Some understanding of recent inelastic neutron scattering on Ce-pyrochlores: Signatures of electric monopoles
Pengwei Zhao Gang V. Chen






gangchen.physics@gmail.com, away from HKU.
^1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
^2Collaborative Innovation Center of Quantum Matter, 100871, Beijing, China


We revisit the pyrochlore spin liquid materials Ce_2Sn_2O_7 and Ce_2Zr_2O_7
by examining the existing experiments. We continue to rely on the special properties of 
the dipole-octupole nature of the Ce^3+ moment. The inelastic neutron scattering (INS)
measurement in the octupolar U(1) spin liquid selects the (gapped) spinon continuum, 
and thus has suppressed spectral weights below the energy threshold of two spinon gaps. 
This measurement, however, includes all other emergent excitations at lower energies in 
the dipolar U(1) spin liquid, in particular, the gapless photon and the continuum of the electric monopoles.    
Although the electric monopole continuum is weakly gapped (compared to the larger spinon gap),    
the energy scale is actually close to the gauge photons, and the spectrum largely overlaps with the photons.
Due to the background dual π flux for the electric monopoles, the density of states is enhanced 
at lower energies, creating peak structures. This can be contrasted with the linearly suppressed 
spectral weight of the gauge photons at low energies. We propose that the electric monopole continuum 
should be mostly responsible for the low-energy spectrum in the INS measurement 
in the dipolar U(1) spin liquid. With these understanding and calculation, 
we discuss the available experimental results and predict further experiments 
for Ce_2Sn_2O_7 and Ce_2Zr_2O_7. 


Some understanding of recent inelastic neutron scattering on Ce-pyrochlores:
 Signatures of electric monopoles
    Gang V. Chen^1,2
    April 29, 2024
==============================================================================================================





There has been a continuous interest in the pyrochlore quantum spin liquid <cit.>. 
It is an exotic quantum state that is understood quite well theoretically and 
at the same time has a strong experimental connection with the pyrochlore 
quantum spin ice materials <cit.>. 
After the intensive exploration of the Tb-pyrochlores, 
Yb-pyrochlores and Pr-pyrochlore <cit.>,
 the recent activities were about the Ce-based pyrochlore magnets,
 where the Ce^3+ local moments were identified as the dipole-octupole (DO) 
doublets and related to distinct symmetry-enriched spin liquids <cit.>. 
There are now three different Ce-based pyrochlore compounds,  
Ce_2Sn_2O_7, Ce_2Zr_2O_7 
and Ce_2Hf_2O_7 <cit.>.
Most experiments were performed on the first two compounds so far. 
In this work, we review some of the existing experiments,
mostly the inelastic neutron scattering (INS) results, and aim to provide some insights 
from the theoretical perspective to the understanding of the experiments. 



Since our major point is about an important corner of the low-energy excitation 
spectra that distinguishes the dipolar U(1) spin liquid and the octupolar U(1) spin liquid, 
we simply state the results here. In short, the INS measurement
could detect all the emergent excitations for the dipolar U(1) spin liquid. In particular, due to
the intrinsic π dual U(1) gauge flux, the electric monopole continuum stands out 
at low energies and creates the peak structures on top of the suppressed photon intensity. 
In reality, the whole INS spectrum contains the peaks 
from the electric monopole continuum at low energies and the peaks 
from the spinon continuum at higher energies, in addition to the linearly 
suppressed photon modes <cit.> at low energies. 
The presence of the electric monopole continuum 
for the dipolar U(1) spin liquid for the DO doublets
in the INS measurement were actually claimed 
in our previous work <cit.>. 
Because many results were scattered in different parts in the previous works, 
the relation with the later experimental results has not yet been fully synthesized and clearly made.
To better serve the community, 
we fulfill this task by reviewing the existing experiments and performing expanded calculations 
based on the existing experimental progress. 
Arriving these results requires the combination of the microscopic nature of the DO doublet,
the effective spin model, and the experimental measurement. To make the presentation 
of this work self-contained, we will start from the basics before embarking 
on a bit new outcomes. 







We begin with the dipole-octupole (DO) doublet on the pyrochlore lattice. 
Each state of this doublet is a one-dimensional irreducible representation 
of the D_3d point group, and is transformed into the other 
by the time reversal operation <cit.>. 
Thus, they are special type of Kramers' doublet 
whose degeneracy is protected by the time reversal symmetry. More specially, 
their wavefunctions are linear superpositions of the local J^z = 3m/2 
(with m an odd integer) where the z direction is along the local 111 direction 
at each magnetic ion. As it is demanded by the point group symmetry, 
this doublet generally occurs among the crystal field states of all spin-orbit-coupled 
local-J moments with J > 1.  If the local moment J happens to be an odd 
multiplier of 3/2 such as J=9/2 for Nd^3+ 
in Nd_2Zr_2O_7 <cit.> 
and 
Nd_2Sn_2O_7 <cit.> 
and J=15/2 for Dy^3+ in Dy_2Ti_2O_7 <cit.>, 
an easy-axis anisotropy would favor the DO doublet as the crystal field ground state. 
For others J's, this way to obtain the DO doublet as the crystal field ground state 
does not seem to apply. 



We are particularly interested in the Ce-pyrochlores with J=5/2.
The possible spin liquid was first experimentally proposed for Ce_2Sn_2O_7 <cit.>. 
The DO doublet nature of ground state doublet for the Ce^3+ ion was clarified 
in the theoretical work of Ref. PhysRevB.95.041106, 
as well as the connection to the pyrochlore ice U(1) spin liquids. 
The key insight of the selective spectroscopic measurements 
for the spinon continuum in the octupolar U(1) spin liquid 
by the INS spectroscopy
was also made over there <cit.>. 
Later developments were achieved for both Ce_2Zr_2O_7 and Ce_2Sn_2O_7, 
especially Ce_2Zr_2O_7 due to the single-crystal 
samples <cit.>. 
The relevance of the π-flux U(1) spin liquid (or U(1)_π spin liquid)  
was then raised, and the translational symmetry enrichment on top of 
the point-group symmetry enrichment, and the symmetry fractionalization 
were discussed <cit.>. 




Later theoretical works have also made progress to examine the distinction 
between different symmetry enriched 
U(1) spin liquids <cit.>. The recent thermodynamics and model calculation were even able to extract the parameters 
of the exchange couplings <cit.>. 
The results are different for Ce_2Zr_2O_7 and Ce_2Sn_2O_7 that are 
likely located in different U(1) spin liquid ground states of the underlying XYZ spin model. 
More recent INS measurements indeed found different
excitation spectra for Ce_2Zr_2O_7 and Ce_2Sn_2O_7 <cit.>. 
Since very low-temperature thermodynamics may contain some uncertainty in the measurements and fitting,
we rely mostly on the neutron scattering measurements for the major reasoning. 


To understand the INS experiments, we start from the spin model for the Ce-pyrochlores <cit.>, 

    H    =   ∑_⟨ ij ⟩[ J_x τ^x_i τ^x_j + J_z τ^z_i τ^z_j + J_xz (τ^x_i τ^z_j + τ^z_i τ^x_j ) 
    
                + J_y τ^y_i τ^y_j ] -∑_i  (n̂·ẑ_i) h τ_i^z ,

where τ^x and τ^z transform as a magnetic dipole moment, τ^y transforms a magnetic octupole moment,
 hn̂ is the external magnetic field that only couples to τ^z linearly, and ẑ_i is the local z direction. 
 The crossing J_xz term can be eliminated by a rotation around y direction in the pseudospin-τ 
 space with S^x_i = τ^x_i cosθ - τ^z_i sinθ, S^z_i = τ^x_i sinθ  + τ^z_i cosθ, 
 S^y_i= τ^y_i, and the resulting spin model is of the XYZ form <cit.>, 
 
    H_XYZ   =   ∑_⟨ ij ⟩J̃_x S^x_i S^x_j + J̃_y S^y_i S^y_j + J̃_z S^z_i S^z_j 
     
           -∑_i  (n̂·ẑ_i) h   ( cosθ S^z_i -  sinθ S^x_i).

where J̃'s are related to the J's via the pseudospin rotation.

 
 INS spectroscopy.—Based on the qualitative and powerful symmetry reasoning, 
 we know that, the model in Eq. (<ref>) 
without the field has two symmetry-enriched U(1) spin liquids from the perspective of the point group. 
For a dominant and antiferromagnetic J̃_z or J̃_x, a (conventional) dipolar U(1) spin liquid 
is realized. In contrast, for a dominant and antiferromagnetic J̃_y, a novel octupolar U(1) spin liquid is realized. 
It was observed that <cit.>, because the magnetic field 
or the neutron spin only couples to the transverse 
components relative to S^y in the octupolar U(1) spin liquid, the inelastic neutron scattering only 
measures the spinon continuum (see Tab. <ref>). 
An energy threshold of two-spinon gap 
is needed in order to observe the spinon continuum in the neutron scattering measurement. 
With this insight, the spinon continuum spectroscopy for both 0-flux and π-flux octupolar 
U(1) spin liquids were studied <cit.>.
Moreover, 
due to this selective linear coupling for the dipolar spin components, 
a regular unpolarized INS may already function as a polarized INS,
and a polarized INS can be a unnecessary luxury. 


Let us re-examine the dipolar U(1) spin liquid for a dominant and antiferromagnetic J̃_x. 
If J_xz is weak (or equivalently, θ is small), the behaviors in the neutron scattering 
measurement in this limit should not be very different from the octupolar U(1) spin liquid. 
In this regime, the coupling to the neutron spin is primarily via S^z component, and the 
coupling to S^x is suppressed by sinθ. 
The intensity of the S^x-S^x correlation, that includes the electric monopole continuum and the gauge photon, 
is suppressed by sin^2 θ. 
The INS measurement
cannot differentiate this dipolar U(1) spin liquid in this regime from the octupolar U(1) spin liquid,
regardless of whether further translational symmetry enrichment with 0 or π flux 
for the spinons is considered. From the perspective of S^z component,
the hidden S^x and S^y components actually have the same effect. 
This can be understood via a rotation of the spin S_i around the z direction by π/2
that switches S^x and S^z. For the same reason, thermodynamics like specific heat 
and magnetic susceptibility cannot differentiate them.  
Thus, in the above mentioned measurements, the dipolar U(1) spin liquid 
for a dominant and antiferromagnetic J̃_x
is not very different from the octupolar U(1) spin liquid. 
We note that, in Ref. Bhardwaj_2022, a finite linear Zeeman coupling to τ^x 
is introduced in Eq. (<ref>), and this would be more-or-less equivalent to a finite θ. 
The presence of this extra coupling could in principle 
make the experimental behaviors of the above two phases different. Our 
understanding here is restricted to the regime with the vanishing τ^x Zeeman coupling
of Eq. (<ref>). 



About Ce_2Sn_2O_7.—The recent INS measurement in the powder sample of 
Ce_2Sn_2O_7 actually observed a large spectral weight at ∼ 0.045meV, 
and the spectral weights are gradually suppressed below this peak energy. 
From the above explanation, this is either consistent with the 
expectation from the octupolar U(1) spin liquid, or compatible with the dipolar U(1) spin liquid 
for a dominant and antiferromagnetic J̃_x and a small θ. 
On the positive side, the measured excitation continuum for both cases is 
interpreted as the spinon continuum for the former, and the predominantly spinon continuum for the latter. 
The further structures in the momentum and/or energy domain of the spinon continuum 
could differentiate the distinct translation symmetry enrichments, i.e. the 0 or π
gauge flux for the spinons <cit.>. 
It is tempting to mention that, recent experiments of Ref. 2024PhRvX..14a1005Y 
on both single-crystal and powder Ce_2Sn_2O_7 
samples with a different preparation condition concluded with a dominant and antiferromagnetic J̃_x 
and an intermediate θ (≈ 0.19 π). The authors suggested that, Ce_2Sn_2O_7 
is located in the dipolar spin ice regime, but the ground state is ordered in the all-in all-out state. 
Even if the ground state is not ordered, an intermediate θ (≈ 0.19 π) for a dipolar U(1) spin liquid
with a dominant and antiferromagnetic J̃_x would imply a significant
spectral weight due to the electric monopole continuum and the gauge photon
at energies below the two-spinon gap. This is clearly different from the inelastic neutron scattering 
result in Ref. poree2024fractional. 
The discrepancy between Ref. 2024PhRvX..14a1005Y and Ref. poree2024fractional 
might arise from the sample preparation <cit.>
and remains to be resolved. 




About Ce_2Zr_2O_7.—In contrast, the recent INS measurement in the single-crystal sample   
of Ce_2Zr_2O_7 observed enhanced spectral weights at low energies, and the large 
spectral intensity extends down to zero energy within the experimental reach <cit.>. 
Without invoking any other features from the experiments, we think this piece of
experimental information alone can be understood from the following three possibilities. 

The first possibility is that, the low-energy spectral weights arise from the spinon continuum with 
a small gap, and the system is either in the octupolar U(1) spin liquid or in the dipolar U(1) spin liquid 
for a dominant and antiferromagnetic J̃_x and a small θ. 
If the fitted exchange parameters in Tab. <ref> are used, however, the system is clearly
not in the regime with weakly gapped spinon excitations. Given the thermodynamic results were used in the fitting, 
 this possibility may be ruled out. 






The second possibility is that, the system is in a dipolar U(1) spin liquid with an intermediate 
θ such that sin^2 θ and cos^2 θ are of the same order. 
This can allow for a dominant and antiferromagnetic J̃_z, 
or a dominant and antiferromagnetic J̃_x, or both. 
The third possibility is that, the system is in a dipolar U(1) spin liquid with a dominant and 
antiferromagnetic J̃_z and a small θ. For the third possibility, the spinon 
continuum at high energies should be suppressed from sin^2 θ. 



For the second and third possibilities, the low-energy spectral weights below the two-spinon 
gap arise from the electric monopole continuum and the gauge photon. But since the spectral 
weight of the gauge photon is intrinsically suppressed at low energies, thus the low-energy 
spectral weights for these two possibilities are mostly given by the electric monopole continuum. 
This is the fundamental difference of our interpretation from Ref. gao2024emergent. 


Electric monopole continuum.—We here focus on and work out the details 
for the low-energy electric monopole continuum that is shared by these two possibilities. 
Unlike the spinons that hop on the diamond lattice formed by the tetrahedral centers of the 
pyrochlore lattice, the electric monopoles reside on the dual diamond lattice whose bonds
penetrate through the hexagonal plaquette centers of the diamond lattice for the spinons. 
If one sticks to the context of quantum spin ice, the spinons are excitations out of the ice manifold,
while the photon and the electric monopoles are excitations that are built up from the ice manifold
and occur in a much lower energy scale than the spinons (see Fig. <ref>). The spinon has a classical analogue 
as the classical magnetic monopole, 
the photon and the electric monopoles are purely of quantum origin and have no classical analogue. 
Moreover, the electric monopoles experience a π dual U(1) gauge flux on the dual diamond lattice. 
Thus, the translation symmetry is fractionalized for the electric monopoles. 





The Hamiltonian describing the electric monopole hopping on the dual diamond lattice is given as <cit.>, 

    H_m = -t∑_⟨ r r' ⟩ e^-2π iα_rr'Φ_r^†Φ_r'^ - μ∑_rΦ_r^†Φ_r^,

where Φ_r^† (Φ_r) represents the monopole creating (annihilation) operator, and
t and μ denote the hopping and chemical potential of the monopoles, respectively. 
The position r spans the dual diamond lattice, 
comprising the A and B sublattices. The dual gauge link 
α_rr', capturing the π dual U(1) gauge flux due to the effective spin-1/2 nature of the moment, 
is chosen as
2πα_r,r+e_μ = ξ_μQ·r, where r∈ sublattice A and t+e_μ∈ sublattice B. 
Here, e_μ (μ=0,1,2,3) represents the nearest-neighbor vectors 
connecting the two sublattices (see Appendix <ref>). 
To fix the gauge, we choose (ξ_0,ξ_1,ξ_2,ξ_3)=(0,1,1,0) and Q=2π(1,0,0). 
Diagonalizing Eq. (<ref>), we obtain four monopole bands

    Ω^ζ_η = tζ[4 + 2η(3+C_xC_y-C_xC_z+C_yC_z)^1/2]^1/2 - μ,

where C_μ=cos q_μ (μ=x,y,z), ζ=±, and η=±. Due to the background 
π dual U(1) gauge flux, the monopole bands are then defined over a “magnetic" Brillouin zone that is half
of the crystal Brillouin zone.

As previously discussed, the two-monopole continuum predominantly contributes to
 the INS signal in the low-energy regime. This signal is expected to correlate 
 with the two-monopole continuum density of states, ρ^(2)(k,ω), 
 regardless of a specific form factor. 
With the momentum and energy conservation, the density of states is given by

    ρ^(2)(k,ω) ∝∑_q,ζ_1,ζ_2,η_1,η_2δ[ω - Ω^ζ_1_η_1(q) - Ω^ζ_2_η_2(k - q - Q)],

where k and ω represent the total momentum and energy of the two monopoles, respectively. 
The offset Q originates from the spatial periodicity of the gauge potential. 
Due to the periodicity of ρ^(2)(k,ω), however, this offset does not influence the spectrum. 
Specifically, at the Γ point, the momenta of the two monopoles cancel out. 
Given the symmetry of the monopole bands under reflection with respect to the Ω=-μ 
plane and inversion q→-q, the continuum spectrum features a flat band near ω=-2μ. 
Consequently, a δ-function-like peak is observable in the two-monopole continuum density of states at the Γ point (see Fig. <ref>(a)).

In the following, we set t as the energy unit and μ=-3t to prevent the electric monopole condensation <cit.>. 
We normalize ρ^(2)(k,ω) by its maximal value. Given the energy overlapping between the four monopole bands, 
the two-monopole continuum spans continuously from a lower to an upper edge, with a spectrum width of 8√(2)t. 
Fig. <ref> illustrates the edge shapes along the high-symmetry line 
(for the definition of high-symmetry points, see Appendix <ref>). 
Additionally, Fig. <ref>(b) and 
(c) display the density of states at two representative high-symmetry points.

































 






 



 








 
 

 



 











 


Discussion—We discuss the fitted parameters for Ce_2Zr_2O_7 from Tab. <ref>. 
For the parameters in Ref. 2022PhRvX..12b1015S, since θ∼ 0,
as we have discussed, one cannot distinguish the octupolar U(1) spin liquid with a dominant J̃_y
and the dipolar U(1) spin liquid with a dominant J̃_x. 
For both phases, the INS measurement can only probe the spinon continuum at high energies. 
Thus, this set of parameters is inconsistent with the neutron results in Ref. gao2024emergent. 
For the parameters in Ref. Bhardwaj_2022, even though an additional linear Zeeman coupling to τ^x in Eq. (<ref>)
is introduced, the parameters for the exchange simply favor an octupolar U(1) spin liquid, for which
the INS measurement can only probe the spinon continuum at high energies. 

For the parameters in Ref. gao2024emergent, the system is in the dipolar U(1) spin liquid.
 As θ =0.12 π is in the intermediate regime, all excitations should show up 
in the INS measurement, where the low energy part should be mostly contributed 
by the electric monopole continuum. This is qualitatively compatible with the expectation. 
On a bit more quantitative side, if one uses the fitted couplings of Ref. gao2024emergent,
the energy scale for the electric monopoles and photons is approximately set by the ring exchange 
J_ring = 12 (J^±)^3/J̃_x^2∼ 0.02meV, while the spinon energy scale is set 
by J̃_x and renormalized by J_±. Thus, we expect that, 
probably below about 0.05meV, the INS spectrum is contributed by photons and electric monopole continuum, and 
mostly by the electric monopole continuum. The detailed structures of the continuum require a high 
energy resolution to resolve. Otherwise, the continuous excitations are simply crowded up at low energies (see Appendix <ref>).
Roughly above about 0.05meV, the INS spectrum is contributed by spinon continuum combined with other low-energy 
excitations (photon, electric monopoles). 



The above analysis, however, has one caveat. If a linear Zeeman coupling to τ^x 
is introduced in Eq. (<ref>) as Ref. Bhardwaj_2022,
 a dipolar U(1) spin liquid with a dominant J̃_z 
or J̃_x with θ∼ 0 could not be simply ruled out. About the calculation, the caveat is that
 we treat the electric monopole at the quadratic level and the dual U(1) gauge field 
as a static background. According to Ref. PhysRevLett.124.097204, 
the threshold spinon-pair production could be abruptly enhanced due to the coupling 
to the dynamic gauge field. Similar features would occur for the threshold electric monopole 
production, giving rise to a large density of states at low energies than our current treatment. 



It is interesting to vision the effect of a weak magnetic field for a dipolar U(1) spin liquid 
if Ce_2Zr_2O_7 realizes it. 
A weak magnetic field polarizes the effective Ising component and modifies the dual gauge 
flux away from π for the electric monopoles. This has two consequences. First, the 
electric monopoles develop Hofstadter-like band structures, and the continuum will have 
more modulations in the energy-momentum domain <cit.>. 
The abundance of the electric monopole continuum at low energy should also be accessible 
through the μSR measurement at zero/weak field <cit.>. 
Second, the electric monopole band will have a nontrivial Berry curvature distribution, generating 
the monopole thermal Hall effect at low temperatures <cit.>. 
If Ce_2Sn_2O_7 realizes the octupolar U(1) spin liquid, no such effect is expected. 
In general, the very low energy scale of the monopole physics for the rare-earth magnets makes
the detailed measurement difficult, and this calls for the candidate d electron systems
with higher energy scales and the numerical calculations <cit.>. 
So far, one candidate d electron magnets for the DO doublets is Cd_2Os_2O_7, but it is ordered <cit.>. 



In summary, we propose the low-energy continuous excitations in the recent inelastic neutron scattering 
measurements in Ce_2Zr_2O_7 is mostly contributed from the electric monopole continuum.
Although the gapless gauge phonon is present, its intensity should be suppressed
by at low energies. While the fitted parameters in Ref. gao2024emergent 
are qualitatively compatible with our expectation from the INS scattering 
measurement, our interpretation is different from Ref. gao2024emergent. 
We further demonstrate the structures of the electric monopole continuum in the energy 
and momentum domains. 


Acknowledgments.—We thank a recent conversation with Bin Gao, Radu Coldea and Bruce Gaulin,
and the previous collaboration with Pengcheng Dai, Brian Maple, Emilia Morosan, Andriy Nevidomskyy and Sang-Wook Cheong.  
This work is supported by the National Science Foundation of China with Grant No. 92065203,
 by the Ministry of Science and Technology of China with Grants No. 2021YFA1400300,
 and by the Fundamental Research Funds for the Central Universities, Peking University.





    


§ LATTICE STRUCTURE

The physical spin operators live in a Pyrochlore lattice, which is formed by corner-shared tetrahedra. A diamond lattice can be constructed by connecting the centers of the tetrahedra, on which a U(1) lattice gauge theory is defined. A further duality transformation yields the theory of monopoles, which live in the dual lattice of the diamond lattice: the dual diamond lattice. It consists of two sublattices. The relative vector between these two sublattices is δ=1/2(1,1,1). Each sublattice is a face-centered cubic (fcc) lattice. Each site of the dual diamond lattice has four nearest neighbors, whose relative vectors are e_0=1/4(1,1,1), e_1=1/4(1,-1,-1), e_2=1/4(-1,1,-1), and e_3=1/4(-1,-1,1). The lattice vectors of the underlying Bravais lattice are chosen to be a_1=1/2(0,1,1), a_2=1/2(1,0,1), and a_3=1/2(1,1,0). Thus, the three lattice vectors of the reciprocal lattice are defined as b_1=2π(-1,1,1), b_2=2π(1,-1,1), and b_2=2π(1,1,-1). As described in the main text, monopole bands in the π-flux background has an enhanced periodicity, which leads to a smaller Brillouin zone spanned by b_i'=b_i/2. To reveil the spectrum, it is nature to consider the high-symmetry points defined in Tab. <ref> in the smaller Brillouin zone.







§ MORE DETAILS ON THE SPECTRUM


 
The peak structure of the two-monopole continuum DoS appears to be insensitive to the total momentum. To validate this observation, we compute the momentum-integrated DoS, given by ∫dk,ρ^(2)(k,ω). In Figure <ref>(b), the momentum-integrated DoS exhibits seven peaks, with their positions centered around the highest peak located at ω=-2μ=6. This peak structure can be elucidated by examining the DoS of the four monopole bands. Each monopole band contributes a peak to the one-monopole DoS, and their collective effect manifests as the observed peak structure in the two-monopole continuum DoS.





While the spectrum presents fine structures, these might not be readily discernible in INS experiments. 
Being excitations within the ice manifold, monopoles operate on a relatively small energy scale, 
making their spectrum width potentially compatible with the energy resolution of INS. 
Consequently, a broadened peak is anticipated rather than a sharp one. To account for finite energy resolution, 
we simulate it using a Gaussian function with a standard deviation equal to 10% of the spectrum width. 
Figure <ref> displays the continuum density of states at three high-symmetry points: 
Γ, X, and K.

To detect the electric monopole continuum, one potential strategy could involve synthesizing both an octupolar U(1) 
spin liquid material and a dipolar U(1) spin liquid material with identical parameters. In the case of the octupolar U(1) spin liquid, 
INS measures only the gapped spinon continuum. By subtracting the spinon continuum signal of the octupolar U(1) spin liquid 
from the INS signal of the dipolar U(1) spin liquid, the remaining signal would predominantly reflect the monopole continuum and gauge photon. 
This differential approach may provide a robust method to isolate and detect the electric monopole continuum signal experimentally.


 


