# Quantum effects on unconventional pinch point singularities

Nils Niggemann<EMAIL_ADDRESS>Dahlem Center for Complex Quantum
Systems and Fachbereich Physik, Freie Universität Berlin, D-14195 Berlin,
Germany  Helmholtz-Zentrum Berlin für Materialien und Energie, D-14109 Berlin,
Germany  Department of Physics and Quantum Center for Diamond and Emergent
Materials (QuCenDiEM), Indian Institute of Technology Madras, Chennai 600036,
India  Yasir Iqbal  Department of Physics and Quantum Center for Diamond and
Emergent Materials (QuCenDiEM), Indian Institute of Technology Madras, Chennai
600036, India  Johannes Reuther  Dahlem Center for Complex Quantum Systems and
Fachbereich Physik, Freie Universität Berlin, D-14195 Berlin, Germany
Helmholtz-Zentrum Berlin für Materialien und Energie, D-14109 Berlin, Germany
Department of Physics and Quantum Center for Diamond and Emergent Materials
(QuCenDiEM), Indian Institute of Technology Madras, Chennai 600036, India

###### Abstract

Fracton phases are a particularly exotic type of quantum spin liquids where
the elementary quasiparticles are intrinsically immobile. These phases may be
described by unconventional gauge theories known as tensor or multipolar gauge
theories, characteristic for so-called type-I or type-II fracton phases,
respectively. Both variants have been associated with distinctive singular
patterns in the spin structure factor, such as multifold pinch points for
type-I and quadratic pinch points for type-II fracton phases. Here, we assess
the impact of quantum fluctuations on these patterns by numerically
investigating the spin $S=1/2$ quantum version of a classical spin model on
the octahedral lattice featuring exact realizations of multifold and quadratic
pinch points, as well as an unusual pinch line singularity. Based on large
scale pseudo fermion and pseudo Majorana functional renormalization group
calculations, we take the intactness of these spectroscopic signatures as a
measure for the stability of the corresponding fracton phases. We find that in
all three cases, quantum fluctuations significantly modify the shape of pinch
points or lines by smearing them out and shifting signal away from the
singularities in contrast to effects of pure thermal fluctuations. This
indicates possible fragility of these phases and allows us to identify
characteristic fingerprints of their remnants.

Introduction.— A particularly fascinating physical situation arises when a
system of interacting spins realizes an emergent gauge theory, which is one of
the defining properties of a quantum spin liquid [1]. Various different types
of gauge theories may be realized in such phases. For example, quantum spin
ice represents a variant of a quantum spin liquid, where an emergent U(1)
gauge theory on a pyrochlore lattice establishes an astonishing analogy to
three-dimensional electromagnetism including emergent photons and an effective
fine-structure constant [2, 3]. The key ingredient enabling these non-trivial
properties is the gauge constraint which, in the charge-free sector of a U(1)
gauge theory, takes the form of a Gauss law
${\bm{\nabla}}\cdot{\bm{E}}(\bm{r})=0$.

Meanwhile, generalizations of the standard U(1) gauge theories have become a
new focus of theoretical investigations where the vector form of the Gauss-law
is replaced by a tensor structure [4, 5, 6, 7], e.g.
$\sum_{\mu\nu}\partial_{\mu}\partial_{\nu}E_{\mu\nu}(\bm{r})=0$, known as
tensor gauge theories describing so-called fracton spin liquids [8, 9]. The
most remarkable consequence of this generalization is that, besides the
effective charge of a quasiparticle, multipole moments of charges become
conserved quantities giving rise to excitations with fractionalized mobility
[10]. Two cases can be distinguished [11]: In type-I fracton phases [12, 13,
14], described by symmetric tensor gauge theories, the quasiparticles are
either completely immobile or have a residual mobility along subdimensional
manifolds. Otherwise, in type-II fracton phases [15, 16, 17] all
quasiparticles are completely immobile. In the associated multipolar gauge
theories the Gauss law contains derivatives of different orders restricting
charge configurations to certain fractal patterns [18, 19, 20, 21].
Remarkably, fracton phases also attract interest in fields such as quantum
information [22, 23] and high energy physics [24, 25, 26, 27].

Recently, important steps have been undertaken to bring the rather abstract
theoretical research on fracton phases closer to the established field of
quantum magnetism and to experiments. For example, it has been found that
type-I fracton phases manifest themselves in multifold pinch-points [6] in the
spin structure factor [Fig. 2(a)], generalizing the famous twofold pinch
points known from conventional U(1) spin liquids [Fig. 1(c)]. Likewise, type-
II fracton phases have been argued to be associated with quadratic pinch
points [Fig. 3(d)] where contour lines exhibit a characteristic parabolic
shape [28]. On a different front, a class of simple classical spin models have
been identified [29] which give straightforward access to classical spin
liquids described by tensor gauge theories and to unconventional pinch points
in the spin structure factor. However, it is an open but experimentally
relevant question how stable these phases are under modification from the
ideal situations in which they are defined, e.g., by allowing for quantum
fluctuations.

In this letter, we study the effects of quantum fluctuations on the ground
state and finite-temperature phases of the classical spin model in Ref. [29]–
the so-called octochlore model – whose three dimensional octahedral lattice is
realized in rare-earth antiperovskites [30, 31]. This model represents a
showcase example for exotic classical spin liquids: Apart from known twofold
and multifold pinch points we identify exact realizations of quadratic pinch
points [28] as well as unconventional pinch line singularities [32]. We add
quantum fluctuations to the system by promoting it from a classical
($S\to\infty$) to a quantum $S=1/2$ Heisenberg model which is then numerically
treated via two powerful quantum many-body techniques, the pseudo fermion and
the pseudo Majorana functional renormalization group. Overall, we find that
exotic pinch point features are drastically affected by quantum fluctuations
and appear more fragile compared to conventional twofold pinch points.

Figure 1:  (a) Octochlore model: Differently weighted sites in Eq. (2) are
indicated by different colors. (b) Phase diagram of the model from Ref. [29].
The labels A, B, C [with parameters $(\alpha,\beta)$ given in the inset]
indicate the locations of multifold pinch points, quadratic pinch points and
pinch lines as shown in Figs. 2, 4 and 3, respectively. (c) Spin structure
factor of a twofold pinch point at $\alpha=\beta=0$ for the classical and
quantum model. (d) Spin structure factor $\mathcal{S}(\mathbf{q})$ along
circular paths indicated in (c) normalized to their maxima.

Unconventional gauge theories from an octochlore model.— The octahedral
lattice consists of corner-sharing octahedra and is defined by simple cubic
lattice vectors $\mathbf{a}_{m}\in\left\\{(1,0,0),(0,1,0),(0,0,1)\right\\}$
together with a three site basis $\mathbf{b}_{m}=\mathbf{a}_{m}/2$. The
Hamiltonian of the octochlore model [29] is constructed as the sum of squared
vectors $\mathbf{M}_{\textrm{oct},\alpha\beta}$ over all elementary octahedra

$\displaystyle H$
$\displaystyle=\frac{J}{2}\sum_{\textrm{oct}}{\mathbf{M}^{2}_{\textrm{oct},\alpha\beta}},$
(1)

where $\mathbf{M}_{\textrm{oct},\alpha\beta}$ is the sum of spins in a
cluster, weighted by dimensionless parameters $\alpha$, $\beta$,

$\displaystyle\mathbf{M}_{\textrm{oct},\alpha\beta}$
$\displaystyle=\sum_{i\in\textrm{oct}}\mathbf{S}_{i}+\alpha\sum_{i\in\langle\textrm{oct}\rangle}\mathbf{S}_{i}+\beta\sum_{i\in\langle\langle\textrm{oct}\rangle\rangle}\mathbf{S}_{i}.$
(2)

Here, a reference octahedron “oct” is given by the green sites in Fig. 1(a),
while its closest surrounding sites “$\langle\textrm{oct}\rangle$” and further
distant sites “$\langle\langle\textrm{oct}\rangle\rangle$” are colored blue
and cyan, respectively. Henceforth, we set the energy scale such that the
maximal Heisenberg coupling between two spins is equal to one.

For classical spins $\mathbf{S}_{i}$, the system’s extensively degenerate
ground states follow from the constraints
$\mathbf{M}_{\textrm{oct},\alpha\beta}=0$ which constitute discrete versions
of Gauss’s law. These constraints can be expressed in reciprocal space as
$\sum_{m}L_{m}(\mathbf{q})\mathbf{S}_{m}(\mathbf{q})=0$ [29], where $m=1,2,3$
label the sublattices, $\mathbf{S}_{m}(\mathbf{q})$ is the Fourier-transformed
spin on sublattice $m$ and $L_{m}(\mathbf{q})$ is the $m$-th component of the
so-called constraint vector. Normalized constraint vectors
$\tilde{L}_{m}(\mathbf{q})=L_{m}(\mathbf{q})/\sqrt{\sum_{n}(L_{n}(\bm{q}))^{2}}$
can be defined over the entire $\mathbf{q}$ space except at singular points
$\mathbf{q}^{\star}$ where $L_{m}(\mathbf{q}^{\star})=0$ for all $m$. For
isolated points $\mathbf{q}^{\star}$ in momentum space and with
$\tilde{L}_{m}(\mathbf{q})$ defined on the unit sphere $S^{2}$ one can assign
a topological index to the defect configuration $\tilde{L}_{m}(\mathbf{q})$
around $\mathbf{q}^{\star}$ defined by the second homotopy group of $S^{2}$,
which is the Skyrmion number $Q$ [33, 34]. As demonstrated in Ref. [29] non-
trivial $Q\neq 0$ give rise to pinch points at $\mathbf{q}=\mathbf{q}^{\star}$
in the equal-time spin structure factor
$\mathcal{S}(\mathbf{q})\equiv\langle\mathbf{S}(-\mathbf{q})\cdot\mathbf{S}(\mathbf{q})\rangle$,
where $|Q|=1$ is associated with twofold pinch points. Furthermore, expanding
$L_{m}(\mathbf{q})$ in powers of $\mathbf{q}$ around $\mathbf{q}^{\star}$
reveals the underlying continuum gauge theory.

The number of such defects and their arrangement in the Brillouin zone yields
a phase diagram spanned by $\alpha$ and $\beta$ featuring 10 distinct
classical spin liquids, see Fig. 1(b). In particular, at points along the
boundary [i.e., point A in Fig. 1(b)] multiple defects with $Q=\pm 1$ merge,
leading to a higher $|Q|>1$ associated with a tensor gauge theory and
multifold pinch points, see Ref. [29]. In addition, we have identified even
richer phenomena at crossing points of several phase boundaries: Point B in
Fig. 1(b) displays a pinch point with purely parabolic contours, recently
predicted to be a hallmark signature of type-II fracton phases [28], while
point C features unusual, one-dimensional manifolds of pinch points, so-called
pinch-lines [32].

Methods— The classical model in Eq. 1 is treated within a standard large-$N$
approach [35], both at zero and finite temperatures, previously found to
correctly encapture the qualitative behaviour for this system [29]. To study
the vastly more complicated quantum $S=1/2$ version, we employ two functional
renormalization group (FRG) approaches that replace spin operators by
fermionic pseudo particles. An established approach at zero temperature is the
so-called pseudo fermion (PF-) FRG [36, 37, 38, 39, 40, 41, 42], in which spin
$S=1/2$ operators are mapped onto two flavors of complex fermions
$f_{i\uparrow},f_{i\downarrow}$ as
$S^{\mu}_{i}=\frac{1}{2}\sum_{a,b\in\\{\uparrow,\downarrow\\}}f^{\dagger}_{ia}\sigma^{\mu}_{ab}f_{ib}$.
At finite temperatures, we apply the pseudo Majorana (PM-) FRG, where we,
instead, represent spins by three flavors $\mu=x,y,z$ of SO(3)-symmetric
Majorana fermions
$\\{\eta^{\mu}_{i},\eta^{\nu}_{j}\\}=\delta_{ij}\delta_{\mu\nu}$ as
$S^{\mu}_{i}=-\frac{i}{2}\sum_{\nu,\sigma}\epsilon_{\mu\nu\sigma}\eta^{\nu}_{i}\eta^{\sigma}_{i}$,
without introducing unphysical states [43, 44, 45]. For both approaches, the
resulting interacting model is treated in the thermodynamic limit 111Please
see Supplemental Material at end of manuscript for methodological details and
a discussion on pinch points. using one-loop FRG. Here, $\sim 10^{8}$ first
order ordinary differential equations are solved numerically as a function of
an artificial Matsubara frequency cutoff $\Lambda$. In the physical limit
$\Lambda\rightarrow 0$, we obtain renormalized fermionic vertex functions well
beyond mean field, from which we calculate the equal-time spin structure
factor $\mathcal{S}(\bm{q})$. Despite the common FRG background, the
approximations associated with a one-loop scheme are different in both
approaches such that one can consider the PFFRG and PMFRG as independent and
complementary techniques. Still, we observe excellent agreement between the
equal-time PMFRG structure factor for the lowest simulated temperatures with
the one obtained from PFFRG at $T=0$ [46].

Twofold pinch points.— Even though not the focus of this work, we start with a
brief discussion of more conventional twofold pinch points with $|Q|=1$,
occurring in the bulk of every phase of Fig. 1(b). At the pinch point
positions $\mathbf{q}=\mathbf{q}^{\star}$, the lowest non-vanishing term in an
expansion of $L_{m}(\mathbf{q})$ is the linear one, and hence, the emergent
continuum Gauss law has the linear form
$\mathbf{\nabla}\cdot\mathbf{E}(\mathbf{r})=\mathbf{q}\cdot\mathbf{E}(\mathbf{q})=0$
where
$\mathbf{E}(\mathbf{q})=\sum_{m}S^{z}_{m}(\mathbf{q})\partial_{\mathbf{q}}L_{m}(\mathbf{q})$
222Derivatives here are taken at $\mathbf{q}=\mathbf{q}^{\star}$ and
$\mathbf{q}$ is meant to be the momentum relative to $\mathbf{q}^{\star}$.
Furthermore, since the different spin components $\mu=x,y,z$ do not couple in
the spin constraint, one may restrict to the $z$-component without loss of
generality.. Under the influence of quantum fluctuations at $T=0$ in the
$S=1/2$ case treated with PFFRG, twofold pinch points show the typical
broadening illustrated in Fig. 1(c) for the case $\alpha=\beta=0$, while the
overall pinch point shape stays rather intact. In particular, we observe the
effects of quantum fluctuations to be analogous to those at a finite
temperature $T\sim 1.3$. This broadening indicates violations of the ice rule
constraint, and is expected as the absolute spin magnitudes
$\mathbf{M}^{2}_{\textrm{oct},\alpha\beta}$ of neighboring octahedra do not
mutually commute and thus fluctuate, i.e.,
$\langle\mathbf{M}^{2}_{\textrm{oct},\alpha\beta}\rangle\neq 0$. Importantly,
the signal at $\mathbf{q}=\mathbf{q}^{\star}$ remains strong and no
indications for magnetic long-range order are observed in the full
$\alpha$-$\beta$ plane [48, 46]. We find these observations to be in direct
analogy with past studies of the closely related nearest neighbor pyrochlore
Heisenberg model [49, 41, 44, 50, 51, 52, 53, 54, 55, 56].

Figure 2: (a-d): Spin structure factor $\mathcal{S}(\bm{q})$ in the
$hhl$-plane for a multifold $Q=-7$ pinch point found at location A
($\alpha=-\frac{11}{10}$, $\beta=\frac{9}{5}$) in the phase diagram of Fig. 1.
(e): $\mathcal{S}(\mathbf{q})$ for the two paths indicated in (a), normalized
to its maximum value along each path. (f): As in (e) but comparing classical
thermal and quantum fluctuations along the paths in (b) and (c). The paths are
counterclockwise and the start point $\varphi=0$ is indicated by a marker.

Multifold pinch points.— A vanishing linear term in an expansion of
$L_{m}(\mathbf{q})$ around $\mathbf{q}=\mathbf{q}^{\star}$ is associated with
multifold pinch points [6, 7, 29]. An instructive example occurs at
$\alpha=-11/10$, $\beta=9/5$ and $\mathbf{q}^{\star}=(\pi,\pi,\pi)$, with a
topological index $Q=-7$ and six lobes of large intensity in the $hhl$ plane,
see Fig. 2(a) and Ref. [29]. We identify a gauge constraint of third rank
$\sum_{\mu\nu\sigma}q_{\mu}q_{\nu}q_{\sigma}E_{\mu\nu\sigma}(\mathbf{q})=0$
where
$E_{\mu\nu\rho}(\mathbf{q})=\sum_{m}S^{z}_{m}(\mathbf{q})\partial_{q_{\mu}}\partial_{q_{\nu}}\partial_{q_{\rho}}L_{m}(\mathbf{q})$,
implying conserved scalar charge, dipole and quadrupole moments. Figure 2
shows the impact of both quantum and thermal fluctuations on this multifold
pinch point.

The value of $\mathcal{S}(\mathbf{q})$ along circular paths around the pinch
point illustrates the presence of the singularity: For the exact gauge theory
in the classical $T=0$ model, it retains the same strong angular dependence
for arbitrarily small radii, see Fig. 2(e). Thermal fluctuations induce a
rather featureless broadening and the angular dependence of the signal
vanishes at small distances from the pinch point, see full red line in Fig.
2(f). Interestingly, the effects of quantum fluctuations are very different.
In addition to a broadening, quantum fluctuations add a shift of spectral
weight away from the pinch point origin in favor of soft maxima at
incommensurate positions, effectively tearing apart the pinch point. We note
that this observation stands in stark contrast to the case of twofold pinch
points shown in Fig. 1(c), for which quantum fluctuations appear to act
similarly to thermal ones.

Figure 3: Pinch point with parabolic contours in the $hhl$-plane found at
point B ($\alpha=0,\beta=-1$). (a): Classical large-$N$ result for
temperatures $T=0$ (left half) and $T=1.4$ (right half). (b) Quantum model at
$T=1.4$ and (c) at $T=0$ obtained from PMFRG and PFFRG in the low cutoff
limit, respectively. (d-f): Magnifications of the regions indicated by black
squares in the upper panel together with black contour lines.

In order to physically interpret the data in Fig. 2, two types of quantum
effects need to be distinguished. First, the aforementioned broadening of
pinch points rather indicates the destruction of the underlying gauge theory.
However, a second well-known quantum effect consistent with a gauge theory is
the formation of gapless photon modes with dispersion $\omega({\bm{q}})$,
resulting from an emergent conjugate vector potential $A({\bm{r}})$. These
photon modes give rise to an extra factor $\omega({\bm{q}})$ in the spin
structure factor (i.e.,
$\mathcal{S}({\bm{q}})\rightarrow\omega({\bm{q}})\mathcal{S}({\bm{q}})$)
suppressing the signal at the singularity due to $\omega({\bm{q}}^{\star})=0$
[57, 6, 28]. To test whether the weight distribution in Fig. 2(c) contains
possible signatures of such a modulation, we note that the mere multiplication
of an exact pinch point with an isotropic factor
$\omega({\bm{q}})\sim|{\bm{q}}-{\bm{q}}^{\star}|^{\gamma}$ (or, for that
matter, any function $\omega(|{\bm{q}}-{\bm{q}}^{\star}|)$) 333In the most
common types of U(1) gauge theories including rank-1 and rank-2 versions the
photon dispersion is linear, i.e., $\gamma=1$, as long as charges are scalar
[6]. leaves the singularity intact such that $\mathcal{S}({\bm{q}})$ along
rings around the pinch point, normalized to its maximum on each path, would
remain unchanged upon decreasing the radius of the rings. However, the dashed
red and blue graphs in Fig. 2(f) illustrating the normalized signal along the
ring-like paths in Fig. 2(c) are qualitatively very different and, hence, our
results seem incompatible with an emergent photon mode. While it is possible
that the ground state is described by a different gauge theory (i.e., with an
emergent electric field given by a more complex function of spin operators),
we deem it questionable whether fractonic phenomena that have been associated
with these spectroscopic features still occur in the $S=1/2$ limit of the
Heisenberg model.

Quadratic pinch points.— A further generalization occurs if the gauge
constraint contains derivatives of different orders as is characteristic for
multipolar gauge theories describing type-II fracton phases. This gives rise
to quadratic pinch points in the spin structure factor where lobes of strong
intensity follow contour lines of the form $q_{\parallel}\propto
aq_{\perp}^{2}$ with $q_{\parallel}$ and $q_{\perp}$ being two perpendicular
momentum space directions and $a$ is the lattice constant (which is set to one
here). The mixing of derivatives causes the lattice constant to explicitly
appear in these spectroscopic patterns which is a direct manifestation of the
ultraviolet-infrared mixing described in recent literature [28].

Strikingly, we have identified such quadratic pinch points in the classical
octochlore model at $\alpha=0$, $\beta=-1$ and $\mathbf{q}^{\star}=(0,0,\pi)$.
The effective gauge theory in this case contains first derivatives along the
$z$-direction, as $\partial_{q_{z}}L_{3}(\mathbf{q})\neq 0$, while for the
perpendicular $x$, $y$ directions
$\partial_{q_{x}}L_{m}(\mathbf{q})=\partial_{q_{y}}L_{m}(\mathbf{q})=0$ for
$m=1,2,3$ and the lowest non-vanishing contribution comes from second
derivatives. The resulting quadratic pinch point in classical large-$N$ [Fig.
3(d)] has a shape which is similar to predictions from the U(1) Haah code
[28]. The effect of finite temperatures in large-$N$ only amounts to a
broadening near $\mathbf{q}^{\star}$ while retaining the quadratic shape and
the strong signal around $\mathbf{q}^{\star}$. This is to be contrasted with
PMFRG at the same temperature where the signal is reduced near
$\mathbf{q}^{\star}$ and quadratic contours are no longer discernible. This
trend continues down to $T=0$ where the spin-structure factor appears even
more strongly reduced around $\mathbf{q}^{\star}$. Again however, this result
seems incompatible with emergent photons, see Ref. [46].

As a side remark, the model with $\alpha=0$, $\beta=-1$ also hosts a fourfold
pinch point [7], see Fig. 3(a) at $\mathbf{q}^{\star}=\mathbf{0}$, associated
with a trace-full rank-2 tensor gauge constraint. With the observed reduction
of $\mathcal{S}(\mathbf{q})$ and the formation of a local minimum at
$\mathbf{q}^{\star}=\mathbf{0}$ in the quantum model as $T\rightarrow 0$, this
is another example reflecting the strong impact of quantum fluctuations on
exotic pinch points.

Figure 4: (a-c): Temperature dependent spin structure factor for a pinch line
at $\alpha=-\frac{1}{2}$, $\beta=1$ [C in Fig. 1(b)] in the $[111]$ direction.
Panels (d-f) show a cut through the pinch line, here given by $q_{z}=0.5\pi$
as indicated by the solid dark red line in panels (a-c).

Pinch-lines.— Points $\mathbf{q}^{\star}$ of vanishing constraint vector are
not necessarily isolated in momentum space but can form one-dimensional
manifolds. This situation has previously been studied in Ref. [32] where the
phenomenon has been dubbed a pinch line. Such patterns exhibit conventional
twofold pinch points in all planar cuts through the pinch line. For the
classical pyrochlore model investigated in Ref. [32] an underlying gauge
constraint linear in the derivatives but with a tensor structure has been
identified and a possible relevance for the pyrochlore material Tb2Ti2O7 [59]
has been pointed out.

We have found an analogous feature in the octochlore model at
$\alpha=-\frac{1}{2}$ and $\beta=1$ where pinch lines run along $[111]$ and
symmetry related directions in momentum space. The lowest non-vanishing
derivatives of $L_{m}(\mathbf{q})$ at $\mathbf{q}^{\star}$ are first order
derivatives perpendicular to the pinch lines, in agreement with the pyrochlore
model of Ref. [32]. Since the topological defect is now line-like and
observing that the normalized constraint vector $\tilde{L}_{m}(\mathbf{q})$
avoids two opposite points on the unit sphere $S^{2}$ [46], the topological
index is given by the integer vortex winding number $w$. We find $|w|=1$ and
consequently, twofold pinch points in planar cuts through the line defect, see
bottom panel of Fig. 4 depicting cuts at $q_{z}=0.5\pi$. Thermal fluctuations
in the classical model [Fig. 4(a), right] shift spectral weight towards the
pinch lines such that they become visible in the $hhl$ plane as well defined,
broadened lines of constant strong signal.

For the corresponding quantum model, similar observations to the previous
cases can be made, such as a re-distribution of spectral weight away from the
pinch line when temperature is lowered, as shown in the bottom panel of Fig.
4. It is again worth contrasting this behavior with conventional twofold pinch
points representing isolated point defects where quantum fluctuations are not
seen to significantly reduce the signal at $\mathbf{q}=\mathbf{q}^{\star}$,
see the example in Fig. 1(c).

Discussion.— We have identified the classical octochlore model as an exquisite
physical platform for studying exotic spectroscopic features, such as
multifold and quadratic pinch points as well as pinch lines, all associated
with unconventional gauge theories. Numerical studies which systematically
investigate the impact of quantum fluctuations on the corresponding classical
spin liquids are, however, lacking so far. In our endeavor to fill this gap,
we treat the quantum spin $S=1/2$ model employing state-of-the-art PFFRG and
PMFRG methods. We find a recurring theme in our results: Multifold pinch
points, quadratic pinch points and pinch lines all undergo a significantly
different modification under quantum fluctuations than conventional twofold
pinch points, showing a reduction of $\mathcal{S}(\mathbf{q})$ at
$\mathbf{q}^{\star}$ that is also at variance compared to the effects of pure
thermal fluctuations in the classical case. This also implies that the absence
of unconventional pinch points in an experimentally measured spin structure
factor does not necessarily exclude the realization of a higher-rank $U(1)$
gauge theory in the corresponding classical system.

From a methodological perspective, here, we benefit from the fact that our
octochlore model has SU(2) spin symmetry which simplifies the application of
PFFRG and PMFRG enormously. A (numerically more challenging) continuation of
our present work could be to lift the SU(2) symmetry by considering an Ising
version of the octochlore model supplemented with small transverse couplings,
thus realizing an analogous situation as in quantum spin ice models. This will
help identifying the fate of exotic pinch point singularities along a
continuous classical-to-quantum interpolation. Thus, our results strongly
motivate new avenues in the investigations of these exotic pinch points under
quantum fluctuations, which appear to have a more significant impact compared
to twofold-pinch points. Furthermore, our work sets the stage for determining
the microscopic wave functions describing these resulting quantum phases, and
whose correlation functions give rise to the static structure factors obtained
here [60].

Acknowledgements.— We thank Owen Benton and Roderich Moessner for insightful
discussions. N. N. and J. R. acknowledge support from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation), within Project-ID
277101999 CRC 183 (Project A04). N. N. thanks IIT Madras for funding a three-
month stay through an International Graduate Student Travel award which
facilitated completion of this research work. J. R. thanks IIT Madras for a
Visiting Faculty Fellow position under the IoE program during which part of
the research work and manuscript writing were carried out. Y. I. acknowledges
support from DST, India through MATRICS Grant No. MTR/2019/001042, CEFIPRA
Project No. 64T3-1, ICTP through the Associates Programme and from the Simons
Foundation through grant number 284558FY19. This research was supported in
part by the National Science Foundation under Grant No. NSF PHY-1748958, IIT
Madras through the Institute of Eminence (IoE) program for establishing the
QuCenDiEM group (Project No. SP22231244CPMOEXQCDHOC), the International Centre
for Theoretical Sciences (ICTS), Bengaluru, India during a visit for
participating in the program “Frustrated Metals and Insulators” (Code:
ICTS/frumi2022/9). N. N. acknowledges usage of the JUWELS cluster at the
Forschungszentrum Jülich and the Noctua2 cluster at the Paderborn Center for
Parallel Computing (PC2). Y. I. acknowledges the use of the computing
resources at HPCE, IIT Madras.

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## Emergent gauge theories

The purpose of this suppelementary section is to provide a more in-depth
introduction to the construction of emergent gauge theories on the octochlore
model. As discussed in Ref. [29], the classical ground state constraint of the
octochlore model can be written as

$\displaystyle\sum_{i\in c}\eta_{i}\bm{S}_{i}$ $\displaystyle=0\quad\forall
c,$ (S1) $\displaystyle\eta_{i}$ $\displaystyle=\begin{cases}1,\quad
i\in\textrm{oct}\\\ \alpha,\quad i\in\langle\textrm{oct}\rangle\\\ \beta,\quad
i\in\langle\langle\textrm{oct}\rangle\rangle\\\ \end{cases},$ (S2)

where $c$ is the cluster of octahedra shown in Fig. 1 of the main text. In
reciprocal space, the constraint can be expressed using a constraint vector
$L_{m}(\mathbf{q})$

$\displaystyle L_{m}(\mathbf{q})$ $\displaystyle=\sum_{i\in m\in
c}\eta_{i}e^{\imath\mathbf{q}(\bm{r}_{c}-\bm{r}_{i})}$ (S3)
$\displaystyle\sum_{m=1}^{n_{u}}L^{*}_{m}(\mathbf{q})\bm{S}_{m}(\mathbf{q})$
$\displaystyle=0\quad\forall\mathbf{q},$ (S4)

where $n_{u}=3$ is the number of sites per unit cell, $\bm{r}_{c}$ indicates
the position of the center of the cluster $c$ and $\bm{r}_{i}$ the position of
site $i$. As all spin components are equivalent, henceforth, we only consider
the $z$-component. Even though the dimension of $L_{m}$ is given by the number
of sublattices and can in principle be of arbitrary dimension, here we shall
label its three components as $L_{x},L_{y},L_{z}$ for notational convenience.
Equation S4 implies that the vector $S^{z}_{m}$ is orthogonal to the
constraint vector $L_{m}(\mathbf{q})$. Hence, the spin structure factor
$\mathcal{S}(\mathbf{q})\equiv\frac{1}{n_{u}}\sum_{m,n}\langle
S^{z}_{m}(-\mathbf{q})S^{z}_{n}(\mathbf{q})\rangle$ can be approximated at
zero temperature by summing over all elements of the matrix projecting out
$\tilde{L}_{m}(\mathbf{q})$ [48, 29]. This explains the appearance of pinch
points whenever $L_{m}(\bm{q})=\bm{0}$ and the projector becomes singular. The
effective gauge theory is then given by expanding $L_{m}(\mathbf{q})$ to
leading order around the location of a pinch point $\mathbf{q}^{\star}$,
corresponding to a coarse graining of the system. If the lowest non-vanishing
contribution is of first order, we obtain

$\displaystyle\sum_{m}\sum_{\mu}\left.\frac{\partial
L^{*}_{m}}{\partial\tilde{q}_{\mu}}\right|_{\tilde{\mathbf{q}}=0}\tilde{q}_{\mu}S^{z}_{m}(\tilde{\mathbf{q}})\equiv\sum_{\mu}\tilde{q}_{\mu}E_{\mu}(\tilde{\mathbf{q}})=0,$
(S5)

where $\tilde{\mathbf{q}}=\mathbf{q}-\mathbf{q}^{\star}$. This is a simple
Gauss’ law $\nabla\cdot\bm{E}=0$ in reciprocal space. The emergent gauge field
$E_{\mu}(\tilde{\mathbf{q}})=\sum_{m}\frac{\partial
L^{*}_{m}}{\partial\tilde{q}_{\mu}}|_{\tilde{\mathbf{q}}=0}S^{z}_{m}(\tilde{\mathbf{q}})$
in this example is of rank-1 U(1) type. An interesting special case emerges
when the gradient of the constraint vector also vanishes. In this case, the
effective gauge field becomes a higher rank tensor which may depend on terms
such as $\frac{\partial^{2}L^{*}_{m}}{\partial q_{\mu}\partial q_{\nu}}$. We
now consider more explicit examples found on the octochlore model, for which
the constraint vector can be written as

$\bm{L}(\mathbf{q})=\begin{pmatrix}2\cos\left(\frac{q_{x}}{2}\right)\left[2\alpha(\cos(q_{y})+\cos(q_{z}))+2\beta\cos(q_{x})+1-\beta\right]\\\
2\cos\left(\frac{q_{y}}{2}\right)\left[2\alpha(\cos(q_{x})+\cos(q_{z}))+2\beta\cos(q_{y})+1-\beta\right]\\\
2\cos\left(\frac{q_{z}}{2}\right)\left[2\alpha(\cos(q_{x})+\cos(q_{y}))+2\beta\cos(q_{z})+1-\beta\right]\end{pmatrix}.$
(S6)

Twofold pinch point.— First, consider the simple special case
$\alpha=\beta=0$. In this case we have
$\bm{L}=2\left(\cos(q_{x}/2),\cos(q_{y}/2),\cos(q_{z}/2)\right)$ which
vanishes only at the pinch point $q^{\star}=(\pi,\pi,\pi)$ (and equivalent
positions). Here, we find
$\partial_{q_{\mu}}L_{m}(\mathbf{q}^{\star})=-\delta_{\mu m}$, i.e., the
underlying gauge structure can be described by an emergent rank-1 gauge field,
as expected.

As argued in the main text, the impact of quantum fluctuations on the
unconventional pinch point features differs significantly from conventional
ones, such as in the pyrochlore lattice. Here, we demonstrate that we find the
same for the octochlore lattice by choosing $\alpha=\beta=0$. Figs. Fig. S1
and Fig. 1(c,d) of the main text show that for the octochlore model quantum
fluctuations only lead to a broadening of the pinch points, while crucially,
no relative reduction of spectral weight at the original pinch point location
is observed. In particular, it can be seen that the peak at the former pinch
point becomes narrower as temperature is lowered, in agreement with classical
expectations, analogous to previous findings on the pyrochlore lattice [44].
As argued in the main text, this observation of broadening is analogous to the
effect of thermal fluctuations

Figure S1: Conventional twofold pinch point at $\alpha=\beta=0$. The
arrangement and description of the individual panels corresponds to that in
Fig. 3 of the main text.

Fourfold pinch point.— For $\alpha=0,\beta=-1$, $\bm{L}(\mathbf{q})$
simplifies to

$\bm{L}(\mathbf{q})=\begin{pmatrix}4\cos\left(\frac{q_{x}}{2}\right)(1-\cos(q_{x}))\\\
4\cos\left(\frac{q_{y}}{2}\right)(1-\cos(q_{y}))\\\
4\cos\left(\frac{q_{z}}{2}\right)(1-\cos(q_{z}))\end{pmatrix}.$ (S7)

We find that all components of $\bm{L}(\mathbf{q})$ and its first derivatives
vanish at $\mathbf{q}^{\star}=(0,0,0)$. The first nonzero contributions are
all of second order

$\left.\partial_{q_{\mu}}\partial_{q_{\nu}}L_{m}\right|_{\mathbf{q}=\mathbf{q}^{\star}}=4\delta_{\mu\nu}\delta_{\mu
m}$ (S8)

As the underlying gauge field is a rank-2 tensor
$E_{\mu\nu}(\mathbf{q})=4\delta_{\mu\nu}S^{z}_{\mu}(\mathbf{q})$, the
structure factor displays a fourfold pinch point. A gauge theory of this form
$\partial_{\mu}\partial_{\nu}E_{\mu\nu}=0$ implies the existence of
quasiparticle excitations with conserved dipole moment [5]. These so-called
_fractons_ are thus immobile unless grouped together to form pairs or larger
clusters.

Quadratic pinch point.— At the same location in the phase diagram
($\alpha=0,\beta=-1$), we also find a pinch point with purely parabolic
contours at $\mathbf{q}^{\star}=(0,0,\pi)$. As pointed out by Hart et. al. in
Ref. [28], such a pinch point arises from the presence of mixed derivatives in
the Gauss law constraint and is a signature of a type-II fractonic phase.
Indeed, we verify that for the present case, $\bm{L}$ has a nonzero first
derivative only in the $q_{z}$ direction:

$\displaystyle\left.\partial_{q_{\mu}}L_{m}(\mathbf{q})\right|_{\mathbf{q}=\mathbf{q}^{\star}}$
$\displaystyle=\begin{cases}-4\quad\mu=m=z\\\ 0\quad\text{else}\end{cases}$
(S9)
$\displaystyle\left.\partial_{q_{\mu}}\partial_{q_{\nu}}L_{m}(\mathbf{q})\right|_{\mathbf{q}=\mathbf{q}^{\star}}$
$\displaystyle=\begin{cases}4\quad\mu=\nu=m\text{ and }m=x,y\\\
0\quad\text{else}\end{cases}.$ (S10)

Figure S2 shows a comparison between thermal and quantum fluctuations on a
quadratic pinch point. In contrast to the twofold pinch point shown in Fig.
1(c,d) of the main text, the effects of quantum and thermal fluctuations are
quite distinct, leading to a suppression of the structure factor around the
pinch point. As discussed in the main text, for systems with emergent photon
excitations, a similar suppression is also observed [57]. Here, the structure
factor simply acquires a prefactor from the dispersion of a photon with the
speed of light $c$.
$\omega(\tilde{\mathbf{q}})=c\sqrt{\tilde{q}^{2}_{z}+\left(\tilde{q}_{x}^{2}+\tilde{q}_{y}^{2}\right)^{2}}$,
where [57]

$\mathcal{S}(\tilde{\mathbf{q}})\rightarrow\omega(\tilde{\mathbf{q}})\coth\left(\frac{\omega(\tilde{\mathbf{q}})}{2T}\right)\mathcal{S}(\tilde{\mathbf{q}}).$
(S11)

Figure S2: Difference between thermal and quantum fluctuations for quadratic
pinch point at $\alpha=0,\beta=-1$. Figure S3: Quadratic pinch point at $T=0$
and $T=0.8$. Dispersion-corrected structure factor $\mathcal{S}(\mathbf{q})$
obtained from large $N$ and Eq. S11 in (a) and (c) in comparison of to PFFRG
(b) and PMFRG (d).

In Fig. S3, the effect of such a modification is considered in comparison to
the findings from PFFRG. The resulting structure factors are clearly
qualitatively distinct. While PFFRG is formally employed at zero temperature,
the influence of its finite cutoff is often similar to a finite temperature.
To investigate this possibility, a rough estimate of the structure factor can
be obtained from large $N$ at finite temperature by applying the correction
from the photon dispersion in Eq. S11. Assuming that $c=1$ (in units of the
normalized lattice constant and energy scale), the result is shown in panel
(c) of Fig. S3. As in the case of quantum spin ice, this returns some spectral
weight back to the pinch point. Although the intensity at the pinch point
depends on microscopic details such as the value of $c$, we notice that the
position of the maxima remains indifferent, following the original parabolic
contour. This contrasts our numerical observations so that the presence of
emergent photons remains unlikely.

Figure S4: Vortex winding number for pinch line. Left: Spin structure factor
from large-$N$ at $T=0$ in a cut through the pinch line, with loops of
different radius. Right: For each loop, the normalized constraint vector
describes a path winding once around the same axis. Sufficiently close to the
pinch point, this path is of circular shape.

Multifold pinch point.— At $\alpha=-11/10,\beta=9/5$, an even higher rank
gauge theory of the form
$\partial_{\mu}\partial_{\nu}\partial_{\rho}E_{\mu\nu\rho}=0$ emerges, since
both the first and the second order derivatives of the constraint vector
vanish at the pinch point $\mathbf{q}^{\star}=(\pi,\pi,\pi)$. Explicitly, we
find for the first component of the constraint vector

$\displaystyle\partial_{q_{x}}^{3}L_{x}(\mathbf{q}^{\star})$
$\displaystyle=-54/5,$
$\displaystyle\partial_{q_{x}}\partial_{q_{y}}^{2}L_{x}(\mathbf{q}^{\star})$
$\displaystyle=\partial_{q_{x}}\partial_{q_{z}}^{2}L_{x}(\mathbf{q}^{\star})=11/5,$
(S12)

while other derivatives such as
$\partial_{q_{x}}^{2}\partial_{q_{y}}L_{x}(\mathbf{q}^{\star})$ are zero.
Consequently, not only monopole and dipole moments, but also the quadrupole
moment are conserved. This feature is characterized by a skyrmion winding
number of $Q=-7$ and displays a sixfold pinch point when cut through the $hhl$
plane.

Pinch line.— As mentioned before, singularities in the structure factor are
present at points $\mathbf{q}^{\star}$ where $L_{m}(\mathbf{q}^{\star})=0$.
Inspecting Eq. S6, we immediately observe that a pinch point can always be
found at $(\pi,\pi,\pi)$, where the prefactors $\cos(q_{\mu}/2)$ vanish. Other
pinch points emerge at more complicated positions determined by a delicate
balance between the parameters $\alpha,\beta$ and the wavevector $\mathbf{q}$.
In the generic case, for a fixed set of $\alpha,\beta$, the requirement that
all three components of $\bm{L}$ have to vanish leads to three equations,
determining the positions of the pinch points $\mathbf{q}^{\star}$ uniquely
(up to point group symmetries). However, for appropriate $\alpha$ and $\beta$,
one or more components can become equivalent, leading to a line-like manifold
of singularities.

Figure S5: Difference between thermal and quantum fluctuations for pinch-line
at $\alpha=-\beta=1$ .

A simple example is found by setting $q_{z}=\pi$, such that $L_{z}=0$. One can
see that whenever $q_{x}=q_{y}$, the first two components of $L_{m}$ become
equivalent:

$\bm{L}=\begin{pmatrix}2\cos\left(\frac{q_{x}}{2}\right)\left[\cos\left(q_{x}\right)\left(2\alpha+2\beta\right)+1-2\alpha-\beta\right]\\\
2\cos\left(\frac{q_{x}}{2}\right)\left[\cos\left(q_{x}\right)\left(2\alpha+2\beta\right)+1-2\alpha-\beta\right]\\\
0\end{pmatrix}$ (S13)

These two components vanish for all $q_{x}=q_{y}$ for $\alpha=-\beta=1$,
resulting in a line of pinch points, or pinch-line. This pinch line can be
characterized by a topological, winding number. Here,
$\tilde{L}_{m}(\mathbf{q})=L_{m}(\mathbf{q})/\sqrt{\sum_{n}{L_{n}^{2}(\mathbf{q})}}$
is traced on the unit sphere as one moves along a closed loop around the pinch
line. Since the corresponding paths on the unit sphere avoid two opposite
poles (see Fig. S4), the topological index can be defined as the corresponding
winding number. For each point on the pinch line, the winding number takes the
same integer value as long as the loop does not contain or intersect any other
pinch point (where $\tilde{L}(\mathbf{q})$ is singular). Figure S5 shows that
the effects of quantum and thermal fluctuations for one of the pinch points
within the manifold. Being of twofold nature, the effects of quantum
fluctuations are again, relatively similar to thermal ones, although a shift
of spectral weight away from the pinch point center is again observed.

Finally, we note that it is also possible for all three components of
$L_{m}(\mathbf{q})$ to become equivalent, leading to a surface in reciprocal
space for which $L_{m}(\mathbf{q})=0$. One such example is found at
$\alpha=\beta=2$. The resulting surface of vanishing $L_{m}(\mathbf{q})$ is
displayed in Fig. S6. Although this surface can be found spectroscopically as
a narrow local maximum of the structure factor at finite temperature or under
the inclusion of quantum fluctuations, becoming infinitely thin as
fluctuations decrease, no pinch points are visible since any cut always
contains a one-dimensional sub-manifold of this surface.

Figure S6: Surface with vanishing constraint vector for $\alpha=\beta=2$.
Figure S7: Selected PFFRG flows: Flows of the maximal zero-frequency Matsubara
susceptibility obtained from PFFRG for an exemplary selection of points along
boundaries and the bulk of phases in the phase diagram. No feature or flow
breakdown is observed, indicating absence of magnetic ordering.

## Methodological Details

For this work, standard PFFRG and PMFRG implementations were used, further
methodological details are found in [36, 43, 44]. In both cases, a large set
of ordinary differential equations are computed for vertices with up to $N=64$
positive Matsubara frequencies (well beyond convergence) and correlations up
to $L=10$ nearest neighbor bonds using an adaptive Runge-Kutta scheme. For the
largest temperatures $T>1$, a smaller number of $N=40$ was used instead, as
convergence is reached even more rapidly. We note that while the FRG’s maximal
length of spin correlations $L$ is a numerical requirement, the shape of the
structure factor converges already before $L=10$. As a result, the finite
width of pinch points, (and, accordingly, the deviation from dipolar
correlations) is not a numerical artifact but due to a physically finite
correlation length. In such cases, where the physical correlation length is
smaller than the numerical length $L$, results obtained via FRG correspond to
the thermodynamic limit. The standard output of these calculations is the
magnetic susceptibility in Matsubara frequency space

$\chi_{ij}(\imath\nu_{n})=\int_{0}^{\beta}d\tau\langle
S^{z}_{i}(\tau)S^{z}_{j}(0)\rangle e^{\imath\nu_{n}\tau}.$ (S14)

In PFFRG, magnetic order is indicated by a sharp feature in the Fourier
transform of the $\nu=0$ component of Eq. S14 for the flow of the order-
defining momentum $\mathbf{q}$. The absence of such order for a variety of
points in the phase diagram is demonstrated in Fig. S7. In all other results
discussed in this work, we instead consider the equal time structure factor
which is obtained as $\mathcal{S}(\mathbf{q})\equiv\langle
S^{z}(-\mathbf{q})S^{z}(\mathbf{q})\rangle=\frac{T}{N_{\textrm{sites}}}\sum_{n}\sum_{ij}\chi_{ij}(i\nu_{n})e^{\imath\bm{q}\bm{R}_{ij}}$,
where $\bm{R}_{ij}$ is the displacement vector between two sites for better
comparison with the large-$N$ approximation. Despite the methodological
differences, we observe good qualitative agreement between the structure
factor from PMFRG at very low temperature and PFFRG as demonstrated in Fig. S8
for the $Q=-7$ multifold pinch point.

Figure S8: Spin structure factor from PMFRG at $T=0.2$ (a) and PFFRG at $T=0$
(b) for the multifold pinch point at $\alpha=-\frac{11}{10}$,
$\beta=\frac{9}{5}$. The color scale is normalized to the extrema of the
susceptibility and the relative location of the contours is indicated.

In the PMFRG’s underlying $SO(3)$ Majorana representation, a local constant of
motion $\theta_{i}=-2i\eta^{x}_{i}\eta^{y}_{i}\eta^{z}_{i},\
\theta^{2}_{i}=1/2$ can be used to derive an exact Ward identity relating the
two-point Green’s function to the local four-point vertex as
$\langle\eta^{z}_{i}(\tau)\eta^{z}_{i}(0)\rangle=2\langle\eta^{x}_{i}(\tau)\eta^{y}_{i}(\tau)\eta^{x}_{i}(0)\eta^{y}_{i}(0)\rangle$.
In a finite truncation of the flow equations, this identity is only fulfilled
approximately and can therefore be used as a rigourous accuracy check. In this
work, the relative violation of this Ward identity is remains below $\sim
10\%$ for all temperatures down to $T=0.2$.