NMR evidence of spin supersolid and Pomeranchuk effect behaviors in the triangular-lattice antiferromagnet Rb$_2$Ni$_2$(SeO$_3$)$_3$

Through high-field $^{85}$Rb NMR measurements on the triangular-lattice antiferromagnet Rb$_2$Ni$_2$(SeO$_3$)$_3$, the study provides evidence for a magnetic up-up-down phase and two distinct gapless regimes consistent with spin supersolid phases, while revealing a unique negative slope in the UUD-V phase boundary driven by strong low-energy spin fluctuations.

The Gist

Imagine a crowded dance floor where everyone is trying to find the perfect spot to dance, but the room is shaped like a triangle. In this triangle, if two people stand next to each other, they want to face opposite directions (like a magnet's North and South pole). But because of the triangular shape, it's impossible for everyone to be happy at the same time. This is called frustration, and it's the playground for the scientists in this paper.

The researchers studied a special crystal called Rb₂Ni₂(SeO₃)₃. Inside this crystal, tiny magnetic atoms (spins) are arranged in these frustrating triangles. They wanted to see what happens when they squeeze these atoms with a powerful magnetic field (up to 26 Tesla, which is about 500,000 times stronger than a fridge magnet) and cool them down.

Here is the story of what they found, explained simply:

1. The "Up-Up-Down" Dance (The UUD Phase)

Usually, when you cool down a magnet, the atoms line up neatly. But in this triangle, they can't. Instead, they found a specific pattern where two atoms point "up" and one points "down" (like a group of three friends where two agree and one disagrees).

• The Analogy: Imagine a trio of dancers. Two are spinning clockwise, and one is spinning counter-clockwise. This specific "Up-Up-Down" pattern is a known state in physics, but finding it in this specific material was a key part of the puzzle.

2. The "Super-Dancers" (Spin Supersolids)

The most exciting discovery was finding two types of "Spin Supersolids."

• What is a Supersolid? In the real world, a solid is stiff (like ice), and a liquid flows (like water). A supersolid is a magical state where something is both rigid and fluid at the same time.
• In the Crystal: The atoms form a rigid, ordered pattern (like a solid), but they can also "flow" or wiggle freely without losing energy (like a superfluid).
• The Discovery: The researchers found two different "dance floors" for these supersolids:
  • The Y Phase: Happens at low magnetic fields.
  • The V Phase: Happens at high magnetic fields.
  • The Evidence: They used a technique called NMR (think of it as a super-precise MRI for atoms) to listen to the atoms. They heard that the atoms were moving freely (gapless) even while locked in a pattern, proving they were "super-dancers."

3. The "Pomeranchuk Effect" (The Magic Trick)

This is the most counter-intuitive part of the paper, and the authors compare it to a famous physics trick called the Pomeranchuk Effect.

• The Normal Rule: Usually, if you squeeze a gas, it gets hotter. If you cool a liquid, it freezes into a solid. You expect the "messier" state (liquid/gas) to be at higher temperatures and the "ordered" state (solid) to be at lower temperatures.
• The Magic Trick: In this crystal, the scientists found the opposite!
  • The "V Phase" (which is more ordered and complex) appeared at higher temperatures.
  • The "UUD Phase" (which is simpler) appeared at lower temperatures.
• Why? It's all about entropy (disorder).
  • Think of the "V Phase" as a room full of people who are dancing wildly. Even though they are in a pattern, they have so much "wiggle room" (quantum fluctuations) that they are actually more disordered (higher entropy) than the rigid "UUD Phase."
  • Because the "V Phase" has more hidden disorder, nature prefers it when it's warmer. As you cool it down, the atoms lose that wiggle room and snap into the rigid "UUD" pattern.
  • The Result: When they increased the magnetic field, the boundary between these two phases tilted backwards. Usually, higher fields push things to lower temperatures, but here, the boundary went the other way. It's like if you turned up the heat on a stove, and your ice cube suddenly froze instead of melting.

Summary of the Journey

1. The Setup: They took a frustrated triangular magnet and squeezed it with a giant magnet.
2. The Observation: They saw the atoms switch between different dance patterns (phases).
3. The Surprise: They found "Supersolids" (rigid yet flowing) in two different zones.
4. The Twist: They found a "Pomeranchuk Effect" where the more complex, ordered state exists at higher temperatures than the simpler state, driven by the hidden "wiggles" of the atoms.

Why does this matter?
This isn't just about crystals; it's about understanding how nature behaves when things are pushed to the limit. It shows that "order" and "disorder" can swap roles in unexpected ways. This could help scientists design new materials for magnetic refrigeration (super-efficient cooling) or even future quantum computers, where controlling these "wiggles" is key.

Technical Summary

Here is a detailed technical summary of the paper "NMR evidence of spin supersolid and Pomeranchuk effect behaviors in the triangular-lattice antiferromagnet Rb2Ni2(SeO3)3".

1. Problem Statement

Frustrated magnetic systems, particularly triangular-lattice antiferromagnets (TLAFMs), are known for hosting exotic quantum phases such as spin liquids, magnetic plateaus, and spin supersolids (SS). While spin supersolidity (simultaneous breaking of spin rotational and lattice symmetries) has been observed in $S=1/2$ systems, its existence and behavior in $S=1$ bilayer TLAFMs remain an open question. Specifically, it is unclear whether these systems can host multiple supersolid phases and how quantum and thermal fluctuations compete to stabilize them. Furthermore, the thermodynamic behavior of these phases, particularly regarding entropy-driven transitions analogous to the Pomeranchuk effect (where an ordered phase exists at higher temperatures than a disordered one due to higher entropy), has not been experimentally confirmed in magnetic systems.

2. Methodology

The authors employed $^{85}$Rb Nuclear Magnetic Resonance (NMR) spectroscopy to investigate the material Rb$_2$Ni$_2$(SeO$_3$)$_3$, an $S=1$ bilayer triangular-lattice antiferromagnet.

• Experimental Conditions: Measurements were conducted in longitudinal magnetic fields up to 26 T and temperatures ranging from low Kelvin to room temperature.
• Key Probes:
  • NMR Spectral Lines: Analyzed for line splitting and spectral weight ratios to determine magnetic symmetry breaking and spin configurations (e.g., Up-Up-Down patterns).
  • Spin-Lattice Relaxation Rate ($1/T_1$): Used to probe low-energy spin fluctuations, distinguish between gapped and gapless excitations, and identify phase boundaries.
  • Knight Shift ($K_n$): Measured to determine hyperfine coupling constants and verify the relationship between local magnetization and bulk susceptibility.
• Analysis: The data was used to construct a comprehensive magnetic phase diagram ($H$-$T$ plane) and compare experimental spectra with theoretical simulations of various magnetic orders (Paramagnetic, UUD, Y, and V phases).

3. Key Contributions

• Discovery of Multiple Supersolid Phases: The study provides definitive evidence for two distinct gapless spin supersolid phases (Y phase at low fields and V phase at high fields) surrounding an intermediate Up-Up-Down (UUD) magnetic plateau phase in an $S=1$ system.
• Observation of a Magnetic Pomeranchuk Effect: The authors identify a counterintuitive phase boundary where the supersolid V phase (higher symmetry breaking) is stabilized at higher temperatures than the UUD phase (lower symmetry breaking). This is attributed to an entropy-driven mechanism analogous to the Pomeranchuk effect in liquid $^3$He.
• Characterization of Fluctuations: The work elucidates the role of strong low-energy spin fluctuations in stabilizing the V phase and the nature of the transition from a gapless supersolid to a gapped UUD state.

4. Key Results

A. Phase Diagram Construction

The study establishes a complex phase diagram with four distinct regions:

1. Paramagnetic (PM) Phase: High temperature.
2. UUD Phase: An intermediate field/temperature region characterized by a 1/3 magnetization plateau (fully realized above 16 T).
3. Y Phase (Low Field Supersolid): Exists below the UUD phase at low fields ($H < 16$ T) and low temperatures.
4. V Phase (High Field Supersolid): Exists between the UUD phase and the high-field PM phase at higher fields ($H > 16$ T).

B. Spectral Evidence

• Low Field (11.5 T): NMR spectra show a transition from a single line (PM) to a split structure at $T_N \approx 7.5$ K (UUD phase), and further splitting at $T_Y \approx 3$ K (Y phase). The spectral weight ratios (1:3:2) confirm the UUD spin configuration. Below 3 K, the gapless nature of the Y phase is confirmed by power-law behavior in $1/T_1 \sim T^3$.
• High Field (24 T): Spectra reveal a transition at $T_N \approx 6.3$ K (PM to V phase) and a second transition at $T_U \approx 3.5$ K (V to UUD phase). The V phase exhibits a three-peak structure, while the UUD phase shows a four-peak structure with a spectral weight ratio consistent with the 2:1 spin population.

C. Spin Dynamics and Gaps

• Gapless Excitations: In both the Y and V phases, $1/T_1$follows a power-law dependence ($T^3$), indicating gapless Goldstone modes associated with broken continuous symmetries (characteristic of supersolids).
• Gapped Excitations: In the UUD phase (low $T$, high $H$), $1/T_1$follows an activated behavior ($e^{-\Delta/k_B T}$), confirming the opening of an energy gap$\Delta$. The gap increases with the magnetic field, reaching a maximum of ~15 K at 23 T.

D. The Pomeranchuk Effect

• Negative Slope ($dT/dH < 0$): The phase boundary between the V phase and the UUD phase exhibits a negative slope in the $H-T$ diagram. This means that upon cooling at a constant high field, the system transitions from the V phase (higher symmetry breaking) to the UUD phase (lower symmetry breaking).
• Entropy Mechanism: This counterintuitive behavior is explained by the Clausius-Clapeyron relation ($dT/dH = -\Delta M / \Delta S$).
  • The UUD phase has a hard energy gap, leading to exponentially suppressed entropy ($S_U \propto e^{-\Delta/T}$) at low $T$.
  • The V phase, being a supersolid with broken spin rotational symmetry, possesses Goldstone modes contributing to entropy ($S_V \propto T^2$).
  • Consequently, $\Delta S = S_V - S_U > 0$ at low temperatures. Since $\Delta M > 0$ (magnetization increases with field), the condition $\Delta S > 0$ forces $dT/dH < 0$.
  • This implies the V phase has higher entropy than the UUD phase at low temperatures, stabilizing it at higher temperatures—a magnetic analogue of the Pomeranchuk effect.

5. Significance

• Extension of Supersolidity: This work extends the observation of spin supersolidity from $S=1/2$ to $S=1$ bilayer systems, demonstrating that interlayer couplings and geometric frustration can stabilize complex supersolid states.
• Thermodynamic Insight: It provides the first experimental evidence of a magnetic Pomeranchuk effect, where the competition between out-of-plane order (gapped) and in-plane order (gapless) leads to an entropy-driven phase transition. This challenges the naive expectation that higher symmetry breaking always corresponds to lower temperature stability.
• Potential Applications: The large entropy change associated with this effect suggests potential applications in magnetocaloric cooling (magnetic refrigeration), as the system exhibits a significant magnetocaloric effect near the phase boundary.
• Theoretical Validation: The findings validate theoretical models of frustrated Ising magnets and provide a concrete platform for studying the interplay of quantum and thermal fluctuations in emergent quantum states.