Anderson localisation in spatially structured random graphs

This paper investigates Anderson localisation on high-dimensional graphs with spatially structured, distance-dependent hopping, revealing that increasing the hopping range shifts the localisation transition to stronger disorder and can ultimately eliminate the localised phase entirely, while confirming a direct transition between delocalised and localised states without an intervening multifractal phase.

The Gist

Imagine a giant, chaotic party where everyone is trying to find their way around, but the room is filled with obstacles. In the world of quantum physics, this "party" is a system of particles (like electrons), the "room" is a graph (a network of connected points), and the "obstacles" are random impurities or disorder.

The big question physicists ask is: Will the particles get stuck in one corner of the room (localization), or will they roam freely everywhere (delocalization)?

This paper, titled "Anderson localisation in spatially structured random graphs," by Bibek Saha and Sthitadhi Roy, explores a new, more realistic version of this party. Here is the story in simple terms:

1. The Setup: A New Kind of Room

Traditionally, physicists studied two extreme types of rooms:

• The "Hallway" Room: You can only talk to the people standing right next to you. If you want to go further, you have to walk step-by-step. This is the standard "short-range" model.
• The "Telepathy" Room: Everyone can talk to everyone else instantly, with equal strength. This is the "fully connected" model.

The authors asked: What if the room is somewhere in between? What if you can talk to people far away, but it gets harder the further they are?

They created a new model called ExpRRG. Imagine a giant tree-like structure (a Random Regular Graph). Usually, you can only whisper to your immediate neighbors. In this new model, you can shout to people far away, but your voice gets quieter the further they are. The "loudness" of your shout drops off exponentially with distance.

2. The Competition: Noise vs. Connection

There are two forces fighting in this room:

• The Noise (Disorder): Imagine random walls or static that confuse the particles, trying to trap them in one spot.
• The Connection (Hopping): The ability of particles to jump from one spot to another.

In the old models, if the noise was strong enough, the particles would always get stuck. But in this new "long-range" model, the authors discovered a surprising twist.

3. The Big Discovery: The "Super-Connection" Effect

The authors found that if you make the "shout" (the connection) reach far enough (increasing the range), something magical happens.

Analogy: Imagine you are trying to hide in a forest (the localised phase).

• Short-range: If you can only walk to the next tree, a strong wind (disorder) can easily trap you in a small clearing.
• Long-range: But if you can suddenly jump to any tree in the forest, even the ones miles away, the wind can't trap you anymore. No matter how strong the wind gets, your ability to jump far away keeps you moving.

The Result: There is a "critical range." If the particles can jump far enough, they can never be trapped, even if the disorder is incredibly strong. The "localised" phase simply ceases to exist. The system becomes "super-connected" and stays fluid.

4. The Mystery of the "Ghost Phase"

In many complex systems, when particles switch from being stuck to moving freely, they pass through a weird "middle ground" called a multifractal phase.

• Metaphor: Think of it like a foggy twilight. The particles aren't fully stuck, but they aren't fully free either. They are "ghostly," existing in a strange, fractal pattern that is hard to pin down.

Many previous theories suggested this "foggy twilight" phase should exist in these high-dimensional graphs. However, the authors' computer simulations and math showed no foggy twilight.

Instead, the transition is a sharp switch.

• State A: Stuck tight (Localised).
• State B: Running wild (Delocalised).
• The Switch: It happens instantly. There is no middle ground.

This is a big deal because it suggests that in these specific types of networks, the "ghostly" phase doesn't survive. The system is either fully trapped or fully free.

5. Why Does This Matter?

This isn't just about abstract math. This research helps us understand Many-Body Localization (MBL).

• The Real World Connection: MBL is a state where a complex quantum system (like a bunch of interacting atoms) refuses to reach thermal equilibrium (it doesn't heat up or cool down evenly). This is crucial for building quantum computers, which need to keep information stable without it getting scrambled by heat.
• The Insight: The Fock space (the mathematical space where these quantum states live) looks like a high-dimensional graph. By understanding how particles move on these graphs with long-range connections, we get better clues about how to protect quantum information from chaos.

Summary

The authors built a bridge between two extreme models of quantum movement. They found that:

1. Distance matters: If particles can reach far enough, they become impossible to trap, no matter how messy the environment is.
2. No middle ground: The transition from "stuck" to "free" is direct and sharp, skipping the weird "fractal" middle phase that many expected.
3. New Physics: This helps us understand the limits of quantum chaos and how to keep quantum systems stable.

In short, they showed that in a sufficiently connected world, you can't hide from the chaos, but you also can't get trapped by it. You just keep moving.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the physics of Anderson localization in high-dimensional disordered systems, specifically on Random Regular Graphs (RRGs). While the standard Anderson model on RRGs (short-range hopping) and fully connected models with uniform hopping (e.g., Rosenzweig-Porter model) are well-studied, there is a gap in understanding systems with spatially structured, long-range hopping.

In finite-dimensional systems, power-law decaying hopping amplitudes induce rich phenomena like robust multifractality and cooperative shielding. The authors aim to generalize this to infinite-dimensional graphs (RRGs) by introducing a model where hopping amplitudes decay exponentially with the graph distance. This creates a competition between:

1. The exponential growth of the number of neighbors at a given distance (a geometric property of RRGs).
2. The exponential decay of the hopping amplitude.

The central questions are:

• How does the hopping range ($\xi$) affect the localization phase diagram?
• Does a multifractal phase (an intermediate non-ergodic delocalized phase) exist between the localized and delocalized phases, as seen in some other high-dimensional models?
• What is the nature of the transition (critical behavior)?

2. Model Definitions

The authors introduce two specific models, collectively termed ExpRRG, defined on an RRG backbone with connectivity $K$. The distance $r_{xy}$ between sites $x$ and $y$ is defined as the shortest path length on the RRG skeleton.

• ExpRRG (Deterministic Hopping):
  The hopping amplitude $t_{xy}$ is deterministic and decays exponentially with distance:
  $$t_{xy} = t e^{-(r_{xy}-1)/\xi}$$
  Here, $\xi$ is the characteristic length scale of the hopping range.

• ExpRRG-RH (Random Hopping):
  The hopping amplitudes are random variables drawn from a Gaussian distribution with zero mean and a standard deviation that decays exponentially:
  $$t_{xy} \sim \mathcal{N}(0, \theta_{r_{xy}}^2), \quad \text{where } \theta_r = t e^{-(r-1)/\xi}$$
  This model introduces the possibility of "weak links" (vanishingly small hoppings), which in other contexts (like randomly weighted Erdős-Rényi graphs) are known to stabilize multifractal phases.

The Hamiltonian includes an onsite disorder potential $\epsilon_x \sim \mathcal{N}(0, W)$, where $W$ is the disorder strength.

3. Methodology

The study employs a combination of numerical exact diagonalization (ED) and analytical renormalized perturbation theory.

• Numerical Diagnostics:

  • Level Statistics: Mean level-spacing ratio $\langle r \rangle$ to distinguish between Wigner-Dyson (delocalized/ergodic) and Poisson (localized) statistics.
  • Inverse Participation Ratios (IPRs): Generalized IPRs $I_q = \sum |\psi_x|^{2q}$ to analyze multifractality. The scaling exponent $\tau_q$ (where $I_q \sim N^{-\tau_q}$) determines the nature of the state.
  • Correlation Functions: Both average ($C_{avg}$) and typical ($C_{typ}$) spatial correlations of eigenstate amplitudes to probe the heterogeneity of wavefunctions.
  • Finite-Size Scaling: Analysis of how these quantities drift with system size $N$ to extrapolate to the thermodynamic limit.

• Analytical Approach:

  • Renormalized Perturbation Series (RPS): Based on the locator expansion, the authors compute the local self-energy $\Sigma_x(\omega)$.
  • Self-Consistent Mean-Field Theory: They solve for the typical value of the imaginary part of the self-energy ($\Delta_{typ}$), which acts as an order parameter for localization.
  • Upper Limit Approximation (ULA): Used to estimate corrections to the mean-field critical disorder.

4. Key Results

A. The Localization Phase Diagram

The phase diagram is plotted in the $(\xi, W)$ plane (hopping range vs. disorder strength).

• Effect of $\xi$: Increasing the hopping range $\xi$ shifts the localization transition to stronger disorder ($W_c$ increases).
• Critical Threshold: There exists a critical length scale $\xi_c = 2/\ln(K-1)$. Beyond this value, the kinetic energy term becomes super-extensive (scaling as $N^2$) and dominates the disorder potential (scaling as $N$). Consequently, the localized phase ceases to exist entirely, even for arbitrarily strong disorder. The system remains delocalized.
• Agreement: The critical lines derived from numerical ED and analytical self-consistent mean-field theory show excellent agreement.

B. Absence of a Robust Multifractal Phase

A major finding is the direct transition between the delocalized (ergodic) and localized phases.

• No Intermediate Phase: Unlike randomly weighted Erdős-Rényi graphs or certain power-law banded matrices, the ExpRRG and ExpRRG-RH models do not host a robust multifractal phase in the thermodynamic limit.
• Mechanism: In the ExpRRG-RH model, despite the presence of random hoppings that could create weak links, the all-to-all connectivity of the underlying RRG skeleton ensures that the number of paths between any two clusters diverges as $N \to \infty$. The probability that all links between two clusters are simultaneously weak is exponentially suppressed. Thus, even a single resonant link is sufficient to delocalize the state, destroying the conditions required for multifractality.
• Critical Multifractality: Multifractality is only observed at the critical point (where the multifractal exponent $q^* \to 1/2$), but not as an extended phase.

C. Nature of the Transition (Kosterlitz-Thouless-like)

The scaling analysis of IPRs reveals a Kosterlitz-Thouless (KT)-like transition, characterized by two-parameter scaling:

• Delocalized Side: Exhibits volumic scaling (scaling with system volume $N$) with a diverging volume scale $\Lambda \sim \exp(b|\delta|^{-\nu})$. This suggests the delocalized phase is a patchwork of non-ergodic volumes.
• Localized Side: Exhibits linear scaling (scaling with $\ln N$) with a diverging length scale $\xi_q \sim |\delta|^{-\nu}$.
• Critical Exponents: The transition is governed by a correlation length that diverges as a power law on the localized side and an essential singularity (exponential divergence) on the delocalized side, consistent with KT physics.
• Distinct Scaling for $q$:
  • For $q > q^*$ (probing rare, high-amplitude peaks), the behavior is dominated by the average correlation function.
  • For $q < q^*$ (probing typical, low-amplitude structure), the behavior is dominated by the typical correlation function.
  • The parameter $q^*$ approaches $1/2$ at the critical point from the localized side and jumps to $1$ in the delocalized phase.

D. Average vs. Typical Correlations

The authors highlight a crucial distinction between average ($C_{avg}$) and typical ($C_{typ}$) correlation lengths ($\zeta_{avg}$ and $\zeta_{typ}$).

• In the localized phase, wavefunctions are supported on rare branches of the graph. $C_{avg}$ is dominated by these rare, high-amplitude branches, while $C_{typ}$ reflects the vanishingly small amplitudes on typical branches.
• At the critical point, $\zeta_{avg}$ approaches a finite value $\zeta_c = 1/\ln(K-1)$, while $\zeta_{typ}$ also approaches a finite critical value. The divergence of the scaling lengths controls the approach of these correlation lengths to their critical values.

5. Significance and Implications

• Generalization of Anderson Models: The work successfully generalizes power-law banded random matrix models to infinite-dimensional graphs, showing that exponential decay in hopping on an RRG is the infinite-dimensional analog of power-law decay in finite dimensions.
• Resolution of the Multifractality Debate: The results clarify that the existence of a multifractal phase is not guaranteed in high-dimensional disordered systems; it depends critically on the topology and the suppression of "weak links." The fixed RRG topology suppresses the weak-link mechanism necessary for robust multifractality.
• Connection to Many-Body Localization (MBL): Since MBL on the Fock-space graph can be mapped to an unconventional Anderson problem on a high-dimensional graph, these findings suggest that the specific nature of the hopping (spatial structure vs. random weights) dictates whether an intermediate multifractal phase exists. The absence of a multifractal phase in these specific models implies that the MBL transition might be a direct transition in certain limits, or that more complex constructions (e.g., Levy-distributed matrix elements) are needed to capture the multifractal features observed in MBL.
• Universality Class: The identification of a KT-like transition with two-parameter scaling reinforces the universality of Anderson transitions on high-dimensional graphs, linking single-particle localization to the MBL problem.

In summary, the paper establishes that introducing spatially structured long-range hopping on random regular graphs leads to a rich phase diagram where the hopping range controls the stability of the localized phase, but the underlying graph topology suppresses the formation of a robust multifractal phase, resulting in a direct, Kosterlitz-Thouless-like Anderson transition.