Characterization of Thermalization Behaviour in a Generalized Aubry-André Model

This study investigates the ergodic-to-many-body localization transition in an interacting generalized Aubry-André model by employing the Frobenius norm of the adiabatic gauge potential to construct a phase diagram and utilizing finite-size scaling and spectral analysis to characterize the stability of the critical point and the scaling of Thouless time.

The Gist

Imagine a crowded dance floor where everyone is trying to move to the music. In a perfect, chaotic party (what physicists call an ergodic state), everyone eventually mixes with everyone else, and the energy spreads out evenly. But sometimes, the music gets weird, or the room gets too crowded, and people get stuck in their own little corners, refusing to mix. This is called localization.

This paper investigates a specific type of "weird music" in a quantum system—a model called the Generalized Aubry-André (GAA) model. The researchers wanted to understand exactly when and how the system switches from a chaotic, mixing party to a stuck, localized one, especially when there are "mobility edges" (zones where some people can still dance while others are stuck).

Here is a breakdown of their findings using everyday analogies:

1. The Setup: A Deterministic Dance Floor

Unlike a real party where the music might be random and unpredictable, this system uses a quasi-periodic pattern. Think of it like a dance floor with a repeating but never-exactly-the-same pattern of lights. It's not random chaos, but it's not a simple loop either. The researchers added "interactions," meaning the dancers (particles) bump into each other, making the dance floor more crowded and complex.

2. The Tools: How They Measured the Chaos

To figure out if the party is chaotic or stuck, the researchers used three main "thermometers":

• The Gap Ratio (The "Personal Space" Check):
  They looked at the distance between the energy levels of the dancers. In a chaotic system, dancers respect each other's personal space (level repulsion), keeping a specific distance. In a stuck system, they don't care about spacing (random spacing). By measuring this, they could map out where the transition happens.

  • Finding: As they tweaked a control knob called $\alpha$ (which changes the shape of the light pattern), the system became more likely to get stuck (localized) even with less "disorder" (less crazy lighting).

• The Spectral Form Factor (The "Echo" Test):
  This measures how long it takes for the system to "settle down" and thermalize (reach a steady state). They looked at something called the Thouless time.

  • Analogy: Imagine shouting in a cave. If the echo comes back quickly, the cave is small and simple (thermalized). If the echo takes forever or never settles, the cave is a maze (localized).
  • Finding: In the "stuck" phase, the time it takes to settle down became incredibly long—sometimes longer than the age of the universe (in their mathematical terms). This confirmed that the system was truly failing to thermalize.

• Fidelity Susceptibility (The "Sensitivity" Test):
  This is the paper's main innovation. They asked: "If we nudge the system just a tiny bit (like a gentle breeze), how much does the dance pattern change?"

  • Analogy: In a chaotic party, a gentle breeze might make a few people stumble, but the whole dance floor shifts easily. In a stuck, frozen party, a gentle breeze might do nothing, or if it hits a specific weak spot, it could cause a massive, unpredictable collapse.
  • Finding: They found that this "sensitivity" spikes right at the moment the system transitions from chaotic to stuck. It acts like a perfect alarm bell for the phase transition.

3. The Big Discovery: The "Drifting" Boundary

The most tricky part of this research is that they are studying finite systems (small dance floors) and trying to guess what happens in an infinite one (the thermodynamic limit).

Usually, when you make the dance floor bigger, the point where the transition happens shifts. The researchers used a mathematical technique called cost-function minimization (essentially finding the "best fit" line) to see if they could predict the transition point for an infinite system.

• The Twist: They found that the "sensitivity" tool (Fidelity Susceptibility) was much better at predicting a stable transition point than the other tools.
• The Result: While other methods suggested the transition point kept drifting wildly as the system got bigger, the sensitivity tool showed that the transition point was actually quite stable and predictable, especially for certain settings of the control knob ($\alpha$).

4. The Conclusion

The paper concludes that by using this "sensitivity" tool (based on something called the Adiabatic Gauge Potential), they can more accurately map out the boundary between a chaotic, thermalizing quantum system and a frozen, localized one.

They found that:

1. Changing the shape of the potential (the $\alpha$ parameter) makes the system much more prone to getting stuck.
2. The "sensitivity" of the system to tiny changes is a powerful way to spot the exact moment the system freezes.
3. This method helps stabilize the prediction of where the transition happens, even as the system size grows, giving a clearer picture of the "infinite" behavior of these quantum materials.

In short, they built a better "seismograph" to detect the exact moment a quantum system stops dancing and starts freezing, revealing that the rules governing this freeze are more stable than previously thought when using the right measurement tool.

Technical Summary

1. Problem Statement

The paper addresses the challenge of characterizing the transition from the ergodic phase to the Many-Body Localized (MBL) phase in quasi-periodic systems. While Random Matrix Theory (RMT) effectively describes quantum chaos in disordered systems, the universal features governing the critical transition in quasi-periodic models (like the Aubry-André model) remain elusive.

• Specific Gap: Existing diagnostics (e.g., adjacent gap ratio, spectral form factor) often struggle to precisely determine the stability of the phase boundary with respect to system size in the thermodynamic limit.
• Target System: The authors focus on the Generalized Aubry-André (GAA) model with interacting spinless fermions. This model is unique because it features a mobility edge (where extended and localized states coexist at different energies) and is experimentally realizable in cold-atom setups.
• Key Question: How does the introduction of the generalized parameter $\alpha$ affect the thermalization behavior, critical disorder strength ($\lambda^*$), and the nature of the phase transition (power-law vs. BKT-type)?

2. Methodology

The authors employ a multi-faceted numerical approach using Exact Diagonalization (ED) for system sizes up to $L=18$ (Hilbert space dimensions up to $\sim 4.8 \times 10^4$).

A. The Model

The Hamiltonian describes spinless fermions with nearest-neighbor interactions ($V$) in a quasi-periodic potential:
$$H = -t \sum_{\langle ij \rangle} a^\dagger_i a_j + \lambda \sum_i C_i n_i + V \sum_{\langle ij \rangle} n_i n_j$$
The potential coefficient is $C_i = \frac{2 \cos(2\pi q i + \phi)}{1 - \alpha \cos(2\pi q i + \phi)}$, where $\alpha \in [0, 1)$ is the generalized parameter. When $\alpha=0$, it reduces to the standard AA model.

B. Diagnostic Tools

1. Adjacent Gap Ratio ($\langle r \rangle$): Used to distinguish between GOE statistics (ergodic, $\langle r \rangle \approx 0.53$) and Poissonian statistics (localized, $\langle r \rangle \approx 0.39$).
2. Spectral Form Factor (SFF, $K(\tau)$): The Fourier transform of the two-point correlation function. Used to extract the Thouless time ($t_{Th}$), which characterizes the thermalization timescale.
3. Fidelity Susceptibility / Adiabatic Gauge Potential (AGP): The core novel contribution. The authors calculate the Frobenius norm of the AGP (equivalent to fidelity susceptibility $\chi_n$) to measure the sensitivity of eigenstates to infinitesimal gauge deformations.
   • They analyze two types of perturbations: Local ($H_1 = n_{L/2}$) and Extensive ($H_1 = \sum n_i n_{i+1}$).
   • They analyze the logarithmic average $\zeta = \langle \log(\chi_n) \rangle$ and the scaled susceptibility $F = \exp(\zeta)/2^L$.

C. Finite-Size Scaling Analysis

To determine the stability of the critical point in the thermodynamic limit, the authors perform a cost-function minimization analysis.

• They test two correlation length ansatzes: Power-law ($\xi \propto |\lambda - \lambda^*|^{-\nu}$) and Berezinskii-Kosterlitz-Thouless (BKT) ($\xi \propto \exp(\nu/\sqrt{|\lambda - \lambda^*|})$).
• They test four functional forms for the critical disorder $\lambda^*$ dependence on system size $L$: constant, linear drift ($\lambda_0 + \lambda_1 L$), inverse linear, and inverse logarithmic.
• The "best fit" is determined by minimizing the cost function $C_X$ for data collapse.

3. Key Results

A. Phase Diagram and Gap Ratio

• Ergodic to MBL Transition: As the disorder strength $\lambda$ increases, the system transitions from ergodic to localized.
• Effect of $\alpha$: Increasing the generalized parameter $\alpha$ suppresses ergodicity. The critical disorder strength $\lambda^*$ decreases as $\alpha$ increases, eventually vanishing as $\alpha \to 1$. This indicates that the GAA model becomes more susceptible to localization with higher $\alpha$.
• Interaction: Increasing interaction strength $V$ increases the critical disorder $\lambda^*$.

B. Spectral Form Factor and Thouless Time

• Thouless Time ($t_{Th}$): In the ergodic regime ($\lambda < \lambda^*$), $t_{Th}$ is much smaller than the Heisenberg time ($t_H$). As the system approaches the MBL phase or enters the mobility edge regime, $t_{Th}$ increases significantly, approaching or even exceeding $t_H$.
• Scaling: For large $\lambda$ (strong disorder), $t_{Th} > t_H$, indicating a breakdown of ergodicity and the presence of a mobility edge where thermalization is extremely slow or absent.

C. Fidelity Susceptibility (AGP) Behavior

• Three Regimes: The scaled fidelity susceptibility $F$ exhibits three distinct regimes: ergodic (scaling with system size), glassy/intermediate (peaked structure), and localized (saturation).
• Peak Drift: The peak of $F$ (indicating the transition) drifts towards lower $\lambda$ values as system size $L$ increases, consistent with the phase diagram derived from $\langle r \rangle$.
• Local vs. Extensive: The extensive operator shows a larger peak magnitude than the local operator, reflecting the global nature of the perturbation.

D. Finite-Size Scaling and Critical Exponents

• Data Collapse:
  • For the adjacent gap ratio ($\langle r \rangle$), the best data collapse is achieved using a BKT-type correlation length with a linear drift of $\lambda^*$ with system size ($\lambda^* = \lambda_0 + \lambda_1 L$).
  • For the scaled fidelity susceptibility ($F$), the best data collapse is achieved using a Power-law correlation length ($\xi_0$) with a linear drift of $\lambda^*$.
• Critical Exponents: The critical exponent $\nu$ is found to be $\sim 0.5$ for $\alpha=0, 0.3$ and $\sim 0.4$ for $\alpha=0.7$. This violates the Harris-Luck criterion ($\nu \ge 2/d$), a common feature in MBL transitions.
• Stability of $\lambda^*$: The AGP-based analysis ($F$) shows a significantly reduced system-size dependence for the critical value $\lambda^*$ compared to the gap ratio. The slope of the linear drift ($\lambda_1$) is much smaller for $F$ ($\sim 0.0036$) than for $\langle r \rangle$ ($\sim 0.04$), suggesting that AGP provides a more robust estimator for the thermodynamic limit critical point.

4. Significance and Contributions

1. Novel Diagnostic Application: This is one of the first systematic applications of the Adiabatic Gauge Potential (AGP) framework to characterize the MBL transition in a quasi-periodic model with a mobility edge.
2. Resolution of Phase Boundary Stability: The study demonstrates that while traditional metrics (like $\langle r \rangle$) show strong finite-size effects, the AGP-based fidelity susceptibility offers a more stable estimation of the critical disorder strength in the thermodynamic limit.
3. Characterization of Mobility Edge: The analysis of the Thouless time confirms that the GAA model exhibits complex thermalization dynamics where $t_{Th}$ can exceed the Heisenberg time, a signature of the mobility edge and the coexistence of phases.
4. Scaling Universality: The work highlights that different observables may follow different scaling laws (BKT vs. Power-law) near the transition, emphasizing the need for multi-faceted diagnostic approaches in quantum chaos studies.

Conclusion

The paper successfully maps the thermalization behavior of the interacting GAA model. By combining spectral statistics with the novel AGP framework, the authors provide a refined phase diagram and demonstrate that the fidelity susceptibility is a superior tool for minimizing finite-size effects when estimating the critical parameters of the ergodic-to-MBL transition in quasi-periodic systems.