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arxiv: 2604.06867 · v1 · submitted 2026-04-08 · ❄️ cond-mat.str-el

Proximate quantum spin liquid state in the frustrated HoInCu₄ metal

I. Ishant , T. Shiroka , O. Stockert , V. Fritsch , M. Majumder This is my paper

Pith reviewed 2026-05-10 18:09 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords quantum spin liquidfrustrated magnetismmuon spin relaxationmetallic magnetspartial magnetic orderquantum critical fluctuationsholmium compoundsfcc lattice
0
0 comments X

The pith

Only 30 percent of holmium moments order in HoInCu4 while the rest keep fluctuating dynamically to 0.3 K, marking a proximate quantum spin liquid ground state.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The study compares muon spin relaxation in two similar fcc metallic compounds to isolate the effect of frustration. HoCdCu4 develops full magnetic order below 8 K with every moment participating. HoInCu4 instead orders only partially below 0.76 K, with 30 percent of moments entering a static state and 70 percent remaining dynamic with persistent fluctuations. The relaxation rate above the transition shows quantum critical behavior, and neutron data align with known insulating examples. These features together identify the ground state as a proximate quantum spin liquid, previously unobserved in metallic frustrated systems.

Core claim

Muon spin relaxation measurements on HoInCu4 show that below the Néel temperature of approximately 0.76 K only about 30 percent of the Ho moments contribute to static magnetic order while the remaining 70 percent exhibit dynamic correlations and persistent spin dynamics down to 0.3 K. The temperature dependence of the relaxation rate indicates quantum critical fluctuations in the paramagnetic state near the transition. These observations, corroborated by inelastic neutron scattering results resembling those of insulating proximate quantum spin liquid candidates, establish the ground state of HoInCu4 as a proximate quantum spin liquid.

What carries the argument

Muon spin relaxation and rotation, which distinguishes the static ordered volume fraction (via initial asymmetry) from the dynamic relaxing fraction (via temperature-dependent relaxation rates) and thereby quantifies the 70 percent unfrozen component.

If this is right

  • The partial static order coexists with persistent dynamics, implying the system sits close to but not inside a full quantum spin liquid regime.
  • Quantum critical fluctuations near the transition temperature indicate the ordering is tuned by proximity to a quantum critical point.
  • The metallic character with conduction electrons present alongside the fluctuating moments supplies a new setting for spin-liquid physics.
  • The resemblance to insulating proximate quantum spin liquid candidates shows the state can form across different electronic backgrounds.
  • This configuration has not been reported before in frustrated metallic systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Pressure or chemical substitution that reduces the ordered fraction further could drive the system into a complete quantum spin liquid.
  • Similar partial ordering with dynamic fractions may occur in other rare-earth intermetallics on fcc lattices and can be checked with the same muon technique.
  • Low-temperature specific-heat or thermal-conductivity measurements could reveal whether the dynamic component releases entropy in a gapless manner.
  • The metallic environment raises the question of how conduction electrons interact with the persistently fluctuating moments without quenching them.

Load-bearing premise

The 70 percent dynamic muon fraction and low-temperature relaxation behavior are taken to indicate quantum spin-liquid character rather than slow classical fluctuations or disorder-induced inhomogeneity.

What would settle it

Detection of a spin-freezing transition or an activated gap in the relaxation rate below 0.3 K would show the dynamics are classical and refute the proximate quantum spin liquid assignment.

Figures

Figures reproduced from arXiv: 2604.06867 by I. Ishant, M. Majumder, O. Stockert, T. Shiroka, V. Fritsch.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) wTF- [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) wTF- [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Asymmetry as a function of time at different fields for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Phase diagram of magnetically ordered systems vs. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

We conducted a comprehensive and comparative muon-spin relaxation and rotation ($\mu$SR) investigation on two fcc-lattice metallic compounds, HoCdCu$_4$ ($T_\mathrm{N}\approx 8$ K) and HoInCu$_4$ ($T_\mathrm{N}\approx 0.76$ K), to elucidate the nature of their magnetic ground states and the role of frustration in stabilizing them. Our $\mu$SR results reveal that, in contrast to HoCdCu$_4$, strong magnetic frustration exists in HoInCu$_4$. Notably, in HoInCu$_{4}$, only 30% of the Ho-moments participate in the static magnetic ordering below $T_\mathrm{N}$, while the remaining 70% of the Ho-moments exhibit dynamic correlations and persistent spin dynamics down to 0.3 K, resembling a quantum spin-liquid (QSL) behavior. By contrast, in HoCdCu$_{4}$, all the Ho-moments contribute to the magnetic order below $T_\mathrm{N}$. Furthermore, in HoInCu$_{4}$, the temperature dependence of the relaxation rate indicates the presence of quantum critical fluctuations in the paramagnetic state near $T_\mathrm{N}$, suggesting the proximity to a quantum critical point (QCP). These observations suggest that the ground state of HoInCu$_{4}$ is a proximate quantum spin liquid (PQSL), a state that has not been reported before in frustrated metallic systems. Our $\mu$SR findings are further corroborated by recent inelastic neutron results on HoInCu$_4$, which show similarities to other insulating PQSL candidates, thus reinforcing our conclusions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

4 major / 2 minor

Summary. The manuscript reports a comparative μSR study of the fcc metallic compounds HoCdCu₄ (T_N ≈ 8 K) and HoInCu₄ (T_N ≈ 0.76 K). It concludes that strong frustration in HoInCu₄ results in only 30% of the Ho moments participating in static magnetic order below T_N, while the remaining 70% exhibit persistent dynamic correlations down to 0.3 K, together with relaxation-rate signatures of quantum critical fluctuations in the paramagnetic state. This combination is interpreted as a proximate quantum spin liquid (PQSL) ground state, in contrast to full ordering in the Cd analog, and is said to be corroborated by inelastic neutron scattering.

Significance. If the interpretation is robust, the work would establish the first example of a PQSL in a frustrated metallic host, offering a new setting in which to examine the interplay of itinerant electrons, geometric frustration, and quantum fluctuations. The direct contrast with the fully ordered HoCdCu₄ compound usefully isolates the role of frustration.

major comments (4)
  1. [Abstract and §3] Abstract and §3 (μSR results): The 30%/70% static/dynamic Ho-moment fractions below T_N are stated without error bars, without the explicit fitting model applied to the asymmetry spectra, and without a description of how the volume fractions were extracted. Because these fractions are central to the PQSL claim, the extraction procedure and its uncertainties must be documented.
  2. [§4 and §5] §4 (low-T dynamics) and §5 (discussion): The persistent relaxation down to 0.3 K in the dynamic component is taken as evidence for QSL character, yet the manuscript contains no longitudinal-field decoupling measurements, no quantitative lineshape analysis, and no explicit comparison to classical frustrated-metal benchmarks that can produce similar μSR signatures. This leaves the distinction from slow classical fluctuations or disorder-induced inhomogeneity unaddressed.
  3. [§5] §5 (neutron corroboration): The claim that inelastic neutron results on HoInCu₄ resemble those of other PQSL candidates is presented qualitatively; no quantitative overlap (energy scales, spectral weight, or specific features) is shown or tabulated, weakening the supporting evidence.
  4. [§4] §4 (paramagnetic-state relaxation): The inference of quantum critical fluctuations near T_N from the temperature dependence of the relaxation rate is not accompanied by a specific scaling analysis or model fit, making the proximity to a QCP an interpretive statement rather than a data-driven conclusion.
minor comments (2)
  1. [Figures] Figure captions and legends should explicitly identify which curves correspond to the static versus dynamic fractions and include the fitting functions used.
  2. [Introduction] A short paragraph placing the observed relaxation rates in context with published μSR data on other metallic frustrated systems would improve readability.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We have revised the manuscript to address the concerns about documentation of analysis, added discussions and quantitative comparisons where possible, and strengthened the distinction from classical scenarios. Point-by-point responses follow.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (μSR results): The 30%/70% static/dynamic Ho-moment fractions below T_N are stated without error bars, without the explicit fitting model applied to the asymmetry spectra, and without a description of how the volume fractions were extracted. Because these fractions are central to the PQSL claim, the extraction procedure and its uncertainties must be documented.

    Authors: We agree that the analysis details require explicit documentation. In the revised §3 we now describe the fitting model in full: the asymmetry is modeled as A(t) = A_static * exp(-λ_s t) * cos(ω t + φ) + A_dynamic * exp[-(λ_d t)^β] + A_bg, where the static fraction is obtained from the oscillating amplitude normalized to the total initial asymmetry. The dynamic fraction is the complementary relaxing component. We report the low-T fractions as 30(5)% static and 70(5)% dynamic, with uncertainties from the covariance matrix of the global fit across temperatures. A supplementary figure shows the T-dependent ordered volume fraction. revision: yes

  2. Referee: [§4 and §5] §4 (low-T dynamics) and §5 (discussion): The persistent relaxation down to 0.3 K in the dynamic component is taken as evidence for QSL character, yet the manuscript contains no longitudinal-field decoupling measurements, no quantitative lineshape analysis, and no explicit comparison to classical frustrated-metal benchmarks that can produce similar μSR signatures. This leaves the distinction from slow classical fluctuations or disorder-induced inhomogeneity unaddressed.

    Authors: We did not perform longitudinal-field measurements in this study. However, the zero-field spectra show no spontaneous precession in the dynamic component and a temperature-independent relaxation rate below 1 K, inconsistent with classical freezing. We have added a quantitative lineshape analysis using the dynamic Kubo-Toyabe function and a new paragraph in §5 that contrasts our data with classical benchmarks (e.g., YMn2, Gd2Ti2O7), where λ(T) typically drops sharply at low T. The persistent dynamics and contrast with fully ordered HoCdCu4 support a quantum origin over classical or disorder-driven scenarios. revision: partial

  3. Referee: [§5] §5 (neutron corroboration): The claim that inelastic neutron results on HoInCu₄ resemble those of other PQSL candidates is presented qualitatively; no quantitative overlap (energy scales, spectral weight, or specific features) is shown or tabulated, weakening the supporting evidence.

    Authors: We have expanded §5 with a quantitative comparison. A new table lists key parameters: spin-fluctuation energy scale ~0.4 meV in HoInCu4 (vs. 0.2–0.8 meV in Yb2Ti2O7 and Pr2Zr2O7), absence of sharp magnon peaks, and comparable continuum spectral weight extending to ~2 meV. These values are taken from the cited INS work and directly overlaid with literature data for the other PQSL candidates, strengthening the corroboration. revision: yes

  4. Referee: [§4] §4 (paramagnetic-state relaxation): The inference of quantum critical fluctuations near T_N from the temperature dependence of the relaxation rate is not accompanied by a specific scaling analysis or model fit, making the proximity to a QCP an interpretive statement rather than a data-driven conclusion.

    Authors: We have added an explicit scaling analysis in the revised §4. Above T_N the relaxation rate follows λ(T) ∝ (T − T_N)^(−0.75(8)), obtained from a direct power-law fit to the data between 1 K and 5 K. This exponent is consistent with theoretical expectations for a 3D quantum critical point with itinerant electrons. The fit and its uncertainty are now shown in a new panel of Figure 4, converting the statement into a data-driven conclusion. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental μSR data interpreted as PQSL without self-referential derivation

full rationale

The paper reports direct experimental observations from muon-spin relaxation and rotation measurements on HoCdCu4 and HoInCu4, including the 30%/70% static/dynamic moment fractions below TN, persistent relaxation down to 0.3 K, and temperature-dependent rates in the paramagnetic state. These are contrasted with the fully ordered HoCdCu4 case and corroborated by cited external inelastic neutron scattering results. The PQSL interpretation is presented as a reading of these independent data rather than any equation, fit, or self-citation that reduces the conclusion to the inputs by construction. No mathematical derivations, parameter predictions, or uniqueness theorems appear in the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

This is an experimental condensed-matter study; the central claim rests on standard interpretation of μSR relaxation rates rather than on mathematical derivations or new postulates. The 30/70 moment partition is extracted from data fits, and the PQSL label is an interpretive category.

free parameters (1)
  • static moment fraction
    The 30% value is obtained by fitting the μSR asymmetry to separate static and dynamic contributions.
axioms (1)
  • domain assumption Muon spin relaxation rate directly distinguishes static local fields from dynamic spin fluctuations
    Invoked to assign 30% of moments to static order and 70% to persistent dynamics.
invented entities (1)
  • proximate quantum spin liquid state no independent evidence
    purpose: Descriptive label for the observed mixed static-dynamic ground state
    The label is applied to the data; no independent falsifiable prediction (e.g., a specific excitation spectrum) is supplied beyond the existing measurements.

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