Scaling analysis and renormalization group on the mobility edge in the quantum random energy model

This paper employs renormalization group analysis to demonstrate that the quantum Random Energy Model exhibits robust scaling behaviors and localization-delocalization transitions analogous to those on expander graphs, revealing an unconventional ergodic phase at zero energy density and a disorder-independent universality class.

The Gist

The Big Picture: Getting Lost in a Maze

Imagine you are in a giant, infinite maze. You are trying to walk from one side to the other.

• The "Ergodic" (Free) State: If the maze is open and clear, you can wander everywhere. You explore the whole space. This is like a normal, healthy system where energy spreads out freely.
• The "Localized" (Stuck) State: If the maze is filled with random, chaotic obstacles (disorder), you might get stuck in a small corner. You can't escape your immediate neighborhood. In physics, this is called localization.

The paper investigates a specific type of quantum maze called the Quantum Random Energy Model (QREM). The scientists wanted to understand: Under what conditions does the particle get stuck, and when does it roam free?

They used a powerful mathematical tool called the Renormalization Group (RG). Think of RG as a "zoom-out" camera. Instead of looking at every single step you take, the camera zooms out to show the general trend of your journey. Does your path eventually lead you to the exit (ergodic), or does it trap you in a dead end (localized)?

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The Three Scenarios Discovered

The researchers looked at the maze under three different conditions and found three very different stories.

1. The Center of the Spectrum: The "Infinite Party"

The Setup: Imagine you are at the exact center of the energy spectrum (zero energy density). Think of this as a massive, high-energy party where everyone is dancing.
The Finding: No matter how messy the room gets (how much "disorder" or obstacles you add), the party never stops. The dancers (particles) always find a way to mix with everyone else.
The Analogy: It's like a crowded dance floor. Even if people bump into each other randomly, the music is so loud and the energy so high that no one gets stuck in a corner. The system is always ergodic (always moving).
The Twist: The scientists noticed something weird. As the system gets bigger, the dancers seem to move faster than the room expands for a while before settling down. It's like a "false start" where the crowd surges forward before realizing the room is huge. This suggests a very complex, subtle way of moving that doesn't fit the standard textbook rules.

2. The Edges of the Spectrum: The "Quiet Library"

The Setup: Now, imagine moving away from the center of the energy spectrum to the edges. This is like moving from the loud dance floor to a quiet library.
The Finding: Here, the story changes. If you add enough random obstacles (disorder), the particles do get stuck. A "Mobility Edge" appears. This is a border line: on one side, you can move; on the other, you are frozen.
The Analogy: In the library, if the shelves are arranged randomly enough, you might get trapped in a specific aisle and never find the exit. The transition from "moving" to "stuck" here behaves exactly like a famous type of maze called an Expander Graph (a highly connected but random network). It's a clean, predictable transition.

3. The "Magic Rescaling": Changing the Rules

The Setup: The scientists asked: "What if we change the rules of the game?" Specifically, what if we adjust the "strength" of the disorder based on the size of the system?
The Finding: By applying a specific mathematical "rescaling" (making the disorder weaker as the system gets bigger), they could force the "Infinite Party" (the center of the spectrum) to behave like the "Quiet Library."
The Analogy: Imagine the dance floor was previously too chaotic to get stuck. But if you magically make the obstacles "lighter" as the room gets bigger, suddenly the dancers can get stuck.
The Surprise: Even though they changed the rules to force a transition, the type of transition remained the same. It still looked like the "Expander Graph" transition. This tells us that the fundamental nature of the maze (its "universality class") is robust. It doesn't care about the tiny details of how you set up the obstacles; the big picture remains the same.

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Why Does This Matter?

This paper is important for two main reasons:

1. Solving a Mystery: For decades, physicists have debated whether certain quantum systems can truly get "stuck" (Many-Body Localization) or if they always eventually mix. This paper provides a clear map showing exactly where the "stuck" zones are and where the "free" zones are.
2. The "Zoom-Out" Camera Works: They proved that the Renormalization Group (the zoom-out camera) is a reliable tool. Even when the system behaves strangely (like the "overshooting" at the center), the camera can still predict the long-term behavior.

The Takeaway Metaphor

Think of the Quantum Random Energy Model as a giant, multi-story hotel.

• The Lobby (Center of Spectrum): It's always a bustling, chaotic party. No matter how many people bump into each other, everyone keeps moving. You can never get stuck here.
• The Upper Floors (Finite Energy): It's quieter. If the furniture is arranged randomly enough, you can get trapped in a room and never leave.
• The Renormalization Group: This is the manager looking at the hotel from a helicopter. The manager realizes that even if you rearrange the furniture (rescale the disorder), the rules of how people get stuck or move remain the same. The hotel's fundamental nature doesn't change, even if the decorations do.

The paper confirms that while the "party" at the center is always safe, the "quiet floors" have a real danger of getting stuck, and the mathematical tools used to predict this are robust and reliable.

Technical Summary

1. Problem Statement and Context

The paper addresses the challenge of understanding the localization-delocalization transition in many-body quantum systems, specifically focusing on the Quantum Random Energy Model (QREM).

• The Challenge: While Anderson localization (single-particle) and Many-Body Localization (MBL) are well-studied, the existence of stable MBL phases and the nature of the transition (critical exponents, scaling laws) remain debated due to severe finite-size effects in numerical simulations and the lack of a robust analytical scaling theory for interacting systems.
• The Model: The QREM is a toy model for MBL. It retains the Fock-space structure of many-body hopping (a hypercubic lattice) but features uncorrelated disorder. Unlike standard Anderson models on expander graphs (like Random Regular Graphs, RRG), the QREM has a unique phase diagram:
  • At zero energy density ($E=0$), the system is fully ergodic regardless of disorder strength.
  • At finite energy density ($E \neq 0$), a mobility edge exists, separating localized and delocalized phases.
• The Gap: Previous perturbative approaches (Forward Scattering Approximation - FSA) suggest a mobility edge but fail to capture non-perturbative effects (loops) inherent in the hypercubic geometry. A rigorous Renormalization Group (RG) scaling analysis for the QREM, particularly regarding the nature of the ergodic fixed point and the universality class of the transition, was lacking.

2. Methodology

The authors employ a combination of exact diagonalization (ED) and Renormalization Group (RG) scaling analysis to reconstruct the beta functions of spectral observables.

• Numerical Approach:
  • They perform exact diagonalization of the QREM Hamiltonian for system sizes up to $L=16$ (Hilbert space dimension $N=2^{16}$).
  • Observables:
    1. Rescaled Gap Ratio ($\phi$): Derived from the spectral gap ratio $r$, rescaled to range from 0 (Poisson/Integrable) to 1 (Wigner-Dyson/Chaotic).
    2. Fractal Dimension ($D$): Derived from the Shannon entropy of eigenstates, serving as an order parameter for localization ($D=1$ for ergodic, $D=0$ for localized).
• RG Framework:
  • They define the beta function $\beta \equiv d \ln A / d \ln N$, where $A$ is the observable ($\phi$ or $D$) and $N$ is the system size.
  • They analyze the flow of these observables to determine if the system flows to a localized fixed point ($A=0$) or an ergodic fixed point ($A=1$).
  • Scaling Hypotheses: They test between One-Parameter Scaling (1PS) (typical of finite-dimensional Anderson transitions) and Two-Parameter Scaling (2PS) (typical of Anderson transitions on infinite-dimensional graphs like RRGs, involving a line of fixed points).
• Disorder Rescaling: To probe the $E=0$ case, they introduce a rescaling of the disorder strength $W \to W / (\sqrt{L} \log L)$, inspired by perturbative expansions on Bethe lattices, to see if a transition can be induced.

3. Key Contributions and Results

A. Behavior at Zero Energy Density ($E=0$)

• No Transition in Natural Scaling: In the "natural" scaling of the QREM (where disorder scales as $\sqrt{L}$ to maintain extensivity), the RG flow for all disorder strengths $W$ consistently flows toward the ergodic phase ($D \to 1, \phi \to 1$). There is no localization transition at $E=0$.
• Unconventional Ergodic Fixed Point: The approach to the ergodic fixed point is characterized by an overshoot where the fractal dimension $D(L)$ temporarily exceeds 1 for intermediate system sizes before converging to 1. This suggests the wavefunction support grows faster than the Hilbert space dimension at finite sizes.
• Induced Transition via Rescaling: When the disorder is rescaled ($W \to W/\sqrt{L}\log L$), a localization transition emerges at $E=0$.
  • This transition is described by Two-Parameter Scaling (2PS).
  • The RG flow reveals a line of fixed points on the negative $\beta$ axis, analogous to the Anderson transition on Random Regular Graphs (RRG) and the random-field XXZ chain.
  • The critical disorder $W_c \approx 1.95$ (numerical) matches well with analytical predictions ($W_c \approx 2.25$) derived from mapping the QREM to an RRG.

B. Behavior at Finite Energy Density ($E \neq 0$)

• Mobility Edge: A localization transition occurs naturally without any disorder rescaling.
• Universality Class: The transition exhibits 2PS behavior, identical to the Anderson model on expander graphs (RRG).
  • The critical line is characterized by a potential $V(\alpha) \propto -e^{n\alpha}$ with exponent $n \approx 1$.
  • The critical disorder $W_c$ is found to be $\approx 20$ (at $E = E_{max}/4$), significantly higher than the FSA prediction ($W_{FSA} \approx 3.5$), highlighting the importance of non-perturbative loop effects.
  • The fractal dimension scales as $D \sim L^{-2/n}$ on the critical line, consistent with RRG results.

C. Robustness of Universality

• The paper demonstrates that the universality class of the QREM localization transition (2PS, line of fixed points) remains unchanged whether the transition is induced by disorder rescaling at $E=0$ or occurs naturally at $E \neq 0$.
• This confirms that the RG flow is independent of microscopic details (like the specific scaling of disorder) and is governed by the underlying graph structure (hypercube vs. expander).

4. Significance and Implications

• Resolution of Finite-Size Ambiguities: The study clarifies the nature of the ergodic phase in the QREM, showing that the absence of a transition at $E=0$ is a robust feature, not an artifact of finite-size effects.
• Validation of 2PS: It provides strong numerical evidence that the Anderson transition on infinite-dimensional graphs (and by extension, the MBL transition in mean-field models) is governed by Two-Parameter Scaling, distinct from the One-Parameter Scaling of finite-dimensional systems.
• Insight into MBL: The results offer a new perspective on the Many-Body Localization transition. The fact that the QREM (a mean-field model) shares the same universality class as the XXZ chain (a 1D interacting system) suggests that the critical behavior of MBL transitions may be universal across different dimensions and interaction types, provided the effective connectivity is high.
• Methodological Advance: The paper demonstrates the power of reconstructing beta functions from numerical data to distinguish between 1PS and 2PS scenarios, a crucial step for understanding critical phenomena where standard finite-size scaling often fails or yields misleading "fake" critical points.

In summary, the paper establishes that the QREM serves as a rigorous mean-field limit for localization transitions, exhibiting a 2PS universality class analogous to expander graphs, and clarifies the conditions under which ergodicity is preserved or broken in the thermodynamic limit.