localization in disordered systems

This paper establishes that classical-wave localization in three-dimensional disordered systems constitutes a distinct constrained disorder class governed by the acoustic sum rule, fundamentally differing from standard electronic diagonal disorder and sharing key qualitative features with off-diagonal disorder, thereby necessitating a corrected unified framework for understanding localization in photonic and acoustic media.

Original authors: Stefano Mossa, Giancarlo Ruocco, Walter Schirmacher

Published 2026-06-29

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CC BY 4.0

Original authors: Stefano Mossa, Giancarlo Ruocco, Walter Schirmacher

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 a crowded dance floor where everyone is trying to move to the music. In a perfectly organized room, the dancers (waves) can glide smoothly across the floor. But if you start throwing random obstacles around—like chairs, tables, or people of different weights—the dancers might get stuck in one spot, unable to move freely. This "getting stuck" is called Anderson localization.

For decades, scientists have been trying to understand how this happens in two very different worlds:

The Quantum World: Electrons (tiny particles) moving through a messy material.
The Classical World: Sound waves or light waves moving through a messy material (like fog or a bumpy road).

The big question this paper answers is: Are these two worlds actually the same, or are they fundamentally different?

Here is the breakdown of what the authors found, using simple analogies.

1. The Old Map Was Wrong (The "Potential" Trap)

For a long time, scientists used a "map" to translate the physics of sound waves into the physics of electrons. They treated the messy material as if it were just a landscape with hills and valleys (a "potential").

The Analogy: Imagine you are trying to predict how a ball rolls down a hill. The old map said, "Just look at the shape of the hill."
The Problem: The authors show that for sound waves, the "hill" changes shape depending on how fast the ball is rolling. The old map was mathematically flawed because it tried to force a square peg (sound waves) into a round hole (electron hills). It was like trying to navigate a city using a map of a different city that looked similar but had different street rules.

2. The New Map: The "Modulus" Approach

The authors fixed the map. Instead of looking at hills, they looked at the stiffness and weight of the floor itself.

The Analogy: Think of a trampoline. If you put a heavy weight on one side, the whole trampoline sags differently. The new map accounts for how the entire structure reacts to the weight, not just the shape of the dip.
The Result: When they used this correct map, they found that sound waves behave very differently than the old map predicted. Specifically, they found a huge region of "stuck" (localized) sound waves at very high frequencies that the old map completely missed.

3. The "Acoustic Sum Rule": The Unbreakable Chain

This is the most important discovery. In the quantum world (electrons), you can mess with the "hills" (where the electron sits) and the "bridges" (how it jumps) completely independently. You can make a hill high without changing the bridge.

The Analogy: Imagine a chain of people holding hands. If one person pulls their hand away (disorder), the person next to them must adjust their grip to keep the chain from breaking. You can't change one link without affecting the others.
The Physics: In sound waves, the laws of physics (specifically the "acoustic sum rule") force the "hills" and "bridges" to be linked. If the mass of a particle changes, the stiffness of the connection must change to keep the system stable.
The Consequence: Because of this "unbreakable chain," sound waves cannot be treated as simple electrons. They form a unique category of disorder that doesn't exist in the standard electron models.

4. Who Gets Stuck? (The Phase Diagrams)

The authors drew "maps" showing where waves get stuck (localized) and where they keep moving (extended) based on how messy the material is.

The Old Electron Model (Diagonal Disorder): Imagine a crowd where people are randomly told to stand still. As the chaos increases, everyone eventually gets stuck, starting from the edges and moving to the center.
The New Sound Wave Model: Imagine a crowd where the floor is bumpy.
- Low Frequencies (Slow, deep sounds): These waves are like a slow, heavy march. Because of the "unbreakable chain" rule, they never get stuck. They can always find a way through, no matter how messy the room gets. They are protected.
- High Frequencies (Fast, sharp sounds): These waves are like a frantic sprint. They do get stuck, but only at the very top end of the speed range.
- The Surprise: The authors found that as the messiness gets extreme, the "stuck" zone for high-speed waves grows huge, swallowing up almost all the fast sounds, while the slow sounds remain free.

5. Why This Matters

The paper concludes that we have been looking for the wrong thing.

The Misconception: We thought sound localization was just a "weaker version" of electron localization.
The Reality: Sound localization is a different beast entirely. It is governed by a conservation law (the chain rule) that protects slow waves and traps fast waves in a way electrons don't experience.

In a nutshell:
If you want to stop sound or light from moving through a messy 3D material (like making a perfect soundproof room or trapping light), you shouldn't just look for random bumps. You need to look for materials where the stiffness and weight vary wildly together. The old way of thinking (treating it like electrons) was missing a massive zone where high-speed waves get trapped, and the new way of thinking reveals exactly where to look.

The authors also noted that when trying to measure this, counting how many people are dancing (the "participation ratio") is a more reliable way to see who is stuck than trying to guess the average mood of the crowd (the "typical density of states"), especially when the crowd is very chaotic.

Technical Summary: Classical versus Quantum Anderson Localization in Disordered Systems

Problem Statement
The paper addresses the long-standing theoretical question regarding the relationship between Anderson localization in disordered classical wave systems (acoustic and electromagnetic) and the standard electronic Anderson model. While a common mapping exists in the literature that equates classical wave equations with spatially fluctuating mass density or dielectric permittivity to the Schrödinger equation with a random potential (the "potential-type" or PT approach), the authors argue this mapping is mathematically unjustified. The PT approach introduces an effective disorder potential that depends explicitly on the eigenvalue (frequency squared), thereby altering the spectral problem. Furthermore, classical wave systems obey the acoustic sum rule—a conservation law linking diagonal and off-diagonal matrix elements due to translational invariance—which is absent in standard electronic models where diagonal and hopping disorders are independent. The central problem is to determine the correct formulation for classical waves and to establish whether their localization properties correspond to electronic diagonal disorder (DD), off-diagonal disorder (ODD), or constitute a distinct constrained disorder class.

Methodology
The authors employ a unified eigenvalue framework to compare four distinct disorder classes within a common mathematical structure ( $\sum_j H_{ij}\phi_j = \lambda \phi_i$ ):

Electronic Diagonal Disorder (DD): Standard Anderson model with random on-site energies.
Electronic Off-Diagonal Disorder (ODD): Random hopping amplitudes with uniform on-site energies.
Classical Mass Disorder (MD): Scalar waves with fluctuating mass density (using the correct modulus-type or MT formulation).
Classical Force-Constant Disorder (FCD): Scalar waves with fluctuating spring constants.

The study utilizes exact diagonalization of real-symmetric matrices on simple-cubic lattices ( $L=10$ to $26$) with extensive disorder averaging. To characterize localization, three complementary diagnostics are employed:

Typical Density of States (TDOS): Sensitive to the spatial fragmentation of eigenstates; vanishes in localized regions.
Participation Ratio (PR): Estimates the effective number of sites occupied by an eigenmode; used to identify mobility edges via finite-size scaling.
Adjacent-Gap Level Statistics: The ratio of consecutive level spacings ( $\langle r \rangle$ ) distinguishes between Wigner-Dyson statistics (extended) and Poisson statistics (localized).

Mobility edges are determined by identifying the boundaries where these diagnostics indicate a transition from extended to localized states, allowing for the construction of comprehensive localization phase diagrams in the disorder–eigenvalue plane.

Key Contributions and Results

Rejection of the Potential-Type (PT) Mapping: The authors demonstrate that the PT approach, which maps classical waves to an electronic model with diagonal disorder, is mathematically flawed because the disorder term depends on the eigenvalue. Consequently, the PT formulation defines a different eigenvalue problem than the original wave equation.
Validation of the Modulus-Type (MT) Formulation: The correct formulation for classical waves (dividing the wave equation by the local mass or permittivity) reveals that disorder enters via the inverse of the material parameter. This corresponds to a problem with off-diagonal disorder in the electronic analogy, but with a crucial constraint: the acoustic sum rule correlates diagonal and off-diagonal matrix elements.
Distinct Constrained Disorder Class: The study establishes that classical-wave systems do not map directly to either standard electronic DD or ODD models. Instead, they constitute a unique "constrained disorder class." The acoustic sum rule protects low-frequency (long-wavelength) acoustic modes from localization, a feature absent in unconstrained electronic models.
Localization Phase Diagrams:
- Electronic DD: Exhibits the conventional Anderson scenario where increasing disorder causes localized states to invade the spectrum from both band edges until complete localization occurs at the band center.
- Electronic ODD: Localized states appear only near band edges, while a robust extended region persists around the band center even at strong disorder.
- Classical FCD and MD: Both exhibit behavior qualitatively similar to ODD regarding the persistence of an extended central sector. However, they display a strong acoustic asymmetry: localization develops predominantly at high frequencies, while low-frequency modes remain extended.
- Specific Finding for Mass Disorder (MD): Using the MT formulation, the authors identify a broad, high-frequency localized regime that grows as disorder approaches its physical limit ( $\Delta m \to 2$ ). This regime is entirely absent in the PT phase diagram. The MD phase diagram is qualitatively different from previous results based on the PT approach.
- Mechanical Instability (FCD): In the FCD model, strong disorder can lead to negative eigenvalues ( $\omega^2 < 0$ ), indicating mechanical instability. The study finds that unstable modes can be localized, while the central spectral region remains extended, creating a complex topology with coexisting stable localized, stable extended, and unstable localized sectors.

Significance and Claims
The paper claims to provide a unified perspective on localization in quantum and classical wave systems, correcting a fundamental misconception in the field regarding the mapping between classical waves and electronic diagonal disorder. The primary significance lies in demonstrating that:

The acoustic sum rule is a critical organizing principle that fundamentally alters the topology of localization phase diagrams, preventing a direct correspondence with standard electronic Anderson models.
The modulus-type (MT) formulation is the necessary framework for analyzing classical wave localization, as the potential-type (PT) approach yields qualitatively incorrect phase diagrams (specifically missing the high-frequency localized sector in mass-disordered systems).
Localization diagnostics must be interpreted with care; the authors find that the participation ratio, corroborated by level statistics, provides a more reliable determination of mobility edges in regimes of strong disorder and near band edges compared to the typical density of states, which can be numerically unstable due to finite-size effects and geometric averaging.

The authors conclude that Anderson localization in three-dimensional classical media should be sought in systems with strong "modulus-type" disorder (large contrasts in mass density or permittivity/permeability) rather than systems modeled as weak potential perturbations. They assert that classical waves represent a distinct class of localization problems governed by conservation laws, rather than imperfect realizations of the electronic Anderson model.