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# Line-Graph Approach to Spiral Spin Liquids

Shang Gao<EMAIL_ADDRESS>Neutron Scattering Division, Oak Ridge
National Laboratory, Oak Ridge, Tennessee 37831, USA  Materials Science &
Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA  Ganesh Pokharel  Materials Science & Technology Division, Oak
Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  Department of
Physics & Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
Andrew F. May  Materials Science & Technology Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  Joseph A. M. Paddison  Materials
Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA  Chris Pasco  Materials Science & Technology Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  Yaohua Liu
Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA  Keith M. Taddei  Neutron Scattering Division, Oak Ridge
National Laboratory, Oak Ridge, Tennessee 37831, USA  Stuart Calder  Neutron
Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA  David G. Mandrus  Materials Science & Technology Division, Oak
Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  Department of
Physics & Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
Department of Material Science & Engineering, University of Tennessee,
Knoxville, Tennessee 37996, USA  Matthew B. Stone  Neutron Scattering
Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
Andrew D. Christianson  Materials Science & Technology Division, Oak Ridge
National Laboratory, Oak Ridge, Tennessee 37831, USA

###### Abstract

Competition among exchange interactions is able to induce novel spin
correlations on a bipartite lattice without geometrical frustration. A
prototype example is the spiral spin liquid, which is a correlated
paramagnetic state characterized by sub-dimensional degenerate propagation
vectors. Here, using spectral graph theory, we show that spiral spin liquids
on a bipartite lattice can be approximated by a further-neighbor model on the
corresponding line-graph lattice that is non-bipartite, thus broadening the
space of candidate materials that may support the spiral spin liquid phases.
As illustrations, we examine neutron scattering experiments performed on two
spinel compounds, ZnCr2Se4 and CuInCr4Se8, to demonstrate the feasibility of
this new approach and expose its possible limitations in experimental
realizations.

###### pacs:

Introduction.— A spiral spin liquid (SSL) is an exotic correlated paramagnetic
state of sub-dimensional degeneracy, meaning that the propagation vectors
$\bm{q}$ of the ground states form a continuous manifold, or spiral surface,
in a dimension that is reduced from the original system Bergman  _et al._
(2007); Lee and Balents (2008); Mulder  _et al._ (2010); Zhang and Lamas
(2013); Niggemann  _et al._ (2020); Attig and Trebst (2017); Balla  _et al._
(2020, 2019); Lee and Balents (2008); Chen (2017); Yao  _et al._ (2021); Huang
_et al._ (2022); Buessen  _et al._ (2018); Liu  _et al._ (2020); Iqbal  _et
al._ (2018). Similar to geometrically frustrated magnets Bramwell and Gingras
(2001); Balents (2010), a SSL may host topological spin textures Gao  _et al._
(2017, 2020); Shimokawa  _et al._ (2019) and quantum spin liquid states Mulder
_et al._ (2010); Zhang and Lamas (2013); Niggemann  _et al._ (2020); Balz  _et
al._ (2016); Biswas and Damle (2018); Pohle  _et al._ (2021). What
differentiates a SSL from a conventional frustrated magnet is the sub-
dimensional degeneracy, which induces highly distinctive dynamics since the
spins are confined to fluctuate collectively as nonlocal spirals Yan and
Reuther (2022). Recent calculations on a square lattice reveal that the low-
energy fluctuations in a SSL may behave as topological vortices in momentum
space Yan and Reuther (2022), leading to an effective tensor gauge theory with
unconventional fracton quadrupole excitations that are deeply connected to
theories of quantum information, elasticity, and gravity Pretko (2017a, b);
Pretko and Radzihovsky (2018); Yan (2019); Nandkishore and Hermele (2019);
Pretko  _et al._ (2020).

To date, bipartite lattices have been the primary avenue through which SSLs
are studied. This is because the ground state degeneracy on a bipartite
lattice can be exact, so that all spin spirals with $\bm{q}$ over the spiral
surface have exactly the same energy Niggemann  _et al._ (2020). Although this
degeneracy stabilizes the SSL down to very low temperatures Bergman  _et al._
(2007); Mulder  _et al._ (2010), it also imposes a strong constraint on real
materials because most of the known bipartite-lattice compounds are dominated
by the nearest-neighbor interactions $J_{1}$ Tristan  _et al._ (2005); T.
Suzuki and H. Nagai and M. Nohara and H. Takagi (2007); Krimmel  _et al._
(2009); Matsuda  _et al._ (2010); MacDougall  _et al._ (2011); Zaharko  _et
al._ (2011); Nair  _et al._ (2014); MacDougall  _et al._ (2016); Ge  _et al._
(2017); Chamorro  _et al._ (2018); Tsurkan  _et al._ (2021); Haraguchi  _et
al._ (2019); Abdeldaim  _et al._ (2020); Otsuka  _et al._ (2020); Wessler  _et
al._ (2020). Even for the established model compounds where the second-
neighbor interactions $J_{2}$ are relatively strong Gao  _et al._ (2017);
Guratinder  _et al._ (2022); Gao  _et al._ (2022), the degeneracy over the
spiral surface is only approximate due to the existence of further
perturbations Iqbal  _et al._ (2018). This degeneracy lifting results in an
approximate SSL state at elevated temperatures where thermal fluctuations
overcome the slight energy difference among the spirals.

Inspired by recent density functional theory (DFT) calculations for the
breathing pyrochlore lattice compounds Ghosh  _et al._ (2019), here we seek
the realization of an approximate SSL, i.e. a SSL with an approximate
degeneracy, beyond the bipartite lattices. According to the Luttinger-Tisza
theory Luttinger and Tisza (1946); Luttinger (1951), the degeneracy of a SSL
model is encoded in the minimum manifold of the interaction matrix. Using
graph theory Knauer and Knauer (2019); Kollár  _et al._ (2020), we show that
the $J_{1}$-$J_{2}$ model on a bipartite lattice shares the same minimum
manifold with a $J_{1}$-$J_{3}$ model on the corresponding line-graph lattice,
where $J_{3}$ denotes the third-neighbor interaction. Thus an approximate SSL
state is achieved in the latter case when $J_{3}$ is sufficiently strong,
which greatly expands the range of materials that may support a SSL state.
This line-graph approach to SSL is vetted through neutron scattering
experiments performed on two Cr-based chalcogenide spinels ZnCr2Se4 and
CuInCr4Se8.

Line-graph approach.— Our starting point is a Heisenberg model on a
$l$-regular lattice, where $l$ counts the number of the nearest-neighbor (NN)
sites. In the presence of a uniform NN exchange interaction $J_{1}$, the
coupled spins form a undirected graph $G=(V,D)$, with $V$ denoting the set of
vertices (i.e. the spin sites) and $D$ denoting the set of edges (i.e. the NN
bonds). Two vertices $i$ and $j$ are called adjacent if the graph contains an
edge $e=\\{i,j\\}$, and the adjacency matrix is defined as $A(G)_{ij}=1$ for
$\\{i,j\\}\in D$ and 0 otherwise. Following this definition, the spin
Hamiltonian, $\mathcal{H}=J_{1}\sum_{\langle
ij\rangle}\bm{S}_{i}\cdot\bm{S}_{j}$ with $\langle ij\rangle$ denoting the NN
bonds, can be expressed through the adjacency matrix as
$\mathcal{H}=\frac{1}{2}J_{1}A(G)_{ij}\bm{S}_{i}\cdot\bm{S}_{j}$. Therefore,
according to the Luttinger-Tisza theory Bergman  _et al._ (2007); Niggemann
_et al._ (2020), the classical ground state of $\mathcal{H}$ can be determined
from the eigensolution of $A(G)$, of which the eigenvalues are defined as the
spectrum of the graph, denoted as $\sigma(G)$.

Algebraic graph theory defines that the related graphs should have related
spectra Knauer and Knauer (2019); Kollár  _et al._ (2020). Of interest here is
the line graph $L_{G}$ Mielke (1991); Tasaki (1992), whose vertices correspond
to the edges of the root graph $G$ and are adjacent if the original edges
share a vertex:
$L_{G}=(D,\\{\\{e,e^{\prime}\\}\,|\,e\,\cup\,e^{\prime}\neq\emptyset,e\neq
e^{\prime}\\})$. For a regular bipartite graph $G$, a convenient way to define
its line graph is to select the vertices at the midpoints of the original
edges. As illustrated in Fig. 1, for the honeycomb ($l=3$) and diamond ($l=4$)
lattices that are the prototype hosts of the SSL, their line graphs form the
kagome and pyrochlore lattices, respectively. According to graph theory Knauer
and Knauer (2019); Kollár  _et al._ (2020), the spectra of $G$ and $L_{G}$ are
related by

$\sigma(L_{G})=(-2)^{m-n}\,\cup\,\sigma(G)+l-2\,\textrm{,}$ (1)

where $m$ ($n$) is the total number of edges (vertices) of the root graph $G$.
In reciprocal space, Eq. (1) indicates the existence of flat eigenbands on
$L_{G}$ with a degeneracy of $l-2$, which has been the focus of many recent
studies Kollár  _et al._ (2020); Ma  _et al._ (2020); Chiu  _et al._ (2022);
Nakai and Hotta (2022). More importantly, it reveals that the non-flat
eigenbands of $\sigma(L_{G})$ and $\sigma(G)$ share the same dispersion up to
a constant of $l-2$, and their eigenvectors are related by the incident matrix
as discussed in Ref. Mizoguchi  _et al._ (2018).

Such a spectrum correspondence can be immediately verified for Heisenberg
models. On a regular lattice of $g$ sublattices and $N$ primitive cells, the
interaction matrix of a $J_{1}$-only model

$\mathcal{J}^{\alpha\beta}_{1}(\bm{q})=\frac{J_{1}}{N}\sum_{\begin{subarray}{c}i\in\alpha,j\in\beta\\\
ij\in\langle
ij\rangle\end{subarray}}\exp{[-i\bm{q}\cdot(\bm{r}_{i}-\bm{r}_{j})]}$ (2)

is a $g\times g$ hollow matrix with zero diagonal elements. The eigenbands
$\nu(\bm{q})$ for $J_{1}<0$ are shown as solid lines in Fig. 1(b) for the
honeycomb and kagome lattices, and in Fig. 1(d) for the diamond and pyrochlore
lattices. The minimum of the eigenbands $\nu_{\textrm{min}}$ has been
subtracted for comparison. Analytical expressions for the eigenbands are
presented in the Supplemental Material sup , which includes additional Refs.
Buessen ; Hukushima and Nemoto (1996); Cameron  _et al._ (2016); Jendrzejewska
_et al._ (2021); Ye  _et al._ (2018); Rodriguez-Carvajal (1993); Arnold  _et
al._ (2014); Farhi  _et al._ (2014); Calder  _et al._ (2018); Toth and Lake
(2015); Aroyo  _et al._ (2011). Aside from the top flat bands in blue color,
the two dispersive bands $\nu_{\pm}(\bm{q})-\nu_{\textrm{min}}$ on the kagome
(pyrochlore) lattice overlap exactly with those on the honeycomb (diamond)
lattice, which is a direct consequence of spectral graph theory.

Figure 1: (a) The line graph of a bipartite honeycomb lattice (large spheres
linked by orange bonds) is a kagome lattice (small spheres linked by gray
bonds). $J_{1}$ is the nearest-neighbor exchange coupling. Black dashed (red
solid) arrows indicate the second-neighbor coupling $J_{2}$ (third-neighbor
couplings $J_{3a}$ and $J_{3b}$) over the honeycomb (kagome) lattice. (b)
Eigenbands of the interaction matrix of Heisenberg models on the honeycomb and
kagome lattices. Orange solid (dashed) lines are the two eigenbands for the
$J_{1}$ model ($J_{1}$-$J_{2}$ model with $J_{2}/J_{1}=-0.25$) on the
honeycomb lattice, which overlap with the two lower eigenbands of the $J_{1}$
model ($J_{1}$-$J_{3}$ model with $J_{3}/J_{1}=-0.25$) on the kagome lattice
shown in blue solid (dashed) lines. (c-d) Similar correspondence exists
between the diamond and pyrochlore lattices. The interaction matrices for the
$J_{1}$ model ($J_{1}$-$J_{2}$ model with $J_{2}/J_{1}=-0.3$) on the diamond
lattice and the $J_{1}$ model ($J_{1}$-$J_{3}$ model with $J_{3}/J_{1}=-0.3$)
on the pyrochlore lattice share two same eigenbands shown by the overlapping
orange and blue solid (dashed) lines.

This eigenband correspondence is maintained under the addition of certain
further-neighbor interactions. For the $J_{1}$-$J_{2}$ model on a bipartite
lattice, $J_{2}$ couples the spins of the same sublattices. Therefore, its
contribution to $\mathcal{J}(\bm{q})$ is a diagonal matrix
$\mathcal{J}_{2}(\bm{q})$ that commutes with $\mathcal{J}_{1}(\bm{q})$,
leading to a $\bm{q}$-dependent shift $\gamma_{G}(\bm{q})$ of the eigenbands
with $\gamma_{G}(\bm{q})=(J_{2}/Ng)\sum_{ij\in\langle\\!\langle
ij\rangle\\!\rangle}\exp{[-i\bm{q}\cdot(\bm{r}_{i}-\bm{r}_{j})]}$, where
$\langle\\!\langle ij\rangle\\!\rangle$ are the second-neighbor bonds sup .
Similar conclusions can be drawn for the $J_{1}$-$J_{3}$ model on the line-
graph lattices, as $J_{3}$ (including both $J_{3a}$ and $J_{3b}$) also couples
spins of the same sublattices sup . Since $J_{2}(G)$ on the root-graph lattice
and $J_{3}(L_{G})$ on the line-graph lattice share the same exchange paths as
compared in Fig. 1(a,c), we expect, in the case of $J_{2}(G)=J_{3}(L_{G})$,
the same dispersive eigenbands up to a constant on the root- and line-graph
lattices. This correspondence is illustrated by the dashed lines in Fig.
1(b,d).

Approximate SSL.— An approximate SSL can be realized on the line-graph
lattices through the eigenband correspondence. Following the results of the
$J_{1}$-$J_{2}$ model on the bipartite lattices Bergman  _et al._ (2007);
Niggemann  _et al._ (2020), it is clear that for the $J_{1}$-$J_{3}$ model on
a line-graph lattice the eigenband minima of $\mathcal{J}(\bm{q})$ will form a
degenerate manifold in reciprocal space at $|J_{3}/J_{1}|>1/(2l)$, where $l$
is the number of the NN sites on the root-graph lattice. Here $J_{1}$ and
$J_{3}$ should be FM and AF, respectively, as the additional flat bands on
line graphs remove the FM-AF duality of a bipartite lattice. Since the equal
moment constraint over the eigenvectors of $\mathcal{J}(\bm{q})$ is not always
satisfied for $\bm{q}$ over the minimum manifold sup , the SSL realized
through the line-graph approach is approximate and needs to be stabilized by
thermal fluctuations. As discussed in the Supplemental Material sup , the
degeneracy breaking over the spiral surface is relatively weak, which
contrasts the strong modulation in the previously studied half-moon patterns
Rau and Gingras (2016); Mizoguchi  _et al._ (2018); Hering  _et al._ (2021);
Kiese  _et al._ (2022). This weak degeneracy breaking on the line-graph
lattices leads to a stable SSL state in a wide temperature regime that is
comparable to that on the bipartite lattices.

Before making comparisons to the experimental data, we further generalize the
SSL model by incorporating a breathing distortion on the line-graph lattice so
that the NN interactions are modulated alternately as $J_{1}$ and
$J_{1}^{\prime}$ Hirschberger  _et al._ (2019); Gao  _et al._ (2019); Ghosh
_et al._ (2019). For the breathing pyrochlore lattice shown in Fig. 2(a), the
eigenbands of $\mathcal{J}(\bm{q})$ can be solved as sup :

$\displaystyle\nu_{1,2}$
$\displaystyle=J_{3}\kappa(\bm{q})\pm\sqrt{4(J_{1}-J_{1}^{\prime})^{2}+J_{1}J_{1}^{\prime}|\eta(\bm{q})|^{2}}+(J_{1}+J_{1}^{\prime})\textrm{,}$
$\displaystyle\nu_{3,4}$
$\displaystyle=J_{3}\kappa(\bm{q})-(J_{1}+J_{1}^{\prime})\,\textrm{,}$

where $\eta(\bm{q})=\sum_{n=1}^{4}\exp(-i\bm{q}\cdot\bm{d}_{n})$ with
$\bm{d}_{n}$ denoting the 4 bonding vectors around each spin site sup  and
$\kappa(\bm{q})=|\eta(\bm{q})|^{2}-4$. Assuming $J_{1}<J_{1}^{\prime}<0$ and
$J_{3}>0$, an approximate SSL state is realized for
$\frac{1}{4}\frac{J_{1}J_{1}^{\prime}}{|J_{1}+J_{1}^{\prime}|}<J_{3}<\frac{1}{4}\frac{J_{1}J_{1}^{\prime}}{|J_{1}-J_{1}^{\prime}|}$.
The corresponding phase diagram is presented in Fig. 2(b). Representative
spiral surfaces in the approximate SSL phase at $|J_{3}/J_{1}|=0.2$ are shown
in Fig. 2(c-e) for $J_{1}^{\prime}/J_{1}=0.8$, 0.6, and 0.5, respectively. The
surfaces are identical to those on the diamond lattice Bergman  _et al._
(2007). Similar conclusions on the breathing kagome lattice are presented in
the Supplemental Material sup .

ZnCr2Se4 with a regular pyrochlore lattice.— The Cr-based chalcogenide spinels
present nearly ideal model compounds to demonstrate the proposed line-graph
approach to SSLs. In these systems, $J_{1}$ is FM due to the 90∘ superexchange
path, while $J_{2}\sim 0$ due to negligible orbital overlap Yaresko (2008);
Tymoshenko  _et al._ (2017). As the first example, we study the short-range
spin correlations in ZnCr2Se4, where the Cr3+ ($S=3/2$) ions form a regular
pyrochlore lattice.

Single crystals of ZnCr2Se4 were grown using the chemical vapor transport
method sup . Figure 3 summarizes the diffuse neutron scattering results
measured on CORELLI at the Spallation Neutron Source (SNS), Oak Ridge National
Laboratory (ORNL) sup . With the statistical chopper, the elastic channel of
our CORELLI data has an average energy resolution of about 0.8 meV for an
incident neutron energy range of 13 to 33 meV. At 20 K, below the Néel
transition temperature $T_{N}\sim 22$ K, magnetic Bragg peaks indexed by
$\bm{q}=(0,0,0.47)$ are observed (see Fig. 3(a)). This is consistent with the
helical ground state reported previously Hemberger  _et al._ (2007);
Yokaichiya  _et al._ (2009); Zajdel  _et al._ (2017), with the weak ring-like
scattering mainly arising from the low-energy magnon excitations Tymoshenko
_et al._ (2017); Inosov  _et al._ (2020). At 30 K, above $T_{N}$, magnetic
Bragg peaks are replaced by broad diffuse scattering with a spherical shape
(see Fig. 3(b-d)), evidencing the emergence of an approximate SSL state where
gapped excitations are replaced by quasielastic fluctuations sup .

Figure 2: (a) The breathing pyrochlore lattice is composed of corner-sharing
tetrahedra of two different sizes, which results in alternating $J_{1}$ and
$J_{1}^{\prime}$ couplings over the gray and orange tetrahedra, respectively.
The exchange paths of the $J_{2}$, $J_{3a}$, $J_{3b}$, and $J_{4}$
interactions are also indicated. (b) Phase diagram on the breathing pyrochlore
lattice with ferromagnetic $J_{1}$ and $J_{1}^{\prime}$ interactions together
with antiferromagnetic $J_{3}$ interactions assuming $J_{3a}=J_{3b}$. The
shaded area indicates the region where an approximate SSL phase can be
stabilized by thermal fluctuations. Contour lines for constant
$|\eta(\bm{q})|$ are shown in the approximate SSL phase. Along these lines,
the spiral surface stay the same. Location of ZnCr2Se4 (ZCS) with
$J_{1}=J_{1}^{\prime}$ and $|J_{3}/J_{1}|=0.15$ is indicated by the yellow
dot. Inset shows the degenerate manifold (blue lines) for
$J_{3}>\frac{1}{4}\frac{J_{1}J_{1}^{\prime}}{|J_{1}-J_{1}^{\prime}|}$ out of
the SSL regime. (c-e) Characteristic spiral surfaces with quasi-degenerate
energies in the first Brillouin zone at $|J_{3}/J_{1}|=0.2$ and
$J_{1}^{\prime}/J_{1}=0.8$ (c), 0.6 (d), and 0.5 (e). These points are
indicated on the phase diagram in (b) using the panel labels.  Figure 3:
Quasistatic spin correlations in ZnCr2Se4 measured on CORELLI at (a) 20 K and
(b-d) 30 K. Slices are along the ($hk2$) plane in panels (a,b), ($hhl$) plane
in panel (c), and (1$\overline{1}$1) plane in panel (d). The right sub-panel
in (a) shows the complete spiral surface for the $J_{1}$-$J_{3}$ model on a
pyrochlore lattice with $J_{3}/J_{1}=-0.15$. The flat planes correspond to the
slice directions as indicated by the colored plaquette at the top left corner
in each panel. The right sub-panels in (b-d) are the calculated diffuse
scattering patterns for the fitted $J_{1}$-$J_{2}$-$J_{3}$-$J_{4}$ model.
Solid line is the spiral surface predicted by the $J_{1}$-$J_{3}$ model with
$J_{3}/J_{1}=-0.15$. The same linear intensity scale is employed in all
panels.  Figure 4: (a) DC magnetic susceptibility of CuInCr4Se8 measured in a
$1\times 10^{4}$ Oe field (circles). Red line shows the Curie Weiss fit at
temperatures between 200 and 380 K. The fitted magnetic moment and Weiss
temperature are $\mu_{\textrm{eff}}=4.6$ $\mu_{B}$/Cr and 97(2) K,
respectively. The inset shows the frequency dependence of the real part of the
ac susceptibility measured in a 10 Oe field. Imaginary part of the ac
susceptibility is presented in the Supplemental Material sup . (b) Difference
between the neutron diffraction intensity at 0.25 and 25 K measured on HB-2A.
The dashed line is a guide to the eyes. Positions of the characteristic wave
vectors are indicated by arrows. The full width at half maximum (FHWM) of the
closest nuclear reflection is indicated by the horizontal bar. Inset shows the
calculated neutron diffuse scattering pattern in the ($h,k,0$) plane at $T=20$
K for $J_{3a}=0.07$ meV (see text). (c-e) Circles show the $Q$-dependence of
the equal-time spin correlations at $T=40$ (c), 100 (d), and 200 K (e).
Results of the SCGA fits using the $J_{1}$-$J_{1}^{\prime}$-$J_{3a}$-$J_{3b}$
model are shown by the red solid lines. Insets are the calculated neutron
diffuse scattering patterns in the ($h,k,0$) plane for $J_{3a}=0.07$ meV at
the corresponding temperatures (see text).

Assuming a Heisenberg model with exchange interactions up to the fourth
neighbors ($J_{4}$), we fit the diffuse scattering data using the self
consistent Gaussian approximation (SCGA) method sup . The calculated slices in
Fig. 3(b-d) reproduce the experimental data. The coupling strengths are fitted
as $J_{1}=-2.86(8)$, $J_{2}=0.00(1)$, $J_{3}=0.48(1)$, and $J_{4}=-0.057(1)$
meV, which is very close to the values determined by inelastic neutron
scattering (INS) Tymoshenko  _et al._ (2017); Inosov  _et al._ (2020) and
indicates marginal changes in the coupling strengths across the phase
transition. The weak strengths of the $J_{2}$ and $J_{4}$ interactions allow a
direct verification of the line-graph approach that is based on a
$J_{1}$-$J_{3}$ model. The solid circles on top of the calculated patterns in
Fig. 3(b-d) are the contours of the spiral surface predicted by the
$J_{1}$-$J_{3}$ model with $J_{3}/J_{1}=-0.15$. The contours capture the
shapes of the diffuse scattering patterns. As compared in the Supplemental
Material sup , the existence of the FM $J_{4}$ interaction slightly reduces
the radius of the spiral surface while strongly modulates the scattering
intensities.

CuInCr4Se8 with a breathing pyrochlore lattice.— Until now, we have assumed a
uniform $J_{3}$ interaction. In real materials, the $J_{3}$ exchange paths on
a line-graph lattice can be different as indicated in Fig. 1. This may lead to
different coupling strengths and destabilize the SSL state. As the second
example, we study the spin correlations in the breathing pyrochlore lattice
compound CuInCr4Se8 Duda  _et al._ (2008), which has been proposed as a SSL
candidate from DFT calculations Ghosh  _et al._ (2019).

A polycrystalline sample of CuInCr4Se8 was synthesized through the solid state
reaction method sup . INS experiments were performed on SEQUOIA at the SNS,
ORNL. Neutron diffraction experiments were performed on HB-2A at the high flux
isotope reactor (HFIR), ORNL. As is consistent with the previous report Duda
_et al._ (2008), magnetic susceptibility shown in Fig. 4(a) suggests a spin
glass-like transition at $T_{f}\sim 15$ K with a clear frequency dependence.
This is confirmed in the neutron diffraction results presented in Fig. 4(b),
where only broad magnetic features are observed down to 0.25 K. The weak
features at $Q\sim 0.41$ and 0.72 Å-1 can be indexed by
$\bm{q}=(0.48,0.48,0)$, while their intensities exhibit a history dependence
as expected for a spin glass state.

Figures 4(c-e) present the equal-time spin correlations obtained by
integrating the INS spectra from [$-20$, 20] meV sup . Assuming a
$J_{1}$-$J_{1}^{\prime}$-$J_{3a}$-$J_{3b}$ Heisenberg spin model, we fit the
diffuse scattering data by the SCGA method sup ; Pokharel  _et al._ (2020).
The fitted results are shown in Fig. 4(c-d) as solid lines. The fitted
coupling strengths are $J_{1}=-1.6(2)$, $J_{1}^{\prime}=-5.4(3)$,
$J_{3a}=0.1(1)$, and $J_{3b}=0.6(1)$ meV, where the different strengths for
$J_{3a}$ and $J_{3b}$ are necessary for a satisfactory fit. As shown in the
inset of Fig. 4(b-e), the spin correlations in the ($hk0$) plane does not
follow a circular shape, implying the absence of a SSL state due to the
unequal $J_{3a}$ and $J_{3b}$ interactions.

The fitted parameter set provides an explanation for the glassy ground state
in CuInCr4Se8. $J_{3a}$, being the weakest interaction in the
$J_{1}$-$J_{1}^{\prime}$-$J_{3a}$-$J_{3b}$ model, is however the most crucial
parameter in determining the exact length of the LRO $\bm{q}=(q,q,0)$. Within
the standard deviation of $J_{3a}$, $q$ varies from 0 at $J_{3a}=0$ up to
$\sim 0.4$ at $J_{3a}=0.2$ meV. Therefore, the broad diffuse scattering at
$Q\sim 0.25$ Å-1, which is tentatively indexed as $(0.3,0.3,0)$ in Fig. 4(b),
may arise from a finite distribution in $J_{3a}$ due to a tiny amount of
structural defects. Weaker features indexed by $\bm{q}=(0.48,0.48,0)$ may be
stabilized by local structural distortions considering its proximity to the
commensurate $(\frac{1}{2},\frac{1}{2},0)$ position Lee  _et al._ (2008); Gao
_et al._ (2018). Studies of a single crystal sample will help verify the
proposed mechanism.

Conclusion.— We have presented a line-graph approach to achieve approximate
SSLs on non-bipartite lattices, which allows the experimental exploration of
SSLs in a broader family of compounds. Besides ZnCr2Se4 studied in this work,
chalcogenide spinels like ZnCr2S4 Yokaichiya  _et al._ (2009) and HgCr2S4
Tsurkan  _et al._ (2006) are worth further investigations as an incommensurate
helical ground state has been observed. The approximate SSLs on the line-graph
lattices also provide a new platform to explore field-induced topological spin
textures since the helicity of the single-$\bm{q}$ component is maintained Gao
_et al._ (2020). Such a mechanism may account for the magnetic skyrmions in
the breathing kagome-lattice compound Gd3Ru4Al12 Hirschberger  _et al._
(2019). On the theoretical side, it remains an open question whether the
approximate SSLs may evolve into quantum spin liquids under sufficient quantum
fluctuations. It is also interesting to explore whether the fracton physics
predicted for the exact SSLs Yan and Reuther (2022) may survive to some extent
in the approximate SSLs.

###### Acknowledgements.

We acknowledge helpful discussions with Jyong-Hao Chen. This work was
supported by the U.S. Department of Energy, Office of Science, Basic Energy
Sciences, Materials Sciences and Engineering Division. This research used
resources at the Spallation Neutron Source (SNS) and the High Flux Isotope
Reactor (HFIR), both are DOE Office of Science User Facilities operated by the
Oak Ridge National Laboratory (ORNL). G.P. acknowledges support from the
Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF9069.

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Supplemental Materials for ‘Line-Graph Approach to Spiral Spin Liquids’

##  I Numerical solution of the interaction matrix

As discussed in the main text, the spectrum correspondence between the root
and line graphs can be directly verified for Heisenberg spin models. In this
section, we present the analytical expressions for the interaction matrices
and eigenvalues of the representative models.

Figure S1: The honeycomb (a) and kagome (b) lattices with the sublattices
indicated. The bonding vectors $\bm{d}_{1}$, $\bm{d}_{2}$, and $\bm{d}_{3}$ on
the honeycomb lattice are shown in panel (a). Exchange paths for the $J_{2}$
($J_{3a}$ and $J_{3b}$) interactions on the honeycomb (kagome) lattice are
shown as dashed lines. In the presence of a breathing distortion, the exchange
interactions over the neighboring triangles on the kagome lattice will have
two different strengths of $J_{1}$ and $J_{1}^{\prime}$ as indicated in panel
(b).

###  I.1 $\boldsymbol{J_{1}-J_{2}}$ model on a honeycomb lattice

The $J_{1}$-$J_{2}$ Heisenberg model on a bipartite honeycomb lattice (see
Fig. S1(a)) is a well-established model that hosts SSL Mulder  _et al._
(2010); Niggemann  _et al._ (2020). Here we present the main results for
comparison. Following the definition in the main text, the interaction matrix
$\mathcal{J}(\bm{q})$ of the spin Hamiltonian $\mathcal{H}=J_{1}\sum_{\langle
ij\rangle}\bm{S}_{i}\cdot\bm{S}_{j}+J_{2}\sum_{\langle\langle
ij\rangle\rangle}\bm{S}_{i}\cdot\bm{S}_{j}$ on a honeycomb lattice can be
expressed as

$\displaystyle\mathcal{J}(\bm{q})=J_{1}\begin{pmatrix}0&\zeta\\\
\zeta^{*}&0\end{pmatrix}+J_{2}\gamma\mathbb{1}\text{,}$ (S1)

where the off-diagonal variable is
$\zeta(\bm{q})=\sum_{n=1}^{3}\exp(-i\bm{q}\cdot\bm{d}_{n})$ with bonding
vectors $\bm{d}_{1}=\frac{1}{3}(\bm{b}-\bm{a})$,
$\bm{d}_{2}=\bm{d}_{1}+\bm{a}$, and $\bm{d}_{3}=\bm{d}_{1}-\bm{b}$ shown in
Fig. S1(a). The diagonal variable is $\gamma(\bm{q})=|\zeta(\bm{q})|^{2}-3$.
The eigenvalues of $\mathcal{J}(\bm{q})$ can be solved as

$\nu_{\pm}(\bm{q})=J_{2}\gamma(\bm{q})\pm|J_{1}\zeta(\bm{q})|.$ (S2)

A SSL is realized in the regime of $|J_{2}/J_{1}|>1/6$. As discussed in Ref.
Niggemann  _et al._ (2020), the eigenvectors of $\mathcal{J}(\bm{q})$ have
equal components over the two sublattices, which indicates an exact degeneracy
over the spiral surface according to the Luttinger-Tisza theory.

###  I.2 $\boldsymbol{J_{1}-J_{3}}$ model on a kagome lattice

For the $J_{1}$-$J_{3}$ Heisenberg model on a kagome lattice (see Fig. S1(b)),
the interaction matrix is

$\displaystyle\mathcal{J}(\bm{q})=J_{1}\begin{pmatrix}0&\xi_{12}&\xi_{13}\\\
\xi_{12}&0&\xi_{23}\\\
\xi_{13}&\xi_{23}&0\end{pmatrix}+J_{3}\gamma\mathbb{1}\text{,}$ (S3)

where the off-diagonal variables are $\xi_{12}(\bm{q})=2\cos(\pi q_{h})$,
$\xi_{13}(\bm{q})=2\cos(\pi q_{k})$, and $\xi_{13}(\bm{q})=2\cos(\pi q_{h}+\pi
q_{k})$. The diagonal variable $\gamma(\bm{q})$ is the same as that in Eq.
(S1) for the honeycomb lattice. The eigenvalues can be solved as

$\displaystyle\nu_{1,2}=J_{3}\gamma(\bm{q})\pm|J_{1}\zeta(\bm{q})|+J_{1},$
(S4) $\displaystyle\nu_{3}=J_{3}\gamma(\bm{q})-2J_{1}.$ (S5)

Here $\zeta(\bm{q})$ is the same as that defined in Eq. (S1). It is
immediately clear that when $J_{3}$ on the kagome lattice equals $J_{2}$ on
the honeycomb lattice, the dispersion of $\nu_{1,2}$ is exactly the same as
$\nu_{\pm}$ on the honeycomb lattice up to a constant of $J_{1}$, which is a
direct consequence of the spectral graph theory.

In spite of the similar eigenvalues, it can be shown numerically that the
eigenvectors of $\mathcal{J}(\bm{q})$ on the kagome lattice generally have
close but unequal components over the three sublattices, which contrasts the
equal component eigenvectors on the honeycomb lattice. This means the
degeneracy over the spiral surface on the the kagome lattice is not exact,
leading to an approximate SSL state that is only stabilized by thermal
fluctuations.

The approximate nature of the SSL state on the kagome lattice can also be seen
from the $\bm{q}$-dependence of the weight factor
$\sum_{\mu,\nu}[\psi^{*}(\bm{q})]_{\mu}[\psi(\bm{q})]_{\nu}$, where $\psi$ is
the eigenfunction for the lowest eigenband of the interaction matrix Mizoguchi
_et al._ (2018). Figure S2 compares the weight factor for the kagome- and
honeycomb-lattice models with the same frustration ratio of 0.25. Like the
diffuse scattering patterns calculated by the SCGA method shown in Fig. S2(a)
and (c), the weight factor for on the kagome and honeycomb lattices shown in
Fig. S2(b) and (e) are very similar to each other, thus justifying the
existence of a SSL state in the former model. However, as compared in Fig.
S2(c) and (f) for line cuts along the $(h,h)$ direction, the weight factor of
the kagome model exhibits an additional $\bm{q}$-dependence near the Brillouin
zone boundary. Such a $\bm{q}$-dependence contrasts the flatter weight factor
of the honeycomb model and indicates the approximate nature of the SSL state
on the kagome lattice.

Figure S2: (a) Spin correlation pattern of the kagome-lattice model with
$J_{1}=-1$ and $J_{3}=0.25$ at inverse temperature $\beta=6$. The first
Brillouin zone is outlined by black lines. Intensities are shown in linear
scale. (b) Pseudocolor map of the weight factor calculated from the
eigenvectors of the kagome model. (c) Weight factor along the $(h,h)$
direction. (d-f) Similar plots as (a-c) for the honeycomb-lattice model with
$J_{1}=-1$ and $J_{2}=0.25$.  Figure S3: Phase diagram on the breathing kagome
lattice with ferromagnetic $J_{1}$ and $J_{1}^{\prime}$ interactions together
with antiferromagnetic $J_{3}$. The shaded area indicates the region of the
approximate SSL phase. Contour lines for constant $|\zeta(\bm{q})|$ are shown
in the approximate SSL phase. Inset shows the representative spiral surface in
reciprocal space at $|J_{3}/J_{1}|=0.204$ (red), 0.306 (yellow), and 0.369
(blue) with $J_{1}^{\prime}$ fixed at $0.6J_{1}$, which correspond to
$|\zeta(\bm{q})|^{2}=3$, 1, and 0.5, respectively.

###  I.3 $\boldsymbol{J_{1}-J_{1}^{\prime}-J_{3}}$ model on a breathing kagome
lattice

On the breathing kagome lattice, the hollow matrix term in the interaction
matrix needs to be revised to incorporate the different coupling strengths
$J_{1}$ and $J_{1}^{\prime}$ over the neighboring triangles. The interaction
matrix of the $J_{1}$-$J_{1}^{\prime}$-$J_{3}$ Heisenberg model on a breathing
kagome lattice can be expressed as

$\displaystyle\mathcal{J}(\bm{q})=J_{1}\begin{pmatrix}0&\varepsilon_{12}&\varepsilon_{13}\\\
\varepsilon_{12}^{*}&0&\varepsilon_{23}\\\
\varepsilon_{13}^{*}&\varepsilon_{23}^{*}&0\end{pmatrix}+J_{3}\gamma\mathbb{1}\text{,}$
(S6)

where

$\displaystyle\varepsilon_{12}(\bm{q})$ $\displaystyle=J_{1}\exp(i\pi
q_{h})+J_{1}^{\prime}\exp(-i\pi q_{h}),$ (S7)
$\displaystyle\varepsilon_{13}(\bm{q})$ $\displaystyle=J_{1}\exp(-i\pi
q_{k})+J_{1}^{\prime}\exp(i\pi q_{k}),$ (S8)
$\displaystyle\varepsilon_{23}(\bm{q})$ $\displaystyle=J_{1}\exp(-i\pi
q_{h}-i\pi q_{k})+J_{1}^{\prime}\exp(i\pi q_{h}+i\pi q_{k}).$ (S9)

The definition of $\gamma(\bm{q})$ is the same as that in Eq. (S1). The
eigenvalues can be solved as

$\displaystyle\nu_{1,2}=J_{3}\gamma(\bm{q})\pm\sqrt{\frac{9(J_{1}-J_{1}^{\prime})^{2}}{4}+J_{1}J_{1}^{\prime}|\zeta(\bm{q})|^{2}}+\frac{J_{1}+J_{1}^{\prime}}{2},$
(S10) $\displaystyle\nu_{3}=J_{3}\gamma(\bm{q})-(J_{1}+J_{1}^{\prime}).$ (S11)

Assuming $J_{1}<J_{1}^{\prime}<0$ and $J_{3}>0$, it can be shown that an
approximate SSL phase is stabilized in the parameter regime of

$\displaystyle\frac{1}{3}\frac{J_{1}J_{1}^{\prime}}{|J_{1}+J_{1}^{\prime}|}<J_{3}<\frac{1}{3}\frac{J_{1}J_{1}^{\prime}}{|J_{1}-J_{1}^{\prime}|},$
(S12)

and the phase diagram is summarized in Fig. S3. the eigenvalue minimum
exhibits a continuous degeneracy. The spiral surface is defined by

$\displaystyle|\zeta(\bm{q})|^{2}=\frac{J_{1}J_{1}^{\prime}}{4J_{3}^{2}}-\frac{9}{4}\frac{(J_{1}-J_{1}^{\prime})^{2}}{J_{1}J_{1}^{\prime}}.$
(S13)

Typical spiral surfaces at $|J_{3}/J_{1}|=0.204$, 0.306, and 0.369 with
$J_{1}^{\prime}$ fixed at $0.6J_{1}$ are shown in the inset of Fig. S3.
Similar to the results on a reagular kagome lattice, this SSL has approximate
degeneracy.

###  I.4 $\boldsymbol{J_{1}-J_{2}}$ model on a diamond lattice

For the $J_{1}$-$J_{2}$ Heisenberg model on a bipartite diamond lattice (see
Fig. S4(a)), the interaction matrix is

$\displaystyle\mathcal{J}(\bm{q})=J_{1}\begin{pmatrix}0&\eta\\\
\eta^{*}&0\end{pmatrix}+J_{2}\kappa\mathbb{1}\text{,}$ (S14)

where the off-diagonal variable is
$\eta(\bm{q})=\sum_{n=1}^{4}\exp(-i\bm{q}\cdot\bm{d}_{n})$ with bonding
vectors $\bm{d}_{1}=\frac{1}{4}(\bm{a}+\bm{b}+\bm{c})$,
$\bm{d}_{2}=\frac{1}{4}(\bm{a}-\bm{b}-\bm{c})$,
$\bm{d}_{3}=\frac{1}{4}(-\bm{a}+\bm{b}-\bm{c})$, and
$\bm{d}_{4}=\frac{1}{4}(-\bm{a}-\bm{b}+\bm{c})$ as indicated in Fig. S4(a).
The diagonal variable is $\kappa(\bm{q})=|\eta(\bm{q})|^{2}-4$. The
eigenvalues can be solved as

$\displaystyle\nu_{\pm}=J_{2}\kappa({\bm{q}})\pm|J_{1}\eta(\bm{q})|.$ (S15)

It has been established that a SSL state with exact degeneracy emerges in the
regime of $|J_{2}/J_{1}|>1/8$ Bergman  _et al._ (2007).

###  I.5 $\boldsymbol{J_{1}-J_{3}}$ model on a pyrochlore lattice

For the $J_{1}$-$J_{3}$ Heisenberg model on a pyrochlore lattice (see Fig.
S4(b)), the interaction matrix is

$\displaystyle\mathcal{J}(\bm{q})=J_{1}\begin{pmatrix}0&\varrho_{12}&\varrho_{13}&\varrho_{14}\\\
\varrho_{12}&0&\varrho_{23}&\varrho_{24}\\\
\varrho_{13}&\varrho_{23}&0&\varrho_{34}\\\
\varrho_{14}&\varrho_{24}&\varrho_{34}&0\end{pmatrix}+J_{3}\kappa\mathbb{1}\text{,}$
(S16)

where the off-diagonal variables are $\varrho_{12}(\bm{q})=2\cos(\frac{\pi
q_{h}+\pi q_{l}}{2})$, $\varrho_{13}(\bm{q})=2\cos(\frac{\pi q_{h}+\pi
q_{k}}{2})$, $\varrho_{14}(\bm{q})=2\cos(\frac{\pi q_{k}+\pi q_{l}}{2})$,
$\varrho_{23}(\bm{q})=2\cos(\frac{\pi q_{k}-\pi q_{l}}{2})$,
$\varrho_{24}(\bm{q})=2\cos(\frac{\pi q_{h}-\pi q_{k}}{2})$,
$\varrho_{34}(\bm{q})=2\cos(\frac{\pi q_{h}-\pi q_{l}}{2})$. The diagonal
variable $\kappa(\bm{q})$ is the same as that in Eq. (S14) for the diamond
lattice. The eigenvalues of $\mathcal{J}(\bm{q})$ can be solved as

$\displaystyle\nu_{1,2}$
$\displaystyle=J_{3}\kappa({\bm{q}})\pm|J_{1}\eta(\bm{q})|+2J_{1},$ (S17)
$\displaystyle\nu_{3,4}$ $\displaystyle=J_{3}\kappa({\bm{q}})-2J_{1}.$ (S18)

Figure S4: The diamond (a) and pyrochlore (b) lattices with the sublattices
indicated. The bonding vectors $\bm{d}_{1}$, $\bm{d}_{2}$, $\bm{d}_{3}$, and
$\bm{d}_{4}$ on the diamond lattice are shown in panel (a). Exchange paths for
the $J_{2}$ ($J_{3a}$ and $J_{3b}$) interactions on the diamond (pyrochlore)
lattice are shown as dashed lines. In the presence of a breathing distortion,
the exchange interactions over the neighboring tetrahedra on the pyrochlore
lattice will have two different strengths of $J_{1}$ and $J_{1}^{\prime}$.

Similar to the results on the kagome lattice, numerical solutions show that
the eigenvectors generally have unequal components over the four sublattices.
Therefore, the SSL on the pyrochlore lattice has approximate degeneracy rather
than exact degeneracy as that on the diamond lattice.

###  I.6 $\boldsymbol{J_{1}-J_{1}^{\prime}-J_{3}}$ model on a breathing
pyrochlore lattice

On a breathing pyrochlore lattice, the hollow matrix term in the interaction
matrix needs to be revised to incorporate the different coupling strengths
$J_{1}$ and $J_{1}^{\prime}$ over the neighboring tetrahedra. Defining

$\displaystyle\tau_{ab}(\bm{q})$ $\displaystyle=J_{1}\exp(\pi q_{a}+\pi
q_{b})+J_{1}^{\prime}\exp(-\pi q_{a}-\pi q_{b})$ (S19)
$\displaystyle\tau_{\overline{ab}}(\bm{q})$ $\displaystyle=J_{1}\exp(\pi
q_{a}-\pi q_{b})+J_{1}^{\prime}\exp(-\pi q_{a}+\pi q_{b})\textrm{,}$ (S20)

the interaction matrix of a $J_{1}$-$J_{1}^{\prime}$-$J_{3}$ Heisenberg model
on a breathing pyrochlore lattice can be expressed as

$\displaystyle\mathcal{J}(\bm{q})=J_{1}\begin{pmatrix}0&\tau_{hl}&\tau_{hk}&\tau_{kl}\\\
\tau_{hl}^{*}&0&\tau_{\overline{kl}}&\tau_{\overline{hk}}^{*}\\\
\tau_{hk}^{*}&\tau_{\overline{kl}}^{*}&0&\tau_{\overline{hl}}^{*}\\\
\tau_{kl}^{*}&\tau_{\overline{hk}}&\tau_{\overline{hl}}&0\end{pmatrix}+J_{3}\kappa\mathbb{1}\text{,}$
(S21)

where $\kappa(\bm{q})$ is the same as that in Eq. (S14) for the diamond
lattice. The eigenvalues can be solved as

$\displaystyle\nu_{1,2}$
$\displaystyle=J_{3}\kappa(\bm{q})\pm\sqrt{4(J_{1}-J_{1}^{\prime})^{2}+J_{1}J_{1}^{\prime}|\eta(\bm{q})|^{2}}+(J_{1}+J_{1}^{\prime})$
(S22) $\displaystyle\nu_{3,4}$
$\displaystyle=J_{3}\kappa(\bm{q})-(J_{1}+J_{1}^{\prime}).$ (S23)

Here $\eta(\bm{q})$ is the same as that defined in Eq. (S14). Assuming
$J_{1}<J_{1}^{\prime}<0$ and $J_{3}>0$, an approximate SSL phase is realized
in the parameter regime of

$\displaystyle\frac{1}{4}\frac{J_{1}J_{1}^{\prime}}{|J_{1}+J_{1}^{\prime}|}<J_{3}<\frac{1}{4}\frac{J_{1}J_{1}^{\prime}}{|J_{1}-J_{1}^{\prime}|}.$
(S24)

The spiral surface in reciprocal space is defined by

$\displaystyle|\eta(\bm{q})|^{2}=\frac{J_{1}J_{1}^{\prime}}{4J_{3}^{2}}-\frac{4(J_{1}-J_{1}^{\prime})^{2}}{J_{1}J_{1}^{\prime}}.$
(S25)

The phase diagram and representative spiral surfaces are presented in Fig. 2
of the main text. As on the regular pyrochlore lattice, the degeneracy in this
SSL state is approximate.

Figure S5: (a) Circular markers are the critical temperature of the
$J_{1}$-$J_{3}$ model on a regular pyrochlore lattice as a function of
$|J_{3}/J_{1}|$. $J_{1}$ and $J_{3}$ are assumed to be FM and AFM,
respectively. Empty (solid) squares indicate the crossover transition between
the paramagnetic and SSL states obtained from the maximum of the broad hump in
magnetization for $J_{3}=0.2$ and 0.25 (heat capacity for $J_{3}=0.14$). The
broad hump in magnetization for $J_{3}=0.14$ is arising from a precursory FM
phase instead of the crossover transition. (b-g) Temperature dependence of the
magnetization ($|M|$) and heat capacity ($C_{V}$) per spin for
$|J_{3}/J_{1}|=0.1$ (b, c), 0.14 (d, e), and 0.6 (f, g). These three parameter
sets are representative for the regions shown in black, red, and blue over the
phase diagram in panel (a).

##  II Stability of the approximate SSL phase

Although the degeneracy of the SSL state on a non-bipartite lattice is only
approximate, the energy difference among the spiral configurations over the
spiral surface is small, thus ensuring a robust SSL state down to low
temperatures. As an illustration, we perform classical Monte Carlo simulations
for the $J_{1}$-$J_{3}$ model on a pyrochlore lattice. Our simulations are
implemented with the SpinMC.jl package Buessen  that employs a parallel
tempering scheme Hukushima and Nemoto (1996) to facilitate thermal
equilibration. Figure S5 summarizes the LRO transition temperature
$|T_{c}/J_{1}|$ as a function of $|J_{3}/J_{1}|$, which is calculated on a
supercell of $16\times 4^{3}$ spins with up to $5\times 10^{5}$ thermalization
sweeps and $4\times 10^{6}$ measurement sweeps. Three characteristic regimes
are observed from the temperature dependence of the magnetization ($|M|$) and
heat capacity ($C_{V}$) per spins as summarized in Fig. S5(b-g): At
$|J_{3}/J_{1}|<=0.13$ (black), the system exhibits a FM ground state; At
$|J_{3}/J_{1}|>=0.17$ (blue), the system exhibits a AFM ground state with
propagation vector along the (100) direction; A transitional regime with
strong FM fluctuation is observed at $0.13<|J_{3}/J_{1}|<0.17$ (red).

The most important observation in our Monte Carlo simulations is that the
transition temperature $T_{c}$ in the SSL regime is rather low. Comparing to
the $J_{1}$-$J_{2}$ model on a diamond lattice where LRO is induced by an
order-by-disorder transition Bergman  _et al._ (2007), $T_{c}$ of the
$J_{1}$-$J_{3}$ model on a pyrochlore lattice is only $\sim 2$ times higher.
Considering the difference in the number of nearest-neighbor bonds, which is 4
and 6 on the diamond and pyrochlore lattices, respectively, we expect very
similar stability regime of the SSL states.

Self-consistent Gaussian approximation (SCGA) calculations were performed to
compare the diffuse scattering patterns in the SSL state on the diamond and
pyrochlore lattices. Implementation details for our calculations can be found
in Ref. Gao  _et al._ (2022). Figure S6 compares the diffuse scattering
patterns in the $(hk0)$ plane for the $J_{1}$-$J_{3}$ model on a pyrochlore
lattice and the $J_{1}$-$J_{2}$ model on a diamond lattice. Both the ratio of
$|J_{3}/J_{1}|$ and $|J_{2}/J_{1}|$ are fixed at 0.25. Similar patterns are
observed on the pyrochlore and diamond lattice although the degeneracy in the
former case is only approximate.

Figure S6: (a-c) Diffuse scattering pattern in the ($hk0$) plane for the
$J_{1}$-$J_{3}$ model with $|J_{3}/J_{1}|=0.25$ at inverse temperature
$\beta=7$ (a), 5 (b), and 3 (c). (d-f) Similar calculations for the
$J_{1}$-$J_{2}$ model with $|J_{2}/J_{1}|=0.25$ at $\beta=14$ (a), 10 (b), and
6 (c). Intensities are shown in linear scale.

##  III Comparison to the half-moon patterns

Recent theoretical studies reveal that the pyrochlore and kagome lattices may
exhibit half-moon scattering patterns with sub-dimensional degeneracy Rau and
Gingras (2016); Mizoguchi  _et al._ (2018); Hering  _et al._ (2021); Kiese
_et al._ (2022). These works focus on a $J_{1}-J_{2}-J_{3a}$ model with
$J_{2}=J_{3a}$ and all couplings being AFM. As a contrast, our line-graph
model utilizes FM $J_{1}$, which establishes a different parameter regime to
look for sub-dimensional degeneracy and novel spin correlations. Furthermore,
our line-graph approach, like the bipartite-lattice models, has the advantage
of generating a complete circular ring or sphere in reciprocal space. This
contrasts the half-moon patterns discussed for $J_{1}-J_{2}-J_{3a}$ model
where the scattering ring is separated into two halves. A complete ring
indicates an emergent U(1) symmetry in momentum space and is essential in
realizing tensor gauge theory Yan and Reuther (2022).

##  IV Sample preparation

Polycrystalline samples of ZnCr2Se4 were synthesized by solid state reaction.
Stoichiometric amounts of Zn (99.99%), Cr (99.95%), Se (99.99%), purchased
from Alfa Aesar, were ground together inside a glove box and then pressed into
a 0.5 inch diameter pellet. The resulting pellet was slowly heated to 600 ∘C
and soaked for 10 h. Then the powder was heated to 950 ∘C and held at that
temperature for 7 d. The pellet was ground again in Ar-atmosphere and then
reheated at 950 ∘C for 3 d. The phase purity of the powder was confirmed by
x-ray diffraction collected on a Bruker D2 Phaser diffractometer.

Single crystals of ZnCr2Se4 were grown using the chemical vapor transport
(CVT) method with iodine as the transport agent. Approximately 10 mass percent
of iodine was mixed with the polycrystalline sample of ZnCr2Se4 and then
sealed in a quartz tube of 19 mm internal diameter. The tube was placed inside
a two-zone furnace horizontally and a temperature gradient of approximately
100 ∘C was maintained with the temperature of 950 ∘ at the hot end. After
eight weeks of growth period, single crystals of average mass $\sim$20 mg were
grown at the cold end side of the sealed tube.

A polycrystalline sample of CuInCr4Se8 was synthesized by directly reacting
high purity elements in an SiO2 ampoule sealed under vacuum. The lowest purity
starting material was the Cr (powder) at 99.99 % metal basis purity. The
ampoule was heated at 50 ∘C/h to 700 ∘C for 12 h, then heated to 1100 ∘C at 50
∘C/h for 24 h. The resulting material was crushed, pressed into a pellet,
sealed under vacuum in SiO2 and annealed at 1100 ∘C for an additional 24 h.
Following this, the product was again ground in air, pressed into a pellet and
annealed in a vacuum-sealed SiO2 ampoule for 7 d at 800 ∘C followed by cooling
to room temperature at 200 ∘C/h. The resulting powder was examined with powder
x-ray diffraction in a PANalytical X’Pert Pro MPD with Cu K$\alpha_{1}$
radiation.

##  V Sample quality

###  V.1 ZnCr2Se4

The good quality of our ZnCr2Se4 crystal was confirmed through single crystal
x-ray diffraction experiments performed on a Rigaku XtaLAB PRO diffractometer
at the Spallation Neutron Source (SNS), Oak Ridge National Laboratory (ORNL).
Altogether 5135 reflections, among which 180 are independent, were collected
at room temperature with the molybdenum $K$ radiation ($\lambda=0.71073$ Å).
Refinement of the dataset was performed using the FullProf software Rodriguez-
Carvajal (1993).

Figure S7 compares the observed and calculated squared structure factor
$F^{2}$ assuming the $Fd\overline{3}m$ space group. The lattice constant is
determined to be $a=10.5097(4)$ Å. The Zn, Cr, Se ions occupy the
$8a\,(\frac{1}{8},\frac{1}{8},\frac{1}{8})$,
$16d\,(\frac{1}{2},\frac{1}{2},\frac{1}{2})$, and $32e\,(x,x,x)$ Wyckoff
sites, respectively. The refined Se position is $x=0.2594(1)$. The goodness-
of-fit factors are $R_{F}=1.1$ % and $R_{F^{2}}=2.3$ %. All sites are fully
occupied within errors.

Figure S7: Comparison between the observed and calculated squared structure
factor $F^{2}$ for ZnCr2Se4.

###  V.2 CuInCr4Se8

The good quality of our CuInCr4Se8 sample was verified through neutron
diffraction measurements performed on the HB2A powder diffractometer at the
High Flux Isotope Reactor (HFIR), ORNL. Experimental details can be found in
the following section of “Neutron Scattering Experiments”. Figure S8 present
the refinement results of the neutron diffraction data collected at $T=150$ K.
With the cubic space group $F\overline{4}3m$ (#216), the lattice constant of
CuInCr4Se8 was refined to be $a=10.5550(1)$ Å. The Cr ions occupy the $16e$
$(x,x,x)$ Wyckoff sites with $x=0.3669(6)$. The Cu and In ions occupy the $4a$
$(0,0,0)$ and $4d$ $(\frac{3}{4},\frac{3}{4},\frac{3}{4})$ sites,
respectively. The Se ions occupy two $16e$ $(x,x,x)$ sites with $x=0.1322(2)$
and 0.6110(3). The overall goodness-of-fit factor is $\chi^{2}=2.60$, with
$R_{F}=4.8$ % for the CuInCr4Se8 phase. Bragg peaks at 62∘ and $80.5^{\circ}$
are due to aluminum scattering from the sample holder. Additional Bragg peaks
from the impurity phase Cr2Se3 were identified with a refined weight fraction
of 3.8 %. Releasing the site occupancy on the Cu and In sites in the
CuInCr4Se8 phase does not improve $\chi^{2}$, which suggests negligible Cu-In
anti-site disorder in our sample.

Figure S8: Rietveld analysis of the neutron diffraction data measured at 150 K
on HB-2A for a CuInCr4Se8 powder. Data points are shown as red circles. The
calculated pattern is shown as the black solid line. Green vertical bars from
top to bottom indicate the positions of the Bragg peaks for CuInCr4Se8,
aluminum, and Cr2Se3, respectively. The blue line at the bottom shows the
difference of measured and calculated intensities.

##  VI Magnetic susceptibility

###  VI.1 ZnCr2Se4

Magnetization measurements on a ZnCr2Se4 crystal were carried out with a
Quantum Design Magnetic Property Measurement System. The measuring field was
applied perpendicular to the (111) direction. The magnetic susceptibility
($\chi$) and inverse magnetic susceptibility ($\chi^{-1}$) measured in 100 Oe
are shown in Fig. S9. $\chi(T)$ shows a sharp drop around 21 K, consistent
with an AFM LRO transition. The observed transition temperature ($T_{N}$
$\sim$ 21 K) is consistent with previous literature Hemberger  _et al._
(2007); Cameron  _et al._ (2016); Jendrzejewska  _et al._ (2021). The zero-
field-cooled and field-cooled magnetic susceptibilities overlap with each
other. $\chi^{-1}(T)$ shows linear temperature dependence, Curie-Weiss
behavior, above 200 K. Fitting in the temperature range of [200, 300] K yields
an effective moment of 3.4 $\mu_{B}$ and a Curie-Weiss temperature of $\sim
85$ K, both being comparable with the previous reports Hemberger  _et al._
(2007); Jendrzejewska  _et al._ (2021).

Figure S9: (a) Magnetic susceptibility ($\chi$) and inverse magnetic
susceptibility ($\chi^{-1}$) of ZnCr2Se4 measured in 100 Oe. The curves with
red and pink squares represent the field-cooled and zero field-cooled magnetic
susceptibilities, respectively. Note that these curves are nearly
indistinguishable on this figure except for temperatures below 20 K. The blue
color solid line represents a Curie-Weiss fit to $\chi^{-1}(T)$ above 200 K.

###  VI.2 CuInCr4Se8

The magnetization of a powder sample of CuInCr4Se8 was characterized using a
Quantum Design Magnetic Property Measurement System and the ac magnetic
susceptibility was measured using a Quantum Design Physical Property
Measurement System with further ac measurements performed in a Quantum Design
MPMS3.

Figure S10 presents the real and imaginary parts of the ac magnetic
susceptibility for CuInCr4Se8 measured in a 10 Oe field at frequencies of 10,
100, 1000, and 10000 Hz. The real part is also shown in Fig. 4(a) of the main
text. The freezing temperature increases with increasing frequencies,
exhibiting a typical spin glass behavior.

Figure S10: Real (a) and imaginary (b) parts of the ac magnetic susceptibility
for CuInCr4Se8 measured in a 10 Oe field.

##  VII Neutron scattering experiments

Diffuse neutron scattering experiments for ZnCr2Se4 were performed on the
CORELLI elastic diffuse scattering spectrometer Ye  _et al._ (2018) at the
SNS, ORNL. Like a typical time-of-flight (TOF) direct geometry spectrometer,
the energy resolution depends on the incident neutron energy $E_{i}$. For
$E_{i}=13\sim 33$ meV that is related to our scattering range, the average
energy resolution is estimated to be $\sim 0.8$ meV. A crystal (mass $\sim 80$
mg) was aligned with the $c$ axis vertical. A helium flow cryostat was
employed to reach temperatures $T$ down to $\sim 1.4$ K. Measurements below
$T_{N}$ were performed at 20 K to suppress strong Bragg scattering. Data were
acquired by rotating the sample in 1∘ steps, covering a total range of 30∘.
Data reduction and projection were performed using the MANTID software Arnold
_et al._ (2014). SCGA fits of the diffuse scattering data utilize the combined
simulated annealing and simplex optimization methods as implemented in the
iFit package Farhi  _et al._ (2014); Gao  _et al._ (2022).

Neutron diffraction experiments on CuInCr4Se8 were performed on the HB-2A
diffractometer at the HFIR, ORNL Calder  _et al._ (2018). About 5 g powder
sample was filled in an aluminum sample can. An incident neutron wavelength of
2.41 Å was selected by the Ge-113 monochromator, and the collimators are set
as open-open-12′. A 3He closed cycle refrigerator (CCR) was employed to reach
temperatures down to 0.25 K. Counting time was 12 hrs at each temperature.
Additional measurements were performed using an orange cryostat with a base
temperature of 1.5 K.

Inelastic neutron scattering experiments on CuInCr4Se8 were performed on the
SEQUOIA spectrometer at SNS, ORNL. The same powder sample as was used for the
diffraction measurements was placed in an aluminum can. A CCR was employed to
reach temperatures down to 5 K. The incident neutron energy was selected at
$E_{i}$ = 10, 25, and 50 meV, with a corresponding Fermi chopper frequency of
180, 240, 360 Hz, respectively, in the high-resolution setup. Data from an
empty sample can were subtracted as the background. Data reduction was
performed using the MANTID software Arnold  _et al._ (2014). SCGA fits of the
integrated equal-time correlations utilize the combined simulated annealing
and simplex optimization methods as that for ZnCr2Se4.

Figure S11: (a) Calculated diffuse scattering pattern in the ($hk2$) plane at
$T=30$ K for the $J_{1}$-$J_{2}$-$J_{3}$-$J_{4}$ model on a pyrochlore
lattice. The magnetic form factor of the Cr3+ ion is applied. The parameter
set is the same as that in Fig. 3 of the main text, which includes a nonzero
$J_{4}=-0.057$ meV. (b,c) Diffuse scattering patterns expected at zero $J_{4}$
(b) and positive $J_{4}=0.057$ meV (c). Intensities are shown in linear scale.

##  VIII Perturbations from $J_{4}$ in ZnCr2Se4

As discussed in the main text, the slight shrinking of the spiral surface and
the intensity modulation over the spiral surface is mainly due to the
perturbation of a FM $J_{4}$ interaction. Figure S11 compares the diffuse
pattern in the ($hk2$) plane calculated by the SCGA method for FM, zero, and
AFM $J_{4}$ perturbations. Without $J_{4}$, intensity over the scattering ring
is only weakly modulated by the Cr3+ magnetic form factor. While for FM and
AFM $J_{4}$ perturbations, stronger intensities are observed along the (100)
and (110) directions, respectively. The former case is observed in ZnCr2Se4,
while the latter case is similar to the AFM $J_{3}$ perturbation in MnSc2S4
Gao  _et al._ (2017).

Figure S12: (a-c) Diffuse scattering patterns for a square-lattice model with
couplings $J_{1}=-1$, $J_{2}=2J_{3}=0.5$, and $J_{4}=0.05$. The inverse
temperatures are $\beta=2$ (a), 1 (b), and 0.5 (c). Intensities are shown in
linear scale. (d-f) Low-energy INS spectra for the same square-lattice model
at inverse temperatures of $\beta=2$ (d), 1 (e), and 0.5 (f). The scanning
path in reciprocal space is a quarter of the circular ring with $\theta$
defined in panel (a). Intensities are shown in log scale.

It is noteworthy that although the perturbation of $J_{4}$ induces a magnetic
long-range order with dispersive magnon excitations as previously studied
using INS Tymoshenko  _et al._ (2017), the excitation gap observed along the
spiral surface will disappear at temperatures above $T_{N}$ due to incoherent
spin excitations, which justifies an emergent SSL state. As an illustration,
we calculate the spin correlations of a simplified SSL model on a square
lattice with $J_{1}=-1$, $J_{2}=2J_{3}=0.5$, and a perturbational
$J_{4}=0.05$. In the case of $J_{4}=0$, this model exhibits a perfect spiral
spin liquid phase Yan and Reuther (2022). The presence of AFM $J_{4}$
perturbations induces a tendency to a spiral long-range order with
$\bm{q}=(0.25,0)$ at $\beta\approx 1.2$. Figure S12(a-c) summarizes the
diffuse scattering patterns in the $(h,k)$ plane calculated by the SCGA method
at $\beta=2$, 1, and 0.5. Figure S12(d-f) summarizes the corresponding low-
energy INS spectra calculated by the molecular dynamics method Pohle  _et al._
(2021) using a superlattice of $80\times 80$. The scanning path is along the
scattering ring in reciprocal space as indicted by the red curve in Fig.
S12(a). In the nearly ordered phase at $\beta=2$, gapped excitations are
observed along the scattering ring at wavevector transfers away from the
ordered $\bm{q}$ positions. However, at $\beta=1$ and 0.5 in the paramagnetic
phase, the excitation gap disappears, meanwhile the diffuse scattering pattern
become a complete circular ring. In ZnCr2Se4, we expect the weak excitation
gap along the spiral surface observed in the ordered phase Tymoshenko  _et
al._ (2017) to be similarly removed at temperatures above $T_{N}$, so that the
quasielastic spin dynamics become dominant over the entire spiral surface in
the SSL state.

Figure S13: (a) Calculated magnon dispersion in the field-induced FM phase of
ZnCr2Se4. The coupling strengths are the same as reported in Ref. Tymoshenko
_et al._ (2017), and a 6 T field is applied along the $c$ axis. (b-d)
Constant-energy slices of the magnon spectra at $E=0.03$ (b), 0.13 (c), and
0.23 meV (d) with an integration width of 0.02 meV. Intensities are shown in
linear scale.

##  IX Field-induced excitations in ZnCr2Se4

The magnon excitations in ZnCr2Se4 have been studied using INS below $T_{N}$
in a magnetic field along the $c$ axis Inosov  _et al._ (2020). It was
observed that in a 6 T field, the magnon bands soften almost isotropically in
reciprocal space, leading to ring-like scattering at low energies. By assuming
a FM long-range ordered ground state Hemberger  _et al._ (2007), the main
features of the low-energy excitations can be reproduced by linear spin wave
calculations Toth and Lake (2015) as shown in Fig. S13. Especially, the
experimentally observed scattering ring Inosov  _et al._ (2020), including the
four empty holes around the long-range order $\bm{q}$, is captured in the
constant-energy slice at $E$ = 0.13 meV shown in Fig. S13(c). The four empty
holes arise from the dispersion of the softened magnons, and the strong
intensities around $\bm{q}=\\{0,0,0.47\\}$ at lower energies shown in Fig.
S13(b) indicate that the quasielastic dynamics is dominated by fluctuations of
discrete $\bm{q}$ vectors. Although both observations contradict the scenario
of a field-induced SSL state in ZnCr2Se4, the rather close energy difference
over the spiral surface (compared to the total magnon bandwidth of $\sim 33$
meV) still suggests magnetic field as a possible route to SSLs, which is a
very interesting route for future research.

Figure S14: INS spectra $S(Q,\omega)$ measured on SEQUOIA with an incident
neutron energy of $E_{i}=25$ meV. The measuring temperatures are $T=40$ K (a),
100 K (b), and 200 K (c).

##  X INS spectra for CuInCr4Se8

Figure S14 presents the INS spectra $S(Q,\omega)$ measured on SEQUOIA with an
incident neutron energy of $E_{i}=25$ meV. The measuring temperatures are
$T=40$ K (a), 100 K (b), and 200 K (c). Empty can data collected at the
corresponding temperatures have been subtracted as the background. The equal-
time spin correlations shown in Fig. 4(c-e) of the main text are obtained by
integrating the INS spectra from $[-20,20]$ meV.

##  XI General guidelines for SSL search in real materials

Several general guidelines for the realization of spiral spin liquids on the
line-graph lattices can be drawn from the example compound ZnCr2Se4 and other
known candidates. Firstly, the favorable electron configurations should
minimize the single-ion anisotropy so that spiral correlations along all
directions can be stabilized. Examples include some instances of the d3 and d5
configurations. Secondly, since the line-graph approach relies on
ferromagnetic $J_{1}$ and antiferromagnetic $J_{3}$ couplings, qualitative
analysis of the coupling signs through the Goodenough-Kanamori rule should be
applied as an initial check for promising compounds. For example, in ZnCr2Se4,
the $J_{1}$ couplings are ferromagnetic because of the $90^{\circ}$ Cr3+-O2--
Cr3+ super-exchange path, while the $J_{3}$ couplings are antiferromagnetic
because the dominant super-exchange terms are those between two empty or two
half-filled orbitals. The $J_{2}$ strength, which should be negligible in the
line-graph approach, is difficult to predict in an empirical way as it
involves competing $e_{g}$-$t_{2g}$ and $t_{2g}$-$t_{2g}$ exchange terms.
Considering the weak $J_{2}$ strength in ZnCr2Se4 has been successfully
reproduced by DFT calculations Yaresko (2008), we expect similar calculations
can be employed for compounds of interests to guide further experimental
studies.

Aside from empirical conjectures and theoretical calculations, magnetic orders
determined from previous neutron diffraction experiments (for example
cataloged at the Bilbao crystallographic server Aroyo  _et al._ (2011)) may
also inform the material search. This is because the ordering transition in
spiral spin liquids, no matter due to the order-by-disorder effect or further-
neighbor perturbations, often leads to a spiral magnetic structure. Therefore,
examining line-graph lattice compounds known with a spiral long-range order
will be a promising strategy to identify additional examples of materials
hosting a spiral spin liquid.

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