Long-range correlation and the spin conductivity in the XXZ chain from ballistic macroscopic fluctuation theory

Using ballistic macroscopic fluctuation theory, this paper demonstrates that superdiffusive spin transport in the critical regime of the spin-1/2 XXZ chain is driven by long-range correlations scaling as $1/N$, while also establishing that spin conductivity is proportional to $1/T$ at high temperatures with a divergent proportionality constant under specific magnetization conditions.

The Gist

The Big Picture: A Crowd of Spinning Tops

Imagine a very long line of people (let's say an infinite line) standing shoulder-to-shoulder. Each person is holding a spinning top. In physics, these tops are called "spins." Usually, if you push one person, the effect ripples down the line like a wave.

This paper studies what happens when these tops are part of a special, highly organized system (the XXZ chain) that is "critical"—meaning it's in a delicate state where tiny changes can have huge effects. The researchers wanted to understand how "spin" (magnetism) moves through this line, specifically looking at conductivity (how easily the spin flows) and correlations (how much one person's spin affects someone far away).

The Experiment: The Magnetic Slope

To test this, the researchers set up a scenario:

1. The Setup: They applied a magnetic field that created a "slope" of magnetization. Imagine the people on the left side of the line are holding their tops tilted one way, and the people on the right are tilting them the other way, with a smooth gradient in the middle.
2. The Release: At time zero, they suddenly cut off the magnetic field.
3. The Reaction: The line of tops starts to wobble and adjust to the new reality. The researchers watched how the "spin current" (the flow of magnetic influence) fluctuated as the system tried to find a new balance.

The Key Discovery: The "Long-Range Whisper"

In normal materials, if you push someone at the start of the line, the person at the very end doesn't feel it immediately or strongly; the effect dies out quickly. This is like a whisper that fades after a few people.

However, this paper found something strange in this specific quantum system. Even though the line is infinitely long, the researchers discovered a "long-range correlation."

• The Analogy: Imagine that in this special line of people, if Person A whispers, Person Z (who is miles away) doesn't just hear a faint whisper; they hear it with surprising clarity. The connection between them doesn't fade; it scales in a very specific way (proportional to $1/N$, where $N$ is the length of the line).
• The Result: This "whisper" across the entire line is what drives the movement of the spin. It's not just local shoving; it's a coordinated, long-distance dance.

The Temperature Twist: Hot and Wild

The researchers looked at what happens when the system is very hot (high temperature).

• The Finding: As the temperature goes up, the ability of the spin to conduct (flow) changes. Specifically, the conductivity is proportional to $1/T$ (inverse temperature).
• The Divergence: Here is the most surprising part. At a specific "isotropic point" (where the rules of the game are perfectly symmetrical), the researchers found that the constant governing this flow diverges.
  • The Analogy: Imagine trying to measure the speed of a river. Usually, the speed is a fixed number. But at this specific point, the "speed" calculation blows up to infinity. It suggests that the spin isn't just flowing; it's flowing in a super-diffusive way. It's moving faster and more chaotically than standard diffusion would predict, driven by those long-range "whispers."

Why Does This Matter? (According to the Paper)

The paper argues that this "super-diffusive" behavior (the infinite conductivity at the limit) is driven directly by the long-range correlations.

• The Mechanism: The long-range correlation acts like a giant, invisible net connecting every part of the chain. When the system is disturbed, this net pulls the whole system into motion simultaneously, rather than step-by-step.
• The Scaling: The paper suggests that at the isotropic point, the way the spin spreads over time follows a unique mathematical rule (scaling as $N^{3/2}$ with a logarithmic correction). This is different from standard diffusion (which scales as $N^2$) and even different from the famous "KPZ" scaling (which describes how surfaces grow, like a pile of sand).

Summary in One Sentence

By using a new theory called "Ballistic Macroscopic Fluctuation Theory," the authors showed that in a specific quantum chain, spin flows incredibly fast and strangely because every part of the chain is "talking" to every other part over vast distances, a phenomenon that becomes infinitely strong at high temperatures and perfect symmetry.

Technical Summary

Technical Summary: Long-range correlation and the spin conductivity in the XXZ chain from ballistic macroscopic fluctuation theory

Problem Statement
This paper investigates the fluctuating spin current in the critical regime of the spin-1/2 XXZ chain, specifically focusing on the role of long-range spin correlations in driving transport. While equilibrium states at finite temperatures exhibit exponentially decaying correlations, non-equilibrium states generated by coarse-grained hydrodynamic approximations are known to develop long-range correlations scaling as $1/\ell$ (where $\ell$ is the wavelength). The authors aim to determine if the spin conductivity in the XXZ chain is driven by these $1/N$-scaled long-range correlations and to analyze the behavior of the spin conductivity in the high-temperature limit ($T \to \infty$ or $\beta \to 0$), particularly at the isotropic point ($\Delta = 1$).

Methodology
The study employs the Ballistic Macroscopic Fluctuation Theory (BMFT), a framework designed to describe fluctuations and correlations in many-body systems with multiple conserved charges. The specific approach involves:

1. Initial State Setup: The system is initialized in a state with a spatially varying magnetization concentration induced by an inclined magnetic field $h_0(x/N)$, which is subsequently removed at $t=0$. This setup creates a non-equilibrium state where the magnetization evolves.
2. Generalized Hydrodynamics (GHD): The time evolution of the system is described using GHD equations for quasiparticle densities $\rho^{(\lambda)}_j(x,t)$. The authors utilize the Euler-scaled equations of motion, neglecting diffusive currents in the ballistic scale, to track the evolution of dressed charges and density ratios $\eta^{(\lambda)}_j(x,t)$.
3. Correlation Function Calculation: The spin correlation function is calculated within the BMFT framework. By introducing auxiliary fields to enforce the Euler equations, the authors derive the equal-time two-point spin correlation function. They demonstrate that this function scales as $O(1/N)$, indicating the emergence of long-range correlations in the non-equilibrium state.
4. Spin Conductivity Derivation: The spin conductivity $\sigma(\beta)$ is defined via the time-integrated current-current correlation function. The authors establish a direct relation between $\sigma(\beta)$ and the integral of the $1/N$-scaled long-range spin correlation function.
5. High-Temperature Limit Analysis: The limit $\beta \to 0$ is analyzed using the Thermodynamic Bethe Ansatz (TBA) equations. The authors examine the behavior of the proportionality constant $\sigma_0 = \lim_{\beta \to 0} \sigma(\beta)/\beta$ for various anisotropy parameters $\Delta = \cos(\pi/p_0)$.

Key Contributions and Results

• Derivation of Long-Range Correlation: The paper analytically demonstrates that in the time evolution of an infinite spin chain starting from a long-wavelength magnetization profile, the equal-time two-point spin correlation function scales as $1/N$. This confirms that the ballistic transport regime is accompanied by long-range correlations of conserved charges.
• Relation to Spin Conductivity: A rigorous relation is established showing that the spin conductivity $\sigma(\beta)$ is proportional to the integration of this $1/N$-scaled long-range correlation function.
• High-Temperature Behavior: In the limit $T \to \infty$ ($\beta \to 0$), the spin conductivity is found to be proportional to $\beta$ (i.e., $\sigma(\beta) \sim \sigma_0 \beta$).
• Divergence at the Isotropic Point: The paper calculates the constant $\sigma_0$ and finds that it diverges in the case where the one-particle magnetization is infinitely large. Specifically, for the isotropic point (XXX chain, $\Delta = 1$), the constant $\sigma_0$ scales as $O((\ln p_0)^2)$, where $p_0 \to \infty$.
• Superdiffusive Transport: The divergence of $\sigma_0$ at the isotropic point is interpreted as evidence of superdiffusive spin transport. The authors link this to the Einstein relation connecting conductivity and diffusion constants.
• Dynamic Scaling Analysis: The paper discusses the dynamic scaling at the isotropic point. While previous studies suggested a Kardar-Parisi-Zhang (KPZ) scaling of $x \sim t^{2/3}$, the authors' analysis of the divergence suggests a dynamical relationship of $T \sim N^{3/2}/\ln N$. This implies that the spin transport in the XXX chain is enhanced beyond the standard KPZ scaling by a logarithmic correction.

Significance and Claims
The paper claims to provide an analytic approach to understanding dynamic scaling in spin transport at the isotropic point, complementing numerical and simulation-based studies. By utilizing BMFT, the authors show that the superdiffusive nature of spin transport in the XXZ chain at the isotropic point is driven by the $1/N$-scaled long-range spin correlations. The divergence of the high-temperature proportionality constant serves as a quantitative indicator of this superdiffusion. The work clarifies the mechanism behind the anomalously fluctuating charge currents in integrable systems and offers a theoretical basis for the dynamic scaling exponents observed in the XXX chain, suggesting a deviation from pure KPZ universality due to logarithmic corrections.