🔬 condensed matter

Long-lived local coherences at high temperature from frozen domain walls

This paper demonstrates that local coherences in spin chains with quantum group symmetry, such as the XXZ and Fredkin models, can exhibit anomalously long-lived algebraic decay at high temperatures due to a mechanism involving frozen domain walls and a derived Mazur bound, contrasting with the typically rapid decay expected in generic chaotic systems.

Original authors: Sarang Gopalakrishnan, Romain Vasseur

Published 2026-10-16

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CC BY 4.0

Original authors: Sarang Gopalakrishnan, Romain Vasseur

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the chaotic world of quantum physics, where countless particles interact in a frantic, unpredictable dance, memory is usually a fleeting thing. If you disturb a small part of a hot, disordered system, the rest of the system quickly notices and scrambles the information, causing any local order to vanish almost instantly. This rapid loss of memory is the norm for what scientists call "transverse" disturbances—changes that flip the internal state of a particle, like turning a tiny magnet upside down. In most chaotic environments, especially at high temperatures, these flips are measured and erased so quickly that their signal fades away exponentially, disappearing before it can be tracked. This makes it nearly impossible to observe quantum coherence, the delicate state where particles act in unison, in everyday conditions. However, a new study suggests that under very specific rules, this rapid erasure can be dramatically slowed, allowing these quantum memories to persist for surprisingly long times, decaying only gradually rather than vanishing.

Researchers Sarang Gopalakrishnan and Romain Vasseur have uncovered a mechanism that allows these local quantum coherences to survive anomalously long in certain one-dimensional chains of spins, even when the system is hot and chaotic. They focused on a special class of materials governed by a hidden symmetry known as a quantum group symmetry. While standard symmetries in physics often lead to predictable, flowing behaviors like heat diffusion, this specific quantum symmetry creates a unique environment where the system's internal structure protects local flips from being immediately measured by the rest of the chain. The team showed that instead of fading away exponentially, the signal from these flips decays much more slowly, following a power law. While a rigorous mathematical bound proves the signal cannot decay faster than the inverse square of time, the actual decay in specific models is even slower. For instance, in the XXZ spin chain with strong interactions, the signal drops off as the inverse square root of time. This means that if you wait twice as long, the signal is not gone; it is merely weaker by a factor of roughly 1.4, a stark contrast to the rapid disappearance seen in typical chaotic systems.

To understand how this protection works, the researchers used a visual tool called the "height representation." Imagine the chain of spins as a landscape where an upward-pointing spin is a step up and a downward-pointing spin is a step down. In this landscape, a local flip creates a small valley or a peak. In a generic system, this feature would be instantly smoothed out by the surrounding chaos. But in these specific quantum chains, the landscape contains "frozen" regions where the terrain is locked in place, and the only things that can move are isolated steps or "valleys" that can wander through the frozen ground. When a researcher flips a spin to create a valley in this frozen terrain, the rest of the system cannot immediately see or measure this new valley because the frozen ground acts as a shield. The valley remains invisible to the rest of the chain as long as it stays the deepest point in its immediate neighborhood. It is only when the valley wanders far enough to encounter a deeper valley elsewhere that the system finally "measures" it and destroys the coherence.

The researchers demonstrated that this mechanism holds true for both perfectly ordered systems and chaotic ones, provided they obey the quantum group symmetry. They calculated the decay rate for a specific model known as the XXZ spin chain, particularly in a limit where the interactions between spins are very strong. In this scenario, they found that the signal decays according to a precise mathematical rule, dropping off as the inverse square root of time, multiplied by a specific numerical factor. This result was confirmed through extensive computer simulations on chains containing up to a thousand sites, showing a perfect match between their theoretical prediction and the simulated data. They also explored what happens when the system is not perfectly balanced, such as when there is a net magnetic pull in one direction. In these cases, the protective effect vanishes, and the system reverts to the standard, rapid decay seen in ordinary chaotic materials, proving that the long-lived coherence is a fragile property dependent on a precise balance of conditions.

This discovery challenges the assumption that quantum coherence is impossible to maintain in hot, chaotic environments without extreme isolation. The authors argue that the key is not the absence of chaos, but the presence of a specific structural rule that locks certain parts of the system into a frozen state, shielding local changes from immediate detection. While this effect has been proven for one-dimensional chains, the researchers note that it is not a universal feature of all integrable systems; in some similar models without this specific symmetry, the signal still vanishes rapidly. Their work suggests that if scientists can engineer materials or quantum circuits that mimic these specific symmetries, they might be able to create systems where quantum information persists for much longer than currently expected, even at high temperatures. The study provides a rigorous mathematical bound and numerical evidence that this slow decay is real, offering a new pathway to understanding how quantum order can survive in a noisy world.

Technical Summary: Long-lived local coherences at high temperature from frozen domain walls

Problem Statement
In generic chaotic many-body systems at finite temperature, local operators that do not overlap with hydrodynamic modes (conserved densities or their derivatives) are expected to lose memory rapidly. Specifically, "transverse" operators—those that change a conserved charge, such as the spin raising operator $\sigma^+_x$ in a $U(1)$ -symmetric system—create local coherences between distinct charge sectors. In high dimensions, these coherences decay exponentially. In one dimension, they typically exhibit stretched-exponential decay, driven by slow dephasing in low-density regions. The central problem addressed is whether there exists a mechanism, even in chaotic systems, that can slow this decay further to a power law, thereby preserving local quantum coherence anomalously long at high temperatures. Recent numerical suggestions indicated that quantum group symmetries might facilitate this, but the underlying mechanism remained unexplained.

Methodology
The authors investigate spin-1/2 chains obeying a quantum group $U_q(\mathfrak{sl}_2)$ symmetry, constructed from Temperley–Lieb generators. The study employs a multi-faceted approach:

Height Representation: The spin configurations are mapped to a height field $h(x)$ , where $\downarrow$ and $\uparrow$ spins correspond to down and up steps. In this representation, the quantum group symmetry acts non-trivially on local minima (valleys) of the height profile.
Mazur Bound Analysis: The authors derive a lower bound on the infinite-time average of the transverse spin autocorrelation function $C(t) \equiv \langle \sigma^-_x(t) \sigma^+_x(0) \rangle$ . They utilize the polar decomposition of the quantum group raising generator $E$ to define a conserved charge $R$ (the "crystal raising operator" in the $q \to \infty$ limit). This operator commutes with the Hamiltonian and acts locally on product states with a local height minimum.
Locality and Non-Undershoot Assumption: To extend the finite-size Mazur bound to finite-time decay in the thermodynamic limit, the authors invoke the Lieb–Robinson bound. They assume a "non-undershoot" condition: that the time-averaged correlator in a finite window does not fall below its late-time plateau before boundary effects become significant. This assumption is supported by numerical checks.
Hydrodynamic and Kinetic Arguments: For chaotic models, the authors decompose the survival of the correlator into the survival probability of a local height minimum and its return probability. They analyze specific models, including the Fredkin spin chain and the XXZ chain in various anisotropy limits, using exact diagonalization, dynamical typicality, and Monte Carlo sampling of free-fermion determinants.

Key Contributions and Results

Algebraic Decay Bound: The authors show that for integrable and chaotic spin chains with quantum group symmetry, the transverse autocorrelation function $C(t)$ decays at most as $1/t^2$ . This bound is derived from the quantum group symmetry and locality, contingent on a physically motivated and numerically verified "non-undershoot" assumption.
Explicit Decay Rates:
- Folded XXZ Chain ( $\Delta = \infty$ ): In the large-anisotropy limit, the authors explicitly compute the decay rate. The mechanism involves the "frozen" domain walls (height minima) which move diffusively. The correlator scales as $C(t) \sim \frac{\sqrt{3}}{18\sqrt{t}}$ . This result is derived by calculating the return probability of a diffusing domain wall and the combinatorial probability ( $\rho = 1/9$ ) that a spin flip modifies the frozen pattern without creating or destroying magnons.
- General Easy-Axis Anisotropy ( $1 < \Delta < \infty$ ): For finite anisotropy, the system supports a hierarchy of stable magnon bound states ("strings"). The decay is governed by the survival of heavy strings. The authors estimate $C(t) \sim t^{-1/2} / \ln^2(t)$ , where the logarithmic correction arises from the competition between the exponentially growing lifetimes of heavy strings and their exponentially decaying densities.
- Chaotic Models (Fredkin and generic $\beta \neq 0$ ): In chaotic systems, the decay is estimated to scale as $C(t) \sim t^{-2/z}$ , where $z$ is the transport exponent (e.g., $z \approx 8/3$ for the Fredkin chain). Numerics confirm that the transverse correlator decays as the square of the standard density autocorrelator, even before the latter reaches its asymptotic regime.
Role of Magnetization: The authors demonstrate that this slow decay is highly sensitive to magnetization. At half-filling (zero magnetization), the height minima are distributed throughout the bulk, allowing for the algebraic decay. However, in ensembles with net magnetization ( $m \neq 0$ ), the height profile develops a tilt, causing minima to concentrate near the boundaries. In this regime, the Mazur bound becomes trivial, and chaotic models revert to generic stretched-exponential decay. Integrable models retain power-law decay but with modified prefactors.

Significance and Claims
The paper claims to identify a specific mechanism—frozen domain walls in the height representation—that allows local coherences to persist anomalously long at high temperatures in systems with quantum group symmetry. Key distinctions emphasized by the authors include:

Non-Hydrodynamic Nature: The slow decay is not due to a hydrodynamic mode. Unlike conserved densities which overlap with hydrodynamic modes and decay algebraically for all densities, the quantum group symmetry restricts the power-law decay strictly to half-filling.
Beyond Integrability: The phenomenon occurs in both integrable (e.g., folded XXZ) and chaotic (e.g., Fredkin) systems, provided they possess the specific quantum group symmetry. This distinguishes the mechanism from standard integrability, where slow decay is often associated with quasi-local conserved charges.
Mechanism of "Invisibility": The core physical insight is that a spin flip at a local height minimum is temporarily "invisible" to the rest of the system. The Hamiltonian restricted to a window commutes with an operator that flips this minimum, and the coherence persists as long as the minimum remains the deepest in its light cone.

The authors conclude that while the slow decay is unexpected even in integrable models (where transverse correlators often decay exponentially, e.g., in the $\Delta=0$ XXZ limit), its presence in chaotic models with diffusive hydrodynamics is particularly surprising. They note that the "non-undershoot" assumption, while physically motivated and numerically robust, is required for the rigorous finite-time bound. The work opens questions regarding the generality of this mechanism in constrained Hamiltonians and higher dimensions, and the potential for experimental realization in quantum circuits with $U_q(\mathfrak{sl}_2)$ symmetry.

Technical Summary: Long-lived local coherences at high temperature from frozen domain walls

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