Signatures of Nonergodicity in Sparse Random Matrices

This paper numerically and analytically investigates sparse random matrices with on-site disorder to demonstrate that the Anderson transition can be identified via ground state statistics, revealing a broad nonergodic regime within the delocalized phase characterized by distinct short- and long-range energy correlations.

The Gist

Imagine you are trying to understand how a massive, chaotic crowd of people behaves. Are they all mixing together freely, chatting with everyone, and moving as one big, fluid group? Or are they stuck in small, isolated pockets, unable to move past their immediate neighbors?

This is the core question physicists ask about quantum systems (like atoms or electrons). When things get messy and full of "impurities" (like a dirty room or a crowded street with obstacles), the system can either stay "ergodic" (everyone mixes) or become "localized" (everyone gets stuck).

This paper investigates a specific type of messy system using Sparse Random Matrices. Here is a simple breakdown of what they did and what they found, using everyday analogies.

1. The Setup: The "Sparse" Party

Imagine a giant party with $N$ guests.

• The "Dense" Party (GOE): In a normal, chaotic party, everyone talks to everyone. If you pick two people at random, there's a high chance they know each other. In physics, this is a "dense" matrix where almost every number is non-zero.
• The "Sparse" Party (sGOE): Now, imagine a party where most people don't know each other. Connections are rare. This is a "sparse" matrix. Most of the numbers in the math are zero; only a few connections exist.
• The "Disorder": To make it realistic, the authors added "on-site disorder." Think of this as giving some guests a heavy backpack or a broken leg. They can't move as freely as others.

The researchers wanted to know: At what point does this sparse, messy party stop mixing and start getting stuck?

2. The Discovery: The "Tipping Point"

They found a very specific "tipping point" (a critical threshold) determined by how sparse the connections are.

• If the party is too sparse: The guests are so isolated that they can't move. The system is Localized. It's like a frozen crowd where everyone is stuck in their own little bubble.
• If the party is just right (but not fully dense): Here is the surprise! The guests can move, but they don't mix perfectly. They wander around, but they never fully explore the whole room. This is called a Nonergodic Extended state.
• If the party is dense: Everyone mixes perfectly. This is the Ergodic state.

The paper proves that this transition happens at a very specific mathematical "tipping point" (when the number of connections drops to a critical level).

3. How They Measured It: The "Ground State" Detective Work

Instead of watching the whole party for hours, they looked at the "Ground State."

• Analogy: Imagine the "Ground State" is the single most relaxed, calmest moment of the party.
• The Test: They checked the statistical "shape" of this calm moment.
  • When the system is Localized, the data looks like a specific curve called the Gumbel distribution (think of a sharp, skewed mountain peak).
  • When the system is Fully Mixed (Ergodic), the data looks like the Tracy-Widom distribution (a very specific, smooth curve seen in chaotic systems).
  • The Result: They found that as they made the connections sparser, the data smoothly shifted from the "Chaotic" shape to the "Stuck" shape right at that critical tipping point.

4. The "Mobility Edge": The Invisible Wall

One of the coolest findings is the Mobility Edge.

• Analogy: Imagine a building with many floors.
  • On the top floors (high energy), the guests are energetic and can run to any room. They are "Extended."
  • On the bottom floors (low energy), the guests are tired and stuck in their rooms. They are "Localized."
  • The Mobility Edge is the invisible floor line where the behavior changes.
• The Finding: Even with the "backpacks" (disorder) added, this invisible line still exists. The system isn't just "all stuck" or "all free"; it has a mix of both, separated by a specific energy level.

5. The "Thouless Energy": The Time to Wander

Finally, they looked at how long it takes for a "disturbance" to spread through the system.

• Analogy: If you drop a pebble in a pond (the system), how long does it take for the ripples to reach the other side?
  • In a perfectly mixed system, the ripples spread instantly.
  • In this sparse, nonergodic system, the ripples spread, but they get "stuck" in certain areas for a long time.
• The Finding: They identified a specific energy scale (called the Thouless energy) that acts like a "speed limit" for how fast information can travel. Below this speed limit, the system behaves chaotically; above it, the system behaves like a random, disconnected mess.

Summary: Why Does This Matter?

This paper is like a map for understanding how complex systems break down.

• Real-world application: This helps us understand things like quantum computers (which need to stay mixed to work) or superconductors (where electrons need to move freely).
• The Big Takeaway: Nature has a "Goldilocks zone" for chaos. If a system is too sparse, it freezes. If it's just right, it enters a weird, "half-mixed" state where things move but don't fully forget where they started. This "half-mixed" state is a new frontier in physics that this paper helps to define.

In short: The authors built a mathematical model of a messy, sparse network, added some "glitches," and proved exactly when and how that network stops behaving like a chaotic crowd and starts behaving like a frozen, isolated group.

Technical Summary

1. Problem Statement and Motivation

The paper investigates the spectral statistics and localization properties of sparse random matrices with on-site disorder. This study is motivated by the prevalence of sparsity in interacting many-body systems (e.g., Heisenberg and Ising models on Fock space graphs) and the need to understand the transition between ergodic (thermalized) and non-ergodic (localized or multifractal) phases.

Specifically, the authors address the Anderson localization-delocalization transition in the context of the Sparse Gaussian Orthogonal Ensemble (sGOE). While previous studies focused on regular graphs (like Random Regular Graphs), this work utilizes Erdős-Rényi-Gilbert (ERG) graphs, which are non-regular and exhibit heterogeneous connectivity. The central question is whether the addition of diagonal disorder (on-site disorder) alters the critical percolation limit for the transition and how non-ergodicity manifests in the ground state and bulk spectrum.

2. Methodology

The authors employ a combination of analytical derivations and extensive numerical simulations:

• Model Definition: The system is modeled as a sparse Gaussian Orthogonal Ensemble (sGOE) defined by the Hamiltonian:
  $$H = H_{\text{Poisson}} + H_{\text{GOE}} \odot A(N, p)$$
  where $A(N, p)$ is the adjacency matrix of an ERG with $N$ nodes and edge probability $p$. $H_{\text{Poisson}}$ introduces diagonal disorder (on-site potential) drawn from a Gaussian distribution, while $H_{\text{GOE}}$ provides the off-diagonal hopping terms. The sparsity is parameterized as $p = N^{-\gamma/2}$, where $\gamma=0$ corresponds to the full GOE limit and $\gamma \to \infty$ to the Poisson limit.
• Numerical Techniques:
  • Ground State: Obtained using the wall-Chebyshev projector method to avoid full diagonalization of large matrices.
  • Density of States (DOS): Estimated using the Lanczos method and the Kernel Polynomial Method (KPM) with Jackson damping to handle Gibbs oscillations.
  • Statistical Estimators: The Girard-Hutchinson estimator is used to compute energy moments stochastically without exact diagonalization.
  • Entanglement: Bipartite von Neumann entanglement entropy is calculated via Schmidt decomposition.
• Analytical Tools: The authors derive exact expressions for the second and fourth energy moments to calculate the shifted kurtosis, which serves as an order parameter for the transition.

3. Key Contributions and Results

A. Ground State Statistics and Localization

• Distribution of Ground State Energy:
  • For $\gamma > 2$ (localized regime), the ground state energy distribution follows the Gumbel distribution (characteristic of uncorrelated Poisson-like spectra).
  • For $\gamma = 0$ (GOE limit), it follows the Tracy-Widom distribution.
  • The transition between these regimes occurs sharply at $\gamma = 2$.
• Multifractality and Entanglement:
  • Inverse Participation Ratio (IPR): For $0 < \gamma < 2$, the ground state exhibits multifractality (non-ergodic extended states), where fractal dimensions $D_q$ depend on $q$ and $0 < D_q < 1$. For $\gamma > 2$, $D_q = 0$, indicating localization.
  • Entanglement Entropy: The bipartite von Neumann entropy scales as $S_{vN} \propto \alpha \ln N$.
    • $\alpha \approx 1/2$ for $\gamma < 2$ (indicating non-ergodic extended states, distinct from the volume-law of fully ergodic states).
    • $\alpha \approx 0$ for $\gamma > 2$ (localized).
  • Conclusion: A quantum phase transition occurs at the critical sparsity $\gamma_c = 2$ (equivalent to the percolation limit $p = N^{-1}$), separating localized states from non-ergodic extended (NEE) states.

B. Density of States (DOS) and Moments

• Shifted Kurtosis: The authors analytically derived the energy moments and defined a shifted kurtosis $K$.
  • $K$ transitions from $0$ (Wigner semicircle, GOE) to $1$ (Gaussian, Poisson).
  • At the critical point $\gamma = 2$, the kurtosis becomes independent of system size, confirming a second-order phase transition.
• DOS Shape: The ensemble-averaged DOS interpolates between the Wigner semicircle law (for $p \to 1$) and a Gaussian distribution (for $p \to 0$). At $\gamma = 2$, the DOS is system-size independent, marking the critical point.

C. Energy Correlations and Mobility Edge

• Short-Range Correlations:
  • The ratio of adjacent level spacings ($r$) transitions from the GOE value ($\langle r \rangle \approx 0.53$) to the Poisson value ($\langle r \rangle \approx 0.39$) as $\gamma$ increases.
  • This crossover confirms the loss of level repulsion (chaos) at $\gamma = 2$.
• Mobility Edge:
  • By analyzing the energy dependence of the level spacing ratio $r(E)$, the authors identify a mobility edge in the delocalized phase ($\gamma < 2$).
  • Low-energy states are localized, while high-energy states are extended. The mobility edge shrinks toward zero energy as $\gamma$ approaches 2.
• Long-Range Correlations (Thouless Energy):
  • Number Variance ($\Sigma^2$): Shows a crossover from logarithmic growth (GOE-like, rigid spectrum) to linear growth (Poisson-like) as the energy gap increases.
  • Power Spectrum: The power spectrum of level fluctuations exhibits a heterogeneous behavior: $P(k) \propto k^{-1}$ (GOE-like) for high frequencies ($k > k_{Th}$) and $P(k) \propto k^{-2}$ (Poisson-like) for low frequencies.
  • Thouless Energy ($E_{Th}$): A characteristic energy scale $E_{Th}$ is identified. Below $E_{Th}$, the system exhibits universal random matrix correlations; above it, correlations vanish. For $\gamma < 2$, $E_{Th}$ is an extensive quantity, indicating a broad non-ergodic regime within the delocalized phase.

4. Significance and Implications

• Robustness of Criticality: The study demonstrates that the critical percolation limit ($p = N^{-1}$) for the localization transition in ERG graphs remains unchanged even in the presence of diagonal disorder.
• Non-Ergodic Extended (NEE) Phase: The work provides strong numerical and analytical evidence for a distinct non-ergodic extended phase in sparse random matrices. In this phase, eigenstates are extended (support over the whole system) but non-ergodic (multifractal, low entanglement entropy compared to volume law).
• Connection to Many-Body Physics: The findings are highly relevant for understanding Many-Body Localization (MBL) and the breakdown of ergodicity in complex quantum systems. The identification of a Thouless energy scale and mobility edges in sparse matrices offers a theoretical framework for interpreting experimental results in cold atoms, ion traps, and superconducting qubits where sparsity is inherent.
• Methodological Advancement: The successful application of wall-Chebyshev projectors and stochastic moment estimators to sparse matrices provides a scalable toolkit for studying large disordered systems where exact diagonalization is infeasible.

In summary, the paper establishes that sparse random matrices with on-site disorder undergo a sharp quantum phase transition at the percolation threshold, characterized by a transition from a multifractal non-ergodic extended phase to a localized phase, with distinct signatures in ground state statistics, DOS shape, and long-range energy correlations.