Many-body spectral transitions through the lens of the variable-range SYK2 model

This paper investigates a power-law decaying quadratic SYK model to demonstrate how single-particle spectral transitions propagate to the many-body regime, revealing that the spectral form factor remains robust under reduced interaction ranges before perturbation theory breaks down and new spectral features emerge.

The Gist

Imagine a giant, chaotic dance floor where thousands of particles (dancers) are constantly bumping into each other. In the ideal world of physics, these dancers can reach out and grab anyone on the floor, no matter how far away they are. This is the famous SYK model, a theoretical playground used by scientists to understand how chaos works in the quantum world and how it might relate to black holes.

However, in the real world, dancers can't reach everyone. They can only grab hands with people nearby. The distance matters: the further away someone is, the harder it is to connect.

This paper asks a simple question: What happens to the chaos when we force the dancers to only interact with their neighbors, and how does that distance change the dance?

Here is the story of their findings, broken down into everyday concepts:

1. The Setup: The "Power-Law" Dance Floor

The researchers created a new version of the dance floor called the variable-range SYK2 model.

• The Rule: The strength of the connection between two dancers depends on the distance between them. If they are close, they dance together strongly. If they are far, the connection is weak, fading away like a signal getting weaker the further you walk from a radio tower.
• The Variable ($\alpha$): They used a knob called $\alpha$ to control how fast this connection fades.
  • Low $\alpha$: The connection fades slowly. Dancers can still reach across the room.
  • High $\alpha$: The connection fades very fast. Dancers can only touch their immediate neighbors.

2. The "Ruler" of Chaos: The Spectral Form Factor (SFF)

To see how the dance is going, the scientists used a special measuring tool called the Spectral Form Factor (SFF). Think of the SFF as a "heartbeat monitor" for the system's energy levels.

• In a perfectly chaotic system, this heartbeat has a very specific, famous shape: it starts high, drops down into a dip (a valley), goes up in a straight ramp (a hill), and then flattens out into a plateau (a flat table).
• This specific shape is the "fingerprint" of chaos. If the fingerprint changes, the nature of the system has changed.

3. The Surprise: The System is Tougher Than Expected

The scientists expected that as soon as they started limiting the dancers' reach (increasing $\alpha$), the chaotic fingerprint would immediately break.

What they found instead:

• The "Stubborn" Phase: When the connection range is reduced a little bit (specifically, when $\alpha$ is less than 0.5), the system is incredibly robust. Even though the dancers can't reach as far, the "heartbeat" of the chaos looks almost exactly the same as the ideal, all-reaching version.
• Why? It turns out that the mathematical "noise" created by the limited connections cancels itself out perfectly. It's like a group of people trying to shout over each other; if they are organized just right, the noise disappears, and the system keeps dancing chaotically.

4. The Tipping Point: When the Dance Breaks

However, once they turned the knob past a critical point ($\alpha \approx 0.5$), the magic stopped working.

• The Dip Gets Deeper: The "valley" in the heartbeat monitor suddenly got much deeper. This is a sign that the system is starting to lose its chaotic nature and is becoming "stuck" or localized.
• The Secondary Plateau: A new, unexpected feature appeared. Before the final flat "table" (the late-time plateau), a second, smaller plateau emerged.
  • Analogy: Imagine the dancers are trying to explore the whole room. In the chaotic phase, they run everywhere. In this new phase, they get stuck in small groups, exploring their immediate area but not the whole room. This "stuck" behavior creates a pause in the heartbeat before they finally settle down.

5. Connecting the Dots: One Dancer vs. The Whole Crowd

The most fascinating part of the paper is how the behavior of the whole crowd (the many-body system) mirrors the behavior of a single dancer (the single-particle limit).

• In the world of single particles, there is a known transition at $\alpha = 0.5$ where a particle goes from being able to roam freely to being stuck in one spot.
• The paper shows that this exact same transition happens for the entire crowd of interacting particles. The "heartbeat" (SFF) of the complex crowd changes in the exact same way the "heartbeat" of a single lonely particle does.

Summary of the Journey

1. Start: You have a chaotic system where everyone connects to everyone.
2. Tweak: You slowly cut the connections so people only talk to neighbors.
3. Result 1 (0 to 0.5): The system doesn't care! It stays chaotic. The "heartbeat" stays the same.
4. Result 2 (0.5 to 1.5): The system starts to break. The "heartbeat" develops a deep dip and a new "stuck" plateau. The chaos is turning into order (localization).
5. Result 3 (Above 1.5): The system becomes fully "integrable" (predictable and non-chaotic), similar to a clockwork machine where every part moves in a fixed pattern.

The Bottom Line:
The paper proves that even in a complex, interacting world of many particles, the rules of "getting stuck" (localization) are surprisingly simple and follow the same rules as a single particle. The "heartbeat" of the system (the SFF) is a reliable tool to spot exactly when the system switches from being a chaotic dance party to a group of isolated, stuck individuals.

Technical Summary

Technical Summary: Many-body spectral transitions through the lens of the variable-range SYK2 model

Problem Statement
The Sachdev–Ye–Kitaev (SYK) model serves as a paradigmatic framework for studying quantum chaos and holographic quantum matter due to its analytical tractability in the thermodynamic limit. However, idealized SYK models assume all-to-all connectivity with perfectly uncorrelated random couplings, a condition rarely met in realistic experimental platforms such as trapped ions, cold atoms, or solid-state systems. These platforms typically exhibit distance-dependent interactions governed by power-law decays. This deviation raises critical questions regarding the robustness of the SYK model's chaotic properties when interactions are constrained to a finite range. Specifically, it remains unclear how transitions known in the single-particle limit (such as Anderson criticality and multifractality in Power-Law Random Banded Matrix (PRBM) models) translate to the many-body system, where fermionic statistics and single-particle mobility edges significantly alter dynamics.

Methodology
The authors introduce and analyze a "variable-range SYK2 model," a quadratic SYK variant where random two-fermion hopping terms are weighted by a distance-dependent function $a(i-j) \propto r^{-\alpha}$. The system consists of $N$ Majorana fermions on a circle with periodic boundary conditions.

To investigate the spectral statistics, the authors focus on the disorder-averaged Spectral Form Factor (SFF), $g(T) = \langle |Z(iT)|^2 \rangle / |Z(0)|^2$. The analysis proceeds via a path-integral formulation:

1. Path Integral Formulation: The partition function is expressed using Grassmannian variables for forward and backward time evolutions. After integrating out the random couplings, collective fields (local two-point functions $G^a_{i}$) and site-dependent auxiliary fields ($\Sigma^a_{i}$) are introduced to render the action quadratic in fermions.
2. Saddle-Point Analysis: The effective action is minimized to find saddle-point solutions. The authors assume a homogeneous saddle point (independent of site position) initially, justified by the restoration of space-translation symmetry upon disorder averaging.
3. Fluctuation Analysis: The stability of the saddle point is examined by analyzing Gaussian fluctuations. The authors identify zero modes arising from the spontaneous breaking of an $SU(2)$ symmetry to $U(1)$ in the symmetry-breaking regime ($|\omega_n| < 2J$).
4. Eigenvalue Spectrum: The analysis relies heavily on the eigenvalues ($\lambda_k$) of the matrix $A$ derived from the interaction profile. The behavior of these eigenvalues determines the number of zero modes and the validity of the perturbative expansion.
5. Numerical Validation: Analytical predictions are compared against numerical simulations for system sizes up to $N=400$, averaging over millions of disorder samples to ensure convergence of the SFF.

Key Contributions and Results

• Robustness of the SFF for $\alpha < 1/2$:
  The study reveals that the SFF remains robust under a considerable reduction of the interaction range. For $\alpha < 1/2$, the system remains in an ergodic phase indistinguishable from the standard SYK model (Gaussian Orthogonal Ensemble). Analytically, this robustness stems from non-trivial cancellations in the one-loop contributions involving the eigenvalues of the interaction matrix. The SFF exhibits the universal structure of a chaotic system: an initial slope, a dip, a linear ramp, and a late-time plateau. The analytical expression derived for this regime matches numerical data with high precision.

• Breakdown of Perturbation Theory at $\alpha \sim 1/2$:
  A critical transition occurs at $\alpha = 1/2$. At this point, the eigenvalues of the interaction matrix undergo a qualitative change, leading to a breakdown of perturbation theory around the uniform saddle point. This coincides with the ergodic-to-non-ergodic transition in the single-particle PRBM limit.

  • Dip Height: As $\alpha$ increases beyond $0.5$, the "dip" in the SFF (the minimum value before the ramp) grows significantly. The authors identify $\alpha \approx 0.501$ as the characteristic point where the system departs from the ergodic behavior, marking the onset of non-ergodicity.

• Emergence of a Secondary Plateau:
  In the regime $1/2 \le \alpha \le 3/2$, a new spectral regime emerges characterized by a "secondary plateau" appearing after early-time oscillations but before the late-time plateau.

  • This feature signals a pre-thermalization regime where the Hilbert space is not fully explored.
  • The height of this secondary plateau increases with $\alpha$, reaching the level of the late-time plateau around $\alpha \sim 3/2$. This progression mirrors the transition from a delocalized non-ergodic phase to a localized super-diffusive phase, and finally to an integrable regime in the single-particle PRBM.

• Connection to Single-Particle Criticality:
  The paper demonstrates that the many-body spectral transitions in the variable-range SYK2 model are directly linked to single-particle localization transitions. The appearance of the secondary plateau and the growth of the dip height serve as reliable markers for these transitions, effectively bridging single-particle criticality (Anderson criticality at $\alpha=1$) with many-body dynamics.

Significance and Claims
The authors claim that their work provides a controlled setting to study the interplay between single-particle criticality and many-body dynamics. By utilizing the SFF, they offer a robust diagnostic tool that can identify ergodic-to-non-ergodic transitions in interacting systems.

The significance of these findings lies in their potential application to the study of Many-Body Localization (MBL), a field where numerical limitations often restrict analysis to small systems and the existence of a thermodynamic limit transition is debated. The variable-range SYK2 model allows for a direct connection between many-body signatures of localization and well-understood single-particle phenomena. Specifically, the dip height and the secondary plateau are proposed as indicators for:

1. The transition between holographic and non-holographic behavior.
2. The onset of pre-thermalization and the emergence of nearly conserved charges.
3. The identification of Anderson criticality in many-body systems.

The paper concludes that while the sharp transition to the integrable regime at $\alpha = 3/2$ (in the PRBM sense) is difficult to resolve numerically due to large fluctuations in the SFF for large $N$, the qualitative disappearance of the ramp and the evolution of the secondary plateau provide strong evidence that the physics of the variable-range SYK2 model is deeply rooted in the single-particle localization transitions of the underlying PRBM.