Planckian bound on the local equilibration time

This paper rigorously establishes a universal lower bound on the local equilibration time of quantum many-body systems, proving that hydrodynamic behavior cannot emerge faster than a dimension-dependent fraction of the Planckian time $\hbar/T$, regardless of the underlying thermalization mechanism or microscopic details.

The Gist

Imagine you have a cup of hot coffee and you drop a single drop of cold milk into it. At first, the milk swirls in chaotic, distinct patterns. But eventually, the milk spreads out until the whole cup is a uniform, lukewarm brown. The time it takes for that chaotic swirl to turn into a smooth, predictable mix is called the equilibration time.

In the world of quantum physics (the physics of the very small), scientists have long suspected that there is a "speed limit" for how fast any system can mix or thermalize. They call this the Planckian bound. It suggests that no matter how you stir your quantum coffee, it cannot mix faster than a specific time dictated by the temperature of the system.

This paper by Marvin Qi, Alexey Milekhin, and Luca Delacrétaz is like a rigorous detective story that finally proves this speed limit exists for a vast class of quantum systems. Here is the breakdown in simple terms:

1. The Mystery: Is There a Universal Speed Limit?

For years, physicists noticed that in many strange materials (like the "strange metals" found in high-temperature superconductors), things seem to mix as fast as physically possible. The formula for this fastest possible speed is roughly:
$$\text{Time} \approx \frac{\hbar}{T}$$
(Where $\hbar$ is a tiny constant of nature, and $T$ is the temperature.)

The big question was: Is this just a coincidence for specific materials, or is it a fundamental law of the universe? Can we prove that nothing can thermalize faster than this?

2. The Definition: When Does "Mixing" Actually Start?

To prove a speed limit, you first need to define exactly when the "mixing" starts. The authors decided to define the equilibration time ($\tau_{eq}$) as the moment when the system stops behaving like a chaotic mess and starts behaving like a fluid.

Think of it like a crowd of people in a stadium:

• Early times: Everyone is running in random directions, bumping into each other. It's chaotic.
• Late times: The crowd starts moving like a fluid wave (a "hydrodynamic" flow). You can predict where the wave will go.

The authors ask: How long does it take for the crowd to stop running randomly and start flowing like a wave?

3. The Detective Work: Using "Time Travel" Math

To prove the limit, the authors used a clever mathematical trick involving complex numbers (numbers that have a "real" part and an "imaginary" part).

Imagine the behavior of the system as a path on a map.

• Real time is the path we can actually observe.
• Imaginary time is a hidden path that exists mathematically but isn't directly observable.

The authors showed that because the laws of quantum mechanics are so strict (specifically, because of "analyticity," which means the math is smooth and continuous), the path in real time is tightly tethered to the path in imaginary time.

They used a mathematical rule (similar to a speed limit sign on a highway) that says: If a function is smooth and bounded in a certain "strip" of the complex plane, it cannot change its value too quickly.

4. The Verdict: The Speed Limit is Real

By applying this rule, they proved that the "rate of change" of the system's correlation (how much the milk drop remembers where it was) cannot exceed a certain value.

Because the system cannot change faster than this mathematical limit, the hydrodynamic behavior (the smooth flow) cannot start too early.

They derived a formula:
$$\tau_{eq} \geq \alpha \times \frac{\hbar}{T}$$
(Where $\alpha$ is a number that depends on the dimension of space, like whether you are in a 1D line, 2D sheet, or 3D room.)

What this means: No matter how strong the interactions are, no matter how chaotic the system is, it cannot become a smooth fluid faster than this Planckian time.

5. Why This Matters (The "So What?")

• It's Universal: This applies to everything from spin chains in a computer chip to the hot soup of particles in the early universe. It doesn't matter if the particles are "quasiparticles" (like little billiard balls) or a messy quantum soup.
• It Explains "Strange Metals": Materials like cuprate superconductors have electrical resistance that goes up linearly with temperature. This paper suggests that's because these materials are hitting the "speed limit" of thermalization. They are mixing as fast as the universe allows.
• It Rules Out "Magic" Materials: You can't engineer a material that thermalizes instantly. There is a fundamental barrier set by the temperature of the universe.

The Analogy of the "Fog"

Imagine a room filled with fog.

• Weak interactions: The fog particles barely bump into each other. It takes a long time for the fog to settle into a uniform layer.
• Strong interactions: The particles bump into each other constantly. The fog settles very quickly.
• The Planckian Bound: The authors prove that even if you make the particles bump into each other infinitely hard, the fog cannot settle instantly. There is a "minimum time" required for the fog to smooth out, dictated by how hot the room is. If the room is hot, the fog settles faster; if it's cold, it takes longer. But it can never be faster than the "Planckian" limit.

Summary

This paper takes a hunch that "nature has a speed limit for mixing" and turns it into a rigorous mathematical proof. They showed that the transition from chaos to order (hydrodynamics) is fundamentally constrained by the temperature of the system. It's a beautiful example of how deep mathematical properties of time and heat set the rules for how our universe behaves.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in quantum statistical physics: Is there a universal lower bound on the local equilibration time ($\tau_{eq}$) of quantum many-body systems?

• The Conjecture: It is widely conjectured that $\tau_{eq}$ cannot be arbitrarily small but is bounded below by the Planckian time scale, $\tau_{Pl} = \hbar/T$. This is motivated by observations in high-temperature superconductors (linear-in-$T$ resistivity) and strongly coupled systems where thermalization occurs at the fastest possible rate allowed by quantum mechanics.
• The Challenge: While the Heisenberg uncertainty principle ($\Delta E \Delta t \gtrsim \hbar$) suggests $\Delta t \sim \hbar/T$ (with $\Delta E \sim T$), making this rigorous for a sharply defined $\tau_{eq}$ has been difficult. Previous attempts often relied on weak-coupling kinetic theory or specific holographic models, lacking a general proof for arbitrary local quantum systems (with or without quasiparticles).
• The Goal: To formally define $\tau_{eq}$ and prove a rigorous lower bound of the form $\tau_{eq} \geq \alpha \hbar/T$ for general local quantum many-body systems, independent of microscopic details or thermalization mechanisms.

2. Methodology

The authors employ a rigorous mathematical approach based on the analytic properties of real-time thermal correlators in the complex time plane.

• Definition of $\tau_{eq}$:
  Instead of relying on quasiparticle scattering rates, the authors define $\tau_{eq}$ as the time scale at which a hydrodynamic description emerges for conserved densities. Specifically, they consider the "two-sided" (or regularized) correlator:
  $$F_{nn}(t) = \text{Tr}(\sqrt{\rho} n(t) \sqrt{\rho} n)$$
  where $\rho = e^{-\beta H}/Z$ is the thermal density matrix and $n$ is a conserved density. $\tau_{eq}$ is defined as the earliest time after which $F_{nn}(t)$ and its derivative agree with the hydrodynamic prediction ($F_{hydro}$) within a small error $\epsilon$.

• Analytic Structure:
  The core of the proof relies on the analyticity of the correlator $F_{AB}(z)$ (where $z = t + i\tau$) in the complex strip $|\text{Im}(z)| \leq \beta/2$. This analyticity encodes the fluctuation-dissipation theorem and the Kubo-Martin-Schwinger (KMS) condition.

• Mathematical Tools:

  1. Schwarz-Pick Theorem: The authors map the half-strip in the complex plane to the unit disk. By constructing a bounded function $f(z)$ derived from the correlator, they utilize the Schwarz-Pick theorem to bound the rate of change of the function.
  2. Phragmén-Lindelöf Principle: Used to extend bounds from the boundary of the complex strip to the interior, ensuring the bound holds for all real times $t$.
  3. Hydrodynamic Constraints: The proof assumes that at late times, the system exhibits hydrodynamic behavior (e.g., diffusion), which imposes specific asymptotic behaviors on the correlator (e.g., positivity of the symmetric Green's function and dominance of the symmetric part over the antisymmetric part).

3. Key Contributions and Results

A. The Rigorous Bound

The paper establishes a universal lower bound on the onset of hydrodynamic behavior:
$$\tau_{eq} \geq \alpha \frac{\hbar}{T}$$
where $\alpha$ is a dimensionless coefficient depending on the spatial dimension $d$ and the type of hydrodynamic mode (diffusion, sound, etc.).

• For Diffusion: In $d$ spatial dimensions, assuming the symmetric Green's function is positive, the bound is:
  $$\tau_{eq} \geq \frac{d}{2\pi} \frac{\hbar}{T}$$
• Generalization: The bound applies to other hydrodynamic behaviors:
  • Super/Sub-diffusion: For scaling $F(t) \sim t^{-d/z}$, the bound becomes $\tau_{eq} \geq \frac{d}{z\pi} \frac{\hbar}{T}$.
  • Sound Modes: The bound applies to the emergence of sound fronts.

B. Independence from Microscopic Details

The bound is universal. It holds for:

• Strongly coupled systems (where perturbative methods fail).
• Systems with or without quasiparticles.
• Systems with inelastic scattering.
• Disordered systems (though the authors note that Brownian motion in disorder can saturate the bound, which is a specific feature of the definition used).

C. Effective Field Theory (EFT) Interpretation

The authors analyze the bound from the perspective of Fluctuating Hydrodynamics EFT.

• They show that while EFTs can be constructed to allow arbitrarily fast hydrodynamic onset ("super-Planckian" theories), such theories violate the analyticity constraints required by unitary quantum mechanics.
• Specifically, KMS symmetry and analyticity impose UV/IR constraints on the Wilsonian coefficients of the EFT, preventing the hydrodynamic scale from being parametrically smaller than $\hbar/T$.

D. Implications for Strange Metals

The paper connects the bound to the linear-in-$T$ resistivity observed in strange metals (e.g., cuprates).

• If a system saturates the Planckian bound ($\tau_{eq} \sim \hbar/T$), causality constraints (Lieb-Robinson bounds) imply a lower bound on the DC resistivity: $\rho_{dc} \gtrsim \frac{m^*}{n} \frac{T}{\hbar}$.
• This provides a theoretical justification for why Planckian thermalization leads to "bad metals" with linear-in-$T$ resistivity, without needing to invoke a quasiparticle picture or a specific transport time.

4. Significance and Impact

1. Formalization of a Conjecture: The paper transforms the heuristic "Planckian conjecture" into a mathematically rigorous theorem for a well-defined class of observables.
2. Universality: It demonstrates that the speed limit of thermalization is a fundamental property of local quantum systems, arising from the interplay between thermodynamics (temperature), quantum mechanics (Planck's constant), and causality (analyticity), rather than specific interaction strengths.
3. Resolution of Strong Coupling Dynamics: It provides a tool to constrain transport and thermalization in regimes where traditional kinetic theory (weak coupling) is inapplicable.
4. Experimental Relevance: The authors suggest that time-resolved near-field spectroscopy or frequency-resolved optical conductivity (specifically $\omega/T$ scaling) can probe this bound experimentally. The observation of $\omega/T$ scaling in cuprates is cited as strong evidence that these materials are close to saturating this fundamental bound.
5. Dimensional Dependence: A novel result is that the bound scales with spatial dimension $d$, implying that quantum thermalizers are inherently slower in higher dimensions.

5. Limitations and Future Directions

• Definition of $\tau_{eq}$: The bound relies on the specific definition of $\tau_{eq}$ as the onset of hydrodynamic agreement. While robust, the authors acknowledge that for free particles in disordered landscapes, the bound can be saturated, which might not reflect "thermalization" in the traditional sense.
• Exponential Decay: The paper notes that non-hydrodynamic exponential decay rates (Liouvillian gaps) are not generically bounded by $T/\hbar$ without additional symmetry assumptions, distinguishing them from hydrodynamic onset times.
• Generalization: While the proof covers diffusion and sound, extending the rigorous bound to other non-hydrodynamic relaxation modes remains an open area for future research.

In summary, this paper provides a foundational proof that hydrodynamics cannot emerge arbitrarily fast in local quantum systems; the onset time is fundamentally limited by the Planckian time scale $\hbar/T$, with the specific prefactor determined by the dimensionality and the nature of the hydrodynamic mode.