Integrability-breaking-induced Mpemba effect in spin chains

This paper demonstrates that weakly broken integrability in spin chains induces a symmetry-restoration Mpemba effect through two distinct mechanisms: an early-time crossing where hotter systems equilibrate faster due to larger phase space, and a late-time crossing where colder systems overtake them because they sustain superdiffusive spin hydrodynamics for a parametrically longer duration in non-integrable models.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Idea: The "Hot Water" Paradox in a New Guise

You've probably heard of the Mpemba effect: the strange observation that, under certain conditions, hot water can freeze faster than cold water. It sounds impossible, but it happens.

Usually, we think of this in terms of cooling down (dissipating heat). But this paper explores a different version of the paradox in the world of spinning magnets (spin chains). Instead of cooling down, these systems are trying to "relax" or "settle down" into a balanced, symmetrical state after being shaken up.

The researchers found that in these magnetic chains, hotter systems can sometimes reach balance faster than colder ones, but for two very different reasons. It's like a race where the rules change halfway through.

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The Setup: The Magnetic Dance Floor

Imagine a long line of dancers (the spins) holding hands.

• The Goal: They want to dance in a perfect circle, facing every direction equally (this is called isotropy).
• The Start: The researchers "break" the symmetry by telling all the dancers to stop looking up and down (the Z-axis) and only look left and right.
• The Race: Once the music starts (the system evolves), the dancers naturally want to return to looking in all directions. The paper measures how long it takes for the "up/down" looking to return to normal.

They tested two types of dance floors:

1. The "Perfect" Floor (Integrable): A floor with perfect rules where dancers move in predictable, non-colliding patterns.
2. The "Messy" Floor (Non-Integrable): A floor where dancers bump into each other, creating chaos and mixing things up.

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Mechanism 1: The "Hot Mess" Advantage (Early Time)

The Analogy: Imagine two groups of people trying to untangle a giant knot of headphones.

• Group A (Cold): They are calm, moving slowly, and carefully.
• Group B (Hot): They are frantic, moving fast, and flailing their arms everywhere.

If the knot is very tight (high symmetry breaking), the frantic group (Hot) might actually untangle it faster at first. Why? Because they have so much energy and movement (phase space) that they scramble the initial mess quickly. The calm group (Cold) is too slow to make much progress.

The Result: In the very beginning of the race, the Hot system often wins. It restores balance faster because its chaos helps it escape the initial "bad" state quickly. This happens on both the Perfect and Messy dance floors.

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Mechanism 2: The "Ghost Runner" Effect (Late Time)

The Analogy: Now, imagine the race continues for a long time.

• On the Perfect Floor: The dancers move in perfect waves. They never crash. They keep moving super-fast (super-diffusive) forever.
• On the Messy Floor: The dancers eventually start bumping into each other.
  • The Hot Dancers: Because they are moving so fast, they crash into each other constantly. These crashes kill their momentum quickly. They slow down and start walking (diffusive) very soon.
  • The Cold Dancers: They move slowly. Because they are slow, they don't crash as often. They can keep running in their special "super-fast wave" mode for a very long time before they finally start walking.

The Twist: Even though the Hot dancers started fast, they burned out. The Cold dancers, who started slow, managed to keep their "super-speed" mode going for much longer. Eventually, the Cold dancers overtake the Hot ones and finish the race first.

The Result: In the Messy (Non-Integrable) systems, the Cold system can win at the end. This is the "Integrability-Breaking Induced Mpemba Effect." It only happens because the "messiness" (breaking integrability) kills the Hot system's speed faster than the Cold system's.

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The Four Possible Outcomes

Depending on how hot the system is and how "messy" the floor is, four things can happen (as shown in the paper's figures):

1. No Crossing: The Hot system starts closer to the finish line and stays ahead. (Hot wins).
2. Early Crossing Only: The Hot system starts far behind but sprints past the Cold one quickly, then slows down. The Cold one never catches up. (Hot wins, but it was close).
3. Double Crossing: The Hot system sprints past the Cold one early, but the Cold system eventually catches up and passes it late. (Hot wins early, Cold wins late).
4. Late Crossing Only (The True Mpemba): The Hot system starts far behind and never catches up early. But because the Cold system keeps its "super-speed" for so long, it eventually overtakes the Hot one. The Cold system wins.

Why Does This Matter?

This paper teaches us that perfect order (integrability) is rare in the real world. In the real world, things are slightly "messy" (integrability is broken).

The authors show that this "messiness" isn't just a nuisance; it creates a new kind of physics. It allows cold systems to maintain their speed longer than hot ones, leading to a counter-intuitive race where the "cold" team wins the long-distance marathon.

In short:

• Hot systems are like sprinters: fast at the start, but they burn out quickly if the track is bumpy.
• Cold systems are like marathoners: slow at the start, but if the track is bumpy, they can keep their rhythm longer than the sprinters.
• The Mpemba Effect is the moment the marathoner passes the sprinter.

Technical Summary

Here is a detailed technical summary of the paper "Integrability-breaking-induced Mpemba effect in spin chains" by Adam J. McRoberts.

1. Problem Statement

The paper investigates the Mpemba effect in the context of closed, non-dissipative classical spin chains. Specifically, it addresses the symmetry-restoration Mpemba effect, a phenomenon where a system initially further from equilibrium (in terms of symmetry breaking) reaches the equilibrium state faster than a system starting closer to equilibrium.

While this effect has been observed in integrable quantum models and various dissipative systems, the paper seeks to understand its behavior in classical spin chains with weakly broken integrability. The core question is how the transition from integrable (superdiffusive) to non-integrable (diffusive) dynamics, driven by the breaking of integrability, influences the equilibration rates of systems at different temperatures.

2. Methodology

Models and Hamiltonians

The author studies two primary classical spin chain models and an interpolating model:

1. Integrable Ishimori Chain: $H = -2J \sum_i \log\left(\frac{1 + \mathbf{S}_i \cdot \mathbf{S}_{i+1}}{2}\right)$.
2. Non-integrable Heisenberg Chain: $H = -J \sum_i (\mathbf{S}_i \cdot \mathbf{S}_{i+1} - 1)$.
3. Interpolating Model: A Hamiltonian parameterized by $\gamma$ (where $\delta = 1-\gamma$ represents the degree of integrability-breaking), which reduces to the Ishimori chain at $\gamma=1$ and the Heisenberg chain as $\gamma \to 0$.

Quench Protocol

• Initial State: Thermal ensembles are generated at specific temperatures ($T$) using heatbath Monte Carlo.
• Symmetry Breaking: The system is quenched out of equilibrium by suppressing the $z$-components of the spins by a factor $\eta$ ($S_z \to \eta S_z$), while rescaling in-plane components to maintain normalization. This breaks rotational isotropy without inducing net magnetization.
• Dynamics: The system evolves under the Hamiltonian of the original (unquenched) ensemble. The equations of motion are derived from the Poisson brackets.

Observable

The degree of isotropy restoration is measured by the quantity $\delta K_z(t)$:
$$\delta K_z(t) = 1 - K_z(t) = 1 - 3 \langle (S_z^i(t))^2 \rangle$$
where the average is over sites and ensemble members. $\delta K_z = 0$ indicates full isotropy. The Mpemba effect is defined as a crossing of equilibration curves where a system with higher initial symmetry breaking (lower $\eta$ or higher $T$) eventually has a lower $\delta K_z(t)$ than a system with lower initial breaking.

3. Key Contributions and Mechanisms

The paper identifies two distinct, independent mechanisms that cause equilibration curves to cross, leading to the Mpemba effect. Depending on system parameters, these mechanisms can reinforce each other, cancel each other out, or act alone.

Mechanism I: Early-Time Crossing (Phase Space Scrambling)

• Origin: Hotter systems possess a larger phase space volume, allowing them to scramble initial conditions more rapidly.
• Behavior: If the hotter system starts with a higher degree of symmetry breaking, it may initially equilibrate faster than the colder system.
• Scope: This mechanism occurs in both integrable and non-integrable systems.
• Mathematical Basis: Derived from the Maclaurin series expansion of the dynamics, yielding an early-time equilibration coefficient $\Gamma(T, \eta)$.

Mechanism II: Late-Time Crossing (Integrability-Breaking-Induced)

• Origin: This is a hydrodynamic effect unique to non-integrable chains.
• Physics:
  • In integrable chains, spin transport is superdiffusive (KPZ scaling, $\delta K_z \sim t^{-2/3}$).
  • In non-integrable chains, asymptotic transport is diffusive ($\delta K_z \sim t^{-1/2}$).
  • However, near the integrable point (small $\delta$) or at low temperatures, solitons (quasiparticles) responsible for superdiffusion have parametrically long lifetimes.
• The Mpemba Mechanism:
  • Hotter systems have a higher density of high-energy solitons. When integrability is broken, these high-energy modes decay rapidly, causing the hot system to transition to slow diffusive behavior quickly.
  • Colder systems have a lower density of these fast-decaying modes and retain a "parametrically long-lived" regime of superdiffusion.
  • Consequently, the colder system maintains a faster equilibration rate (superdiffusive) for a much longer duration than the hotter system, eventually overtaking it.
• Timescale: The crossover time from superdiffusive to diffusive scales as $\tau \sim \delta^{-3}$ (integrability breaking) and $\tau \sim T^{-8}$ (temperature).

4. Results

The study utilizes numerical simulations (Monte Carlo and time evolution) on chains of size $L \approx 8000$ to $32000$ to demonstrate four possible scenarios based on the interplay of the two mechanisms:

1. No Crossing: The hotter system starts closer to isotropy and equilibrates faster throughout; or the cold system transitions to diffusion before crossing.
2. Early-Time Crossing Only: The hot system equilibrates faster initially due to phase space, but both systems transition to diffusion quickly, preventing a late-time reversal.
3. Both Crossings: The hot system wins early (Mechanism I), but the cold system wins late (Mechanism II), resulting in two crossings.
4. Late-Time Crossing Only (Pure Mpemba Effect): The hot system starts further from isotropy and does not win early. However, due to the integrability-breaking mechanism, the cold system's superdiffusive lifetime is long enough to overtake the hot system at late times. This is the "clean" demonstration of the integrability-breaking-induced Mpemba effect.

Key Findings:

• The late-time crossing time $t^*$ is highly sensitive to the degree of integrability breaking $\delta$. As $\delta \to 0$ (approaching integrability), $t^*$ diverges, and the late-time crossing disappears (restoring the pure superdiffusive behavior where no crossing occurs).
• The effect is non-universal; specific combinations of temperature and symmetry-breaking strength are required to observe the late-time crossing.

5. Significance

• New Mechanism for Mpemba Effect: The paper establishes that integrability breaking itself is a distinct mechanism for generating the Mpemba effect, separate from the phase-space scrambling seen in integrable systems.
• Hydrodynamic Timescales: It highlights the importance of parametrically long-lived quasiparticle lifetimes in non-integrable systems. Even though the asymptotic state is diffusive, the pre-asymptotic superdiffusive regime dominates the dynamics for a long time, and this regime is temperature-dependent.
• Experimental Relevance: Since perfect integrability is never realized in physical experiments, this work suggests that the Mpemba effect is a robust phenomenon in real-world spin systems (e.g., magnetic materials) where weak perturbations break integrability.
• Theoretical Insight: It bridges the gap between integrable hydrodynamics (superdiffusion) and standard hydrodynamics (diffusion), showing how the crossover between them can invert the ordering of equilibration rates.

In conclusion, the paper demonstrates that the Mpemba effect in spin chains is a rich phenomenon arising from the competition between initial phase-space scrambling and the temperature-dependent lifetime of solitonic modes in weakly non-integrable systems.