Dipolar-Octupolar Spin Hamiltonian Overview

• Dipolar–octupolar spin Hamiltonian is an effective pseudospin-1/2 model for rare-earth pyrochlores, where two components behave as dipoles and one as an octupole under D3d symmetry.
• It enables an economical nearest-neighbor exchange Hamiltonian that supports diverse phases including all-in–all-out order, moment fragmentation, and U(1) quantum spin ice regimes with 0-flux or π-flux sectors.
• Its symmetry properties and rotated basis formulation lead to clear neutron selection rules, distinguishing dipolar responses from nearly invisible octupolar dynamics in experimental probes.

The dipolar–octupolar spin Hamiltonian is the effective nearest-neighbor pseudospin description for rare-earth pyrochlores whose crystal-field ground state is a dipolar–octupolar Kramers doublet. Its defining feature is that the three pseudospin components do not transform uniformly under the local $D_{3d}$ symmetry: two components transform as magnetic dipoles, while the third transforms as a magnetic octupole. On the pyrochlore lattice of corner-sharing tetrahedra, this symmetry structure permits an unusually economical exchange Hamiltonian, yet supports a wide range of phases, including all-in–all-out order, moment fragmentation, and several $U(1)$ quantum spin ice regimes with either $0$-flux or $\pi$-flux backgrounds (Hosoi et al., 2022, Benton, 2020, Sahu et al., 2023).

1. Local doublets, local frames, and multipolar content

In Ce-, Nd-, and Sm-based pyrochlores, the low-energy magnetic degree of freedom is a pseudospin-$1/2$ defined within a crystal-field Kramers doublet on each rare-earth site. The natural local quantization frame is tied to the four sublattices of the pyrochlore lattice, with local Ising axes along the four $[111]$-type directions, for example

$$\hat z_0=\frac{(1,1,1)}{\sqrt3},\quad \hat z_1=\frac{(1,-1,-1)}{\sqrt3},\quad \hat z_2=\frac{(-1,1,-1)}{\sqrt3},\quad \hat z_3=\frac{(-1,-1,1)}{\sqrt3}.$$

These axes define the site-dependent local frames in which the effective Hamiltonian is written (Hosoi et al., 2022, Gao et al., 6 Jan 2026).

A central property of the dipolar–octupolar doublet is the symmetry distinction among pseudospin components. In one standard convention, used in several Ce-based analyses, $S^x$ and $S^z$ transform as magnetic dipoles, while $S^y$ transforms as a rank-3 magnetic octupole; all three are odd under time reversal (Hosoi et al., 2022, Sahu et al., 2023). In other conventions, especially in some Ce$U(1)$0Hf$U(1)$1O$U(1)$2 and Ce$U(1)$3Sn$U(1)$4O$U(1)$5 analyses, $U(1)$6 is taken as the dipolar component and $U(1)$7 as octupolar components (Porée et al., 2023, Yahne et al., 2022). This reflects the use of different rotated local bases rather than different underlying symmetry content.

The magnetic-field and neutron-selection rules follow directly from this multipolar structure. Dipolar components couple linearly to a uniform magnetic field, while the octupolar component does not couple linearly and is typically inefficiently probed by standard magnetic neutron scattering. In Ce$U(1)$8Zr$U(1)$9O$0$0, the effective Zeeman term is dominated by the local $0$1 projection, and $0$2 is strongly suppressed, so weak fields and neutrons couple predominantly to $0$3 (Hosoi et al., 2022). In Ce$0$4Hf$0$5O$0$6 and hydrothermal Ce$0$7Sn$0$8O$0$9, the working approximation is likewise that only the dipolar $\pi$0 or $\pi$1 component couples linearly to the field (Porée et al., 2023, Yahne et al., 2022).

2. Symmetry-allowed Hamiltonians and rotated-basis forms

The most general nearest-neighbor bilinear Hamiltonian for a dipolar–octupolar doublet on the pyrochlore lattice contains three diagonal exchanges and a single symmetry-allowed off-diagonal $\pi$2–$\pi$3 term,

$\pi$4

The absence of $\pi$5 and $\pi$6 terms follows from the fact that the octupolar component transforms in a different irreducible representation from the two dipolar components (Sahu et al., 2023, Chern et al., 2023).

A uniform rotation about the local $\pi$7-axis diagonalizes this Hamiltonian. Defining

$\pi$8

with

$\pi$9

one obtains the diagonal XYZ form

$1/2$0

where

$1/2$1

$1/2$2

$1/2$3

In this rotated basis, the nearest-neighbor exchanges are bond-independent, and the bond-dependent complex phase factors familiar from conventional anisotropic pyrochlore Hamiltonians are not needed for the DO doublet (Sahu et al., 2023, Benton, 2020).

Several papers further specialize this Hamiltonian. For Ce$1/2$4Zr$1/2$5O$1/2$6, analyses often set $1/2$7, leading to a minimal nearest-neighbor XYZ model

$1/2$8

which is the form used in exact diagonalization, molecular dynamics, and pseudofermion FRG studies (Hosoi et al., 2022, Chern et al., 2023). In octupolar-spin-ice treatments, an equivalent DO–XXZ form is often written with a dominant $1/2$9 term and transverse couplings $[111]$0 and $[111]$1 (Potts et al., 2024, Desrochers et al., 2023).

3. Quantum spin ice regimes and emergent gauge structure

Near an Ising limit, the dipolar–octupolar Hamiltonian supports several $[111]$2 quantum spin ice regimes. The basic distinction is whether the dominant Ising component lies in a dipolar or octupolar channel. In the notation of the rotated XYZ model, octupolar QSI arises when $[111]$3 is the dominant antiferromagnetic coupling, while dipolar QSI arises when $[111]$4 or $[111]$5 is dominant (Hosoi et al., 2022).

The low-energy description is an emergent lattice $[111]$6 gauge theory,

$[111]$7

where the Ising component maps to an emergent electric field and the transverse components generate gauge dynamics. The sign of the effective ring-exchange term $[111]$8 determines the background flux sector: unfrustrated transverse interactions favor $[111]$9-flux, while frustrated transverse interactions favor $$\hat z_0=\frac{(1,1,1)}{\sqrt3},\quad \hat z_1=\frac{(1,-1,-1)}{\sqrt3},\quad \hat z_2=\frac{(-1,1,-1)}{\sqrt3},\quad \hat z_3=\frac{(-1,-1,1)}{\sqrt3}.$$0-flux (Hosoi et al., 2022). Accordingly, four QSI regimes appear in the nearest-neighbor DO model: $$\hat z_0=\frac{(1,1,1)}{\sqrt3},\quad \hat z_1=\frac{(1,-1,-1)}{\sqrt3},\quad \hat z_2=\frac{(-1,1,-1)}{\sqrt3},\quad \hat z_3=\frac{(-1,-1,1)}{\sqrt3}.$$1-O-QSI, $$\hat z_0=\frac{(1,1,1)}{\sqrt3},\quad \hat z_1=\frac{(1,-1,-1)}{\sqrt3},\quad \hat z_2=\frac{(-1,1,-1)}{\sqrt3},\quad \hat z_3=\frac{(-1,-1,1)}{\sqrt3}.$$2-O-QSI, $$\hat z_0=\frac{(1,1,1)}{\sqrt3},\quad \hat z_1=\frac{(1,-1,-1)}{\sqrt3},\quad \hat z_2=\frac{(-1,1,-1)}{\sqrt3},\quad \hat z_3=\frac{(-1,-1,1)}{\sqrt3}.$$3-D-QSI, and $$\hat z_0=\frac{(1,1,1)}{\sqrt3},\quad \hat z_1=\frac{(1,-1,-1)}{\sqrt3},\quad \hat z_2=\frac{(-1,1,-1)}{\sqrt3},\quad \hat z_3=\frac{(-1,-1,1)}{\sqrt3}.$$4-D-QSI (Hosoi et al., 2022, Benton, 2020).

In octupolar formulations with dominant $$\hat z_0=\frac{(1,1,1)}{\sqrt3},\quad \hat z_1=\frac{(1,-1,-1)}{\sqrt3},\quad \hat z_2=\frac{(-1,1,-1)}{\sqrt3},\quad \hat z_3=\frac{(-1,-1,1)}{\sqrt3}.$$5, the same distinction is հաճախ encoded by the sign of $$\hat z_0=\frac{(1,1,1)}{\sqrt3},\quad \hat z_1=\frac{(1,-1,-1)}{\sqrt3},\quad \hat z_2=\frac{(-1,1,-1)}{\sqrt3},\quad \hat z_3=\frac{(-1,-1,1)}{\sqrt3}.$$6: within the GMFT used for DO pyrochlores, $$\hat z_0=\frac{(1,1,1)}{\sqrt3},\quad \hat z_1=\frac{(1,-1,-1)}{\sqrt3},\quad \hat z_2=\frac{(-1,1,-1)}{\sqrt3},\quad \hat z_3=\frac{(-1,-1,1)}{\sqrt3}.$$7 corresponds to $$\hat z_0=\frac{(1,1,1)}{\sqrt3},\quad \hat z_1=\frac{(1,-1,-1)}{\sqrt3},\quad \hat z_2=\frac{(-1,1,-1)}{\sqrt3},\quad \hat z_3=\frac{(-1,-1,1)}{\sqrt3}.$$8-flux QSI and $$\hat z_0=\frac{(1,1,1)}{\sqrt3},\quad \hat z_1=\frac{(1,-1,-1)}{\sqrt3},\quad \hat z_2=\frac{(-1,1,-1)}{\sqrt3},\quad \hat z_3=\frac{(-1,-1,1)}{\sqrt3}.$$9 to $S^x$0-flux QSI (Potts et al., 2024, Zhou et al., 2024). In Ce$S^x$1Hf$S^x$2O$S^x$3, the fitted parameter sets yield $S^x$4, which places the system in the $S^x$5-flux octupolar QSI sector; the ring-exchange scale $S^x$6 is then of order a few $S^x$7 meV (Porée et al., 2023).

The emergent excitations depend on this gauge structure. The standard $S^x$8 description contains gapless emergent photons and gapped spinons. In DO systems, however, the neutron cross section is dominated by the dipolar channel, so the octupolar Ising variable can be nearly invisible even when it is the field that enforces the ice rule. This asymmetry between the internal gauge variable and the observable dipolar channel is one of the characteristic features of the dipolar–octupolar problem (Hosoi et al., 2022, Desrochers et al., 2023).

4. Neutron selection rules and structure-factor diagnostics

Because neutrons couple to magnetic dipoles, the measured response is controlled by the dipolar sector of the DO doublet. The basic dynamical correlator is

$S^x$9

For Ce$S^z$0Zr$S^z$1O$S^z$2, where $S^z$3, the equal-time neutron structure factor is dominated by $S^z$4 correlations,

$S^z$5

This polarization factor is the standard local-frame projection for magnetic neutron scattering (Hosoi et al., 2022).

Classical and quantum calculations produce sharply different signatures. In classical spin ice or classical molecular dynamics, the Ising-sector correlations show sharp pinch points characteristic of divergence-free constraints. In exact diagonalization, these pinch points are substantially smeared by quantum fluctuations of the $S^z$6 gauge field, even in unfrustrated regimes (Hosoi et al., 2022). In the $S^z$7-flux octupolar QSI regime, the equal-time structure factor displays high-intensity peaks at $S^z$8 and $S^z$9 together with rod-like motifs along $S^y$0; these peaks are absent in classical MD and in the other QSI regimes studied in the same work (Hosoi et al., 2022).

The dynamical structure factor provides an even sharper distinction. Exact diagonalization for $S^y$1-O-QSI finds a pronounced high-intensity feature at the $S^y$2 point and a gapped spectrum with $S^y$3 for a representative parameter set, suggestive of gapped spinon excitations; by contrast, $S^y$4-O-QSI and the dipolar-QSI regimes show suppressed intensity at $S^y$5 and much weaker momentum dependence (Hosoi et al., 2022). A later GMFT analysis of octupolar QSI predicted that $S^y$6-O-QSI should exhibit a broad continuum with three distinctive peaks in the momentum-integrated spectrum, arising from two mostly flat spinon bands (Desrochers et al., 2023).

Polarized neutron channels also encode the flux sector. PFFRG calculations for the nearest-neighbor DO XYZ model showed that the $S^y$7-flux QSI regime produces pronounced rod-like features in the spin-flip channel and intensity minima at the centers of the fcc-Brillouin-zone hexagons in the non-spin-flip channel, consistent with polarized-neutron observations on Ce$S^y$8Zr$S^y$9O$U(1)$00 (Chern et al., 2023). This point is important because it rules out the common simplification that all rod-like diffuse scattering in a DO pyrochlore is equivalent: the detailed SF/NSF modulation depends on both the flux background and the role of quantum fluctuations.

5. Material realizations and competing interpretations

Ce$U(1)$01Zr$U(1)$02O$U(1)$03 is the most extensively analyzed DO-QSI candidate. Exact diagonalization and molecular-dynamics comparisons concluded that the quantum structure factor of the $U(1)$04-flux octupolar QSI regime is most compatible with neutron data, and that nearest-neighbor interactions suffice once quantum fluctuations are included, without requiring sizable further-neighbor couplings (Hosoi et al., 2022). PFFRG studies place the experimentally relevant parameter sets inside the $U(1)$05-flux QSI region of the nearest-neighbor $U(1)$06-dominated phase diagram (Chern et al., 2023). Field studies further interpret Ce$U(1)$07Zr$U(1)$08O$U(1)$09 as an octupolar $U(1)$10 QSL that undergoes an Anderson–Higgs transition in sufficiently large fields, with octupolar magnons that are invisible to neutrons but contribute to the heat capacity (Gao et al., 2022).

Ce$U(1)$11Hf$U(1)$12O$U(1)$13 was analyzed by neutron scattering and finite-temperature Lanczos fits, which found best-fit parameter sets with dominant octupolar exchange, significant but smaller dipolar exchange, and small negative $U(1)$14. The resulting phase identification is an octupolar QSI in the $U(1)$15-flux sector, with a low-energy spinon continuum centered at $U(1)$16 meV and a quasi-elastic low-$U(1)$17 dipolar signal. The work described the material as a “quantum multipolar liquid” because both dipolar and octupolar correlations are experimentally visible (Porée et al., 2023).

Ce$U(1)$18Sn$U(1)$19O$U(1)$20 remains more contentious. Earlier powder neutron data on solid-state samples showed high-$U(1)$21 diffuse scattering associated with magnetic octupoles and motivated an octupolar-QSI interpretation. By contrast, a later study of hydrothermally grown single crystals and powders fitted heat capacity, susceptibility, and diffuse neutron data to the nearest-neighbor DO XYZ model and placed the material inside the dipolar all-in–all-out Néel phase, with a finite-temperature proximate dipolar spin-ice regime above an expected low-temperature transition (Yahne et al., 2022). This contrast is a materials issue rather than a formal ambiguity of the Hamiltonian.

Ordered DO pyrochlores show a different use of the same framework. In Nd$U(1)$22Hf$U(1)$23O$U(1)$24, the rotated DO XYZ Hamiltonian with

$U(1)$25

accounts for all-in–all-out order together with a flat, gapped pinch-point mode at $U(1)$26 meV and dispersive half-moon features, which were interpreted as signatures of magnetic fragmentation (Samartzis et al., 2022). In Sm$U(1)$27Ti$U(1)$28O$U(1)$29, inelastic neutron scattering identified a DO Ising doublet, neutron diffraction found all-in–all-out order below $U(1)$30 K with ordered moment $U(1)$31, and $U(1)$32SR showed persistent low-energy spin dynamics, making the material a candidate for moment fragmentation physics (Mauws et al., 2018).

6. Magnetic fields, advanced probes, and current theoretical extensions

Magnetic-field response is highly anisotropic in DO pyrochlores because the field couples strongly only to the dipolar component projected onto the local $U(1)$33 axes. For fields along $U(1)$34, $U(1)$35, and $U(1)$36, the sublattice-dependent projections differ qualitatively, leading to distinct phase diagrams and structure-factor evolution (Zhou et al., 2024, Zhou et al., 19 Feb 2025). In GMFT calculations for Ce$U(1)$37Zr$U(1)$38O$U(1)$39-like parameters, the $U(1)$40-flux QSI is destabilized by comparatively small critical fields, while Ce$U(1)$41Hf$U(1)$42O$U(1)$43-like parameters yield a more robust $U(1)$44-flux QSI and, near the Ising limit under $U(1)$45 field, even allow staggered-flux states such as $U(1)$46 (Zhou et al., 2024, Zhou et al., 19 Feb 2025).

A recent neutron study under $U(1)$47 field introduced a same-temperature high-field subtraction protocol to separate photon and spinon sectors in Ce$U(1)$48Zr$U(1)$49O$U(1)$50. Weak fields of about $U(1)$51 T suppress the low-energy photon weight while leaving the higher-energy spinon continuum robust, albeit hardened; GMFT and exact diagonalization were used to interpret this as spectroscopic demarcation of emergent photons and spinons in a $U(1)$52-flux QSI (Gao et al., 6 Jan 2026). This is a notable development because it uses the selective Zeeman coupling of the DO Hamiltonian as a spectroscopic control parameter.

Nonlinear spectroscopy has also been proposed as a probe of the DO Hamiltonian. A two-dimensional coherent spectroscopy study of octupolar QSI showed that, in the intermediate-temperature window

$U(1)$53

spinons remain quantum coherent but move in an incoherent ice background, producing a broad response. At lower temperature, a sharp rephasing signal emerges and distinguishes $U(1)$54-flux from $U(1)$55-flux QSI (Potts et al., 2024). This suggests that the hierarchy of scales built into the DO Hamiltonian can be accessed by probes other than neutron scattering.

Theoretical generalizations now extend beyond bosonic spinons. A fermionic parton mean-field treatment classified 12 fully symmetric uniform $U(1)$56 QSLs for DO pyrochlores, including four “monopole-flux” states. Several of these states show linear low-temperature specific heat, while others show $U(1)$57 with $U(1)$58 close to $U(1)$59; the work connected these results to the metallic specific-heat response in Nd$U(1)$60ScNbO$U(1)$61 (Sahu et al., 2023). A plausible implication is that the dipolar–octupolar Hamiltonian is not tied to a single parton language: its restricted symmetry structure admits both gauge-theoretic and fermionic-spinon descriptions, with experimentally distinguishable consequences.

Across these developments, the central lesson remains stable. The dipolar–octupolar spin Hamiltonian is minimal at the level of symmetry, but not at the level of phenomenology. Its single octupolar channel, its basis-dependent dipole–octupole decomposition, and its selective coupling to fields and neutrons together generate a phase structure and spectroscopy that differ qualitatively from conventional pyrochlore anisotropic exchange models (Benton, 2020, Hosoi et al., 2022, Zhou et al., 2024).