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Disorder Induced Quantum Griffiths Phase in the Proximate Kitaev Quantum Spin Liquid Na2Co2TeO6

By utilizing 23Na NMR and ab initio calculations, this study demonstrates that intrinsic disorder in Na2Co2TeO6 induces a transition to a quantum Griffiths phase characterized by coexisting gapped and nearly gapless spin excitations, challenging the conventional view that disorder solely destabilizes Kitaev quantum spin liquids.

Original authors: Wassilios Papawassiliou, Nicolas Lazardis, Eunice Mumba Mpanga, Jonas Koppe, Nikolaos Panopoulos, Lydia Gkoura, Romain Berthelot, Michael Fardis, Gaël De Paëpe, Andrew Pell, Georgios Papavassiliou

Published 2026-08-12

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Wassilios Papawassiliou, Nicolas Lazardis, Eunice Mumba Mpanga, Jonas Koppe, Nikolaos Panopoulos, Lydia Gkoura, Romain Berthelot, Michael Fardis, Gaël De Paëpe, Andrew Pell, Georgios Papavassiliou

Original paper licensed under CC BY 4.0 (https://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

Imagine the world of atoms as a giant, bustling dance floor. In most materials, the dancers (which are tiny magnetic particles called "spins") eventually get tired of dancing randomly and decide to line up in perfect, rigid rows, all facing the same direction. This is what we call a magnet, like the one on your fridge. But in a special, exotic corner of physics called "quantum matter," the rules are different. Here, the dancers refuse to line up. Instead, they stay in a constant, frenzied state of motion, even when the temperature drops to near absolute zero. Scientists call this a "Quantum Spin Liquid."

Why does this matter? Because in this liquid state, the dancers don't just move; they split apart. A single dancer can break into two or three pieces that float around independently, carrying only a fraction of the original energy. It's like if a single person at a party could suddenly split into a ghost and a shadow, both dancing on their own. This "fractionalization" is the holy grail for building future super-computers, but it's incredibly fragile. Usually, if you introduce a little bit of messiness or "disorder" into the crystal structure—like a dancer tripping over a shoe—it ruins the whole liquid state and forces the spins to line up anyway.

However, a new study suggests that sometimes, a little messiness doesn't destroy the party; it actually creates a whole new kind of dance. The researchers looked at a specific material, a crystal called Na2Co2TeO6, which is naturally a bit messy inside. They found that this disorder doesn't kill the quantum liquid; instead, it creates a strange, patchwork world where some parts are frozen and gapped, while others are wild and gapless, all living together in a "Griffiths phase."

The Paper: Disorder Induced Quantum Griffiths Phase in the Proximate Kitaev Quantum Spin Liquid Na2Co2TeO6

The scientists behind this study decided to investigate a material called Na2Co2TeO6 (let's call it NCTO for short). This material is famous because its atoms are arranged in a honeycomb pattern, similar to a beehive, which is the perfect stage for a "Kitaev Quantum Spin Liquid." In this ideal scenario, the spins should fractionalize into tiny, ghostly particles called Majorana fermions. But there's a catch: real crystals are never perfect. NCTO has a natural flaw where some sodium atoms are missing or sitting in the wrong spots. This is "disorder."

Usually, scientists think disorder is the enemy. They expect that if you mess up the honeycomb pattern, the delicate quantum liquid will collapse, and the spins will just freeze into a standard magnetic order. But the authors of this paper asked a different question: What if the disorder actually creates something new?

To find out, they used a technique called NMR (Nuclear Magnetic Resonance). Think of NMR as a super-sensitive stethoscope that listens to the heartbeat of the atoms inside the crystal. By cooling the material down from room temperature to just a few degrees above absolute zero, they watched how the spins reacted.

What they found is a tale of two worlds.

When the material was warm (between 100 K and 10 K), the spins behaved exactly as expected for a quantum spin liquid. They were fractionalized, and there was a clear "energy gap," meaning the spins needed a certain amount of energy to get excited. It was like a calm, gapped ocean where waves only form if you throw a big enough stone.

But then, something weird happened as they cooled it below 10 K. The single, calm ocean split into two distinct channels.

The Slow Lane: One group of spins continued to behave like the calm, gapped ocean. They still needed energy to move, preserving the "fractional" nature of the quantum liquid.
The Fast Lane: The other group of spins suddenly went wild. They started behaving like a "gapless" fluid, where even the tiniest nudge could make them move. Their behavior followed a rule called the "Korringa law," which is usually seen in metals, not in these exotic quantum liquids.

This split is the smoking gun for what the authors call a Quantum Griffiths Phase. Imagine a city where some neighborhoods are frozen solid (the slow lane), while right next door, other neighborhoods are in a chaotic, high-energy rave (the fast lane). The disorder in the crystal—the missing sodium atoms—acts like a chaotic architect. It distorts the bonds between the atoms, creating a mosaic. Some areas become "vison-rich" (full of magnetic flux defects) and gapless, while others remain "vison-poor" and gapped.

The paper explicitly rules out the idea that the material simply froze into a standard magnetic order, like a zigzag pattern, at low temperatures. If it had done that, the NMR signals would have looked very different, with sharp, predictable shapes. Instead, the signals were broad and messy, which only a complex, disordered quantum state could explain. They also ruled out the idea that this was just a "spin glass" (a random, frozen mess), because the way the spins relaxed didn't match the patterns of a glass.

The researchers used powerful computer simulations (DFT calculations) to back up their findings. They built a virtual model of the crystal with the missing atoms and let the computer relax the structure. The simulation showed that the missing atoms naturally twisted the bonds between the cobalt atoms by about 3%, which is enough to scramble the magnetic forces by about 30%. This confirmed that the disorder was indeed strong enough to create the patchwork of gapped and gapless regions.

Furthermore, they discovered "ultraslow" spin fluctuations. While the fast lane was chaotic, there were also spins that were moving incredibly slowly, taking microseconds to flip. These slow movements followed an "Arrhenius activation law," meaning they were trying to climb over a small energy hill of about 4.9 K. This is a hallmark of a Griffiths phase, where rare, special regions of the material behave differently from the rest.

In short, the paper suggests that in Na2Co2TeO6, disorder isn't a bug; it's a feature. Instead of destroying the quantum spin liquid, the natural imperfections in the crystal reshape it into a "Griffiths phase." This is a state where the material is a dynamic mix of gapped islands and gapless seas, all coexisting. The authors conclude that this disorder-driven state is a robust, emergent form of fractionalized matter, proving that even in a messy crystal, the strange, beautiful physics of the quantum world can not only survive but thrive in a new, heterogeneous form.

Technical Summary: Disorder Induced Quantum Griffiths Phase in the Proximate Kitaev Quantum Spin Liquid Na2Co2TeO6

Problem Statement
The identification of Kitaev quantum spin liquids (QSLs) in real materials is complicated by the inevitable presence of lattice imperfections, magnetic order, and random bond perturbations. While materials like $\alpha$ -RuCl $_3$ and $\alpha$ -(Na,Li) $_2$ IrO $_3$ have established the phenomenology of proximate Kitaev magnetism, their ground states are typically dominated by long-range magnetic order (e.g., zigzag antiferromagnetism) rather than a pure QSL. A critical open question is whether disorder, traditionally viewed as a destabilizing force, can instead generate qualitatively new forms of fractionalized matter. Na $_2$ Co $_2$ TeO $_6$ (NCTO) presents a unique case: it is a 3d transition metal honeycomb magnet close to the Kitaev limit but possesses substantial intrinsic Na-site occupation disorder. The challenge is to determine if this disorder merely perturbs the system or actively reorganizes the low-energy sector into a distinct fractionalized regime.

Methodology
The authors employed a multi-faceted approach combining experimental nuclear magnetic resonance (NMR) with ab initio modeling:

Experimental NMR: Extensive $^{23}$ $^{23}$ Na solid-state NMR measurements were conducted on polycrystalline NCTO across a temperature range of 3–300 K in magnetic fields of 4.7 T and 9.4 T. Techniques included:
- Magic-Angle-Spinning (MAS) and static lineshape analysis to probe static spin configurations.
- Spin-lattice ( $T_1$ ) and spin-spin ( $T_2$ ) relaxation measurements to characterize spin dynamics.
- $T_1$ - $T_2$ correlation spectroscopy and q-CPMG pulse sequences to resolve heterogeneous relaxation channels and ultraslow fluctuations.
Ab Initio Calculations: Density Functional Theory (DFT) was used to relax a 2 $\times$ 2 $\times$ 2 NCTO supercell incorporating random Na-site occupancies derived from NMR data. This allowed for the calculation of distributions in Co-O bond lengths and Co-O-Co angles.
Spectral Simulation: Ab initio NMR powder patterns were simulated for three candidate ground states: zigzag antiferromagnetic (AFM), triple- $q$ AFM, and a QSL-like state, to compare against experimental low-temperature spectra.

Key Results

Disorder-Induced Bond Randomness: DFT relaxation of the disordered supercell revealed that random Na occupancy induces significant spatial variations in Co-O-Co bond angles and bond lengths (~3–4% variation). Using a phenomenological model, these structural distortions translate to ~30% fluctuations in the Kitaev exchange coupling constants ( $K$ ), effectively creating a random-bond Kitaev system.
QSL-like Ground State: At low temperatures (3 K) and high fields (9.4 T), the experimental static $^{23}$ Na NMR lineshapes are broad, single-peaked, and Gaussian. Ab initio simulations demonstrate that only a QSL model reproduces this spectral shape and width (~40,000 ppm). In contrast, zigzag and triple- $q$ AFM models produce distinct plateau features or edge structures not observed experimentally, ruling out a simple static ordered ground state under these conditions.
Bifurcation of Relaxation Channels: The spin-lattice relaxation rate ( $1/T_1$ $1/ T_{1}$ ) exhibits a distinct temperature dependence:
- 100 K – 10 K: $1/T_1$ follows a concave profile consistent with gapped fractional spin excitations (Majorana fermions), fitting an activated law with a gap $\Delta \approx 35$ K.
- Below 10 K: The relaxation bifurcates into two distinct channels. A slow-relaxing component preserves the gapped, activated behavior. A fast-relaxing component emerges, obeying a Korringa-like law ( $1/T_1 T \approx \text{const}$ ), indicative of nearly gapless, Fermi-liquid-like spin excitations.
Ultraslow Dynamics: Below 10 K, $T_2$ distributions broaden and split, revealing ultraslow spin fluctuations on the microsecond timescale. These fluctuations follow an Arrhenius activation law with an energy barrier $U/k_B \approx 4.9$ K.
Field Independence: The dual-channel relaxation behavior persists at 4.7 T (where a PM-AFM transition occurs at $T_N \approx 27$ K), suggesting that the Griffiths-like dynamics dominate the low-energy sector even in the presence of magnetic ordering patches.

Significance and Claims
The paper claims that NCTO realizes a disorder-driven Quantum Griffiths Phase. Rather than destroying the Kitaev state, the intrinsic Na-site disorder acts as an organizing principle that reshapes the low-energy density of states. The authors propose a "mosaic" picture where the system consists of:

Vison-poor islands: Regions retaining gapped fractional excitations (slow relaxation).
Vison-rich regions: Rare regions where disorder closes the flux gap, generating a continuum of low-energy, nearly gapless states (fast, Korringa-like relaxation).

This work provides direct experimental evidence that disorder can generate a dynamically heterogeneous, Griffiths-type fractionalized phase. It suggests that the coexistence of gapped and gapless fractional excitations in real space is a robust feature of disordered Kitaev systems, offering a new perspective on the stability and nature of quantum spin liquids in the presence of structural randomness. The findings align with theoretical predictions for bond-disordered Kitaev models, where disorder replaces sharp gaps with power-law continua, stabilizing a Griffiths phase that survives down to the lowest temperatures.

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