Unconventional Thermalization of a Localized Chain Interacting with an Ergodic Bath

This paper introduces the interacting Anderson Quantum Sun model to reveal unconventional thermalization regimes that defy standard many-body localization paradigms, specifically identifying phases where volume-law entanglement coexists with intermediate spectral statistics and where Poisson statistics appear alongside sub-volume entanglement growth.

The Gist

Imagine a crowded dance floor. In a normal party (an ergodic system), everyone eventually mixes together, dances with everyone, and forgets who they were when they first walked in. The music (energy) spreads out evenly, and the crowd reaches a state of "thermal equilibrium."

Now, imagine a different party where everyone is glued to their spot, refusing to move or talk to anyone else. They remember exactly where they started and who they were. This is localization. In physics, this is called Many-Body Localization (MBL). It's like a frozen snapshot of chaos where nothing ever settles down.

For a long time, physicists thought these two states were like light switches: you were either fully mixed (thermal) or fully frozen (localized). But this new paper introduces a strange, new kind of party where the rules get weird.

The Setup: The "Quantum Sun" and the "Frozen Chain"

The researchers built a model called the Anderson Quantum Sun (AQS). Think of it as two distinct groups at a massive, infinite party:

1. The "Sun" (The Bath): A small, chaotic group of 3 people who are dancing wildly, mixing with everyone, and acting like a "thermal bath." They represent the heat or energy source.
2. The "Chain" (The Localized Guests): A long line of people standing in a row, far away from the Sun. These people are initially frozen in place (Anderson localization) because of a "disorder" (like a chaotic, unpredictable floor plan) that keeps them stuck.

The twist? The Sun and the Chain are connected by invisible strings. The closer a person in the Chain is to the Sun, the stronger the string pulling them. As you get further down the line, the string gets weaker and weaker (exponentially decaying).

The Discovery: The "In-Between" Zones

The researchers asked: What happens when we turn up the volume of the music (interaction strength) or loosen the floor plan (disorder)?

They expected a simple switch: either the Chain stays frozen, or the Sun's chaos eventually drags the whole Chain into a wild dance. Instead, they found two weird, intermediate zones that break the rules of physics as we know them.

1. The "Ghostly Connection" Zone (Regime B)

• The Analogy: Imagine the Chain is frozen in place (they haven't moved from their spots), so they look localized. However, if you look at their "thoughts" (quantum entanglement), they are actually connected to the whole room in a deep, complex way.
• The Physics: The people in the Chain have sub-volume entanglement. This means they are sharing information with the whole system, but not quite enough to become fully "thermal."
• The Catch: Even though they are sharing information, the "music" (energy levels) still sounds like a frozen, random mess (Poisson statistics).
• The Surprise: Usually, if you are sharing information with the whole room, the music should sound chaotic. Here, the music sounds frozen, but the people are secretly connected. It's like a group of people sitting silently in a room, but somehow, they all know exactly what the person at the other end of the room is thinking.

2. The "Chaotic but Quiet" Zone (Regime C)

• The Analogy: Now, imagine the Chain is fully dancing and mixing with the Sun. They are fully "thermalized." But, strangely, the music they are dancing to still sounds somewhat random and orderly, not fully chaotic.
• The Physics: The Chain has volume-law entanglement (fully mixed), but the energy statistics are intermediate. They aren't fully chaotic (GOE) nor fully frozen (Poisson).
• The Surprise: This is like a room full of people dancing wildly, but the rhythm of the music hasn't fully settled into a chaotic beat yet. It's a "half-mixed" state that shouldn't exist in a simple system.

Why Does This Matter?

For decades, physicists believed that how a system looks (frozen or mixed) and how it sounds (random or ordered) were perfectly linked.

• Frozen = Random sound.
• Mixed = Chaotic sound.

This paper says: "Not so fast!"

They found that you can have a system that is frozen in appearance but secretly connected, or fully mixed but still sounding orderly. This suggests that the path from "frozen" to "mixed" isn't a simple switch; it's a long, winding road with strange rest stops in the middle.

The "Avalanche" Metaphor

In the old view, if you had a small patch of chaos (the Sun), it would slowly spread like a fire, eventually burning down the whole frozen forest (the Chain). This is called an "avalanche."

This paper suggests that sometimes, the fire spreads in a weird way. It might burn the trees (entanglement) without changing the sound of the wind (spectral statistics), or it might change the sound without fully burning the trees. It reveals that the "fire" of thermalization is much more complex and subtle than we thought.

The Bottom Line

The universe might be full of these "in-between" states. Just because something looks frozen doesn't mean it's not connected, and just because something looks mixed doesn't mean it's fully chaotic. The researchers have found a new map of these strange, "unconventional" territories, showing us that the transition between order and chaos is far more interesting than a simple on/off switch.

In short: They found a "ghostly" middle ground where things are frozen but connected, and a "quiet" middle ground where things are mixed but orderly. It's a new chapter in understanding how the quantum world settles down (or refuses to).

Technical Summary

1. Problem Statement and Motivation

The paper addresses a fundamental question in non-equilibrium quantum statistical mechanics: How does an Anderson insulator (a non-interacting localized system) thermalize when coupled to an ergodic bath?

• Context: Conventionally, many-body localized (MBL) systems are expected to exhibit a strict correspondence between spectral statistics (Poisson distribution) and eigenstate properties (Area-law entanglement entropy). Conversely, ergodic systems follow Random Matrix Theory (GOE statistics) and Volume-law entanglement.
• The Gap: Recent studies suggest the MBL phase might be unstable in the thermodynamic limit due to "avalanches" (rare thermal regions destabilizing the localized phase). The Quantum Sun (QS) model was introduced as a toy model to study this, showing a sharp ergodicity-breaking transition. However, the QS model lacks spatial structure in the localized chain.
• Objective: The authors introduce the Anderson Quantum Sun (AQS) model, which couples a spatially extended Anderson chain (with nearest-neighbor interactions) to an ergodic bath. They aim to map the phase diagram to understand the interplay between Anderson localization, interaction strength, and the distance-dependent coupling to the bath.

2. Methodology

The Model: Anderson Quantum Sun (AQS)

The Hamiltonian is defined as:
$$H_{AQS} = H_{QS} + J \sum_{j=1}^{L-1} (S^x_j S^x_{j+1} + S^y_j S^y_{j+1})$$

• The Bath: A "Sun" consisting of $N=3$ spins described by a random Gaussian Orthogonal Ensemble (GOE) matrix, acting as a thermal bath.
• The Chain: A chain of $L$ spins subject to a random magnetic field (disorder strength $W=0.5$).
• Coupling: The bath spins couple to the chain spins via an interaction term that decays exponentially with distance: $V_{AQS} = \alpha^{u_j} = \exp(-u_j/\Lambda_\alpha)$.
• Parameters: The study varies the interaction decay length $\Lambda_\alpha$ (controlled by $\alpha$) and the nearest-neighbor hopping/interaction strength $J$ in the chain.

Numerical Techniques

• Exact Diagonalization (ED): Used for small system sizes ($N+L \leq 13$).
• POLFED Algorithm: A polynomially filtered exact diagonalization method used for larger systems ($13 < N+L \leq 20$).
• Observables:
  • Spectral Statistics: Gap ratio $r = \min(\delta_i, \delta_{i+1}) / \max(\delta_i, \delta_{i+1})$. Values $\langle r \rangle \approx 0.39$ indicate Poisson (localized), while $\approx 0.53$ indicate GOE (ergodic).
  • Entanglement Entropy (EE): Von Neumann entropy $S$ of half-chain bipartitions, normalized by Page entropy $S_P$. Scaling $S \propto L^b$ distinguishes phases: $b=0$ (Area-law), $b=1$ (Volume-law).
  • Correlation Functions: Connected correlations $C_{ij}$ to detect rare events and ergodic instabilities.
  • Multifractal Analysis: Participation entropy ($P_q$) to characterize Fock-space localization.
• Finite-Size Scaling: Critical points ($\alpha_c$) are extrapolated to the thermodynamic limit ($L \to \infty$) using $1/L$ scaling.

3. Key Contributions and Results

The authors identify four distinct regimes in the phase diagram, revealing a breakdown of the standard correspondence between spectral statistics and eigenstate entanglement.

Regime [A]: Standard MBL / Anderson Insulator

• Conditions: Strong disorder, weak coupling ($J$ small, $\alpha$ small).
• Properties: Poisson spectral statistics, Area-law entanglement ($b=0$), short-range correlations.
• Significance: This is the conventional localized phase where the bath fails to thermalize the chain.

Regime [B]: The "Sub-Volume" Anomaly (Key Discovery 1)

• Conditions: Weak to intermediate coupling, moderate disorder.
• Properties:
  • Spectrum: Remains Poisson (suggesting localization).
  • Entanglement: Exhibits Sub-volume law ($0 < b < 1$), meaning entanglement grows faster than Area-law but slower than Volume-law.
  • Correlations: Dominated by rare events and system-wide correlations. The correlation length ratio $\xi_x/\xi_z$ decreases with system size, indicating ergodic instabilities.
• Significance: This regime challenges the ETH paradigm. The system is spectrally localized but possesses extended, non-thermal eigenstates with strong long-range correlations, likely driven by rare resonant regions.

Regime [C]: The "Intermediate" Anomaly (Key Discovery 2)

• Conditions: Stronger coupling, weaker disorder.
• Properties:
  • Spectrum: Intermediate statistics (between Poisson and GOE).
  • Entanglement: Volume-law ($b=1$), indicating fully thermalized eigenstates.
• Significance: This is a "mixed phase space" regime where the system is fully thermal (Volume-law) but retains non-ergodic spectral signatures. This contrasts with toy models (like Rosenzweig-Porter) by explicitly capturing distance-dependent resonances in real space.

Regime [D]: Standard Ergodic Phase

• Conditions: Strong coupling.
• Properties: GOE spectral statistics, Volume-law entanglement, standard ETH behavior.

Phase Diagram Structure

• Transition Separation: The transition in entanglement ($\alpha_{c,S}$) occurs at a significantly lower coupling strength than the transition in spectral statistics ($\alpha_{c,r}$).
• Avalanche Threshold: The boundary between Regime [A] and [B] follows a generalized avalanche threshold: $\Lambda_\alpha + z\xi_{AL} = \zeta^*$, where $\xi_{AL}$ is the Anderson localization length.

4. Significance and Implications

1. Decoupling of Spectral and Eigenstate Properties: The paper provides robust numerical evidence that spectral statistics (Poisson vs. GOE) and entanglement scaling (Area vs. Volume) are not strictly coupled in interacting systems with spatial structure. This contradicts the standard MBL picture where these properties change simultaneously.
2. Nature of the Intermediate Phase: The identification of Regime [B] (Poisson + Sub-volume) and Regime [C] (Intermediate + Volume) suggests that the breakdown of localization is a multistage process. It is not a direct jump from MBL to ETH but involves extended phases dominated by rare resonances and instabilities.
3. Validation of Avalanche Scenarios: The results support the "avalanche" scenario where rare thermal regions destabilize the MBL phase, but they reveal a more complex landscape than previously thought, including a stable sub-volume phase before full thermalization.
4. Experimental Relevance: The authors suggest that ion trap chains could potentially realize the AQS model, offering a platform to observe these unconventional thermalization routes and the separation of spectral and entanglement transitions.

Conclusion

The study of the Anderson Quantum Sun model reveals that the path from localization to ergodicity is far more complex than a simple phase transition. The existence of sub-volume entangled phases with Poisson statistics and volume-law phases with intermediate statistics highlights the role of rare events and spatial resonances in destabilizing Anderson localization. These findings necessitate a revision of the standard classification of quantum phases in disordered interacting systems.