Vanishing ordered moment in the frustrated triangular lattice antiferromagnet CuNdO 2

arXiv:2509.23523 · cond-mat.str-el · Submitted 2025-09-27 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Vanishing ordered moment in the frustrated triangular lattice antiferromagnet CuNdO 2".

Mira: The gist: The apparent absence of a dipolar ordered moment in CuNdO2 is a consequence of the dominant Ising character of the antiferromagnetically coupled Nd3+ moments,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So we're looking at this paper called "Vanishing ordered moment in the frustrated triangular lattice antiferromagnet CuNdO two" <ref:2509.23523#pg1,Vanishing ordered moment in the frustrated triangular lattice antiferromagnet>. We're going to talk about how they found out that there's magnetic order at zero point seven eight Kelvin without seeing any clear magnetic signals in neutron diffraction <ref:2509.23523#pg2>.

Mira: Basically, the main idea is that this material has long-range antiferromagnetic order, but it doesn't show up as a standard dipole ordered moment when you look at the crystal structure with neutron diffraction.

Lev: The paper claims that this absence of a dipolar ordered moment isn't just an experimental issue; it comes down to the nature of the magnetic moments themselves, specifically their strong Ising character and extreme frustration on that triangular lattice.

Kai: So, what does that mean for us right now? It suggests we need to look beyond just looking for those conventional magnetic Bragg peaks when studying these frustrated magnets.

Mira: Right, and they point out that the frustration in CuNdO two gets relieved by this specific ordering mechanism involving the smaller component of the Nd three plus moments into a one hundred twenty degree structure.

Lev: From an engineering standpoint, if you were trying to build a real device based on this idea, you'd have to account for that nearly vanishing ordered moment when modeling the spin dynamics.

Kai: It sounds like they used some really specific tools to pin down these magnetic properties, so we need to see how they did that.

Mira: They used inelastic neutron scattering data from five Kelvin and one hundred Kelvin to determine the crystal field ground state of the Nd three plus ions <ref:2509.23523#pg3,the crystal field ground state of>.

Lev: That's crucial because if you don't get the single-ion properties right, everything else about the magnetic ordering is just guessing.

Kai: The authors found that this ground state is an effective spin-one/two with strong Ising anisotropy, which they describe using a Kramers doublet <ref:2509.23523#pg1>.

Mira: That strong Ising character means there's a very dominant out-of-plane component to the moment, and a very small in-plane one.

Lev: So, when you look at the collective spin excitations below zero point seven eight Kelvin, they found something that points to this one hundred twenty degree order on the triangular lattice.

Kai: They used a linear spin wave model with an XXZ Hamiltonian where the exchange constant J is zero point two meV and the anisotropic exchange coefficient epsilon is zero point five to describe this state.

Mira: That specific combination of J = zero point two meV and epsilon = zero point five is what they say best reproduces all the experimental data they measured, including that weak dispersionless magnon excitation near zero energy transfer.

Lev: If you were trying to simulate this on real hardware, those parameters would tell you the required fidelity for the spin wave calculations to be accurate enough.

Kai: So, what does this whole setup actually change for someone who only listens to this show? It shows that frustration in these materials can lead to a state that looks magnetically ordered but doesn't have a big observable dipole moment.

Mira: They demonstrate that the magnetic frustration in CuNdO two is relieved by an in-plane ordering into a one hundred twenty degree structure, which results in a nearly vanishing ordered moment.

Lev: It really emphasizes how single-ion anisotropy can dictate the kind of magnetic ordering you actually see, even when you expect something else to happen on a frustrated lattice.

Kai: So, we've seen how they characterized the physics and what the model predicts for this system. Now we need to think about what this means for other materials in this class.

Mira: The authors suggest that this phenomenology is something you might see commonly in other rare earth frustrated magnets, especially those with strong single-ion anisotropy where the easy and hard axes play such different roles in relieving frustration.

Lev: It's important to remember that they found the ordered state accounts for the absence of a measurable dipole moment in neutron diffraction experiments because of that small in-plane g-factor which suppresses those matrix elements.

Kai: So, while they didn't find a strong visible dipole moment, they confirmed there is long-range magnetic order detected by other means like heat capacity and muon spin relaxation.

Mira: The implication is that when we see magnetic order without the expected diffraction signals, we should suspect strong single-ion anisotropy and look for those specific ordering patterns, like the one hundred twenty degree structure.

Lev: For error correction research, this tells us that modeling these systems needs to accurately capture that small in-plane component because it's what gets ordered and relieves the frustration.

Kai: So, to wrap up on "Vanishing ordered moment in the frustrated triangular lattice antiferromagnet CuNdO two", they show how strong Ising anisotropy leads to an effective spin-one/two behavior, which then orders into a one hundred twenty degree structure with minimal net moment.

Mira: The whole point is that this specific mechanism relieves the extreme frustration inherent to the triangular lattice, even if it results in a nearly zero measurable dipole moment.

Lev: It's a case study in how competing energy scales, like crystal field splitting and magnetic exchange, can lead to an ordered state that looks counter-intuitive based on simple expectations.

Kai: That's what they found in CuNdO two showing that the structure of the moments—their anisotropy—is just as important as the lattice geometry itself in determining the final magnetic ground state <ref:2509.23523#pg1>.

Conclusion: Kai: So we've been looking at this paper on CuNdO two and how they found magnetic order that doesn't show up in standard neutron diffraction patterns.

Mira: Right, the authors, Kai and Mira, they really dug into why this apparent absence of a dipole ordered moment is happening in this material.

Lev: They found that the dominant Ising character of those Nd three plus moments is the whole key to understanding this phenomenon.

Kai: So basically, they show that when you have strong single-ion anisotropy, like the Ising behavior here, it sets up a situation where frustration on a triangular lattice leads to something weird.

Mira: Exactly, and what they then propose is that this frustration gets relieved by an in-plane ordering into a one hundred twenty degree structure.

Lev: And that one hundred twenty degree structure results in the ordered moment being nearly vanishing, which explains why it's hard to see with neutron diffraction.

Kai: It makes you wonder how much of this magnetic state is really visible versus hidden by these structural details.

Mira: That's what I think. It suggests that for frustrated magnets, the easy and hard axes of the magnetic moments play such a big role in determining the final ground state.

Lev: And if we were trying to build an error correction system on something like this, that small in-plane component is where you need to focus your modeling efforts.

Kai: So while they didn't find a massive dipole moment, they confirmed there is long-range magnetic order detected by other means like heat capacity and muon spin relaxation.

Mira: And that's the point. It shows that we have to look at the physics of single ions before we can fully predict how the whole magnetic system behaves on a frustrated lattice.

Lev: That opens up a lot of possibilities for designing new materials where you can engineer these kinds of magnetic responses deliberately.

Jonathan Gaudet, Dalmau Reig-i-Plessis, Bogeng Wen, Thomas J. Hicken, Jonas A. Krieger, Jan Peter Embs, Hubertus Luetkens, Adam A. Aczel, Stuart A. Calder, Matthew B. Stone, Hae-Young Kee, * and Alannah M Hallas

NIST Center for Neutron Research, National Institute of Standards and Technology · Department of Materials Science and Engineering, University of Maryland · Stewart Blusson Quantum Matter Institute, University of British Columbia · Laboratory for Solid State Physics, ETH Zurich · Department of Physics, University of Toronto · PSI Center for Neutron and Muon Sciences, Villigen PSI · Neutron Scattering Division, Oak Ridge National Laboratory · Canadian Institute for Advanced Research (CIFAR)

cond-mat.str-el

Submitted: 2025-09-27

Updated: 2026-02-07

Journal ref: npj Quantum Materials 11, 29 (2026)

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: The gist: The apparent absence of a dipolar ordered moment in CuNdO2 is a consequence of the dominant Ising character of the antiferromagnetically coupled Nd3+ moments, which experience extreme

Key concepts

Ising Character: This describes the strong single-ion anisotropy of the Nd3+ magnetic moments, meaning they prefer to point along a specific axis (out-of-plane) rather than moving freely in any direction. This strong preference makes the moments behave like classical Ising spins, which is crucial for understanding how frustration manifests and how it is resolved in this material.
Magnetic Frustration: Frustration occurs when competing magnetic interactions prevent all neighboring spins from simultaneously satisfying their lowest energy state. In CuNdO2's triangular lattice, the antiferromagnetic coupling between Nd3+ ions creates this frustration. The system cannot settle into a simple ordered pattern due to the geometry and strong anisotropy.
120° Ordering: The frustration is relieved when the smaller component of the Nd3+ moments orders into a specific 120-degree pattern within the plane of the crystal (the ab-plane). This specific arrangement allows for a long-range magnetic order while resulting in a nearly vanishing net dipole moment, which explains why no large magnetic peaks are seen in neutron diffraction.

Terminology

Frustration — 2× in this paper · explained in 3 episodes

Summary

The gist: The apparent absence of a dipolar ordered moment in CuNdO2 is a consequence of the dominant Ising character of the antiferromagnetically coupled Nd3+ moments, which experience extreme frustration on the triangular lattice, and this frustration is relieved through in-plane ordering into a 120° structure with a nearly vanishing ordered moment<ref:2509.23523#pg2>.

Magnetic Ground State Determination

The investigation of the magnetic ground state of CuNdO2 reveals that long-range magnetic order at TN = 0.78 K is detected by heat capacity, magnetic susceptibility, and muon spin relaxation, but without accompanying magnetic Bragg peaks in neutron diffraction<ref:2509.23523#pg2>. The key ingredient to resolve this apparent contradiction is the consideration of single-ion anisotropy<ref:2509.23523#pg2>. Crystal field analysis reveals that Nd has a dominantly Ising character with a very small in-plane component<ref:2509.23523#pg2>. The extreme frustration of the antiferromagnetically coupled Ising moments is alleviated through the ordering of the significantly smaller in-plane component of the Nd3+ moment<ref:2509.23523#pg2>. Based on linear spin wave modelling, a 120° ordered state for CuNdO2 is proposed with a nearly vanishing magnitude of the ordered moment<ref:2509.23523#pg2>.

Single-Ion Anisotropy and Crystal Field

The single-ion properties of the Nd3+ moments were characterized using inelastic neutron scattering data<ref:2509.23523#pg2>. The spin-orbit ground state manifold of Nd3+ with total angular momentum J = 9/2 has a (2J + 1) = 10-fold degeneracy<ref:2509.23523#pg2>. When subjected to the D3d local crystal electric field (CEF) environment in CuNdO2, this degenerate manifold is expected to split into five Kramers doublets, the lowest energy of which will form the CEF ground state<ref:2509.23523#pg2>. The resulting CEF ground state of the Nd3+ pseudospins in CuNdO2 corresponds to a Kramers doublet described by G.S.±⟩ = ∓a ± 1/2⟩ + b ∓ 5/2⟩ + c ± 7/2⟩<ref:2509.23523#pg4>. This Kramers doublet is described by a Γ4 irreducible representation so it behaves essentially like a S = 1/2 spin degree of freedom<ref:2509.23523#pg5>. The dipole moment associated with this pseudo-spin-1/2 CEF ground state is highly anisotropic<ref:2509.23523#pg5>.

Absence of Magnetic Bragg Peaks

Powder neutron diffraction measurements showed no changes in Bragg peak intensity nor formation of any new commensurate or incommensurate magnetic Bragg peaks due to the onset of long-range magnetic order<ref:2509.23523#pg5>. This absence is reinforced by examining the difference between the two data sets, showing no deviations from zero within the signal-to-noise of the experiment<ref:2509.23523#pg5>. The neutron diffraction data at both temperatures can be fully captured through a purely structural refinement<ref:2509.23523#pg5>. This leads to the conclusion that there are no observable magnetic Bragg peaks in the neutron diffraction pattern of CuNdO2 within its ordered state<ref:2509.23523#pg5>.

Spin Dynamics and Magnon Spectrum

The final experimental evidence related to the magnetic ground state comes from measurements of collective spin excitations with inelastic neutron scattering<ref:2509.23523#pg7>. Below TN, a weakly dispersive excitation centered at an energy transfer close to 0.3 meV is observed, signifying a symmetry breaking of the CEF ground-state doublet<ref:2509.23523#pg7>. The minimal XXZ model with anisotropic exchange together with the small in-plane g-factor successfully captures the key experimental observations, including magnetic long-range order found by specific heat and µSR, and the presence of seemingly dispersionless magnon excitations<ref:2509.23523#pg7>. The best fit to the data is obtained with J = 0.2 meV and ϵ = 0.5, which indicates a 120° antiferromagnetic order on the triangular lattice, with ordered moments confined to the ab-plane<ref:2509.23523#pg7>.

Conclusion on Magnetic Ordering

The resulting CEF ground state of the Nd3+ pseudospins in CuNdO2 corresponds to a strong out-of-plane Ising anisotropy with a vanishing in-plane dipole moment of 0.11(8) µB<ref:2509.23523#pg7>. Despite this strong Ising anisotropy, the magnetic frustration inherent to the triangular lattice is relieved via an in-plane ordering into a 120° spin structure<ref:2509.23523#pg7>. The small in-plane g-factor significantly suppresses the dipole matrix elements for elastic neutron scattering, leading to an undetectably small magnetic Bragg intensity despite the presence of long-range magnetic order<ref:2509.23523#pg7>. The ordered state accounts for the absence of a measurable dipole moment in neutron diffraction experiments<ref:2509.23523#pg7>. This phenomenology is expected to commonly appear in rareearth frustrated magnets, especially those with strong single-ion anisotropy, where the easy and hard axes respectively promote and relieve magnetic frustration<ref:2509.23523#pg2>. The paper synthesizes and thoroughly characterizes the collective magnetism of CuNdO2, which hosts a frustrated triangular lattice of highly anisotropic pseudo-spins S = 1/2<ref:2509.23523#pg7>. The study concludes that the magnetic frustration is relieved via an in-plane ordering into a 120° spin structure<ref:2509.23523#pg7>. All authors discussed the results and commented on the manuscript<ref:2509.23523#pg10>. The data that support the findings of this study are provided in the main text<ref:2509.23523#pg9>. The raw data is available from the corresponding author upon request<ref:2509.23523#pg9>.

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The investigation of CuNdO2, which is a delafossite with a triangular lattice of magnetic Nd3+ ions that are well separated by non-magnetic Cu spacer layers, reveals the strong Ising character of the pseudo-spin 1/2 Nd3+ moments<ref:2509.23523#pg2>. Magnetic susceptibility and heat capacity measurements reveal the onset of long-range antiferromagnetic order at TN = 0.78 K<ref:2509.23523#pg2>. While the magnetic transition is definitively observed with muon spin relaxation, accompanied by the formation of a weakly dispersing spin wave excitation, no dipole-ordered moment is detected with neutron diffraction<ref:2509.23523#pg2>. We show that the apparent absence of a dipolar ordered moment is a consequence of the dominant Ising character of the antiferromagnetically coupled Nd3+ moments, which experience extreme frustration on the triangular lattice<ref:2509.23523#pg2>. Consequently, the frustration in CuNdO2 is relieved through in-plane ordering of the substantially smaller perpendicular component of the Nd3+ moments into a 120° structure, with a nearly vanishing ordered moment<ref:2509.23523#pg2>.

How it works

The magnetic frustration inherent to the triangular lattice is relieved through in-plane ordering of the substantially smaller perpendicular component of the Nd3+ moments into a 120° structure<ref:2509.23523#pg2>. This ordering results in a nearly vanishing ordered moment<ref:2509.23523#pg2>. The analysis is based on linear spin wave modelling of the collective spin excitation, which proposes a 120° ordered state for CuNdO2<ref:2509.23523#pg2>. This model is energetically stabilized by an XXZ spin Hamiltonian with exchange constant J = 0.2 meV and an anisotropic exchange coefficient epsilon = 0.5<ref:2509.23523#pg9>. The resulting magnon bands with J = 0.2 meV and epsilon ≈ 0.5 best reproduces the key experimental features<ref:2509.23523#pg9>.

Improvements for AI systems

Bold Header: Integration of Single-Ion Anisotropy into Magnetic Ground State Models

This improvement allows AI systems to move beyond simple Heisenberg models by incorporating single-ion anisotropy derived from crystal field analysis, which is crucial for accurately describing rare-earth magnetic behavior in frustrated lattices, as the paper states: The extreme frustration of the antiferromagnetically coupled Ising moments is alleviated through the ordering of the significantly smaller in-plane component.

Bold Header: Predictive Modeling of Magnetic Bragg Peak Absence

The AI can be trained to predict and explain experimental discrepancies where long-range order is confirmed by magnetometry but not by neutron diffraction, specifically learning from the conclusion that the apparent absence of a dipolar ordered moment is a consequence of the dominant Ising character of the antiferromagnetically coupled Nd3+ moments.

Bold Header: Quantitative Determination of Spin Wave Parameters

The system can accurately calculate and parameterize effective spin Hamiltonian constants, such as exchange coupling strength J and anisotropy coefficient epsilon, by fitting inelastic neutron scattering data to linear spin wave theory, leading to the specific findings: the best fit to the data is obtained with J = 0.2 meV and ϵ = 0.5.

Bold Header: Simulation of Collective Spin Excitations

The improved system can simulate magnon dispersion relations using the derived XXZ model Hamiltonian, allowing it to predict the magnon modes with J = 0.2 meV and ε ≈ 0.5 best reproduces the key experimental features and characterize their nearly dispersionless nature.

Bold Header: Identification of Hidden Magnetic Order Scenarios

The AI can distinguish between different explanations for conflicting experimental results, specifically evaluating whether observed order is a higher order multipole or if moments are below the detection limit, based on the established CEF ground state character.

Abstract

We investigate the magnetic ground state of CuNdO 2, which is a delafossite with a triangular lattice of magnetic Nd 3+ ions that are well separated by non-magnetic Cu spacer layers. From inelastic neutron scattering measurements of the crystal electric field, we determine the strong Ising character of the pseudo-spin 1/2 Nd 3+ moments. Magnetic susceptibility and heat capacity measurements reveal the onset of long-range antiferromagnetic order at T N=0.78 K. While the magnetic transition is definitively observed with muon spin relaxation, accompanied by the formation of a weakly dispersing spin wave excitation, no dipole-ordered moment is detected with neutron diffraction. We show that the apparent absence of a dipolar ordered moment is a consequence of the dominant Ising character of the antiferromagnetically coupled Nd 3+ moments, which experience extreme frustration on the triangular lattice. Consequently, the frustration in CuNdO 2 is relieved through in-plane ordering of the substantially smaller perpendicular component of the Nd 3+ moments into a 120 structure, with a nearly vanishing ordered moment.

Sources

Sunny.jl: A Julia Package for Spin Dynamics