Dynamic scaling and Family-Vicsek universality in the Hubbard model at infinite temperature

This paper investigates Family-Vicsek scaling of charge, spin, and energy fluctuations in the one-dimensional Hubbard model at infinite temperature, revealing that while integrable systems exhibit ballistic or KPZ transport regimes and broken integrability leads to diffusion, all cases display a universal short-time ballistic growth before entering their respective hydrodynamic scaling windows.

The Gist

Imagine a long, crowded hallway filled with people (electrons) who can move left or right. This hallway represents a one-dimensional chain of atoms in a material called the Hubbard model. The paper investigates how "messiness" or "fluctuations" spread through this hallway when everyone is moving chaotically at maximum speed (infinite temperature).

The researchers are trying to answer a simple question: How does the disorder in a specific section of this hallway grow over time?

To visualize this, think of the "messiness" as the height of a pile of sand or the roughness of a wall being painted. In physics, this is called Family-Vicsek scaling. It's a rulebook that predicts how rough a surface gets based on two things: how big the section you are looking at is, and how much time has passed.

Here is what the paper discovered, broken down into everyday concepts:

1. The Three Types of "Traffic"

The researchers looked at three different things moving through the hallway:

• Charge: The movement of people themselves (electrons).
• Spin: The direction people are facing (up or down).
• Energy: The total activity or "buzz" of the crowd.

They found that how these three things spread depends entirely on the "rules of the game" (the interactions between the people).

2. The Three Scenarios

Scenario A: The Free-Flowing Crowd (No Interactions)
Imagine the people in the hallway don't bump into each other at all. They just walk straight.

• Result: Everything moves in a straight line at a constant speed. This is called Ballistic transport.
• Analogy: Like cars on an empty highway with no traffic lights. If you look at a section of the road, the "messiness" (fluctuations) grows steadily and predictably.
• Who behaves this way? Charge, Spin, and Energy all do this when there are no interactions.

Scenario B: The "Integrable" Crowd (Strict Rules, But Interacting)
Now, imagine the people bump into each other, but they follow a very strict, magical set of rules (mathematical "integrability") that prevents total chaos. They can't just do whatever they want; their movements are highly coordinated.

• Charge & Spin: These two get stuck in a weird, super-diffusive state called KPZ scaling.
  • Analogy: Imagine a crowd trying to form a line, but they keep bumping into each other in a way that creates a "traffic jam" that grows faster than normal but slower than a free-flow. It's like a wave of people moving through a concert venue where everyone is trying to dance in sync but getting in each other's way. The "roughness" grows in a specific, curved pattern.
• Energy: Surprisingly, the energy still moves like the free-flowing crowd (Ballistic).
  • Analogy: Even though the people are bumping into each other, the "buzz" or "heat" of the room still zips through instantly, unaffected by the traffic jam of the people themselves.

Scenario C: The Chaotic Crowd (Broken Rules)
Finally, the researchers broke the magical rules by adding a new, messy interaction (people bumping into their neighbors' neighbors). This destroys the "integrability."

• Result: Everything becomes Diffusive.
• Analogy: This is like a crowded party where everyone is bumping into everyone else randomly. If you drop a drop of dye in the water, it spreads out slowly and spreads out evenly. The "messiness" grows much slower than in the previous scenarios.
• Who behaves this way? Charge, Spin, and Energy all slow down and become diffusive when the rules are broken.

3. The "Microscopic" Surprise

Before any of these long-term patterns (the highway, the traffic jam, or the party) fully set in, the researchers found a very short, initial moment where everything behaves the same way: it grows very fast, like a ball being thrown.

• Analogy: No matter what the rules are later, if you look at the very first split second, the "messiness" shoots up quickly before settling into its long-term rhythm. This is a universal "microscopic regime" that happens before the big picture emerges.

Summary of the Findings

The paper concludes that Integrability (the existence of those strict, magical rules) is the boss.

• If the rules are perfect (Integrable): Charge and Spin get stuck in a "traffic jam" (KPZ), but Energy zooms through (Ballistic).
• If the rules are broken (Non-Integrable): Everything slows down to a slow, random spread (Diffusive).
• If there are no people bumping into each other (Free): Everything zooms through (Ballistic).

The researchers used a clever mathematical tool (Quantum Generating Function) to count these fluctuations without having to track every single person in the hallway, allowing them to see these patterns clearly. They confirmed that the "roughness" of the system follows a universal mathematical law, but the speed at which it grows depends entirely on whether the system is following those strict rules or not.

Technical Summary

Technical Summary: Dynamic Scaling and Family–Vicsek Universality in the Hubbard Model at Infinite Temperature

Problem Statement
This work investigates the transport properties and fluctuation dynamics of the one-dimensional Fermi–Hubbard model at infinite temperature. Specifically, the authors aim to characterize the scaling behavior of charge, spin, and energy fluctuations within subsystems. The study focuses on how these fluctuations evolve from short-time microscopic regimes to long-time hydrodynamic windows, and how these behaviors depend on the system's integrability. The central question is whether the Family–Vicsek (FV) scaling framework, originally developed for classical interface growth, applies to quantum many-body systems and how the universality classes of charge, spin, and energy sectors differ under varying interaction strengths and integrability conditions.

Methodology
The authors employ a Quantum Generating Function (QGF) approach to compute time-dependent cumulants of transferred conserved quantities without reconstructing the full probability distribution.

• Model: The study utilizes the 1D Hubbard Hamiltonian ($H_{\text{Hubb}}$) with nearest-neighbor hopping ($J$) and on-site interaction ($U$). To probe the effects of broken integrability, a next-nearest-neighbor density interaction ($V$) is added ($H = H_{\text{Hubb}} + H_{\text{NNN}}$).
• Initial State: Simulations are performed at infinite temperature, represented by a fully mixed density matrix $\rho_\infty = 1/4^L$, corresponding to half-filling and zero magnetization.
• Numerical Technique: The QGF is evaluated using Matrix Product Operator (MPO) representations and Time-Evolving Block Decimation (TEBD) in Liouville space. The method involves applying a "twist" operator $R(\lambda) = e^{i\lambda Q_\ell}$ to a segment of the chain, evolving the state, and extracting cumulants from the characteristic function.
• Scaling Analysis: The authors analyze the "roughness" $W(\ell, t)$, defined as the square root of the second cumulant of the transferred quantity in a subsystem of size $\ell$. They test the FV scaling ansatz $W(\ell, t) \sim \ell^\alpha f(t/\ell^z)$ to extract the roughness exponent ($\alpha$), growth exponent ($\beta$), and dynamical exponent ($z$).

Key Contributions and Results
The paper establishes a comprehensive mapping of transport regimes in the Hubbard model by analyzing three distinct sectors: charge, spin, and energy.

1. Universal Short-Time Microscopic Regime:
   Across all sectors and parameter regimes (free, integrable interacting, and non-integrable), the authors observe a short-time regime where fluctuations grow ballistically ($W \sim t$). This precedes the emergence of the hydrodynamic scaling window.

2. Charge and Spin Sectors:

   • Free Limit ($U=0, V=0$): Both charge and spin exhibit ballistic transport with dynamical exponent $z=1$ (growth exponent $\beta=1/2$).
   • Integrable Interacting Limit ($U \neq 0, V=0$): The system transitions to the Kardar-Parisi-Zhang (KPZ) universality class. The dynamical exponent becomes $z=3/2$ ($\beta=1/3$), indicating superdiffusive transport.
   • Non-Integrable Limit ($V \neq 0$): Breaking integrability drives the system into a diffusive regime with $z=2$ ($\beta=1/4$).

3. Energy Sector:

   • Integrable Regime ($V=0$): Unlike charge and spin, the energy sector remains ballistic ($z=1$) even in the presence of finite on-site interactions ($U \neq 0$). The authors attribute this to the fact that the energy current does not couple to the non-Abelian charges responsible for the KPZ anomaly in the other sectors.
   • Non-Integrable Regime ($V \neq 0$): Similar to the other sectors, breaking integrability renders the energy transport diffusive ($z=2$).

4. Universality of Roughness:
   The roughness exponent $\alpha$ is found to be universal ($\alpha = 1/2$) across all sectors and regimes, consistent with expectations for one-dimensional systems.

Significance and Claims
The paper claims that the Family–Vicsek scaling framework provides a compact and unified method to organize and distinguish ballistic, anomalous (KPZ), and diffusive transport within the Hubbard model. The primary significance lies in demonstrating that integrability is the controlling factor for long-time dynamics, but its effect is sector-dependent:

• In the integrable Hubbard chain, charge and spin exhibit KPZ scaling, while energy remains ballistic.
• The breakdown of integrability (via $V$) universally restores diffusive behavior across all sectors.

The authors emphasize that their results connect microscopic conservation laws and integrability to emergent hydrodynamics. They note that while previous works focused on noninteracting systems or specific limits, this study systematically compares charge, spin, and energy on equal footing in both integrable and non-integrable interacting regimes. The findings suggest that energy transport follows a distinct route from charge and spin along the integrable line, a nuance captured effectively by the FV scaling analysis. The work is motivated by the availability of cold-atom quantum simulators capable of probing such high-temperature, far-from-equilibrium dynamics.