Spin-liquid-like spin dynamics in the frustrated antiferromagnet TbBO3

Through thermodynamic, muon-spin relaxation, and neutron scattering measurements down to 16 mK, this study reveals that the distorted triangular lattice antiferromagnet TbBO3 exhibits persistent spin-liquid-like dynamics driven by dominant 2D short-range correlations and the interplay of frustration and spin-orbit coupling, despite the absence of long-range magnetic order.

The Gist

Imagine a crowded dance floor where everyone wants to dance with their neighbors, but the music is playing a confusing rhythm that makes it impossible for everyone to find a perfect partner. This is the story of a special material called TbBO3, and the scientists who studied it are trying to figure out why the dancers (atoms) never seem to settle down into a neat, organized line, even when the room gets freezing cold.

Here is the story of their discovery, broken down into simple concepts:

1. The Dance Floor: A Frustrated Triangle

In most magnets, atoms act like tiny compass needles. Usually, when you cool them down, they all line up neatly, pointing in the same direction or alternating perfectly (North-South-North-South). This is called "magnetic order."

But in TbBO3, the atoms are arranged in a triangular pattern. Imagine three friends standing in a triangle, each wanting to hold hands with the person on their left. If Friend A holds hands with Friend B, and Friend B holds hands with Friend C, Friend C is stuck. They can't hold hands with Friend A without breaking the rule that they must hold hands with everyone in a specific way.

This is called geometric frustration. The atoms are "frustrated" because the rules of the triangle make it impossible for them to all be happy at the same time.

2. The Mystery: Why Don't They Freeze?

Usually, when you cool a system down to near absolute zero (the coldest temperature possible), the atoms stop moving and freeze into a solid, organized pattern.

The scientists took TbBO3 down to 16 millikelvin—that is colder than outer space! They expected the atoms to finally stop dancing and freeze into a neat pattern. But they didn't.

• The Analogy: Imagine a chaotic mosh pit at a concert. Usually, when the music stops, everyone freezes in place. But in this material, even when the "music" (thermal energy) stops, the atoms keep spinning and dancing wildly. They never settle down.

3. The "Ghost" Magnetism

The scientists used special tools (like muon spin relaxation and neutron scattering) to look inside the material. They found something strange:

• The atoms are definitely interacting strongly (they are "talking" to each other).
• But they aren't forming a solid, frozen structure.
• Instead, they are in a state of constant, fluid motion.

This is what the paper calls a "Spin Liquid." Think of it like water. In ice (a normal magnet), the water molecules are locked in a rigid grid. In a liquid (this spin liquid), the molecules are still connected and interacting, but they flow and move freely. The "spins" (the magnetic direction) are constantly fluctuating, never settling into a single pattern.

4. The Secret Ingredient: Quantum Magic

Why is this happening? The paper suggests it's due to a mix of two things:

1. Strong Spin-Orbit Coupling: This is like a heavy backpack the atoms are wearing that makes them wobble in a specific, complex way.
2. Quantum Superposition: In the quantum world, particles can be in two states at once. The atoms in TbBO3 are essentially "blending" their ground state with excited states. It's as if the atoms are simultaneously dancing and standing still, creating a "ghost" magnetic moment that keeps them moving.

5. The Universal Connection

The researchers didn't just look at this one material. They compared it to other "spin liquid" candidates (like different types of frustrated magnets). They found a universal pattern.

• The Analogy: It's like finding that different types of cars (a Ferrari, a truck, and a motorcycle) all have the same engine sound when they are idling. Even though they look different, they all share a common "heartbeat" at low temperatures. This suggests that nature has a common recipe for creating these liquid-like magnetic states.

6. The "Fuzzy" Clues

When they fired neutrons (tiny particles) at the material to see how the atoms were arranged, they didn't see sharp, clear lines (which would mean a solid order). Instead, they saw broad, fuzzy smears.

• The Analogy: If you take a photo of a spinning fan, you don't see the blades clearly; you see a blur. That blur tells you the blades are moving fast. The "fuzzy smear" in the data told the scientists that the magnetic spins are constantly shifting and only have short-range connections (like a small group of friends chatting) rather than a long-range order (like a whole stadium chanting in unison).

Why Does This Matter?

This discovery is exciting for the future of Quantum Computing.

• The Problem: Current computers use bits (0s and 1s) that are very fragile. If they get disturbed, the information is lost.
• The Solution: Spin liquids are "topologically protected." Imagine a knot in a string. You can shake the string, but the knot stays tied. Spin liquids are like that knot. Their quantum information is stored in the way the spins are entangled, making it very hard to accidentally break.

The Bottom Line

The scientists found a material where the atoms are so frustrated by their triangular arrangement and so influenced by quantum mechanics that they refuse to freeze, even at the coldest temperatures imaginable. Instead, they remain in a perpetual, liquid-like dance. This "spin liquid" state is a rare and exotic form of matter that could be the key to building the next generation of super-powerful, stable quantum computers.

Technical Summary

1. Problem Statement

The search for Quantum Spin Liquids (QSLs)—states of matter characterized by long-range quantum entanglement, topological order, and the absence of symmetry-breaking magnetic order down to absolute zero—is a central goal in condensed matter physics. While QSL candidates are well-studied in systems with $S=1/2$ spins (often Kramers ions with effective $J_{eff}=1/2$), the behavior of non-Kramers ions (ions with integer total angular momentum $J$) remains less understood.

Specifically, non-Kramers ions typically possess a non-magnetic singlet ground state due to Crystal Electric Field (CEF) splitting. Theoretical questions persist regarding whether such systems can host induced quantum magnetism and spin-liquid-like dynamics through the quantum superposition of the singlet ground state and low-lying excited states, particularly in the presence of geometrical frustration and strong spin-orbit coupling. The authors investigate the distorted triangular lattice antiferromagnet TbBO$_3$ (containing Tb$^{3+}$, a non-Kramers ion with $J=6$) to determine if it stabilizes a spin-liquid-like state despite significant antiferromagnetic exchange interactions.

2. Methodology

The study employs a comprehensive, multi-technique approach combining thermodynamic measurements, local probes, and scattering techniques, supported by theoretical calculations:

• Synthesis & Structural Characterization: Powder X-ray diffraction (XRD) and Neutron Powder Diffraction (NPD) were used to confirm the monoclinic $C2/c$ crystal structure and the absence of anti-site disorder.
• Thermodynamic Measurements:
  • DC/AC Magnetic Susceptibility: Measured down to 0.4 K (DC) and 50 mK (AC) to detect magnetic ordering or spin freezing.
  • Specific Heat ($C_p$): Measured down to 65 mK to identify phase transitions (e.g., $\lambda$-type anomalies) and estimate magnetic entropy.
• Local Probes:
  • Muon Spin Relaxation ($\mu$SR): Zero-field (ZF) measurements down to 16 mK to detect static internal magnetic fields (indicative of long-range order) or spin dynamics.
  • Nuclear Magnetic Resonance (NMR): $^{11}$B NMR measurements to probe local magnetic fields and spin-lattice relaxation rates ($1/T_1$).
• Scattering Techniques:
  • Inelastic Neutron Scattering (INS): Performed on the MARI spectrometer (ISIS) to map low-energy magnetic excitations and dispersion relations.
  • Elastic Neutron Scattering: Used to identify magnetic diffuse scattering.
• Theoretical Support: Density Functional Theory + Hubbard $U$ (DFT+U) calculations and Crystal Electric Field (CEF) modeling were used to interpret exchange interactions and energy levels.

3. Key Contributions and Results

A. Absence of Long-Range Order (LRO) and Spin Freezing

Despite strong antiferromagnetic exchange interactions, the system shows no evidence of magnetic ordering down to the lowest measured temperatures:

• Susceptibility: DC susceptibility shows no anomalies down to 0.4 K. Zero-field-cooled (ZFC) and field-cooled (FC) curves overlap, ruling out spin glass freezing. AC susceptibility shows a broad hump around 0.7 K that is frequency-independent, inconsistent with spin freezing.
• Specific Heat: No $\lambda$-type anomaly is observed down to 65 mK. The residual specific heat shows a broad maximum attributed to short-range correlations and CEF effects rather than a phase transition.
• $\mu$SR: The muon-spin relaxation rate ($\lambda$) shows no oscillations or "1/3 tail" (signatures of static fields) down to 16 mK. Instead, $\lambda$ saturates below 1 K, indicating persistent spin dynamics.
• NMR: The $^{11}$B NMR linewidth broadens at low temperatures but shows no peak splitting or rectangular line shapes, confirming the absence of LRO down to 4 K.

B. Nature of Magnetic Interactions and Frustration

• Exchange vs. Dipolar: Curie-Weiss analysis yields a negative Weiss temperature ($\theta_{CW} \approx -7.6$ K to $-12$ K), indicating dominant antiferromagnetic exchange. However, the frustration ratio suggests that dipolar interactions and spin-orbit coupling play significant roles.
• Induced Moments: The system is analyzed using a control parameter $\xi = m^2 J_{ex} / 2\Delta$. With the measured parameters ($\xi < 1$), the system is predicted to remain disordered. The large effective moment observed at low temperatures suggests that thermal excitation and extended superexchange, rather than just quantum superposition, drive the magnetic behavior.

C. Spin Dynamics and Short-Range Correlations

• Scaling Behavior: The temperature dependence of the $\mu$SR relaxation rate ($\lambda$) for TbBO$_3$ and other spin-liquid candidates (e.g., NaYbO$_2$, SrCr$_8$Ga$_4$O$_{19}$) collapses onto a single universal curve when scaled by a characteristic energy $T^* \approx 20$ K. This suggests a common underlying mechanism driven by short-range correlations among fluctuating moments.
• Mechanism: The relaxation is governed by an Orbach process involving a CEF gap ($\Delta_{CEF} \approx 20-24$ K), confirmed by both $\mu$SR and NMR ($1/T_1$) data.

D. Neutron Scattering Evidence

• Diffuse Scattering: Elastic and low-energy inelastic neutron scattering reveal a broad magnetic diffuse maximum at momentum transfer $Q \approx 1.03$ Å$^{-1}$.
• Correlation Length: Modeling this diffuse scattering with a Warren function yields a short-range correlation length of $\sim 10$ Å, consistent with dominant 2D antiferromagnetic short-range correlations.
• Spin Gap: INS spectra reveal a tiny spin gap of $\sim 3.5$ K in the low-energy excitation spectrum, a feature often associated with topological spin correlations in QSL candidates.

4. Significance

This work provides compelling experimental evidence for spin-liquid-like dynamics in a non-Kramers ion system on a distorted triangular lattice.

1. Beyond $S=1/2$ Systems: It demonstrates that spin-liquid behavior is not exclusive to Kramers ions ($J_{eff}=1/2$) but can emerge in non-Kramers systems (integer $J$) where the ground state is a singlet. The dynamics arise from the admixture of excited CEF states into the ground state, driven by frustration and spin-orbit coupling.
2. Universal Mechanism: The scaling of relaxation rates across diverse materials suggests a generic mechanism for spin-liquid-like states in frustrated magnets, driven by short-range correlations rather than specific lattice symmetries.
3. Topological Implications: The observation of a gapped excitation spectrum and persistent dynamics down to 16 mK supports the existence of a gapped quantum spin liquid or a closely related topological state in TbBO$_3$.
4. Material Design: The study highlights the potential of rare-earth borates with distorted lattices as platforms for realizing exotic quantum states, guiding future research into disorder-free frustrated magnets.

In conclusion, TbBO$_3$ represents a robust candidate for a spin-liquid-like state, stabilized by the interplay of geometrical frustration, strong spin-orbit coupling, and crystal field effects, offering a new avenue for exploring quantum entanglement in non-Kramers systems.