Metallic Magnetism Hints at Exotic Quantum State

Author: Denis Avetisyan

New research suggests the compound HoInCu4 exhibits properties of a quantum spin liquid, a rare phase of matter typically found only in insulators.

The phase diagram elucidates the magnetically ordered states of face-centered cubic lattices as a function of <span class="katex-eq" data-katex-display="false">J_2/J_1</span>, revealing transitions between antiferromagnetic orderings (type II and type III), a quantum critical point, and both quantum spin liquid and proximate quantum spin liquid phases-a landscape in which the compounds HoCdCu4 and HoInCu4 are positioned according to their respective Néel temperatures <span class="katex-eq" data-katex-display="false">T_N</span>. — The phase diagram elucidates the magnetically ordered states of face-centered cubic lattices as a function of $J_2/J_1$ , revealing transitions between antiferromagnetic orderings (type II and type III), a quantum critical point, and both quantum spin liquid and proximate quantum spin liquid phases-a landscape in which the compounds HoCdCu4 and HoInCu4 are positioned according to their respective Néel temperatures $T_N$ .

Evidence from muon-spin relaxation reveals a proximate quantum spin liquid state in the frustrated metallic compound HoInCu4.

The search for quantum spin liquid (QSL) states, exotic phases of matter with persistent quantum entanglement, has largely focused on insulating materials. Here, we report on a comprehensive μSR investigation of the frustrated metallic compound HoInCu$_4$, as detailed in ‘Proximate quantum spin liquid state in the frustrated HoInCu$_4$ metal’. Our results demonstrate that HoInCu$_4$ exhibits characteristics of a proximate quantum spin liquid (PQSL), with only 30% of the Ho moments undergoing static magnetic ordering while the remaining moments display dynamic correlations down to 0.3 K. Does this discovery in a metallic system offer new insights into the stabilization mechanisms of QSL behavior and pave the way for realizing these states in technologically relevant materials?

Unveiling the Quantum Dance of Spin

Conventional magnetism typically arises from the alignment of atomic spins over long distances, creating a predictable and ordered arrangement. However, the principles of quantum mechanics permit a far wider range of magnetic behaviors, extending beyond this classical picture. Quantum mechanics allows for states where spins are highly entangled, exhibiting correlations that persist even without a fixed, long-range order. These exotic states, unlike traditional magnets, don’t simply flip polarity at a specific temperature-the Curie temperature-but instead exhibit more subtle and complex responses to external stimuli. The exploration of these quantum states is driving innovation in materials science, promising novel functionalities and potentially revolutionary technologies that go beyond the limitations of conventional magnetic materials.

The pursuit of Quantum Spin Liquids (QSLs) represents a significant frontier in condensed matter physics, driven by the tantalizing prospect of matter existing in a fundamentally new state. Unlike traditional magnets where electron spins align in an ordered fashion, QSLs exhibit a remarkable form of magnetic behavior – persistent entanglement without long-range order. This means the spins remain constantly fluctuating and correlated, even at absolute zero temperature, behaving as a collective quantum entity. Realizing these states isn’t simply an academic exercise; QSLs are theorized to host emergent properties like fractionalized excitations – quasiparticles with unusual quantum numbers – and could potentially revolutionize technologies ranging from quantum computing to materials science. The challenge lies in identifying or engineering materials where the delicate balance needed to suppress conventional magnetic order and promote this exotic quantum state can be achieved, leading to intense research focused on frustrated magnetic systems and novel material designs.

The pursuit of novel magnetic states often hinges on the concept of magnetic frustration, a phenomenon where competing interactions prevent spins from aligning in a simple, predictable pattern. Imagine a triangle of magnetic moments; if each interaction favors anti-alignment, a single, stable arrangement is impossible, leading to a ‘frustrated’ ground state. This isn’t merely a static impasse; instead, frustration encourages exotic behaviors like spin fluctuations and the formation of entangled quantum states. Materials exhibiting strong magnetic frustration, such as certain triangular lattices or kagome lattices, become prime candidates for hosting Quantum Spin Liquids – phases of matter where spins are highly correlated yet lack long-range magnetic order. These frustrated systems, therefore, aren’t failures of magnetism, but rather pathways to entirely new, and potentially technologically valuable, magnetic phenomena.

Muon spin rotation measurements on HoCdCu4 reveal a transition to long-range magnetic ordering at <span class="katex-eq" data-katex-display="false">T_N = 8K</span>, evidenced by changes in the asymmetry signal and a corresponding temperature-dependent non-magnetic volume fraction. — Muon spin rotation measurements on HoCdCu4 reveal a transition to long-range magnetic ordering at $T_N = 8K$ , evidenced by changes in the asymmetry signal and a corresponding temperature-dependent non-magnetic volume fraction.

Decoding Spin Signals: The Muon’s Perspective

Muon-Spin Relaxation/Rotation (µSR) is a technique leveraging the sensitivity of implanted polarized muons to local magnetic fields. Positive muons, possessing a magnetic dipole moment, act as sensitive probes, precessing at a rate – the Larmor frequency – directly proportional to the strength of the magnetic field at their location. This precession is disrupted by both static and fluctuating magnetic fields, leading to a decay in the muon spin polarization which is measured as a function of time. The rate of this decay provides quantitative information about the magnitude and timescale of magnetic fluctuations, enabling the detection of extremely weak or disordered magnetic phenomena that may be undetectable by other methods. Due to the muon’s relatively weak interaction with matter, it probes the magnetic environment with minimal disturbance, making it suitable for studying a wide range of materials, including complex magnetic systems and those exhibiting unconventional magnetic ordering.

Muon-Spin Relaxation (µSR) measurements performed on HoInCu4, a material theorized to be a Proximate Quantum Spin Liquid (PQSL), demonstrate the absence of long-range static magnetic order. The µSR technique detects local magnetic fields experienced by implanted positive muons, and in HoInCu4, the relaxation rates observed are consistent with a dynamically fluctuating spin environment rather than a frozen, ordered state. This finding indicates that the magnetic moments within the material do not align in a fixed, static pattern, even at very low temperatures, supporting the hypothesis that HoInCu4 may exhibit characteristics of a quantum spin liquid where magnetic excitations are fractionalized and highly entangled.

Neutron diffraction experiments performed on HoInCu4 corroborate the absence of long-range magnetic ordering, providing further support for its classification as a proximate quantum spin liquid (PQSL) candidate. This technique, sensitive to the arrangement of magnetic moments within the material, revealed a diffuse scattering pattern characteristic of disordered spins rather than the sharp, well-defined peaks expected from a magnetically ordered state. The lack of detectable Bragg scattering confirms that, despite the presence of localized magnetic moments from the Ho ions, these moments do not align to form a conventional antiferromagnetic or ferromagnetic structure down to measurement temperatures of approximately 0.76 K, consistent with the predicted behavior of a quantum spin liquid.

Muon-Spin Relaxation/Rotation (µSR) employs multiple measurement techniques to characterize dynamic spin correlations within materials. Zero-field µSR detects static magnetic fields, while longitudinal-field µSR probes the fluctuations perpendicular to the applied field, and transverse-field µSR is sensitive to fluctuations transverse to the field. Application of these techniques to HoInCu4, a candidate Proximate Quantum Spin Liquid, has revealed a very low ordering temperature (TN) of 0.76 K. This low TN, combined with the absence of conventional magnetic order detected via µSR and confirmed by neutron diffraction, suggests the presence of unconventional magnetic behavior and supports the hypothesis that HoInCu4 may exhibit characteristics of a quantum spin liquid.

Muon spin rotation measurements reveal a temperature-dependent non-magnetic volume fraction in HoInCu4, characterized by asymmetry changes in both transverse field (a) and zero-field (b) modes, and the resulting temperature dependence of λ (c) and <span class="katex-eq" data-katex-display="false">f_s</span> (d), with the latter detailed in the inset of Figure 1(c). — Muon spin rotation measurements reveal a temperature-dependent non-magnetic volume fraction in HoInCu4, characterized by asymmetry changes in both transverse field (a) and zero-field (b) modes, and the resulting temperature dependence of λ (c) and $f_s$ (d), with the latter detailed in the inset of Figure 1(c).

The Dance of Frustration and Correlation

Heavy fermion systems, characterized by the hybridization of f-electrons with conduction electrons, exhibit enhanced density of states at the Fermi level due to strong electron correlations. This leads to large effective masses for the electrons and heightened magnetic susceptibility. The strong correlations also promote competing magnetic interactions, increasing the likelihood of forming unconventional magnetic states such as quantum spin liquids (QSLs). Unlike conventional magnetism where spins order at low temperatures, QSLs maintain a disordered spin state even at absolute zero, due to strong quantum fluctuations and topological constraints arising from the interplay of these correlated electrons. This makes heavy fermion materials ideal platforms for investigating and potentially realizing QSL phases, which are of significant interest in condensed matter physics due to their exotic properties and potential applications in quantum technologies.

Magnetic frustration occurs when competing exchange interactions within a material prevent the establishment of a simple, globally ordered magnetic ground state. Specifically, interactions that favor different magnetic alignments – such as antiferromagnetic versus ferromagnetic coupling – can lead to a cancellation of net magnetization and a disordered spin configuration. This frustration is often mitigated by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which, under certain conditions, promotes long-range magnetic order, effectively resolving the frustration. Alternatively, the Kondo effect, involving the screening of localized magnetic moments by conduction electrons, can also suppress magnetic frustration by reducing the strength of the localized moments and diminishing the competing interactions.

Research on heavy fermion compounds, specifically CeRh_1-xPd_xSn, CePd_1-xNi_xAl, and CeRhSn, indicates that manipulating the lattice parameters via chemical substitution-effectively applying chemical pressure-can induce a transition towards a quantum spin liquid (QSL) state. By varying the composition (x) and therefore the interatomic distances, researchers can suppress long-range magnetic order and promote the emergence of correlated quantum fluctuations characteristic of QSLs. This tuning is achieved by altering the balance between competing magnetic interactions, ultimately driving the system towards a state with a macroscopic ground state degeneracy and fractionalized excitations.

Muon Spin Rotation (µSR) with transverse fields (wTF) is utilized to quantify the Non-magnetic Volume Fraction (NMVF) within materials, providing insight into the prevalence of disordered spins. Measurements on HoInCu4 reveal that only approximately 30% of the Ho³⁺ moments participate in magnetic ordering below the Néel temperature (T_N). This is in stark contrast to the isostructural HoCdCu4, where nearly 100% of the Ho³⁺ moments exhibit magnetic ordering under similar conditions. The significantly reduced magnetic ordering in HoInCu4 suggests a substantial fraction of spins remain disordered, consistent with the proposed Quantum Spin Liquid (QSL) state where magnetic moments do not fully condense into a classically ordered pattern.

Beyond the Ideal: Proximate States and Controlled Emergence

Proximate Quantum Spin Liquids (PQSLs) represent an expansion of the conventional Quantum Spin Liquid (QSL) paradigm, observed in materials like K2Ni2(SO4)3 and KYbSe2. Unlike conventional QSLs which exhibit robust, long-range entanglement and fractionalized excitations as inherent ground state properties, PQSLs are characterized by being ‘close’ to magnetically ordered states. This proximity means their behavior is strongly influenced by perturbations, and they often exhibit features of both ordered and disordered phases. The identification of these materials demonstrates that strong correlations and geometric frustration can give rise to novel ground states even without fully realizing the idealized conditions required for ‘pure’ QSL behavior, thus broadening the scope of QSL research to include systems with competing interactions and a greater susceptibility to external influences.

Proximate Quantum Spin Liquids (PQSLs) arise from the interplay of strong electron correlations and geometric frustration within their crystal lattice. These conditions prevent conventional magnetic ordering, even at low temperatures, leading to the formation of novel ground states characterized by fractionalized excitations and long-range entanglement. Unlike traditional magnetically ordered materials, PQSLs exhibit properties such as a continuous spectrum of magnetic excitations and the absence of a clear magnetic ordering temperature. The resulting quantum states are not described by simple classical models, necessitating the application of quantum many-body techniques to understand their behavior and predict their measurable properties. These unique characteristics distinguish PQSLs from conventional quantum spin liquids and open possibilities for realizing exotic quantum phenomena.

Field-Induced Critical Spin Liquids (CSLs) represent a pathway to realizing quantum spin liquid (QSL) behavior in materials that do not exhibit it in their ground state. Application of an external magnetic field alters the exchange interactions and can drive a system into a QSL phase. This induction mechanism bypasses the need for specific material compositions or crystal structures inherently favorable to QSLs. Materials like CePdAl demonstrate this behavior under hydrostatic pressure and applied magnetic fields, transitioning to a CSL state. The ability to manipulate and induce QSL phases through external stimuli expands the scope of QSL research and provides a means to study these exotic states in a wider range of materials.

Certain materials, such as CePdAl, demonstrate Critical Spin Liquid (CSL) behavior when subjected to hydrostatic pressure, offering an alternative pathway to investigate these quantum states beyond zero-field conditions. Supporting evidence for Proximate Quantum Spin Liquid (PQSL) characteristics is found in materials like HoInCu4, where muon-spin relaxation measurements reveal a power-law exponent of 0.26. This value signifies the presence of robust quantum fluctuations within the material, further validating the PQSL ground state and contributing to a broader understanding of correlated electron systems.

The exploration into HoInCu₄’s unusual magnetic behavior reveals a system perpetually testing the boundaries of established magnetic ordering. This metallic compound, exhibiting characteristics of a proximate quantum spin liquid, doesn’t simply conform to expected phases; it exists in a state of dynamic fluctuation, a dance on the precipice of order. As Georg Wilhelm Friedrich Hegel observed, “The truth is the whole.” The researchers, by probing the material’s magnetic response, are attempting to grasp this ‘whole’ – a complex interplay of frustration and quantum effects. The very existence of a quantum spin liquid state in a metallic material challenges the conventional wisdom and suggests a far richer landscape of magnetic phenomena than previously imagined, demanding a constant re-evaluation of fundamental principles.

Where Do We Go From Here?

The observation of proximate quantum spin liquid behavior in a metallic compound-HoInCu₄-forces a reassessment of established boundaries. The conventional wisdom held such states largely confined to insulators; this work suggests the architectural constraints were misidentified, not absolute. The material doesn’t quite achieve the fully realized spin liquid state, existing instead in a frustratingly close approximation. This near-miss, however, is often where the most instructive data resides. The ‘bug’ in this system-its failure to fully transition-confesses the design flaws in current theoretical models.

Future investigations must address the role of metallicity itself. Is the conduction band actively suppressing long-range entanglement, or merely a spectator to the underlying spin frustrations? Furthermore, systematic exploration of similar metallic compounds with varying degrees of frustration is crucial. A single data point, however compelling, only sketches the outline of a larger, potentially radical, phase space. The search isn’t for confirmation, but for the inevitable deviations-the materials that don’t behave as predicted.

Ultimately, the pursuit of quantum spin liquids isn’t about finding a new state of matter, but about dismantling the assumptions that define ‘matter’ in the first place. Each imperfect realization, each tantalizingly close approximation, is a controlled demolition of established paradigms. The real discovery won’t be what exists, but why the universe allows such defiance of simple categorization.

Original article: https://arxiv.org/pdf/2604.06867.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Unveiling the Quantum Dance of Spin

Decoding Spin Signals: The Muon’s Perspective

The Dance of Frustration and Correlation

Beyond the Ideal: Proximate States and Controlled Emergence

Where Do We Go From Here?

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