Octupolar Spin Liquid in Pyrochlore Systems

• Octupolar spin liquid is a multipolar quantum-disordered state where magnetic octupoles dominate, giving rise to unconventional U(1) quantum spin ice with emergent photon and spinon excitations.
• Microscopic models based on dipolar-octupolar doublets reveal anisotropic interactions and a 2-plus-2-minus ice rule that distinguish octupolar systems from traditional spin ices.
• Experimental probes such as neutron scattering and specific heat measurements uncover the hidden octupolar correlations and complex dynamical structure inherent to these quantum spin liquids.

An octupolar spin liquid is a multipolar quantum-disordered state in which the dominant frustrated local degree of freedom is a magnetic octupole rather than a conventional magnetic dipole. In Ce-based dipolar-octupolar pyrochlores, the Ce$^{3+}$ ground-state Kramers doublet is described by a pseudospin-$\tfrac12$ for which $S^x$ and $S^z$ transform as magnetic dipoles while $S^y$ transforms as a magnetic octupole; when the octupolar interaction is dominant and antiferromagnetic, the system can realize an octupolar quantum spin ice, namely a three-dimensional $U(1)$ quantum spin liquid with fractionalized spinons and an emergent photon (Li et al., 2016, Hosoi et al., 2022). The term is also used in distinct settings beyond pyrochlore quantum spin ice, including one-dimensional octupolar Tomonaga-Luttinger liquids and field-induced fragmented or Kitaev-type multipolar liquids (Sato et al., 2011, Lozano-Gómez et al., 11 Sep 2025).

1. Local symmetry and the meaning of “octupolar”

The modern pyrochlore formulation starts from the dipole-octupole doublet. For Ce$^{3+}$ on the pyrochlore lattice, the local moment is a Kramers doublet with an unusual symmetry structure: two pseudospin components behave as magnetic dipoles, while the third behaves as a magnetic octupole. In the generic nearest-neighbor theory for these doublets, the $y$-component is the octupolar one, and this symmetry distinction survives the standard rotation that converts the exchange problem to an XYZ form; the octupolar $y$-component cannot be mapped to an ordinary dipole, and the resulting $U(1)$ spin liquids are therefore symmetry-enriched in a way absent from conventional dipolar quantum spin ice (Li et al., 2016).

This distinction has direct consequences for observables. In an octupolar $\tfrac12$0 spin liquid, the Ising variable of the emergent gauge theory lives in the octupolar channel, so the fundamental low-energy correlations are partly “hidden” from probes that couple linearly to magnetic dipoles. In Ce$\tfrac12$1Sn$\tfrac12$2O$\tfrac12$3, this hidden character was linked to diffuse neutron intensity weighted toward larger scattering vectors, indicating correlated degrees of freedom with a more complex magnetization density than typical magnetic dipoles, together with a continuum of excitations attributed to spinons and an octupolar “2-plus-2-minus” ice rule (Sibille et al., 2019).

2. Microscopic models and the octupolar ice manifold

The generic nearest-neighbor exchange model for dipole-octupole doublets is

$\tfrac12$4

After a rotation about the $\tfrac12$5-axis, this becomes an XYZ model in which $\tfrac12$6 remains octupolar while $\tfrac12$7 are dipolar. If the dominant Ising-like coupling is the antiferromagnetic octupolar term, the low-energy manifold is an octupolar analogue of spin ice: the local constraint is imposed on the octupolar component rather than on a dipolar one. A commonly used simplified octupolar model is

$\tfrac12$8

where the field couples to the dipolar channel and quantum tunneling is generated by the transverse term (Li et al., 2016).

In Ce$\tfrac12$9Sn$S^x$0O$S^x$1, the octupolar limit was formulated even more minimally as $S^x$2, with dominant octupole-octupole coupling $S^x$3 and subdominant dipolar exchange $S^x$4 acting as a quantum perturbation. The resulting manifold obeys a “2-plus-2-minus” rule for $S^x$5, directly paralleling the “2-in-2-out” constraint of ordinary spin ice, and finite quantum terms melt this classical octupole ice into a quantum liquid of magnetic octupoles (Sibille et al., 2019).

3. Emergent gauge structure and phase taxonomy

Within the quantum spin ice regime, the low-energy theory is a compact lattice $S^x$6 gauge theory,

$S^x$7

where $S^x$8 and $S^x$9 are emergent conjugate electric and vector gauge fields. The sign of $S^z$0 selects the background flux sector: $S^z$1-flux for $S^z$2 and $S^z$3-flux for $S^z$4. Because the Ising component can lie either in a dipolar or in an octupolar channel, four quantum spin ice regimes arise (Hosoi et al., 2022).

Ising channel	Flux sector	Label
Dipolar	$S^z$5-flux	0-D-QSI
Dipolar	$S^z$6-flux	$S^z$7-D-QSI
Octupolar	$S^z$8-flux	0-O-QSI
Octupolar	$S^z$9-flux	$S^y$0-O-QSI

For the nearest-neighbor XYZ model with dominant antiferromagnetic octupolar coupling, pseudofermion functional renormalization group identifies four principal phases: $S^y$1-flux QSI, $S^y$2-flux QSI, and all-in-all-out orders along the local $S^y$3 and $S^y$4 axes. In that treatment, the $S^y$5- and $S^y$6-flux QSIs remain quantum disordered down to the lowest RG cutoffs, and the parameter regime relevant to Ce$S^y$7Zr$S^y$8O$S^y$9 lies deep inside the $U(1)$0-flux QSI region (Chern et al., 2023).

Beyond this perturbative taxonomy, parton constructions enlarge the landscape. A Schwinger-boson PSG analysis found 4 symmetric $U(1)$1 QSL classes and 16 symmetric bosonic $U(1)$2 QSL classes, with two gapped $U(1)$3 states closely competing in the frustrated region relevant to Ce$U(1)$4Zr$U(1)$5O$U(1)$6; a fermionic parton mean-field treatment stabilized 12 fully symmetric uniform $U(1)$7 QSLs, including four time-reversal-invariant “monopole-flux” states carrying $U(1)$8 flux through each tetrahedron (Desrochers et al., 2021, Sahu et al., 2023).

4. Spectroscopy, thermodynamics, and diagnostic probes

For neutron scattering, the leading experimental sensitivity is to dipolar correlations, so the equal-time and dynamical responses of octupolar spin liquids are indirect and highly structured. Exact diagonalization and molecular dynamics across all four pyrochlore QSI regimes showed that the quantum structure factor of $U(1)$9-O-QSI has high-intensity peaks at $^{3+}$0 and $^{3+}$1, rod-like motifs with a distinctive intensity modulation, and a dynamical structure factor with strong spectral weight at the $^{3+}$2 point $^{3+}$3; the same calculations reported a gapped spectrum with $^{3+}$4, interpreted as a spinon gap, while the corresponding signatures are absent or reversed in the other QSI regimes and are not reproduced by classical simulations (Hosoi et al., 2022).

Gauge mean-field theory sharpened the dynamical prediction for octupolar quantum spin ice. In the $^{3+}$5-flux octupolar phase, the dynamical spin structure factor is a broad continuum with three distinctive peaks arising from two mostly flat spinon bands; by contrast, the $^{3+}$6-flux octupolar phase yields a continuum with a single broad peak near the upper edge. The same framework reproduces intensity-modulated rod motifs in the energy-integrated neutron signal and links the three-peak structure directly to two-spinon kinematics in the $^{3+}$7-flux state (Desrochers et al., 2023).

A later field-tuned neutron study on Ce$^{3+}$8Zr$^{3+}$9O$y$0 used a same-temperature high-field subtraction protocol to separate low-energy and high-energy contributions. In the rotated basis, the neutron intensity decomposes as

$y$1

with the first term identified as the photon channel, the second as the spinon channel, and the third as photon-spinon mixing. Under a $y$2 field, weak fields of approximately $y$3 T suppress the low-energy photon weight below $y$4 meV while leaving the higher-energy spinon continuum in the $y$5–$y$6 meV range robust, though hardened, providing a spectroscopic demarcation of emergent photon and spinon excitations in a dipolar-octupolar QSL (Gao et al., 6 Jan 2026).

Thermodynamic and spectroscopic probes do not interrogate the same gap. In the octupolar $y$7 spin liquid, specific heat measures all excitations, including the gapless photon, gapped spinons, and gapped magnetic monopoles, whereas uniform susceptibility and Knight shift access the single-spinon gap $y$8, and inelastic neutron scattering and $y$9 access the two-spinon continuum threshold $y$0. This multiplicity relation was proposed as a direct signature of fractionalization in Ce-pyrochlores (Chen, 2023).

Other probes were developed specifically to distinguish octupolar from dipolar quantum spin ice. Magnetostriction under $y$1 field provides a selection rule: dipolar QSI exhibits sharp non-analytic features such as drops and kinks, whereas octupolar QSI shows a smooth, monotonic response with at most a gentle kink as octupolar order is destroyed (Patri et al., 2020). Two-dimensional coherent spectroscopy provides a nonlinear dynamical diagnostic: at intermediate temperatures the response is broad because spinon dynamics are constrained by an incoherent spin background, while at lower temperatures a sharp signal emerges; within the coherent regime, $y$2-flux QSI gives a single sharp diagonal rephasing streak, whereas $y$3-flux QSI yields multiple sharp peaks and streaks reflecting its multi-band spinon structure (Potts et al., 2024).

5. Candidate materials, field response, and competing interpretations

Ce$y$4Sn$y$5O$y$6 is the clearest material realization of a quantum liquid of magnetic octupoles in the cited literature. High-$y$7 diffuse neutron scattering reveals substantial structured intensity at large momentum transfer rather than at low $y$8, low-energy inelastic neutron scattering detects a broad gapped continuum centered near $y$9 meV with width $U(1)$0 meV, and no long-range magnetic order is observed down to $U(1)$1 K. These data were interpreted as evidence for correlated octupolar moments, spinon excitations, and an octupolar “2-plus-2-minus” ice rule on the pyrochlore lattice (Sibille et al., 2019).

Ce$U(1)$2Zr$U(1)$3O$U(1)$4 occupies a more finely balanced part of parameter space. Heat capacity, susceptibility, and polarized neutron diffraction were analyzed using a near-neighbor XYZ Hamiltonian and numerical linked-cluster expansion, yielding a best-fit parameter set $U(1)$5 meV, $U(1)$6 meV, $U(1)$7 meV, and $U(1)$8. That analysis argued for a $U(1)$9 QSL ground state near the boundary between dipolar and octupolar character, with zone-boundary diffuse scattering in the non-spin-flip channel attributed to interactions beyond nearest neighbors (Smith et al., 2021). Subsequent exact-diagonalization work found that the quantum neutron structure factor of $\tfrac12$00-O-QSI is the most compatible among the four QSI regimes with the measured neutron data, while large-field neutron and heat-capacity measurements identified an Anderson-Higgs transition in which spinons condense into a static ferromagnetic ordered state whose octupolar spin waves are invisible to neutrons at leading order but contribute to the heat capacity (Gao et al., 2022, Hosoi et al., 2022).

The magnetic-field response has become a major part of the identification program. Gauge mean-field theory for fields along $\tfrac12$01, $\tfrac12$02, and $\tfrac12$03 predicts distinct evolutions of the equal-time and dynamical structure factors, and for $\tfrac12$04 it allows an additional staggered-flux state $\tfrac12$05; for Ce$\tfrac12$06Zr$\tfrac12$07O$\tfrac12$08, however, the same analysis concluded that only the $\tfrac12$09-flux QSL is realized at accessible fields, while transitions to staggered-flux or $\tfrac12$10-flux states would require unrealistically small $\tfrac12$11 or very large fields (Zhou et al., 2024).

The interpretation of Ce$\tfrac12$12Zr$\tfrac12$13O$\tfrac12$14 is therefore strong but not unique. Schwinger-boson mean-field theory identified two closely competing gapped $\tfrac12$15 QSLs with narrow spinon dispersion whose equal-time and dynamical structure factors reproduce key rod-like and high-symmetry-point features of the neutron data (Desrochers et al., 2021). Independent analyses also emphasized that the nearest-neighbor model does not fully capture the non-spin-flip zone-boundary diffuse scattering, and that similar scattering patterns could in principle arise from still unidentified quantum spin-liquid states; analytical calculations of the $\tfrac12$16-O-QSI structure factors were explicitly noted as lacking (Smith et al., 2021, Hosoi et al., 2022). This suggests that “octupolar spin liquid” in Ce$\tfrac12$17Zr$\tfrac12$18O$\tfrac12$19 denotes a well-motivated but still actively discriminated interpretation rather than a fully closed assignment.

6. Broader variants of octupolar spin liquids

A distinct one-dimensional usage of the term appears in the spin-$\tfrac12$20 $\tfrac12$21-$\tfrac12$22 frustrated chain under magnetic field. There, the octupolar phase is a Tomonaga-Luttinger liquid of condensed three-magnon bound states occurring for $\tfrac12$23. Longitudinal spin and octupolar operators $\tfrac12$24 have algebraic correlations, while transverse single-spin correlations are exponentially decaying because breaking a three-magnon bound state costs a finite energy $\tfrac12$25. The NMR relaxation rate $\tfrac12$26 correspondingly shows an anomalous low-temperature and field dependence, decreasing with decreasing temperature near saturation in contrast to conventional TL liquids (Sato et al., 2011).

Field-induced and higher-dimensional multipolar analogues broaden the concept further. In classical dipole-octupole pyrochlores under $\tfrac12$27 field, an intermediate fragmented spin liquid was identified in which a Kagome-plane $\tfrac12$28 spin liquid coexists with spontaneous $\tfrac12$29 symmetry breaking and partial dipolar polarization; its elastic neutron signature is a set of “shadow pinch points,” namely low-intensity pinch points underlying strong Bragg peaks (Lozano-Gómez et al., 11 Sep 2025). In spin-orbit-coupled $\tfrac12$30 honeycomb Mott insulators, a magnetic field can induce bond-anisotropic quadrupole-octupole interactions and tune the effective non-Kramers doublet to a Kitaev multipolar liquid with fractionalized Majorana excitations (Rayyan et al., 2022).

Octupolar correlations also enter exactly solvable hybrid settings. In a $\tfrac12$31 honeycomb model coupled to itinerant electrons, the on-site octupolar Kondo coupling

$\tfrac12$32

hybridizes conduction electrons with emergent Majorana fermions of the spin liquid, producing spontaneous time-reversal breaking and odd-frequency pairing (Farias et al., 2020). These constructions do not define the pyrochlore octupolar $\tfrac12$33 spin liquid itself, but they show that octupolar frustration supports a wider class of entangled multipolar liquids than the pyrochlore quantum spin ice alone.