Domain coarsening in fractonic systems: a cascade of critical exponents

This paper demonstrates that in fractonic systems where the $m$-th multipole moment of the order parameter is conserved, domain coarsening after a quench follows an anomalously slow growth law of $R(t) \sim t^{1/(2m+3)}$, thereby establishing a new family of non-equilibrium universality classes characterized by a cascade of dynamical critical exponents.

The Gist

Imagine you have a large room filled with people, some wearing red shirts and some wearing blue shirts. At the start, they are mixed up randomly, like a chaotic crowd at a concert. Suddenly, the lights dim, and everyone decides they want to stand next to people wearing the same color.

This process of sorting themselves out into big red groups and big blue groups is called coarsening. In physics, this is how materials (like magnets or alloys) settle down after being heated and then cooled.

Usually, this sorting happens at a predictable speed. But this paper asks a fascinating question: What if there are strict rules about how people can move?

The Three Rules of the Game

The authors study three different "rulebooks" for how these people (spins) can swap places to form groups:

1. The Free-For-All (Glauber Dynamics)

• The Rule: Anyone can change their shirt color instantly if it helps them fit in better.
• The Result: The groups form quickly. It's like a crowd where people can just shout and change their minds. The size of the groups grows as the square root of time ($t^{1/2}$).
• Analogy: Imagine a dance floor where people can instantly switch partners. The dance floor clears up fast.

2. The "No Shirt Changes" Rule (Kawasaki Dynamics)

• The Rule: You can't change your shirt color. You can only swap places with a neighbor. If you are Red, you must stay Red; you just have to find a Red neighbor to swap with.
• The Result: This slows things down. To move a Red person from one side of the room to the other, they have to pass the baton of "Redness" through a long line of people. It's like a game of "pass the parcel." The groups grow slower ($t^{1/3}$).
• Analogy: Imagine a line of people holding hands. To move the person at the front to the back, everyone has to shuffle one step. It takes a lot of shuffling to move the whole line.

3. The "Fracton" Rules (The New Discovery)

• The Rule: This is the wild card. Not only can you not change your shirt, but you also have to keep the balance of the group perfect.
  • Dipole Conservation: If a Red person moves left, a Blue person must move right to keep the center of gravity balanced. You can't just move one person; you have to move a pair.
  • Quadrupole Conservation: It gets even stricter. You might need to move a whole "triangle" of people in a specific pattern just to shift one person.
• The Result: The sorting process becomes agonizingly slow. The paper calls these systems "fractonic" because the particles act like "fractons"—particles that are stuck unless they move in very specific, coordinated groups.

The "Cascade" of Slowness

The authors discovered a beautiful pattern, which they call a "cascade of critical exponents."

Think of the "speed" of the group formation as a ladder.

• Level 0 (Free): You climb the ladder fast.
• Level 1 (Conserved Charge): You climb slower.
• Level 2 (Conserved Dipole): You climb even slower.
• Level 3 (Conserved Quadrupole): You are practically crawling.

The math shows that if you conserve the $m$-th level of balance (where $m=0$ is just total number, $m=1$ is dipole, etc.), the time it takes for the groups to grow follows a specific formula: Time $\sim$ Size$^{2m+3}$.

• If you conserve the Dipole ($m=1$), the groups grow as $t^{1/5}$. That's much slower than the usual $t^{1/3}$.
• If you conserve the Quadrupole ($m=2$), they grow as $t^{1/7}$.

Why Does This Matter?

You might wonder, "If it's so slow, do the groups ever actually form?"

The authors proved that yes, they do, but it takes a very long time. They showed that even with these incredibly strict rules, there is always a way for the "Red" and "Blue" groups to merge and grow, provided the rules allow for moves that span a few steps (not just immediate neighbors).

However, the paper also highlights a "glassy" problem. At the very beginning, the system gets stuck. It's like trying to untangle a knot where you can only pull two threads at once. The system gets stuck in a "frozen" state for a long time before it finally starts moving again.

The Big Picture

This paper is like discovering a new type of traffic jam.

• In normal traffic (Glauber), cars can change lanes freely.
• In conserved traffic (Kawasaki), cars can't change lanes, they can only swap with the car next to them.
• In Fractonic traffic, cars can't move unless a whole convoy moves in a perfect geometric pattern.

The authors found that for every new rule you add to the traffic laws, the traffic moves exponentially slower. This creates a whole new family of "universality classes"—a fancy physics term for "types of behavior."

In simple terms: The more rules you put on how particles can move to organize themselves, the slower the organization happens. And this paper maps out exactly how much slower, revealing a hidden mathematical rhythm to the slowness of the universe.

Technical Summary

1. Problem Statement

The paper investigates the non-equilibrium dynamics of phase-ordering kinetics (coarsening) in systems subject to fractonic mobility constraints.

• Context: In standard statistical mechanics, when a system is quenched from a disordered phase to an ordered phase (e.g., the Ising model), domains of the ordered phase grow over time. The typical domain size $R(t)$ usually follows a power law $R(t) \sim t^{1/z}$, where $z$ is the coarsening exponent.
• Known Cases:
  • Glauber dynamics (non-conserved order parameter): $z=2$ ($R \sim t^{1/2}$).
  • Kawasaki dynamics (conserved total charge/magnetization): $z=3$ ($R \sim t^{1/3}$).
• The Gap: Recent studies on fracton models have revealed systems that conserve not only the total charge (monopole) but also higher spatial moments (dipole, quadrupole, etc.). While these constraints are known to cause subdiffusion and Hilbert space fragmentation at infinite temperature, their impact on low-temperature domain coarsening was largely unexplored.
• Core Question: How does the conservation of the $m$-th multipole moment of the order parameter affect the dynamical critical exponent $z$ and the mechanism of domain growth? Does the kinetic constraint lead to "frozen" dynamics where domains cannot grow indefinitely?

2. Methodology

The authors employ a combination of analytical field theory, heuristic scaling arguments, and large-scale Monte Carlo simulations.

A. Analytical Framework

1. Continuum Field Theory: They generalize the Cahn-Hilliard equation (Model B) to systems with multipole conservation.
   • Standard conservation leads to a current $J \sim \nabla \mu$.
   • $m$-th moment conservation implies the current is a derivative of a higher-order current, leading to a generalized evolution equation:
     $$\partial_t \phi = -(-\nabla^2)^{m+1} \left( \frac{\delta F}{\delta \phi} \right)$$
   • This results in subdiffusion of the order parameter: $\partial_t \phi \sim -(-\nabla^2)^{m+1} \phi$.
2. Scaling Arguments:
   • They apply the Gibbs-Thomson relation ($\mu \sim \kappa$, where $\kappa$ is curvature) to the generalized current.
   • For $m$-pole conservation, the velocity of a domain wall scales as $v \sim \nabla^{2m+1} \mu \sim 1/R^{2m+2}$.
   • Since $v = \partial_t R$, this yields the differential equation $\partial_t R \sim R^{-(2m+2)}$, leading to $R(t) \sim t^{1/(2m+3)}$.
3. Defect Relaxation: A rigorous derivation is provided by analyzing the relaxation rate $\omega(k)$ of a perturbed domain wall with wavevector $k$. They show $\omega(k) \sim k^{2m+3}$, confirming the exponent $z = 2m+3$.
4. Early-Time Corrections: The authors derive corrections for early times where energy-raising processes are suppressed. They show that for $m$-pole conservation, the early-time exponent is effectively $z_{early} = 2m + 3 + 2m = 4m+3$ (e.g., $z=7$ for dipole conservation instead of $z=5$).

B. Numerical Simulations

• Model: 2D Ferromagnetic Ising model on a square lattice.
• Dynamics: Custom Monte Carlo algorithms implementing local updates that strictly conserve the $m$-th multipole moment (e.g., exchanging dipoles for $m=1$).
• Scale: Simulations were pushed to $10^{10}$ Monte Carlo steps to access the asymptotic regime, which is significantly longer than standard coarsening simulations due to the slow dynamics.
• Observables: The domain size $R(t)$ was extracted from the first moment of the structure factor $S(k,t)$ and the equal-time spin-spin correlation function $C(r,t)$.

C. Combinatorial Proof (Ergodicity)

To address whether domains can grow indefinitely (i.e., is the system ergodic?), the authors provide a combinatorial argument. They prove that in the "weakly fragmented" regime (where update ranges are sufficiently large), the largest connected component of the configuration space contains states with domains scaling extensively with system size ($R \sim L$), ensuring that coarsening is not strictly frozen.

3. Key Contributions and Results

A. The Cascade of Critical Exponents

The primary result is the discovery of a new family of dynamical universality classes characterized by a cascade of exponents:
$$z = 2m + 3$$
Where $m$ is the highest conserved multipole moment:

• $m=0$ (Charge conserved, Kawasaki): $z = 3$ ($R \sim t^{1/3}$).
• $m=1$ (Dipole conserved): $z = 5$ ($R \sim t^{1/5}$).
• $m=2$ (Quadrupole conserved): $z = 7$ ($R \sim t^{1/7}$).

B. Mechanism of Growth

The paper elucidates that domain growth in fractonic systems is driven by the subdiffusive transport of the order parameter. Unlike standard diffusion where particles move independently, multipole conservation forces charges to move only as bound clusters (dipoles, quadrupoles), drastically slowing the redistribution of magnetization required to smooth domain walls.

C. Numerical Verification

• Dipole Conservation ($m=1$): Simulations confirmed the presence of an early-time regime with $z \approx 7$ (due to surface diffusion constraints) before crossing over to the asymptotic $z \approx 5$ regime. The data supports the $R(t) \sim t^{1/5}$ scaling, though reaching the pure asymptotic limit requires extremely long times.
• Quadrupole Conservation ($m=2$): Simulations showed even slower growth, with early-time scaling consistent with $z=11$ and asymptotic trends compatible with $z=7$.
• Scaling Collapse: The authors demonstrated that correlation functions $C(r,t)$ collapse onto a single universal curve when rescaled by $R(t)$, validating the dynamical scaling hypothesis even in these highly constrained systems.

D. Ergodicity and Fragmentation

The paper resolves the concern that multipole conservation might freeze the system. While "strongly fragmented" sectors exist (frozen states), the authors prove that for sufficiently long-range updates, the system remains in a weakly fragmented regime where the largest dynamical sector supports arbitrarily large domains.

4. Significance and Implications

• New Universality Classes: This work establishes a systematic classification of non-equilibrium universality classes based on the order of conserved multipole moments, extending the standard Hohenberg-Halperin classification.
• Fracton Physics: It bridges the gap between high-temperature fracton phenomenology (subdiffusion, Hilbert space fragmentation) and low-temperature thermodynamic phenomena (phase ordering).
• Experimental Relevance: The findings are relevant for quantum simulators (e.g., cold atoms in tilted potentials) where dipole conservation can be engineered. The predicted slow coarsening ($t^{1/5}$ or slower) offers a distinct experimental signature to detect emergent fractonic constraints.
• Glassy Dynamics: The interplay between slow transport and energy barriers suggests these systems may exhibit glassy behavior and aging, opening new avenues for studying non-ergodicity in classical and quantum systems.

In summary, the paper demonstrates that conserving higher moments of the order parameter fundamentally alters the kinetics of phase ordering, replacing standard diffusion with subdiffusion and generating a hierarchy of dynamical exponents $z = 2m+3$.