The search for quantum spin liquids (QSL)—exotic forms of matter with fractionalized excitations—has drawn much attention of the physics community. While theories are no longer in short supply, tracking down materials has turned out to be remarkably tricky, in large part because of the difficulty to diagnose experimentally a state with only topological, rather than conventional, forms of order. Pyrochlore systems have proven a particularly promising ground, with the spin ices Dy/Ho2Ti2O71 hosting a classical Coulomb phase2,3,4,5, with subsequent proposals of candidate QSLs in other pyrochlores6. Quantum fluctuations have a tendency to pick an ordered state from the classically degenerate ice manifold via the “order by disorder”7 mechanism, making QSLs extremely rare in nature. However, it is known that certain QSLs exist as a robust phase in realistic theoretical models on the pyrochlore lattice8; and one thus hopes that a material can be found that realizes a “quantum spin ice”.

The recently discovered pyrochlore Ce2Zr2O79,10 shows tremendous promise as the long sought-after QSL/quantum spin ice. Its neutron-scattering response shows a complete lack of magnetic ordering and the presence of a continuum of spectral weight9,10, a characteristic hallmark of fractionalized excitations. Additionally, no signature of an ordering transition at low temperature has been observed in the specific heat. These observations immediately led to the proposals that Ce2Zr2O7 might provide a realization of a QSL.

What makes the cerium-based Ce2Zr2O7 pyrochlore distinct from the earlier studied Yb-based quantum spin-ice candidate Yb2Ti2O711,12,13,14 is the dipolar-octupolar nature15,16,17 of the ground state doublet of the Ce3+ ion, shown schematically in Fig. 1(a). It is given by the \(\left|J=5/2,{m}_{J}=\pm 3/2\right\rangle\) doublet, where the quantization axis z is chosen as the local [111] axis of the cubic lattice9. The transverse components of the angular momentum thus have vanishing expectation values in the ground state, which in turn implies that they are invisible to the spin-flip scattering in the neutron-scattering experiments because ΔmJ = 3 excitations do not couple, to leading order, to the dipolar magnetic moment of neutrons. Using an effective pseudospin 1/2 representation of the ground state doublet15, one of the three components (historically denoted sy) represents the octupolar moment, and the other two components (sx and sz) transform as the the familiar dipole.

The key to the unusual properties of Ce2Zr2O7 is the effective interactions between spin components, both dipolar and octupolar, belonging to the nearest-neighbors Ce ions, which are different from the thoroughly studied dipolar spin ice. A model nearest neighbor (nn) Hamiltonian incorporating all symmetry-allowed spin-spin interactions on the pyrochlore lattice is15:

$${H}_{{\mathrm{nn}}}=\mathop{\sum}\limits_{\langle ij\rangle }{J}_{y}{s}_{i}^{y}{s}_{j}^{y}+[{J}_{x}{s}_{i}^{x}{s}_{j}^{x}+{J}_{z}{s}_{i}^{z}{s}_{j}^{z}+{J}_{xz}({s}_{i}^{x}{s}_{j}^{z}+{s}_{i}^{z}{s}_{j}^{x})],$$

with si again expressed in the local frame, i.e., relative to the local [111] direction on a given Ce site (for a detailed expression in the global frame see Supplementary Note 1). Jx, Jy, Jz, and Jxz are magnetic coupling strengths and 〈ij〉 refers to nearest neighbor pairs of sites.

Fig. 1: Quantum spin liquid in a pyrochlore magnet and its experimental ramifications.
figure 1

a Depiction of a pyrochlore lattice with octupolar components forming 2-in-2-out ice states, and showing representative nearest neighbor (nn) and next-nearest neighbor (nnn) bonds. b Our model parameter sets are shown with black dots, superposed on the mean-field phase diagram of ref. 20. c Dynamical structure factors at T = 0.06 K integrated over the energy range 0.00 to 0.15 meV obtained from classical molecular dynamics (MD) for 8192 spins along with the corresponding comparison with ref. 9 (T = 0.035 K) and ref. 10 (T = 0.06 K, with an additional background subtraction). The plots reported in the published references were adapted for the purpose of comparison with our results. Special points in the Brillouin zone are also shown. d, e The dynamical structure factor as function of energy and momentum using the MD data (for 1024 sites) compared to experiment. The quantum-classical correspondence dictates that the MD data be rescaled23 by the factor βE (β = 1/kBT where kB is the Boltzmann factor, and E is the neutron energy transfer). Panel d is for the momentum (0, 0, l) cross section (Γ0 → X → Γ1) and panel e is the powder average. For panels c, d, e the numerically optimized Hamiltonian (parameter set no. 2 with both nearest neighbor and next-nearest neighbor terms, see Supplementary Table 1) was simulated. Additional Lorentzian convolution of width Γ was applied to mimic the limited experimental resolution, based on values reported in ref. 9 and ref. 10. The color scheme used for the theoretical calculation and the two experiments differs, and in the absence of additional information only the relative variations should be compared. Note that the high intensity around q = (1, 1, ± 1) and (2, 2, − 2) seen in the experimental inset in panel c results from an imperfect subtraction (using high-temperature data) of the nuclear Bragg peaks, which are absent from our theoretical calculations. The same is true in panel e at q ~ 1 Å−1 and q ~ 2 Å−1, where the maxima of intensity originate from imperfect subtraction of nuclear Bragg peaks.

While the form of the Hamiltonian is dictated by symmetry, the values of the parameters, which dictate the location of the material on a theoretical phase diagram, depend on its chemistry. Owing to the multiple competing scales, predicting these numbers accurately from first-principles calculations is known to be challenging and has been the subject of various past and ongoing debates (see for example, refs. 18,19). The route we take is to optimize the parameters in Eq. (1) by quantitatively reproducing results of multiple experiments using a combination of quantum and classical methods. After accomplishing this objective, we appeal to known theoretical predictions and infer the kind of phase Ce2Zr2O7 realizes. We find that all our optimized parameter sets lie in a π-flux U(1) QSL part of a previously constructed mean-field phase diagram20.


Effective Hamiltonian from specific heat and magnetization

By fitting the experimental magnetization and specific heat to (quantum) finite temperature Lanczos method (FTLM) calculations21 (see Fig. 2, Methods and Supplementary Note 2 for more details), we have determined the parameters of the model Hamiltonian in Eq. (1). A crucial result from the modeling perspective is that we have identified the Jy interaction — which acts between the octupolar components — to be the largest term (Jy ≈ 0.1 meV), with two interactions (Jx and Jz) playing a subleading role, and a vanishing Jxz. We have obtained several sets of parameter values (see Supplementary Table 1) within the fitting error bars, shown by the black dots in Fig. 1(b), clustered around Jy = 0.08 ± 0.01 meV, Jx = 0.05 ± 0.02 meV, Jz = 0.02 ± 0.01 meV. We show additional cost function analyses for a wide range of Jx, Jy and Jz in Supplementary Note 3, that illustrate constraints on our fits given the current availability of experimental data. Additionally, since the fitting was performed using a 16-site cluster to reduce the computational cost, we also verified that the parameters adequately describe a 32-site system (see Supplementary Note 4).

Fig. 2: Fitting magnetization, specific heat, and neutron-scattering data to obtain the model Hamiltonian parameters.
figure 2

Results of a two-stage fitting process used to determine the optimal Hamiltonian consistent with all reported experiments. a Specific heat as a function of temperature obtained for four different model parameter sets, vs. experimental data (extracted from ref. 9) for the case of zero field and applied field of 2 T along the [111] direction. b Magnetization obtained from finite temperature Lanczos of the model Eq. (1) on a 16-site cluster (for analysis of finite-size effects, see Supplementary Note 4) as functions of magnetic field applied along the [111] and [110] directions. For clarity, the theoretical data is shown for one parameter set only (set no. 2, see Methods and Supplementary Note 3, also for fits with other parameter sets, of comparable quality) and only the experimental results for magnetization (extracted from ref. 9) with field along [111] direction have been shown. c The predicted zero-field neutron-scattering structure factor S(q), Eq. (4), computed with SCGA, for three different values of Jnnn, with the left panel (Jnnn = 0) corresponding to the nearest-neighbor only model, Eq. (1).

Inelastic neutron spin structure factor

Note that since the octupolar sy moments do not couple to neutron spins to leading order, it is crucial to perform fits to the magnetization and specific heat as described above; attempts to fit solely the dynamical structure factors (see expressions in Supplementary Note 5) measured in inelastic neutron scattering (INS) are less reliable. We do employ the INS data, however, at the second stage of our fitting process. After determining Hnn, we introduce an additional weak next-nearest neighbor (nnn) coupling Jnnn (see Methods and Supplementary Note 1), which likely originates from the magnetic dipole-dipole interaction between Ce ions. Its optimal value Jnnn ~ −0.005Jy ≈ −0.5 μeV was determined by comparing the Self-Consistent Gaussian Approximation (SCGA) prediction of the spin structure factor with INS data (see Fig. 2(c)).

Armed with this complete Hamiltonian (nn and nnn terms), we compute the energy-integrated and energy-resolved momentum-dependent neutron spin structure factor using classical Molecular Dynamics (MD)22,23,24 for N = 1024 and N = 8192 sites. Representative comparisons with previous experiments on Ce2Zr2O79,10 are shown in Fig. 1(c–e). Figure 1(c) shows the characteristic ring-like structure centered around Γ0 = (0, 0, 0) point, with pronounced maxima at q = 1 (in particular q = (0, 0, 1) and \((\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)\) where we use the standard Miller indices (h, k, l) to denote the location in reciprocal space), consistent with experimental observations. Using the quantum-classical correspondence to rescale the MD data23, in Fig. 1(d) we show the results for a one-dimensional cross section in momentum space (Γ0 → X ≡ (0, 0, 1) → Γ1 ≡ (0, 0, 2)). The increased intensity at the X point at low energies is broadly consistent with experimental findings (additionally, see Supplementary Note 6). Figure 1(e) shows our results for the powder averaged case, compared with the experimental data in ref. 10, suggesting an overall agreement of the energy scales over the entire Brillouin zone (the X point corresponds to q ~ 0.6 Å−1 in physical units, where the intensity is highest).

The fact that multiple experimental features are accurately reproduced by a model of dipolar-octupolar interactions between the Ce spin components in Eq. (1) (plus a small Jnnn term) poses the question about what phase corresponds to its ground state. Indeed, the fact that the coupling Jy between the nearest neighbor octupolar moments is antiferromagnetic and by far the largest suggests that the leading behavior is for the corresponding components to form a (classical) 2-in/2-out spin-ice manifold (In Supplementary Note 7, we show additional evidence for the importance of Jy in explaining the experimental data). The presence of non-zero Jx and Jz interactions then adds quantum effects; generically, this opens the possibility of obtaining a quantum spin-ice phase.

Quantum spin-ice phase

The phase diagram for the model Hamiltonian in Eq. (1) was studied in refs. 20,25, using a combination of analytical and mean-field analysis as well as exact diagonalization. In our modeling of the magnetization and specific heat, we have determined four candidate sets of fitting parameters depicted by black dots in Fig. 1(b). All four fall deep into the parameter regime of the π-flux octupolar quantum spin-ice (π-OQSI) phase, based on the phase diagram of ref. 20. In this phase, the emergent gauge field A takes a non-trivial ground state configuration that hosts a flux Φ = π through each hexagon26,27,28,

$${\nabla }_{{{\hexagon}}}\times A\equiv \mathop{\sum}\limits_{i\in \,{{\hexagon}}}{A}_{{{{{\bf{r}}}}}_{i},{{{{\bf{r}}}}}_{i+1}}=\pi ,$$

where ri is the site of the dual diamond lattice, and the spins and A live on the links ri, ri+1 of the pyrochlore lattice, as shown schematically in the inset of Fig. 3(a). This fact follows from the effective U(1) quantum field theory, described in the Supplementary Note 8, with the flux-dependent contribution in the form

$${H}_{{{\mathrm{flux}}}}={J}_{{{\mathrm{ring}}}}\mathop{\sum}\limits_{\,{{\hexagon}}}\cos ({\nabla }_{{{\hexagon}}}\times A),$$

where \({J}_{{{\mathrm{ring}}}} \sim {({J}_{x}+{J}_{z})}^{3}/(64{J}_{y}^{2})\) is positive and thus favors flux Φ = π in each hexagon, resulting in the π-OQSI phase.

Fig. 3: Properties of the octupolar quantum spin-ice state.
figure 3

a Schematic depiction of the energy spectrum of π-OQSI state, with the green band denoting the spin-flip excitations that create a pair of magnetic monopoles (spinons). The blue band represents the dispersive, gapped vison excitations, which are analogous to electric charges. The red line is the hallmark of gauge-neutral photon excitations of the quantum spin-ice, except here the octupolar (rather than dipolar) character renders the photons “invisible” to neutrons. Inset, Representation of the gauge flux Φ through the hexagon formed by 6 corner-sharing tetrahedra on the pyrochlore lattice. b The diagonal pseudospin correlation functions \({\tilde{S}}^{xx}\), \({\tilde{S}}^{yy}\), and \({\tilde{S}}^{zz}\) (see \({\tilde{S}}^{\alpha \beta }({{{\bf{q}}}})\) defined in the main text). Here x, y, z are the local axes (relative to the local \(\hat{{{{\bf{z}}}}}=[111]\) direction). The pinch-point patterns in the \({\tilde{S}}^{yy}\) channel are invisible to neutrons because of the octupolar nature of sy.

Like “conventional” quantum spin ice, π-OQSI has gapless photons and two types of gapped excitations (magnetic and electric charges), in close analogy to Maxwell electrodynamics. Their approximate energy scales are illustrated in Fig. 3(a). The first type of gapped excitations are spinons created in pairs by flipping the octupolar moment on a single site, costing energy \({{{\mathcal{O}}}}({J}_{y})\). These are analogs of the magnetic monopoles in electrodynamics, and observable in the specific heat as a characteristic Schottky peak at energy ~ Jy ≈ 1 K, as our FTLM calculations corroborate in Fig. 2(a). The second type of gapped excitations corresponds to the so-called visons (analogs of electric charges), which are sources of the gauge flux violating the condition in Eq. (2). Note the energy scale for exciting visons at Jring ~ 0.01 K is very low. One therefore expects to find thermally excited visons even at the base temperature of the experiment, so that their gap, if not closed by the quantum dynamics, will not be separately resolved. Rather, visons will strongly interact and mix with the emergent gapless photons, named in analogy to the photons familiar from Maxwell electrodynamics, due to the overlap of their energy scales [cf. Fig. 3(a)].

The energy and temperature scales for observing the photons would correspondingly be very low. More importantly, they would not directly couple to neutrons because of the octupolar nature of the π-OQSI. To illustrate this point, we have computed the pseudospin correlation functions \({\tilde{S}}^{\alpha \beta }({{{\bf{q}}}})=\frac{1}{N}{\sum }_{i,j}{e}^{i{{{\bf{q}}}}\cdot ({{{{\bf{r}}}}}_{i}-{{{{\bf{r}}}}}_{j})}\langle {s}^{\alpha }({{{{\bf{r}}}}}_{i}){s}^{\beta }({{{{\bf{r}}}}}_{j})\rangle\) in Fig. 3(b) by SCGA (see Methods for details). The largest components of this matrix are the diagonal ones \({\tilde{S}}^{xx}\), \({\tilde{S}}^{yy}\) and \({\tilde{S}}^{zz}\). It is the octupolar \({\tilde{S}}^{yy}({{{\bf{q}}}})\) component in the middle panel of Fig. 3(b) that displays the pinch-points characteristic of the spin-ice4,29, and the photons’ gapless dispersion will emanate from the q location of those pinch-points. Crucially however, the sy components of spin, at leading order, do not couple to the magnetic field or to the neutron moment (see Methods, especially Eq. (6)), meaning that the aforementioned pinch-points will not feature in the experiment. Instead, the (energy-integrated) neutron-scattering structure factor

$$S({{{\bf{q}}}})=\frac{1}{N}\mathop{\sum}\limits_{\mu \nu }\left({\delta }_{\mu \nu }-\frac{{q}_{\mu }{q}_{\nu }}{{q}^{2}}\right)\,\mathop{\sum}\limits_{i,j}{e}^{i{{{\bf{q}}}}\cdot ({{{{\bf{r}}}}}_{i}-{{{{\bf{r}}}}}_{j})}\langle {m}^{\mu }({{{{\bf{r}}}}}_{i}){m}^{\nu }({{{{\bf{r}}}}}_{j})\rangle$$

is expressed in terms of true magnetic moments mμ = ∑λgμλsλ that contain the g-factors, whose gμy components are all zero (see Supplementary Note 1). As a consequence, \({\tilde{S}}^{yy}\) drops out of the neutron structure factor, computed in Figs. 1(c) and 2(c) using Monte Carlo and SCGA respectively. The main contributors to the neutron-scattering structure factor are the \({\tilde{S}}^{xx}\) and \({\tilde{S}}^{zz}\) channels convoluted with the neutron-coupling form factor in Eq. (4). As a result, instead of the pinch-points, a sixfold, three-rod-crossing-like pattern is observed. Such a rod pattern is expected for a pyrochlore lattice with nearest-neighbor interactions only30 and is associated with the dispersion of spinons (magnetic monopoles) in the context of quantum spin ice30,31,32.

Upon inclusion of (weak) nnn interactions Jnnn, the pattern of crossing rods deforms into a characteristic ring-like structure that appears in S(q), as shown in the three panels of Fig. 2(c). Phenomenologically fixing the value Jnnn ~ −0.5 μeV matches very well with the experimental INS observations [see Fig. 1(c)]. Thence, while it is tempting to associate the intensity variation along the rods in Fig. 2(c) as the disappearance of pinch-point intensity centered at (1, 1, 1) (and equivalent points), which has been predicted to be the quintessential feature of the dispersive, quantum photon modes in dipolar quantum spin ice33, our analysis rather suggests that these are the consequence of the small nnn interactions between Ce ions that modulate the INS intensity along the [111] direction. As for the emergent photons, while they are indeed expected to be present in the π-OQSI phase, as shown in Fig. 3(a), their octupolar nature turns out to render them much less visible to neutrons, and they can only be detected via weaker, higher-order coupling to neutrons at large momentum transfer34,35, or indirectly, for instance through their contribution to the low-temperature specific heat (at TJring/kB ≈ 0.01 K).


Given the effective microscopic description of Ce2Zr2O7, an interesting question is to ask how the (putative) octupolar quantum spin-ice state responds to the application of an external magnetic field25. While the octupolar sy pseudospin components do not couple linearly to the field and will remain in the 2-in/2-out configuration, the sz components will cant along the field direction (see Eq. (6), where gx ≈ 0). In order to elucidate the experimental consequences, we compute the in-field spin structure factor S(q) within classical Monte Carlo calculations on our model for 8192 sites, shown in Fig. 4. As a function of increasing field along the [001] direction, the ring-like structure in S(q) quickly weakens (Fig. 4b), until eventually disappearing and giving way to sharp Bragg peaks in high fields (Fig. 4c). These predictions are to be compared with future INS data in an applied magnetic field.

Fig. 4: Predictions for static spin structure factor as expected to be measured in INS.
figure 4

The predicted (energy-integrated) neutron-scattering spin structure factor (in arbitrary units) in a zero field and an applied magnetic field along [001] b B = μ0H = 0.01 T and c B = μ0H = 4 T, computed using MC calculations for 8192 sites at T = 0.1 K. The Bragg peaks at 4 T, which have intensity value ~3400, appear in the white regions indicated with arrows.

The fact that the magnetic octupolar degrees of freedom do not couple in the leading order to neutron spin or to the external magnetic field, makes the octupolar spin liquid difficult to detect. Its elusive nature may however prove to be a blessing in disguise, as a reduced coupling to magnetic defects and associated stray magnetic fields—which are known to destabilize the more conventional dipolar spin liquids—is similarly suppressed. Indeed, chemical disorder on magnetic sites is believed to be the leading reason for the failure to observe the quantum spin liquid behavior in, for instance, the herberthsmithite kagome compounds despite their high crystallographic quality36. The fact that neither magnetic order nor spin glassiness is seen in cerium pyrochlores Ce2Zr2O7 and Ce2Sn2O734,37 may be taken as an additional, albeit indirect, evidence of the robustness of the underlying octupolar spin liquid. While the possibility of such a quantum spin liquid has been entertained in seminal theoretical studies before15,20, our present work firmly identifies Ce2Zr2O7 as a very promising host for the π-flux octupolar quantum spin-ice phase. The present study also underscores the importance of carefully fitting multiple experiments, including specific heat and magnetization, in addition to the INS spectra, to determine the effective model Hamiltonian, which otherwise may be plagued with uncertainties affecting the searches and identification of QSLs18,19.


Fitting model Hamiltonian parameters

In addition to the nearest neighbor (nn) Hamiltonian in Eq. (1) we have considered the next-nearest neighbor (nnn) interaction, which in the local basis is given by,

$${H}_{{\mathrm{nnn}}}=\mathop{\sum}\limits_{\langle \langle i,j\rangle \rangle }{J}_{{\mathrm{nnn}}}\left[{s}_{i}^{x}\,{s}_{i}^{y}\,{s}_{i}^{z}\right]\left[\begin{array}{lll}{g}_{x}^{2}&0&{g}_{x}{g}_{z}\\ 0&0&0\\ {g}_{x}{g}_{z}&0&{g}_{z}^{2}\end{array}\right]\left[\begin{array}{l}{s}_{j}^{x}\\ {s}_{j}^{y}\\ {s}_{j}^{z}\end{array}\right].$$

where 〈〈i, j〉〉 refers to nnn pairs of sites.

For the case of an applied external magnetic field, the Zeeman term must also be accounted for. This term involves the coupling of magnetic field to effective spin 1/2 degrees of freedom, which are not the usual and familiar dipoles, and are instead dipolar-octupolar doublets. A key observation is that the octupolar magnetic moments do not couple, to linear order, to the external magnetic field. The octupolar moments can in principle couple to the third power of the magnetic field H3 (ref. 20), however, this effect is expected to be small for the experimentally accessible fields. It can be shown that only the component of the field H along the local z-axis couples to the dipole moment15, as follows:

$${H}_{Z}=-\mathop{\sum}\limits_{i}({{{\bf{h}}}}\cdot {{{{\bf{z}}}}}_{i})({g}_{z}{s}_{i}^{z}+{g}_{x}{s}_{i}^{x}),$$

where h = μBμ0H is the effective magnetic field strength, μB is the Bohr magneton. Note that when projected onto the J = 5/2 multiplet, one expects the Landé g-factor gz = 2.57 and gx = 0. However, if an admixture of higher spin-orbit multiplet (J = 7/2) is present in the ground state doublet, one generically expects a non-zero value of gx (see Supplementary Note 9), which we allow for in our modeling.

As explained in the main text, the Hamiltonian parameters in Eq. (1) were determined using both zero and finite applied magnetic field data. For the specific heat and magnetization, we have performed quantum FTLM calculations on a 16-site cluster. Details of the general technique can be found in ref. 21 and previous application to some pyrochlore systems can be found in ref. 38. Our convergence checks of the method (see Supplementary Note 2) suggest that the statistical errors present in the specific heat and magnetization are smaller than the line width used in Fig. 2. Figure 2(a) shows the temperature dependence of the specific heat and suggests that fitting it is somewhat challenging, both in zero and applied field. Our four distinct parameter sets are generally able to describe specific heat very well at higher temperature T 1 K, however, the lower-temperature behavior is trickier and none of the parameter sets used satisfactorily fit the data especially in the absence of applied magnetic field. We attribute this to the finite-size effects in our FTLM calculations, which become more pronounced at lower temperatures. The magnetization as a function of applied field strength, shown in Fig. 2(b), appears to be less prone to finite-size effects (also see Supplementary Note 4) and is fitted reasonably well, especially at low fields. The discrepancy at high fields is not entirely unexpected, previous experimental reports have suggested a changing g-factor past a μ0H = 4 T field9, an effect not built into our model.

Despite the finite-system size limitations of the numerical calculations, and the finite energy and momentum resolution in INS experiment, all our parameter sets obtained are in agreement with Jy being antiferromagnetic. For three of these sets, we find Jy is large compared to the other two interactions (Jx and Jz) in Eq. (1), as depicted in Fig. 1(b) (also see Supplementary Note 7). A fourth parameter set, obtained by constraining gx to be zero, suggests Jy ≈ Jx > Jz. To further build confidence in our results, we have performed a brute force scan of Jx and Jz for representative fixed values of Jy and constructed a contour map of an appropriately defined cost function (see Supplementary Note 3, more subtleties with the fitting and additional competitive parameter sets are also discussed). Additional future experiments could potentially further constrain the values of these coupling constants.

The second step of our parameter fitting involved the determination of Jnnn, which we found to be small relative to Jy. Despite its smallness, it is responsible for significant reorganization of intensity in the Brillouin zone. Figure 2(c) shows the static structure factor in the (h, h, l) plane computed with SCGA (the details of which will be explained shortly) for representative values of Jnnn. Performing a brute force line search, we determined the optimal value Jnnn = − 5 × 10−4 meV working in steps of 10−4 meV.

The following values of the parameters (which we refer to as set no. 2, see Supplementary Note 3) were used in subsequent calculations (all values are in meV): Jx = 0.0385, Jy = 0.088, Jz = 0.020, Jxz = 0, Jnnn = − 0.0005, with the g-factors gx = − 0.2324, gz = 2.35. The other parameters sets are quoted in full in the Supplementary Table 1.

Details of the Monte Carlo and molecular (spin) dynamics calculations

For the Hamiltonian \({{{\mathcal{H}}}}\equiv {H}_{{\mathrm{nn}}}+{H}_{{\mathrm{nnn}}}+{H}_{Z}\), the dynamical structure factor is computed by integrating the classical Landau–Lifshitz equations of motion

$$\frac{{\mathrm{d}}}{{\mathrm{d}}t}{{{{\bf{S}}}}}_{i}=-{{{{\bf{S}}}}}_{i}\times \frac{\partial {{{\mathcal{H}}}}}{\partial {{{{\bf{S}}}}}_{i}},$$

which describes the precession of the spin in the local exchange field. We carry out all our calculations by transforming our Hamiltonian to the global basis, and have used the label Si to represent spins in this basis (see Supplementary Note 1 for more details on transformations between local and global bases).

Following the protocol adopted in previous work22,23, the initial configuration (IC) of spins is drawn by a Monte Carlo (MC) run from the Boltzmann distribution \(\exp (-\beta {{{\mathcal{H}}}})\) at temperature T = β−1 = 0.06 K. Then, for each starting configuration the spins are deterministically evolved according to Eq. (7) with the fourth order Runge-Kutta method. This procedure, referred to as molecular dynamics (MD) (also see Supplementary Note 5), is repeated for many independent IC (their total number being NIC) and the result is averaged,

$$\langle {S}_{i}^{\mu }(t){S}_{j}^{\nu }(0)\rangle =\frac{{\sum }_{{{\mbox{IC from MC}}}}{S}_{i}^{\mu }(t){S}_{j}^{\nu }(0){\left|\right.}_{{{\mbox{IC}}}}}{{N}_{IC}}.$$

We perform a Fourier transform in spatial and time coordinates to get the desired dynamical structure factor. We work with N = 16L3 sites, where L3 is the number of cubic unit cells, the results in Fig. 1(c) are for L = 8 (N = 8192) and L = 4 (N = 1024) for Fig. 1(d, e). NIC ≈ 104 was used, and each IC was evolved for 500 meV−1 in steps of δt = 0.02 meV−1. The statistical error bars in the classical MD/MC simulations presented are small enough to not change the features of the color plots.

To obtain an estimate of the quantum dynamical structure factor, we used a classical-quantum correspondence, which translates to a simple rescaling of the classical MD data by βE. More details and justification can be found in ref. 23.

Convolution with Lorentzian function to mimic limitations of experimental resolution

Figure 1(d, e) shows significant broadening along the energy axis, an effect not captured to the same extent by the raw (rescaled) MD data. Since the interaction energy scales in the material are small, a fairer comparison between experiment and theory is achieved by modeling the instrument’s energy resolution. Using Γ values in the ballpark suggested by refs. 9,10, we convolved our data with a Lorentzian factor,

$${S}_{{{\mathrm{exp}}}}({{{\bf{q}}}},E)=\frac{1}{\pi }\int S{({{{\bf{q}}}},E^{\prime} )}_{MD}\frac{{{\Gamma }}}{{{{\Gamma }}}^{2}+{(E^{\prime} -E)}^{2}}dE^{\prime}$$

The integral was approximated by a sum over discrete \(E^{\prime}\) points in steps of 0.01 meV.

Details of the SCGA calculations

The Self-Consistent Gaussian Approximation is an analytical method that treats the spin in the Large-N (flavor) limit. Our calculation follows closely ref. 4. In this study, we first treat sx,y,z as independent, freely fluctuating degrees of freedom. The Hamiltonian in momentum space is written as


where \({{{\bf{S}}}}=({s}_{1}^{x},{s}_{2}^{x},{s}_{3}^{x},{s}_{4}^{x},\ldots ,{s}_{3}^{z},{s}_{4}^{z}).\) The interaction matrix HLarge-N is the Fourier transformed interaction matrix that includes the nearest and next-nearest neighbor interactions.

We then introduce a Lagrangian multiplier with coefficient μ to the partition function to get

$${{{\mathcal{Z}}}}=\exp \left(-\frac{1}{2}{\int_{{{\mbox{BZ}}}}}{{\mbox{d}}}{{{\bf{k}}}}\,{{\mbox{d}}}{{{\bf{S}}}}\;{{{\bf{S}}}}\left[\beta {{{{\mathcal{H}}}}}_{{{\mathrm{Large-N}}}}+\mu {{{\mathcal{I}}}}\right]{{{\bf{S}}^T}}\right)$$

in order to impose an additional constraint of averaged spin-norm being one, or

$$\langle {{{{\bf{s}}}}}_{1}^{2}+{{{{\bf{s}}}}}_{2}^{2}+{{{{\bf{s}}}}}_{3}^{2}+{{{{\bf{s}}}}}_{4}^{2}\rangle =1.$$

For a given temperature kBT = 1/β, the value of μ is fixed by this constraint via relation

$${\int_{{{\mbox{BZ}}}}}{{\mbox{d}}}\,{{{\bf{k}}}}\mathop{\sum }\limits_{i=1}^{12}\frac{1}{{\lambda }_{i}({{{\bf{k}}}})+\mu }=\langle {{{{\bf{s}}}}}_{1}^{2}+{{{{\bf{s}}}}}_{2}^{2}+{{{{\bf{s}}}}}_{3}^{2}+{{{{\bf{s}}}}}_{4}^{2}\rangle =1,$$

where λi(k), i = 1, 2, … , 12 are the twelve eigenvalues of \(\beta {{{{\mathcal{H}}}}}_{{{\mathrm{Large-N}}}}\). With μ fixed, the partition function is completely determined for a free theory of S, and all correlation functions can be computed from \({\left[\beta {{{{\mathcal{H}}}}}_{{{\mathrm{Large-N}}}}+\mu {{{\mathcal{I}}}}\right]}^{-1}\).