Noncommuting zero-noise and zero-frequency limits in particle-hole symmetric fluids

This paper demonstrates that in particle-hole symmetric charged fluids, the charge diffusion constant exhibits a discontinuous dependence on noise strength due to a noncommuting zero-noise and zero-frequency limit, where weak noise can induce singular changes like superdiffusion through a mechanism of hydrodynamic recoupling that invalidates standard zero-noise extrapolations.

The Gist

Imagine a busy highway where two types of traffic are moving: fast, sleek race cars (representing sound waves or energy) and slow, heavy delivery trucks (representing electric charge).

In a special kind of fluid, like the electrons in a piece of graphene at a specific balance point, these two types of traffic have a unique relationship. Because of a rule called "particle-hole symmetry," the fast race cars and the slow trucks don't usually bump into each other. The race cars zoom past the trucks without disturbing them. As a result, the trucks move in a predictable, steady way (diffusion), while the race cars zoom along in perfect, straight lines (ballistic motion).

The Big Surprise: The "Zero-Noise" Trap

The researchers in this paper discovered a strange trap that happens when you try to predict how the trucks move by adding tiny amounts of "noise" (random bumps or friction) and then trying to imagine what happens if you remove that noise completely.

Usually, if you add a little bit of friction to a system and then take it away, the system smoothly returns to its original behavior. But here, it doesn't work that way. The paper shows that the behavior of the trucks changes discontinuously.

• The Analogy: Imagine the race cars are so fast they usually just pass the trucks once and never look back.
  • Scenario A (Perfectly Smooth Road): If the road is perfectly smooth (no noise), the race cars zoom by once, and the trucks keep moving steadily.
  • Scenario B (Slightly Bumpy Road): If you add even the tiniest bit of "bumpiness" (noise), the race cars start to slow down and bounce back and forth. Now, instead of passing the trucks once, a single race car might bounce back and hit the same truck over and over again.

This repeated bouncing changes the truck's movement entirely. The paper proves that if you try to calculate the truck's speed by starting with a bumpy road and slowly smoothing it out to zero, you will get a completely wrong answer. The answer you get depends entirely on how you smooth the road (whether you smooth out the energy bumps first or the momentum bumps first).

The Two Weird Outcomes

The paper highlights two specific weird scenarios that happen when you introduce this tiny bit of noise:

1. The "Super-Speed" Truck (Superdiffusion):
   If you keep the energy conservation perfect but add a tiny bit of noise that breaks momentum conservation, the trucks don't just move faster; they move wildly faster. The race cars, now bouncing around, start pushing the trucks in the same direction repeatedly. It's like a crowd of people pushing a stalled car; if they all push in the same rhythm, the car shoots forward. The paper calls this "superdiffusion," and mathematically, the "diffusion constant" (a measure of how fast things spread) actually blows up to infinity.

2. The "Stuck" Truck (Subdiffusion):
   If you do the opposite (keep momentum perfect but break energy conservation), the race cars bounce back and forth in a way that cancels themselves out. They push the truck forward, then backward, then forward again. The truck ends up moving much slower than it should, almost getting stuck. This is called "subdiffusion."

Why This Matters

The main takeaway is a warning for scientists and computer simulators. Many researchers use a technique called "zero-noise extrapolation." They run a computer simulation with a little bit of noise (because real computers have limits) and then try to guess what the result would be with no noise.

This paper says: Don't do that for this specific type of fluid.

If you use that method here, you will get a number that looks reasonable, but it will be completely wrong compared to the true, noise-free reality. The true behavior is a "singular" jump that you can't see if you are just looking at the noisy data.

The "Hydrodynamic Recoupling"

The authors call the mechanism behind this "hydrodynamic recoupling."

• Decoupled: In the perfect world, sound waves and charge are strangers who ignore each other.
• Recoupled: In the noisy world, the noise forces them to interact repeatedly. The sound waves act like a "bath" that the charge is constantly swimming in, getting kicked around in a very specific, long-lasting way.

In Summary
The paper reveals that in certain symmetric fluids, the way charge moves is incredibly sensitive to tiny imperfections. The relationship between "no noise" and "a little noise" is broken. You cannot simply smooth out the noise to find the truth; the truth is a different world entirely where the rules of movement change based on the specific type of noise you introduce.

Technical Summary

Technical Summary: Noncommuting Zero-Noise and Zero-Frequency Limits in Particle-Hole Symmetric Fluids

Problem Statement
In charged fluids exhibiting particle-hole symmetry (e.g., the Dirac fluid in graphene at charge neutrality), charge transport is diffusive even when momentum is conserved, due to the hydrodynamic decoupling of charge fluctuations from ballistically propagating sound waves. While this symmetry-protected diffusion is well-understood in the strictly conserved limit, it remains unclear how this transport regime connects to "regular" diffusion when conservation laws are weakly broken by noise or impurities. Specifically, the behavior of the charge diffusion constant $D$ as the energy ($\gamma_e$) and momentum ($\gamma_p$) relaxation rates are simultaneously tuned to zero is the central question. Standard hydrodynamic intuition suggests a smooth crossover, but the authors investigate whether the zero-noise limit is singular.

Methodology
The authors employ a combination of analytical hydrodynamic theory and numerical simulations:

1. Analytical Framework: They utilize a minimal three-mode hydrodynamic model describing conserved densities of energy ($e$), momentum ($p$), and charge ($n$) in quasi-one-dimensional geometries. The model incorporates particle-hole symmetry, which dictates that charge is odd under charge conjugation while energy and momentum are even.
2. Perturbative Analysis: They introduce relaxation rates $\gamma_e$ and $\gamma_p$ to break conservation laws. The charge diffusion constant is calculated via the Green-Kubo formula, relating it to the integrated autocorrelation function of the charge current. This involves analyzing the stochastic advection of charge packets by sound waves (convective contribution) alongside Fickian diffusion.
3. Limiting Cases: The authors analyze specific limits:
   • Energy-conserving ($\gamma_e=0$): Where momentum relaxes but energy is conserved.
   • Momentum-conserving ($\gamma_p=0$): Where energy relaxes but momentum is conserved.
   • General Ray: Approaching the origin $(\gamma_e, \gamma_p) \to (0,0)$ along arbitrary rays.
4. Numerical Validation: They construct a stochastic gas model that realizes the same linear fluctuating hydrodynamics. In this model, particles carry charge labels, and symmetry-breaking is implemented via stochastic particle splitting/merging (breaking number conservation, acting as energy relaxation) and stochastic velocity sign flips (breaking momentum conservation).

Key Results
The paper establishes that the zero-noise limit of the charge diffusion constant is singular and non-universal:

1. Discontinuous Diffusion Constant: The diffusion constant $D(\gamma_e, \gamma_p)$ does not converge smoothly to the strictly conserved value $D(0,0)$. Instead, as $\gamma_e, \gamma_p \to 0$, $D$ converges to a ray-dependent value determined by the ratio $\gamma_e/\gamma_p$.

   • The general scaling form is $D(\gamma_e, \gamma_p) \approx D_{\text{Fick}} + D_{\text{conv}}(\gamma_e/\gamma_p)^{1/2}$.
   • The convective contribution $D_{\text{conv}}$ diverges as $\gamma_p \to 0$ (with $\gamma_e \neq 0$), leading to superdiffusive transport.
   • Conversely, if $\gamma_e \to 0$ (with $\gamma_p \neq 0$), the convective contribution vanishes, leading to subdiffusive behavior ($X(t) \sim t^{1/4}$).

2. Hydrodynamic Recoupling: The singularity arises from a mechanism termed "hydrodynamic recoupling." In the strictly conserved limit, sound waves and charge fluctuations decouple. When weak relaxation is introduced, sound waves undergo repeated backscattering.

   • If energy is conserved ($\gamma_e=0$), backscattering reverses momentum, causing anticorrelated kicks that suppress diffusion.
   • If momentum is conserved ($\gamma_p=0$), backscattering preserves momentum direction relative to energy flow, causing correlated kicks that enhance diffusion.
   • This creates a long timescale in the current autocorrelation function, making the zero-noise and zero-frequency limits non-commuting.

3. Superdiffusion and KPZ Universality: In the limit where only momentum is conserved ($\gamma_p=0, \gamma_e \neq 0$), the coupled dynamics of momentum and charge are described by a degenerate limit of coupled Burgers equations. This leads to superdiffusive charge transport with a variance scaling as $t^{2/3}$ (Kardar-Parisi-Zhang universality), distinct from the linear hydrodynamic prediction of $t^{3/4}$ at intermediate times.

4. Numerical Confirmation: Simulations of the stochastic gas model confirm the analytical predictions. The time-dependent diffusion constant $D(t)$ exhibits plateaus that depend on the ratio $\gamma_e/\gamma_p$. The data confirms the square-root singularity $D - D_{\text{Fick}} \sim \sqrt{\gamma_e/\gamma_p}$ and the distinct scaling behaviors in the pure momentum-breaking and pure energy-breaking limits.

Significance and Claims
The authors claim that their results demonstrate a fundamental limitation of zero-noise extrapolation techniques used in both classical and quantum simulations to compute transport coefficients.

• Failure of Extrapolation: Because the diffusion constant is a discontinuous function of the noise strength (depending on the path taken to zero noise), extrapolating from finite-noise data to the zero-noise limit yields a value that is unrelated to the true zero-noise diffusion constant.
• Generality: Unlike previous findings of discontinuities in integrable spin chains, this phenomenon is shown to be general, relying only on particle-hole symmetry and hydrodynamic principles, not on integrability or specific noise models.
• Physical Mechanism: The work identifies "hydrodynamic recoupling" as a general mechanism by which weak noise can induce singular changes in transport coefficients, fundamentally altering the nature of charge transport from diffusive to superdiffusive or subdiffusive depending on the symmetry of the perturbation.

The paper concludes that while the specific case of particle-hole symmetric fluids presents an explicit example of this failure, the underlying mechanism suggests that zero-noise extrapolation may be unreliable for predicting dynamical quantities in systems where conserved quantities are coupled via symmetry-protected hydrodynamic modes.