On deforming and breaking integrability

This paper investigates nearest-neighbour deformations of integrable models, specifically the XXZ spin chain, by classifying four distinct deformation types and providing numerical evidence that the onset of chaos and its volume scaling differ significantly between models that are perturbatively integrable versus those that are fully integrable or non-integrable.

The Gist

Imagine you have a perfectly organized, predictable machine. In the world of physics, this is called an integrable model. Think of it like a Swiss Army knife where every tool works perfectly, and you can predict exactly how it will behave forever. It never gets "chaotic" or messy.

Now, imagine you want to tinker with this machine. You add a little bit of glue, a new screw, or a tiny spring. This is called a deformation. Usually, when you mess with a perfect machine, it breaks. It stops being predictable and starts behaving like a chaotic, unpredictable mess (like a pile of loose screws rattling in a box). This is called breaking integrability, and it's the path to quantum chaos.

This paper is a detective story about how different machines break when you tinker with them. The authors found that there isn't just one way to break a machine; there are four distinct ways, and they break at different speeds and in different patterns.

Here is the breakdown of their findings using simple analogies:

1. The Four Ways a Machine Can Break

The authors tested different ways to add a "perturbation" (a small change) to a famous magnetic chain of atoms called the XXZ spin chain. They found four scenarios:

• Type A: The Total Disaster (Generic Breaking)
  • The Analogy: You take a perfect clock and randomly glue a rock to the gears.
  • What happens: The clock stops working immediately. It becomes chaotic right away. This is the most common way things break.
• Type B: The Perfect Upgrade (Exact Preservation)
  • The Analogy: You replace the clock's gears with slightly better, gold-plated gears.
  • What happens: The clock still works perfectly. It's still predictable. Some changes are so compatible with the machine's design that they don't break the rules at all.
• Type C: The "All-or-Nothing" Illusion (Long-Range Deformations)
  • The Analogy: Imagine a magic trick where the clock looks broken if you only look at the first second, but if you watch the whole hour, the gears magically realign and it works again.
  • What happens: If you only look at the first step of your change, it looks like the machine is broken. But if you look at every step of the change all the way through, the machine actually stays perfect. This happens in complex holographic models (related to black holes and string theory).
• Type D: The "Almost" Break (The Star of the Paper)
  • The Analogy: This is the most interesting one. Imagine you have a clock that works perfectly for the first hour. Then, you add a tiny, tricky spring. For the next hour, the clock still works perfectly. But if you try to keep adding more springs or wait longer, the clock eventually jams and becomes chaotic.
  • What happens: The machine looks integrable (perfect) for a while, but it's a "fake" integrability. It's only perfect up to a certain point. The authors created a specific model (called $H_{QInt}$) that does exactly this. It's "quasi-integrable."

2. The Race to Chaos: Who Breaks First?

The authors wanted to see how fast these machines turn from "perfect" to "chaotic" as they added more and more "glue" (increasing the deformation strength).

• The Generic Breaker (Type A): As soon as you add a tiny bit of glue, the machine starts to wobble and become chaotic very quickly. It's like a house of cards; a small breeze knocks it over.
• The Quasi-Integrable Breaker (Type D - $H_{QInt}$): This machine is much more stubborn. You can add a lot of glue, and it keeps working perfectly for a long time. It resists chaos much longer than the generic machine.

3. The "Volume" Mystery (Size Matters)

Here is the coolest part of the discovery. The authors looked at how the size of the machine (the number of atoms, or "volume") affects when it breaks.

• Small Machines: In a small clock, it's hard to tell the difference between the "stubborn" one and the "weak" one.
• Big Machines: As the machines get bigger, the difference becomes huge.
  • The Generic Breaker becomes chaotic very fast as it gets bigger.
  • The Quasi-Integrable Breaker is in the middle. It's not as stubborn as the "perfect" ones, but it's much tougher than the "generic" ones.

The authors found a mathematical "speed limit" for when chaos starts.

• Generic breakers follow a rule like $1/L^3$ (they break very fast as size increases).
• The Quasi-Integrator follows a rule like $1/L^{2.5}$.
• The Metaphor: Imagine two runners. One is a sprinter who trips immediately (Generic). The other is a marathon runner who stumbles but keeps going for a long time (Quasi-Integrable). The paper found a third runner who is somewhere in between—stumbling more than the sprinter but falling less than the marathon runner. This "middle ground" is a new discovery.

4. Why Should We Care?

In the real world, nothing is perfectly integrable. Everything has some tiny imperfections.

• If a system is integrable, it doesn't "thermalize" (it doesn't reach a steady, hot equilibrium). It keeps remembering its past.
• If a system is chaotic, it forgets its past and becomes a hot, messy soup (thermalizes).

The Quasi-Integrable models are the "Goldilocks" zone. They remember their past for a very long time before finally giving in to chaos. This is crucial for understanding:

1. Quantum Computers: How long can we keep quantum information before it gets messed up by noise?
2. New Materials: Some materials might act like these "stubborn" machines, staying ordered for a long time even when they should be chaotic.
3. Black Holes: The math used here is similar to what physicists use to study black holes, helping us understand how information might get trapped or released.

Summary

The paper is like a study of how different toys break when you shake them. They found that some toys break instantly, some never break, and some have a "superpower" where they look like they are breaking but actually hold together for a surprisingly long time. This "superpower" toy (the quasi-integrable model) behaves in a unique way that is different from both the fragile toys and the indestructible ones, offering a new middle ground for understanding how the universe moves from order to chaos.

Technical Summary

1. Problem Statement

Integrable models possess extensive symmetries and conserved charges that allow for exact solvability and prevent thermalization (relaxing instead to a Generalized Gibbs Ensemble). Generic perturbations break this integrability, leading to quantum chaos and thermalization. However, the transition between integrability and chaos is not binary; there exists a spectrum of "integrability breaking."

The paper addresses the classification of nearest-neighbour deformations of integrable models, specifically the XXZ spin chain. It seeks to distinguish between:

1. Trivial breaking: Random perturbations that immediately destroy integrability.
2. Exact preservation: Deformations that remain integrable to all orders.
3. Long-range integrability: Deformations that are only integrable if all orders of the deformation parameter are included (e.g., holographic long-range models).
4. Perturbative (Quasi-) integrability: Deformations that preserve integrability to a specific order (e.g., first order) but cannot be extended to an integrable model at higher orders.

The authors aim to characterize these classes analytically and investigate their physical consequences, specifically the onset of quantum chaos and spectral statistics.

2. Methodology

Analytical Framework: The Boost Operator

The authors utilize the boost operator formalism to classify deformations.

• Setup: Starting with an integrable Hamiltonian $H^{(0)}$, they introduce a perturbation $H = H^{(0)} + \epsilon H^{(1)} + \epsilon^2 H^{(2)} + \dots$.
• Conserved Charges: They construct the next conserved charge $Q_3$ using the boost operator acting on the deformed Hamiltonian density.
• Integrability Condition: They impose the commutativity condition $[Q_2(\epsilon), Q_3(\epsilon)] = 0$ order by order in $\epsilon$.
  • At $O(\epsilon)$, this yields linear constraints on the deformation parameters.
  • At $O(\epsilon^2)$, this yields quadratic constraints on the first-order parameters and linear constraints on second-order parameters.
• R-Matrix/Lax Operator: To verify integrability, they attempt to construct an R-matrix (via the Sutherland equation) or a Lax operator. If the deformation equations cannot be satisfied at higher orders without forcing parameters to zero (returning to a known integrable case), the model is deemed "quasi-integrable."

Numerical Analysis: Spectral Statistics

The authors perform exact diagonalization on spin chains of varying lengths ($L \in [16, 21]$) to study the transition to chaos.

• Models Compared:
  • $H_{QInt}$: A quasi-integrable model (integrable to $O(\epsilon)$ but not beyond) containing specific combinations of XYZ, total spin, and Dzyaloshinskii–Moriya (DMI) terms.
  • $H_{dXYZ}$: A generic integrability-breaking model (XYZ chain + DMI term).
• Indicators:
  • Level Spacing: Distribution of energy level spacings ($s_k$) fitted to the Brody distribution to extract the parameter $\omega_B$ (0 for Poisson/integrable, 1 for Wigner-Dyson/chaotic).
  • Critical Coupling ($\epsilon_c$): The deformation strength where $\omega_B = 0.5$.
  • Volume Scaling: Analyzing how $\epsilon_c$ scales with system size $L$ (e.g., $\epsilon_c \sim L^{-b}$).
  • Eigenvector Entropy: Measuring the delocalization of eigenstates in the basis of the unperturbed Hamiltonian to test the Eigenstate Thermalization Hypothesis (ETH).

3. Key Contributions

A. Classification of Deformations

The paper rigorously identifies four distinct types of deformations for the XXZ spin chain:

1. Integrability Breaking: Generic deformations that break integrability immediately.
2. Exactly Integrable: Deformations that can be extended to all orders (e.g., adding a $\sigma^z \otimes \sigma^z$ term to XXX to get XXZ).
3. All-Order Integrable (Long-Range): Deformations that require infinite series of terms to maintain integrability (common in AdS/CFT).
4. Perturbatively Integrable (Quasi-Integrable): A novel class identified in this work. These models satisfy integrability conditions up to $O(\epsilon)$ but fail at $O(\epsilon^2)$ unless parameters are tuned to trivial cases.
   • Example Model ($H_{QInt}$): $H_{XXZ} + \epsilon \sum (\sigma^x_i \sigma^x_{i+1} - \sigma^y_i \sigma^y_{i+1}) + \epsilon \sum \sigma^z_i + \epsilon \sum (\sigma^x_i \sigma^y_{i+1} - \sigma^y_i \sigma^x_{i+1})$.
   • Crucially, while individual terms might preserve integrability in isolation, their combination breaks it at second order.

B. R-Matrix and Lax Operator Analysis

The authors demonstrate that for the quasi-integrable model, an R-matrix exists at $O(\epsilon)$ satisfying the Yang-Baxter equation. However, attempting to solve the Sutherland equation at $O(\epsilon^2)$ yields inconsistent equations unless the deformation parameters are set to zero, confirming the model cannot be extended to a full integrable system.

C. Scaling of Critical Coupling

The paper provides numerical evidence for a new scaling regime for the onset of chaos:

• Generic Breaking ($H_{dXYZ}$): Exhibits exponential scaling of the critical coupling ($\epsilon_c \sim e^{-dL}$), characteristic of gapped systems and "crossover" behavior in Fock space.
• Quasi-Integrable ($H_{QInt}$): Exhibits power-law scaling $\epsilon_c \sim L^{-b}$ with an exponent $b \approx 2.5$.
  • This is intermediate between the generic gapless scaling ($b=3$) and the "weak" breaking scaling found in long-range models ($b=2$).
  • This suggests the quasi-integrable model occupies a distinct physical regime between strong and weak integrability breaking.

D. Eigenvector Entropy and Thermalization

• Generic Breaking: Eigenstates rapidly delocalize, and entropy becomes a smooth function of energy (consistent with ETH) even at moderate deformation strengths.
• Quasi-Integrable: Entropy increases more slowly. Crucially, the entropy does not become a smooth function of energy; it retains high variance (RMSD) even at larger deformations. This indicates that the system retains memory of the integrable structure longer and does not thermalize as efficiently as generic chaotic systems.

4. Results

• Four Classes Confirmed: The boost operator formalism successfully classifies deformations into the four predicted categories.
• Intermediate Chaos: The quasi-integrable model $H_{QInt}$ displays a "slower" onset of chaos compared to generic breaking models. The Brody parameter $\omega_B$ increases slowly with $\epsilon$, showing a plateau or slow rise before a rapid transition.
• Scaling Exponent: The critical coupling for $H_{QInt}$ scales as $\epsilon_c \sim L^{-2.64}$ (best fit), confirming an intermediate scaling behavior.
• Non-Universal Thermalization: The quasi-integrable model fails to satisfy the smoothness condition of the ETH (entropy vs. energy) as strongly as the generic chaotic model, suggesting a pre-thermalization phase or quasi-conserved charges that persist longer than in generic systems.

5. Significance

• Theoretical Classification: The paper provides a rigorous algebraic framework for distinguishing between models that are "nearly integrable" in a perturbative sense versus those that are truly integrable or generically chaotic.
• New Universality Class: The discovery of the intermediate scaling exponent ($b \approx 2.5$) suggests the existence of a universality class for "quasi-integrable" systems that is distinct from both generic chaotic systems and weakly broken integrable systems (like long-range deformations).
• Physical Implications: The findings are relevant for understanding thermalization timescales in quantum many-body systems. The quasi-integrable models may exhibit anomalously slow thermalization (scaling as $\tau \propto \epsilon^{-b}$ with $b > 2$), which is relevant for experiments in cold atoms and condensed matter systems where Dzyaloshinskii–Moriya interactions are present.
• Methodological Advance: The combination of the boost operator formalism with high-precision spectral statistics provides a robust tool for probing the "edge" of integrability, bridging the gap between exact solvability and statistical mechanics.