Defining classical and quantum chaos through adiabatic transformations

This paper proposes a unified formalism defining classical and quantum chaos through the complexity of adiabatic transformations, quantified by fidelity susceptibility, which successfully distinguishes between integrable, chaotic non-thermalizing, and ergodic regimes while predicting the universal onset of chaos in coupled spin models.

The Gist

Imagine you are trying to figure out if a system is "chaotic" (wild, unpredictable, and messy) or "orderly" (predictable and calm). In physics, this is a huge question. We know how to spot chaos in a spinning top or a double pendulum, but figuring it out in the quantum world (the world of atoms and particles) is much harder because quantum particles don't follow neat paths like planets do.

This paper proposes a new, universal way to measure chaos that works for both the quantum world and the classical world. They call this measure "Fidelity Susceptibility."

Here is the breakdown using simple analogies:

1. The Core Idea: The "Butterfly Effect" vs. The "Rigid Machine"

Usually, we think of chaos as the "Butterfly Effect": a tiny flap of a butterfly's wing causes a tornado weeks later. In physics, this means if you change the starting conditions just a tiny bit, the whole system goes wildly off course.

However, the authors argue that looking at starting conditions isn't the best way to define chaos in all systems. Instead, they ask a different question:

"If I slightly tweak the rules of the game (the Hamiltonian), how hard is it to keep the system's long-term behavior the same?"

• The Orderly System (Integrable): Imagine a perfectly tuned clock. If you slightly change the tension of a spring, the clock still ticks in a very predictable, smooth way. You can easily adjust the gears to make it tick exactly the same as before. It's easy to "deform" the system without breaking its rhythm.
• The Chaotic System: Imagine a house of cards or a turbulent river. If you change the wind speed just a tiny bit, the whole structure collapses or the river flow changes completely. To make the system behave the same way after the change, you would need to perform a wildly complex, impossible-to-calculate series of adjustments.

The Measure: The authors say that the complexity of the adjustments needed to keep the system stable is the definition of chaos. The more complex the adjustment, the more chaotic the system.

2. The Tool: "Fidelity Susceptibility"

How do we measure this complexity? They use a mathematical tool called Fidelity Susceptibility.

Think of it like a sensitivity test for a musical instrument.

• If you have a perfectly tuned violin (an orderly system), and you slightly loosen a string, the note changes a little, but you can easily retune it. The "sensitivity" is low.
• If you have a chaotic drum kit where the drums are all rattling against each other, and you slightly loosen one screw, the entire sound becomes a mess. To get the sound back to normal, you'd have to re-tune every single drum in a crazy, complex pattern. The "sensitivity" is huge.

In the paper, they calculate this sensitivity by looking at low-frequency noise.

• Orderly systems are quiet at low frequencies (they don't have long-term instability).
• Chaotic systems scream with "noise" at low frequencies. This noise tells us that the system is unstable over long periods.

3. The Big Discovery: Chaos is "Maximal" in the Middle

One of the most interesting findings is where chaos is strongest.

Usually, we think:

• Integrable (Orderly): No chaos.
• Chaotic: Maximum chaos.
• Thermalizing (Ergodic): The system forgets its past and settles into a calm equilibrium (like hot coffee cooling down).

The authors found a "Goldilocks Zone" (an intermediate regime) between the orderly and the thermalized states.

• In this middle zone, the system is maximally chaotic. It is so sensitive to changes that it hasn't yet settled down into a calm, thermal state.
• Think of it like a crowd of people.
  • Orderly: Everyone is standing in neat rows (Integrable).
  • Thermalized: Everyone is sitting down, chatting quietly, and has forgotten who they were standing next to (Equilibrium).
  • Maximally Chaotic: Everyone is running around in a panic, bumping into each other, and the slightest push causes a massive stampede. This is the most chaotic state, and it happens before the crowd settles down.

4. The Quantum vs. Classical Surprise

The paper tested this on a model of two spinning magnets (spins).

• Classical Limit (Big Spins): As the spins get bigger (approaching the classical world), the chaos behaves as expected.
• Quantum Limit (Tiny Spins): Here is the surprise. When the spins are small (quantum), the system is less chaotic than the classical version, even when it should be chaotic.
  • Analogy: Imagine a quantum particle is like a ghost that can be in two places at once. This "ghostly" nature actually protects it from some of the chaos that would destroy a solid object. The system gets "stuck" in a way that prevents it from fully exploring the chaos. This is similar to how a quantum particle can get "trapped" in a specific spot (localization) instead of spreading out.

5. Why This Matters

Previous methods to find quantum chaos (like looking at how fast information spreads or using "Out of Time Order Correlators") often failed or were hard to apply to real-world systems.

This new method (Fidelity Susceptibility) is powerful because:

1. It's Universal: It works for both the quantum world and the classical world using the same logic.
2. It Distinguishes Chaos from Equilibrium: It can tell the difference between a system that is truly chaotic (wild and unstable) and a system that is just "thermalized" (calm and mixed).
3. It's Practical: It relies on looking at how observables (things we can measure) fluctuate over long periods, which is something we can actually calculate and measure.

Summary

The authors invented a new "chaos meter." Instead of asking "How fast do things go wrong?" (which is hard to define in quantum mechanics), they ask, "How hard is it to fix the system after a tiny change?"

If the answer is "It's impossible to fix without a super-complex, wild adjustment," then the system is chaotic. They found that this chaos is strongest in a specific "middle ground" between perfect order and total calm, and that quantum mechanics actually acts as a shield, making small systems less chaotic than their big, classical counterparts.

Technical Summary

1. Problem Statement

The paper addresses the fundamental challenge of defining and distinguishing chaos, ergodicity, and integrability in both classical and quantum systems.

• Limitations of Existing Definitions:
  • Classical Chaos: Traditionally defined via Lyapunov exponents (instability of trajectories to initial conditions) or the breakdown of conservation laws (KAM theorem). However, Lyapunov exponents are ill-defined in quantum mechanics and for macroscopic systems described by probability distributions rather than single trajectories.
  • Quantum Chaos: Often defined via the Eigenstate Thermalization Hypothesis (ETH) or Random Matrix Theory (RMT) statistics (level spacing). However, these are measures of ergodicity (thermalization) rather than chaos itself. A system can be chaotic but non-ergodic (e.g., near integrability or in mixed phase spaces), and conversely, some non-chaotic systems can thermalize.
  • OTOCs and Operator Growth: Out-of-Time-Order Correlators (OTOCs) and Krylov space operator growth are popular probes. The authors argue that OTOCs are experimentally difficult (requiring time reversal) and that operator growth often fails to distinguish integrable from chaotic systems, particularly near the classical limit where both exhibit similar asymptotic behaviors.
• The Core Question: Is there a unified, rigorous framework that defines chaos based on the sensitivity of physical states/trajectories to Hamiltonian deformations, applicable to both quantum and classical regimes, and capable of distinguishing chaotic non-ergodic regimes from fully ergodic ones?

2. Methodology

The authors propose a formalism based on adiabatic transformations and the fidelity susceptibility ($\chi_\lambda$).

• Adiabatic Gauge Potential (AGP):

  • The AGP, denoted $A_\lambda$, is the generator of unitary (quantum) or canonical (classical) transformations that preserve the instantaneous eigenstates (or time-averaged trajectories) of a Hamiltonian $H(\lambda)$ under a parameter deformation $\lambda$.
  • It satisfies the equation $[G_\lambda, H] = 0$ (quantum) or $\{G_\lambda, H\} = 0$ (classical), where $G_\lambda = \partial_\lambda H + i[A_\lambda, H]/\hbar$ (or Poisson bracket equivalent).
  • $A_\lambda$ is formally defined via a time integral with a regularization cutoff $\mu$:
    $$A_\lambda = \frac{1}{2} \int_{-\infty}^{\infty} dt \, e^{-\mu|t|} \text{sgn}(t) \partial_\lambda H(t)$$

• Fidelity Susceptibility ($\chi_\lambda$):

  • Defined as the variance (or connected part) of the AGP: $\chi_\lambda = \langle A_\lambda^2 \rangle_c$.
  • Physically, it quantifies the "complexity" or "cost" of deforming the canonical variables to keep time-averaged trajectories invariant under a small perturbation $\delta \lambda$.
  • Connection to Spectral Functions: $\chi_\lambda$ is directly related to the low-frequency asymptotics of the spectral function $\Phi_\lambda(\omega)$ of the observable $\partial_\lambda H$:
    $$\chi_\lambda = \int_{-\infty}^{\infty} d\omega \frac{\omega^2}{(\omega^2 + \mu^2)^2} \Phi_\lambda(\omega)$$

• Regimes of Scaling:
  The behavior of $\chi_\lambda$ as the cutoff $\mu \to 0$ (or time $T \to \infty$) distinguishes three regimes:

  1. Integrable: $\Phi_\lambda(\omega \to 0) \to 0$ (e.g., $\omega^2$). $\chi_\lambda$ remains finite or grows slowly (power law).
  2. Ergodic/Thermalizing: $\Phi_\lambda(\omega \to 0) \to \text{const}$. $\chi_\lambda \sim \exp(S)$ (extensive).
  3. Maximally Chaotic (Non-Ergodic): $\Phi_\lambda(\omega \to 0) \sim \omega^{-1+\epsilon}$. $\chi_\lambda$ diverges faster than in the ergodic case (e.g., $\sim \mu^{-2}$), indicating maximal sensitivity to perturbations without full thermalization.

• Model System:
  The authors apply this formalism to a model of two coupled spins ($S_1, S_2$) with Hamiltonian:
  $$H = \sum_{\alpha} \left[ -J_\alpha S^\alpha_1 S^\alpha_2 + \frac{1}{2}A_\alpha ((S^\alpha_1)^2 + (S^\alpha_2)^2) \right]$$
  They analyze this system for finite spin $S$ (quantum) and the classical limit $S \to \infty$, varying parameters to tune between integrable, chaotic, and ergodic regimes.

3. Key Contributions

1. Unified Definition of Chaos: The paper establishes that fidelity susceptibility serves as a universal probe for chaos in both classical and quantum systems. It defines chaos not by thermalization (ergodicity) but by the instability of time-averaged trajectories/eigenstates to Hamiltonian deformations.
2. Distinction between Chaos and Ergodicity: The authors demonstrate that the "maximally chaotic" regime (where $\chi_\lambda$ diverges most rapidly) is distinct from the ergodic regime. This intermediate regime is characterized by slow, glassy dynamics and mixed phase spaces, often found near integrable points.
3. Failure of Short-Time Probes: The study shows that high-frequency spectral data (related to short-time operator growth and nested commutators) cannot distinguish between integrable and chaotic systems in the classical limit, as both exhibit exponential decay. Chaos is instead encoded in the low-frequency behavior.
4. Anomalously Large Finite-$S$ Effects: The paper reveals that near integrability, finite quantum spin effects are anomalously large. The convergence to the classical chaotic limit is extremely slow (requiring $S \sim 10^5$), suggesting strong quantum suppression of chaos (analogous to dynamical localization) in weakly non-integrable systems.

4. Key Results

• XXZ Model (Integrable):
  • The spectral function $\Phi_{ZZ}(\omega)$ vanishes as $\omega \to 0$ (scaling as $\omega$ or faster depending on symmetry).
  • The fidelity susceptibility $\chi_\lambda$ is finite (or logarithmically divergent) as $\mu \to 0$, consistent with integrability.
• XYZ Model (Integrable, no continuous symmetry):
  • Despite lacking continuous symmetries, the system remains integrable.
  • $\Phi_{ZZ}(\omega) \sim \omega^2$ at low frequencies.
  • $\chi_\lambda$ remains finite, confirming that the absence of continuous symmetry does not imply chaos.
• Chaotic Model (Broken Integrability):
  • For intermediate integrability-breaking parameters, the system is chaotic but non-ergodic.
  • Spectral Function: $\Phi_{ZZ}(\omega) \sim \omega^{-\gamma}$ with $\gamma \approx 0.625$ (classical) or $\gamma \approx 1$ (near integrability).
  • Fidelity Susceptibility: $\chi_\lambda$ diverges as $\mu^{-(1+\gamma)}$. Near the integrable point, $\chi_\lambda \sim \mu^{-2}$, indicating "maximal chaos."
  • Phase Space Resolution: By analyzing specific trajectories, the authors show that chaotic and regular regions in a mixed phase space have distinct low-frequency spectral tails ($\omega^{-1}$ vs. $\omega^2$), allowing the fidelity susceptibility to resolve local chaos.
• Absence of Ergodicity:
  • Level spacing statistics ($r$-ratio) and microcanonical ensemble comparisons confirm that the chaotic model does not thermalize (does not reach Wigner-Dyson statistics or microcanonical equilibrium) for the parameters studied. It remains in a mixed phase space regime.
• Quantum vs. Classical Convergence:
  • While high-frequency behavior converges quickly, low-frequency behavior near integrability converges very slowly with increasing $S$.
  • Quantum systems exhibit "dynamical localization" effects, suppressing the divergence of $\chi_\lambda$ compared to the classical prediction until $S$ becomes extremely large.

5. Significance

• Theoretical Clarity: The paper resolves the ambiguity between chaos and ergodicity by providing a metric (fidelity susceptibility) that specifically measures trajectory instability rather than thermalization. This aligns better with the intuitive "butterfly effect" (sensitivity to perturbations) than standard ETH-based definitions.
• Experimental Relevance: Unlike OTOCs, which require time-reversal operations, fidelity susceptibility can be inferred from the low-frequency noise (spectral function) of physical observables, which is experimentally accessible.
• New Regime Identification: The identification of a "maximally chaotic" but non-ergodic regime provides a new framework for understanding glassy dynamics, Arnold diffusion, and the crossover between integrable and thermalizing systems.
• Methodological Advancement: The work demonstrates that adiabatic gauge potentials and fidelity susceptibility are robust tools for analyzing complex dynamics in systems with few degrees of freedom, bridging the gap between classical nonlinear dynamics and quantum many-body physics.

In conclusion, Kim et al. propose that chaos is best defined by the complexity of adiabatic transformations required to preserve time-averaged states, quantified by the fidelity susceptibility. This approach successfully unifies the description of chaos across classical and quantum domains and reveals a rich phase diagram where maximal chaos exists in a non-ergodic, intermediate regime.