Timescales for Deep and Full Thermalization

Through extensive numerical studies of a chaotic many-body quantum model, this paper demonstrates that while both deep and full thermalization exhibit exponential relaxation, full thermalization of higher-order correlation functions occurs faster than deep thermalization, where all moments relax at a rate comparable to the standard Eigenstate Thermalization Hypothesis.

The Gist

Imagine you have a sealed, perfectly isolated box containing a chaotic dance of quantum particles. You start them all in a specific, ordered pose. Over time, even though the box is sealed and no energy escapes, the particles interact so wildly that they eventually "forget" their starting pose and settle into a state that looks like a random, hot mess. In physics, we call this thermalization.

For a long time, scientists had a good rulebook for how this happens, called the Eigenstate Thermalization Hypothesis (ETH). Think of this rulebook as a way to predict how a single particle or a simple pair of particles behaves as the system settles down. It's like knowing that if you stir a cup of coffee, the sugar will eventually dissolve evenly.

However, this paper asks: "What happens if we look at the coffee not just as a whole, but by checking the sugar crystals in incredibly complex, multi-layered patterns?" The authors investigate two advanced ways to measure how "mixed up" the system gets. They call these Full Thermalization and Deep Thermalization.

Here is the breakdown of their findings using simple analogies:

1. The Two Ways to Measure "Mixing"

Full Thermalization (The "Complex Pattern" Check)
Imagine you are checking the coffee by looking at how four, five, or six sugar crystals interact with each other simultaneously. This is Full Thermalization. It looks at very complex, high-order connections between particles.

• The Analogy: It's like trying to predict the exact path of a specific leaf in a hurricane by watching how it bumps into other leaves, branches, and the ground all at once.
• The Finding: The authors found that as you look at more complex patterns (higher orders), the system actually settles down faster. The more complex the pattern you check, the quicker it looks random. It's as if the hurricane scrambles the most intricate leaf patterns almost instantly.

Deep Thermalization (The "Snapshot" Check)
Now, imagine you take a photo of just half the coffee cup while the other half is hidden. You take a picture, then another, then another, each time measuring the hidden half in a different way. This creates a collection of "snapshots" (an ensemble) of what the visible half looks like. Deep Thermalization asks: Does this collection of snapshots eventually look like a perfectly random, standard deck of cards?

• The Analogy: It's like taking a thousand photos of a spinning fan. At first, the photos look different depending on when you snapped them. But eventually, if the fan spins long enough, the collection of photos looks exactly like a random blur you'd expect from a fan spinning forever.
• The Finding: The authors found that this "collection of snapshots" takes a longer, steady amount of time to become perfectly random. Unlike the complex patterns in Full Thermalization, getting this collection of snapshots to look perfectly random doesn't get faster just because you look at more complex details. It moves at a consistent, slower pace.

2. The Race: Who Wins?

The main discovery of this paper is a race between these two methods.

• At the start (Simple checks): Both methods take about the same amount of time to settle down. This is the "standard" thermalization we already knew about.
• At the finish line (Complex checks): Full Thermalization wins. The complex patterns of particle interactions become random much faster than the collection of snapshots (Deep Thermalization) becomes random.

The authors describe this as a surprise. You might think that if the system is chaotic enough to scramble complex patterns instantly, it would also scramble the "snapshots" instantly. But it doesn't. The "snapshots" (Deep Thermalization) lag behind.

3. Why Does This Happen?

The paper suggests a reason for this lag. When you do the "Snapshot" check (Deep Thermalization), you are essentially keeping a record of the measurement outcomes from the hidden part of the system. It's like having a referee who keeps a scorecard. The authors suggest that keeping track of this partial information (the measurement outcomes) might actually slow down the process of the visible part becoming perfectly random. The system is "holding onto" some information longer than it does when you just look at the complex particle interactions directly.

4. The "Even-Odd" Quirk

The researchers also noticed a weird quirk when looking at very small systems (like just one or two atoms).

• If they looked at an odd number of snapshots (1, 3, 5), the mixing speed was normal.
• If they looked at an even number of snapshots (2, 4, 6), the mixing was noticeably faster.
  They believe this is a mathematical trick caused by the tiny size of the system, similar to how a coin flip behaves differently than a roll of dice. They don't expect this quirk to happen in larger, more realistic systems.

Summary

In short, this paper compares two ways of checking if a quantum system has "forgotten" its past.

1. Full Thermalization (checking complex particle interactions) gets faster the more complex you look.
2. Deep Thermalization (checking collections of measurement snapshots) stays at a steady, slower pace.

The result is that for complex systems, the "complex patterns" become random much faster than the "collections of snapshots." The system scrambles its internal connections quickly, but it takes a bit longer for the "recorded history" of measurements to look completely random.

Technical Summary

1. Problem Statement

The Eigenstate Thermalization Hypothesis (ETH) provides the standard framework for understanding how isolated quantum systems relax to thermal equilibrium, primarily focusing on the relaxation of local observables and two-point correlation functions. However, recent theoretical and experimental developments have highlighted two distinct extensions of ETH that probe thermalization at higher orders:

1. Full Thermalization: The relaxation of higher-order correlation functions, specifically generalized Out-of-Time-Order Correlators (OTOCs), which characterize the joint statistical properties of matrix elements and quantum chaos (scrambling).
2. Deep Thermalization: The relaxation of the projected ensemble (the ensemble of reduced states in a subsystem obtained via projective measurements on the complement) towards a "state design" (specifically, the Haar random ensemble). This involves the convergence of all higher moments of the projected ensemble, not just the mean (reduced density matrix).

The Core Question: While both concepts describe thermalization beyond standard ETH, their relationship and relative timescales in generic, spatially local many-body quantum systems remain largely unknown. Specifically, does full thermalization occur faster or slower than deep thermalization, and how do these timescales depend on the order of the correlation/moment ($k$)?

2. Methodology

The authors perform extensive numerical studies using a paradigmatic model of chaotic many-body dynamics: the kicked Ising spin chain with random longitudinal fields.

• Model: A 1D chain of $L$ spin-1/2 degrees of freedom governed by a Floquet operator $U = e^{-iH_z}e^{-iH_x}$, where $H_z$ includes nearest-neighbor interactions and random longitudinal fields, and $H_x$ is a transverse field. The parameters are chosen to ensure chaotic dynamics (Wigner-Dyson level spacing statistics).
• System Partitioning: The system is bipartitioned into a local subsystem $A$ (size $L_A$) and its complement $B$.
• Probes:
  • Full Thermalization ($F_k(t)$): The authors define a modified $k$-th order OTOC for a pure initial state $|\psi_0\rangle$ and a local observable $X$:
    $$F_k(t) = \text{Re}\langle \psi_0 | (X(t)X)^k | \psi_0 \rangle$$
    This measures the decay of correlations between time-evolved and static operators.
  • Deep Thermalization ($\Delta_k(t)$): Instead of using the standard Frobenius distance between the projected ensemble and the Haar ensemble (which is computationally expensive and sensitive to all observables), the authors introduce a probe based on the moments of local observables. They define:
    $$\Delta_k(t) = D_k(t) - D_k^{\text{Haar}}$$
    where $D_k(t) = \sum_{z_B} p(z_B) \langle \psi_A(t, z_B) | X_A^{\otimes k} | \psi_A(t, z_B) \rangle$. This measures how the $k$-th moment of the projected ensemble, when tested against local observables, approaches the Haar limit.
• Numerical Approach:
  • Simulations are performed for system sizes $L = 8$ to $20$.
  • Averaging is performed over 100–150 realizations of the random longitudinal fields to suppress sample-to-sample fluctuations.
  • Thermalization rates ($\gamma$) are extracted by performing median regression on the logarithm of the decay curves ($F_k(t)$ and $\Delta_k(t)$) over an intermediate time window, avoiding initial transients and late-time finite-size plateaus.
  • Two subsystem sizes are tested: $L_A=1$ (Pauli matrices) and $L_A=2$ (Gell-Mann matrices).

3. Key Contributions

1. First Comprehensive Comparison: This work provides the first systematic comparison of the timescales for full and deep thermalization in a generic, spatially local many-body system (as opposed to exactly solvable toy models or dual-unitary circuits).
2. Unified Probe Framework: The authors introduce a method to compare both concepts by restricting the analysis to local observables in both the OTOC and the projected ensemble moments, ensuring a fair comparison of "apples to apples."
3. Discovery of Divergent Timescales: They establish that while both processes exhibit exponential relaxation, the dependence on the order $k$ is fundamentally different.

4. Key Results

A. Exponential Relaxation

Both full thermalization ($F_k$) and deep thermalization ($\Delta_k$) exhibit exponential decay towards their equilibrium values (zero for traceless observables) in the thermodynamic limit. The decay is characterized by rates $\gamma_{F,k}$ and $\gamma_{\Delta,k}$.

B. Dependence on Order ($k$)

• Full Thermalization: The relaxation rate increases monotonically with the order $k$.
  • $\gamma_{F,k} > \gamma_{F,1}$ for $k \geq 2$.
  • This implies that higher-order correlations (e.g., 4-point, 6-point functions) decay faster than lower-order ones. Full thermalization accelerates as the order increases.
• Deep Thermalization: The relaxation rates are approximately independent of the order $k$.
  • $\gamma_{\Delta,k} \approx \text{constant}$ for all $k$.
  • There is a minor "even-odd" effect observed for minimal subsystems ($L_A=1$) due to algebraic properties of Pauli matrices ($X^2 \propto I$), but for larger subsystems ($L_A=2$) and generic observables, the rates are uniform across orders.

C. Relative Speed

• At $k=1$, both rates coincide ($\gamma_{F,1} \approx \gamma_{\Delta,1}$), consistent with the standard ETH description where both reduce to the relaxation of the reduced density matrix.
• At $k \geq 2$, Full Thermalization is strictly faster than Deep Thermalization ($\gamma_{F,k} > \gamma_{\Delta,k}$).
  • This means that while higher-order correlations scramble and decay rapidly, the formation of a state design (deep thermalization) via the projected ensemble is a slower process that proceeds at a rate comparable to the standard two-point function.

D. Finite Size Effects

• The saturation values (plateaus) at late times scale as $1/\sqrt{D_B}$ (where $D_B$ is the dimension of the complement), consistent with random matrix theory predictions for typical states.
• The authors confirm that the exponential decay rates are intrinsic properties of the thermodynamic limit, as they stabilize with increasing system size $L$.

E. Observable Dependence

• The thermalization rates are largely independent of the specific choice of local observable (tested across Gell-Mann matrices and Pauli strings), supporting the universality of these timescales.
• The authors note that the standard Frobenius distance $\delta_k(t)$ (which probes all observables) decays sub-exponentially, whereas the moment-based probe $\Delta_k(t)$ (probing only local, permutation-symmetric observables) decays exponentially. This suggests that deep thermalization of local observables is faster than the convergence of the entire projected ensemble to the Haar measure.

5. Significance and Implications

• Hierarchy of Thermalization: The results reveal a hierarchy where "full thermalization" (scrambling of high-order correlations) is a rapid process, while "deep thermalization" (the formation of state designs in the projected ensemble) is a slower, more robust process that persists at the timescale of standard thermalization.
• Information Scrambling vs. State Design: The finding that full thermalization is faster suggests that while quantum information is scrambled (decaying OTOCs) quickly, the specific statistical structure required for the projected ensemble to mimic a Haar random state (state design) takes longer to establish. This may be because the projected ensemble retains partial information about the measurement outcomes in the complement, hindering the "perfect" scrambling required for deep thermalization.
• Theoretical Framework: The work bridges the gap between free probability theory (used for full ETH) and projected ensemble theory. It suggests that while both phenomena are driven by chaotic dynamics, the mechanisms governing their timescales differ significantly at higher orders.
• Experimental Relevance: Since deep thermalization has been observed experimentally (e.g., in quantum simulators), understanding its timescale relative to OTOCs is crucial for designing protocols for Hamiltonian learning and benchmarking quantum devices. The result implies that measuring higher-order OTOCs might reveal chaos faster than verifying the formation of a state design.

In conclusion, the paper demonstrates that in generic chaotic many-body systems, full thermalization accelerates with the order of correlation, whereas deep thermalization proceeds at a constant rate, making full thermalization the faster of the two processes for orders $k \geq 2$.