Dynamic scaling and Family-Vicsek universality in $SU(N)$ quantum spin chains

This paper demonstrates that the Family-Vicsek scaling framework, traditionally used for classical surface growth, universally describes the infinite-temperature dynamics of one-dimensional $SU(N)$ quantum spin chains, revealing distinct ballistic, superdiffusive, and diffusive transport regimes characterized by specific dynamical exponents that are determined by the system's integrability and symmetry properties.

The Gist

Imagine you are watching a crowd of people in a long hallway. In a calm, orderly situation, people might walk in straight lines without bumping into each other. But in a chaotic, crowded party, they jostle, bump, and spread out randomly.

This paper is about studying how "chaos" or "fluctuations" spread through a line of quantum particles (specifically, tiny magnets called spins) at extremely high temperatures. The researchers wanted to see if the rules that govern how rough a surface gets over time (like sand piling up on a beach) also apply to these invisible quantum particles.

Here is a breakdown of their findings using simple analogies:

The Big Idea: The "Roughness" of a Quantum Line

In the physical world, if you watch a surface grow (like snow accumulating or paint drying), it starts smooth and gets rougher over time. Scientists have a famous rule called Family-Vicsek scaling that predicts exactly how fast that roughness grows and how it depends on the size of the area you are looking at.

The authors asked: Does this same rule apply to the invisible "roughness" of quantum spins?
To answer this, they treated the quantum spins like a line of people. They measured how much the "mood" (spin direction) of a specific group of people fluctuated over time. They found that yes, the same mathematical rules apply to quantum particles as they do to classical surfaces.

The Three Types of "Traffic"

The researchers studied two different types of quantum "traffic jams" (models) and found that the behavior changes depending on how the particles interact with each other. They identified three distinct regimes, which they compared to different ways a crowd might move:

1. The Bullet Train (Ballistic Transport):

   • What it is: When the particles don't really interact with each other, they zip down the line in perfect straight lines, like a bullet or a bullet train.
   • The Result: The "roughness" grows very fast. The particles move so efficiently that the disturbance spreads quickly.
   • Analogy: Imagine a hallway where everyone is running in a straight line without stopping. The "noise" of their movement spreads instantly.

2. The Super-Organized Dance (Superdiffusive / KPZ Transport):

   • What it is: This happens when the particles have a very special, perfect symmetry (like a perfect dance routine where everyone knows exactly what the next person will do). This is called "integrability."
   • The Result: The movement is faster than random walking but slower than a bullet train. It follows a specific, complex pattern known as the KPZ (Kardar-Parisi-Zhang) scaling.
   • Analogy: Imagine a line of dancers who are perfectly synchronized. They move together in a wave-like motion that is more efficient than random stumbling but not as straight as a bullet train. This only happens when the "dance rules" (symmetry) are perfectly preserved.

3. The Random Stumble (Diffusive Transport):

   • What it is: This is the most common state. The particles bump into each other randomly, like people in a crowded, chaotic mosh pit.
   • The Result: The "roughness" spreads slowly, following a standard "diffusive" pattern (like a drop of ink spreading in water).
   • Analogy: Imagine trying to walk through a crowded market. You bump into people, change direction, and move slowly. The disturbance spreads slowly and evenly.

The "Magic Switch": Breaking the Rules

The most important discovery in the paper is what happens when you break the perfect order.

• The "Integrability" Switch: In the quantum world, some systems are "integrable," meaning they have perfect mathematical rules that prevent chaos. The researchers found that as long as these perfect rules exist, the system can show the "Super-Organized Dance" (KPZ) behavior.
• The "Chaos" Switch: However, the moment you introduce a tiny bit of imperfection or "break" the symmetry (by adding a small extra interaction between particles), the system immediately loses its special behavior.
• The Result: No matter how you start the system, if you break the perfect rules, it always collapses into the "Random Stumble" (Diffusive) mode. The special, fast-moving patterns disappear, and the system behaves like a standard, messy crowd.

The Two Models They Tested

They tested this on two specific "playgrounds":

1. The XXZ Model (Spin-1/2): Think of this as a line of simple magnets that can point up or down. They found all three traffic types here depending on how the magnets were tuned.
2. The Izergin-Korepin Model (Spin-1): This is a more complex version where the magnets have more options (three states instead of two). They found the same pattern: perfect symmetry leads to the "Super-Organized Dance," but breaking that symmetry leads to the "Random Stumble."

The Takeaway

The paper concludes that the Family-Vicsek scaling is a universal law. It doesn't matter if you are looking at a growing sand dune (classical physics) or a line of quantum magnets (quantum physics). If the system is perfectly ordered, it moves in a special, fast way. But the moment you break that order, it reverts to the standard, slow, random spreading of chaos.

In short: Perfect symmetry allows for special, fast quantum transport, but any imperfection forces the system to behave like a normal, diffusing crowd.

Technical Summary

Technical Summary: Dynamic Scaling and Family-Vicsek Universality in SU(N) Quantum Spin Chains

Problem Statement
The Family-Vicsek (FV) scaling framework is a well-established paradigm for describing the universal dynamics of surface growth and roughness evolution in non-equilibrium classical systems. While universal scaling laws are recognized in quantum systems, the applicability of FV scaling specifically to the dynamics of strongly correlated, interacting quantum many-body systems remains largely unexplored. Previous investigations have largely focused on non-interacting or weakly interacting systems. This work addresses the gap in understanding whether FV scaling holds for strongly correlated quantum spin chains, specifically those with SU(N) symmetries, and how integrability and symmetry breaking influence these scaling behaviors.

Methodology
The authors investigate the infinite-temperature dynamics of one-dimensional SU(N) spin chains, focusing on two primary models:

1. The SU(2) spin-1/2 XXZ model.
2. The SU(3) spin-1 Izergin-Korepin (IK) model.

To overcome the computational challenges associated with accessing large system sizes and long timescales required for robust scaling analysis, the study employs the Quantum Generating Function (QGF) approach. This method involves:

• Calculating the Heisenberg time evolution of a unitary operator $R_\Sigma(\lambda) = \exp(i\lambda\Sigma)$, where $\Sigma$ is a conserved U(1) charge (specifically the total spin in a subsystem).
• Computing the generating function $G_\Sigma(\lambda, t) = \text{Tr}\langle R_\Sigma(\lambda, t)R^\dagger_\Sigma(\lambda, 0) \rangle_{\tilde{\rho}}$ at infinite temperature.
• Extracting the second cumulant of spin fluctuations, $\kappa_{2,\Sigma}(t)$, which serves as the quantum analogue of classical surface roughness ($W$).
• Utilizing the Time-Evolving Block Decimation (TEBD) algorithm via the ITensor library to simulate chains with lengths up to $L=2000$ sites and evolution times up to $t \cdot J = 2000$.

The study systematically varies anisotropy parameters ($\Delta$ for XXZ, $\tilde{\Delta}$ for IK) and introduces integrability-breaking terms (next-nearest-neighbor interactions) to map out different transport regimes.

Key Contributions and Results

1. Validation of FV Scaling in Quantum Systems:
   The authors demonstrate that the quantum analogue of surface roughness, defined as $W_{S^z}(l, t) = \sqrt{\kappa_{2,S^z_l}(t)}$, obeys the FV scaling hypothesis $W(l, t) \sim l^\alpha f(t/l^z)$. The scaling function $f(x)$ exhibits the expected power-law behavior in early times ($t \ll l^z$) and saturation in late times ($t \gg l^z$).

2. Transport Regimes in the XXZ Model:

   • Ballistic Regime ($\Delta < 1$): In the easy-plane regime, the system exhibits ballistic transport with a dynamical exponent $z=1$. The growth exponent is $\beta=1/2$, and the roughness exponent is $\alpha=1/2$.
   • Superdiffusive/KPZ Regime ($\Delta = 1$): At the SU(2) symmetric point, the integrable system displays superdiffusive transport characterized by Kardar-Parisi-Zhang (KPZ) scaling, with $z=3/2$ and $\beta=1/3$.
   • Diffusive Regime ($\Delta > 1$): In the easy-axis regime, transport becomes diffusive-like with $z=2$ and $\beta=1/4$.
   • Integrability Breaking: Introducing next-nearest-neighbor interactions breaks integrability and universally drives the system into the diffusive regime ($z=2$), regardless of the anisotropy value or whether SU(2) symmetry is preserved.

3. Transport Regimes in the IK Model:

   • SU(3) Symmetric Point ($\tilde{\Delta} = 0$): The model exhibits superdiffusive KPZ scaling with $z=3/2$.
   • Symmetry Breaking ($\tilde{\Delta} \neq 0$): Breaking the global SU(3) symmetry down to U(1) induces a transition to diffusive scaling ($z=2$).
   • Integrability Breaking: Similar to the XXZ model, breaking integrability via next-nearest-neighbor interactions forces the system into the diffusive Edwards-Wilkinson (EW) regime ($z=2$).

4. Universality of Exponents:
   Across all regimes and models studied, the roughness exponent remains consistently $\alpha = 1/2$. This is attributed to the infinite-temperature environment where, at long times, uncorrelated spins of random sign populate the subsystem, yielding $|\Delta S^z_l| \sim l^{1/2}$. The dynamical exponent $z$ and growth exponent $\beta$ vary according to the transport regime, satisfying the FV relation $z = \alpha/\beta$.

Significance and Claims
The paper claims that Family-Vicsek scaling is not limited to classical surface growth but extends universally to quantum many-body models with SU(N) symmetry. The study establishes that:

• The QGF method is a powerful tool for extracting cumulants and verifying scaling laws in strongly correlated systems where other methods fail due to system size limitations.
• The dynamical exponent $z$ serves as a robust classifier for transport regimes in quantum spin chains: ballistic ($z=1$), KPZ superdiffusive ($z=3/2$), and diffusive ($z=2$).
• Integrability is a necessary condition for observing superdiffusive (KPZ) transport in these models; breaking integrability universally restores diffusive (Edwards-Wilkinson) behavior.
• The results align with predictions from Generalized Hydrodynamics (GHD) and recent studies on integrable classical spin models, suggesting that the underlying symmetry structure (non-Abelian vs. Abelian) is the crucial determinant of the dynamical universality class in both quantum and classical settings.

The work does not propose new experimental setups or specific future applications beyond the theoretical framework of quantum transport and scaling laws, maintaining a focus on the fundamental universality of these phenomena in quantum statistical mechanics.