# Classification of Classical Spin Liquids: Topological Quantum Chemistry and
Crystalline Symmetry

Yuan Fang  Department of Physics and Astronomy, Stony Brook University, Stony
Brook, New York 11794, USA  Jennifer Cano  Department of Physics and
Astronomy, Stony Brook University, Stony Brook, New York 11794, USA  Center
for Computational Quantum Physics, Flatiron Institute, New York, New York
10010, USA  Andriy H. Nevidomskyy  Department of Physics and Astronomy, Rice
University, Houston, TX 77005, USA  Rice Center for Quantum Materials, Rice
University, Houston, TX 77005, USA  Han Yan (闫寒)  Department of Physics and
Astronomy, Rice University, Houston, TX 77005, USA  Smalley-Curl Institute,
Rice University, Houston, TX 77005, USA

###### Abstract

Frustrated magnetic systems can host highly interesting phases known as
classical spin liquids (CSLs), which feature extensive ground state degeneracy
and lack long-range magnetic order. Recently, Yan and Benton et al. proposed a
classification scheme of CSLs in the large-$\mathcal{N}$ (soft spin) limit
[arXiv:2305.00155, arXiv:2305.19189]. This scheme classifies CSLs into two
categories: the algebraic CSLs and the fragile topological CSLs, each with
their own correlation properties, low energy effective description, and finer
classification frameworks. In this work, we further develop the classification
scheme by considering the role of crystalline symmetry. We present a
mathematical framework for computing the band representation of the flat bands
in the spectrum of these CSLs, which extends beyond the conventional
representation analysis. It allows one to determine whether the algebraic
CSLs, which features gapless points on their bottom flat bands, are protected
by symmetry or not. It also provides more information on the finer
classifications of algebraic and fragile topological CSLs. We demonstrate this
framework via concrete examples and showcase its power by constructing a
pinch-line algebraic CSL protected by symmetry.

##  I Introduction

Figure 1:  (a) The CSL Hamiltonian generally features one or more degenerate
flat bands at the bottom of its spectrum. (b) Algebraic CSLs feature gap
closing points between the bottom flat bands and the dispersive bands. The
band touching point determines the emergent Gauss’s law. $\mathcal{B}_{FB}$ is
the flat band representation, and will be discussed in detail in the main
text. (c) Fragile Topological CSLs (FT-CSLs) have no such gap-closing points,
and are classified by their eigenvector homotopy. (d) The landscape of the CSL
phase diagram consists of FT-CSLs whose boundaries are Algebraic CSLs.

The investigation of magnetic systems lacking long-range order has a rich
history, spanning several decades starting from the exploration of disorder
effects in spin glasses [1, 2] and the proposal of resonating valence bond
states [3, 4], which have become fundamental to contemporary research in
strongly frustrated magnets. A particular concept of interest is that of
classical spin liquids (CSLs), which emerge when spin models exhibit an
extensively degenerate ground state manifold as a consequence of frustration,
and fluctuations among ground states preclude any form of ordering [5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

Despite their general instability to perturbations at absolute zero
temperature ($T=0$), the substantial entropy that CSLs possess at low energies
allows them to dominate the finite temperature physics in a finite region of
model parameters. Moreover, CSLs often serve as ‘parent states’ or an
intermediate temperature regime for quantum spin liquids (QSLs), which arise
when quantum fluctuations introduce dynamics among the classical ground states
[20, 21, 22, 23, 24, 25, 26, 27, 28, 29].

Therefore, understanding and classifying spin liquids is of great importance.
Successful classification schemes for quantum spin liquids have been developed
based on the projective symmetry group [30] and the modern perspective of
gapped QSLs [31, 32]. In contrast, the classification of CSLs has made much
slower progress. Previous works have attempted to classify frustrated
classical spin systems using constraint counting [7], linearization around
given spin configurations [33], supersymmetry-inspired constructions [34], or
identification of topological invariants tailored to specific lattices [17].

Recently, a more general scheme has been proposed for classifying CSLs based
on their energy spectrum [35, 36]. More concretely, the scheme utilizes the
connection between CSLs and physics of flat bands at the bottom of the
Hamiltonian’s spectrum, responsible for the extensive degeneracy of the
classical ground states ((Fig. 1(a)). In this classification scheme, CSLs are
divided into two categories. The first is characterized by a topological
invariant that persists as long as the lowest flat bands in its energy
spectrum remain separated by a gap from the higher dispersive bands. This
category is called “fragile topological” classical spin liquids (FT-CSLs, Fig.
1(b)) since the topological characteristics can be made to disappear by adding
spins to the unit cell without closing the spectral gap. The second category,
called algebraic CSLs (Fig. 1(c)), occupies the boundaries between FT-CSLs
where the spectral gap closes, as illustrated schematically in Fig. 1(d). The
eigenvector configurations around the gap-closing points determine the
emergent Gauss’s law describing the algebraic spin correlations in this
category of CSLs.

In this work, we advance the above classification scheme by investigating the
consequences of crystalline symmetry on the classification of classical spin
liquids, in analogy to how crystalline symmetries enrich the classification of
topological phases. Specifically, motivated by the classification of band
representations in Topological Quantum Chemistry (TQC) [37], we develop a
mathematical framework for determining how the flat bands transform under
symmetry, and elucidate whether symmetry protects the gap closing between the
bottom flat bands and the higher dispersive bands. If the gap closing is
protected, then the symmetry forbids a FT-CSL, which is a significant
constraint for model building or material analysis. Further details of the
symmetry representations yield information on the topological classification
of FT-CSLs and the degeneracy structure of the algebraic CSLs.

Importantly, the present mathematical framework goes beyond how TQC is used to
classify electron band structures. Specifically, the symmetry data of the
lattice spins is generally not sufficient to classify the CSL. An additional
piece of information – the emergent, virtual lattice of the constrainer terms
in the Hamiltonian and their crystalline symmetry properties – is crucial and
must be incorporated into the symmetry analysis.

In the following, we briefly review the constrainer Hamiltonian formalism of
CSLs and its classification in Sec. II. We then give the recipe of the
abstract crystalline symmetry analysis in Sec. III. This recipe is then
applied to two known models on the Kagome model in Sec. IV as a demonstration.
We then introduce a new pinch-line model with symmetry-protected nodal line
degeneracies guided by our insight from the symmetry classification in Sec. V.
Finally we summarize our results in Sec. VI.

##  II Brief review of the constrainer Hamiltonian and CSL classification

We study spin models in the limit of a large number of spin components
$\mathcal{N}$. This is equivalent to adopting a “soft spin” approximation,
where the constraint on the spin length $\mathbf{S}^{2}=1$ is enforced only on
average as $\langle{S}^{2}\rangle=1$, by introducing a Lagrange multiplier or
“chemical potential” to the spins in the Self-Consistant Gaussian
Approximation, a method generalized from the Luttinger-Tisza method [38, 39].
This approximation has been demonstrated to be valid for many Heisenberg
candidate CSLs [11, 40, 41, 8, 42]. The Hamiltonians of these CSLs can be
written in the constrainer form,

$\mathcal{H}=\sum_{\mathbf{R}\in\text{u.c.}}\sum_{I=1}^{M}\,[\mathcal{C}_{I}(\mathbf{R})]^{2}\
,$ (1)

where for a given constraint index $I$, the constrainer
$\mathcal{C}_{I}(\mathbf{R})$ is a scalar that defines a linear combination of
a local cluster of spins inside and around the unit cell located at
$\mathbf{R}$ (see Eqs. (14), (20) for concrete examples). The Hamiltonians we
consider are translationally invariant and consist of sums of squared
constrainers.

A more explicit way to express $\mathcal{C}_{I}(\mathbf{R})$ without referring
to pictures of the lattice is to write

$\begin{split}\mathcal{H}&=\sum_{\mathbf{R}\in\text{u.c.}}\sum_{I=1}^{M}[\mathcal{C}_{I}(\mathbf{R})]^{2}\\\
&=\sum_{\mathbf{R}\in\text{u.c.}}\sum_{I=1}^{M}\left[\sum_{\mathbf{r}}\mathbf{S}(\mathbf{r})\cdot\mathbf{C}_{I}(\mathbf{R},\mathbf{r})\right]^{2}\
.\end{split}$ (2)

Here, $\mathbf{S}(\mathbf{r})=(S_{1},\dots,S_{N})(\mathbf{r})$ is the vector
whose $N$ components are the spins on the $N$ sublattice sites respectively.
For example, $S_{b}(\mathbf{r})$ is the $b$-th sublattice site in the unit
cell labelled by $\mathbf{r}$. The term
$\sum_{\mathbf{r}}\mathbf{S}(\mathbf{r})\cdot\mathbf{C}_{I}(\mathbf{R,\mathbf{r}})$
is the constrainer $\mathcal{C}_{I}(\mathbf{R})$ written in a more explicit
form, that is,

$\displaystyle\mathbf{C}_{I}(\mathbf{0},\mathbf{r})=\left(\begin{array}[]{c}\sum_{j\in\text{1st
sub-lat. sites }}c_{1,j}\delta_{\mathbf{r},\mathbf{a}_{1,j}}\\\
\sum_{j\in\text{2nd sub-lat. sites
}}c_{2,j}\delta_{\mathbf{r},\mathbf{a}_{2,j}}\\\ \vdots\\\
\sum_{j\in\text{$N-$th sub-lat. sites
}}c_{N,j}\delta_{\mathbf{r},\mathbf{a}_{N,j}}\end{array}\right)\ ;$ (7)
$\displaystyle\mathbf{C}_{I}(\mathbf{R},\mathbf{r})=\mathbf{C}_{I}(\mathbf{0},\mathbf{r}-\mathbf{R})$
(8)

encodes exactly the information of how spins are summed together in
$\mathcal{C}_{I}(\mathbf{R})$. The $c_{a,j}$ are numerical coefficients, and
$\delta_{\mathbf{r},\mathbf{a}_{a,j}}$ encodes the spin in the unit cell
$\mathbf{a}_{a,j}$ on sublattice $a$, which is summed with coefficient
$c_{a,j}$.

Consider a system with $N$ sublattice sites and $M$ linearly independent
constrainers per unit cell, where $M<N$. There is an extensive degeneracy for
the ground states in this system since the condition of all constrainers being
zero does not fix all the spins. Consequently, the Hamiltonian spectrum in
momentum space has $N-M$ degenerate flat bands, each corresponding to one set
of these degenerate ground states. Above these flat bands, there are $M$
higher dispersive bands that encode the finite-energy states.

References [35, 36] have discussed in great detail how the structure of the
bottom flat bands and the higher dispersive bands can be used to classify the
CSLs. For simplicity, let us use the case of $M=1$ (one constrainer per unit
cell) to illustrate this scheme. The scenario yields $N-1$ flat bands at zero
energy, which corresponds to spin states obeying the constraint
$\mathcal{C}(\mathbf{R})=0$; we have dropped the index $I$ since there is only
one constrainer per unit cell. Additionally, there is a higher dispersive band
describing spin states violating the constraint. The eigenvector of this
dispersive band, denoted as $\mathbf{T}(\mathbf{q})$, can be expressed
analytically and is precisely the Fourier transform of the constrainer
$\mathbf{C}_{I}(\mathbf{0},\mathbf{r})$. The dispersion of the higher band is
$\omega_{T}(\mathbf{q})=|\mathbf{T}(\mathbf{q})|^{2}$ (see example in Sec.
IV.2 in addition to the detailed formulation in Refs. [35, 36]).

The spectrum of the Hamiltonian encodes the classification of the CSL into one
of two categories, determined by whether the dispersive band has a singular
band touching point with the flat band or not. Within each category, a finer
classification can be made by examining the configuration of eigenvectors
around the gapless point (first category) or its global topology (second
category). In more detail, the classification is as follows.

1\. Algebraic CSL: If a gap closure point exists between the bottom flat bands
and the higher dispersive band, the system is an algebraic CSL with
algebraically decaying spin correlations. Here, the ground states conform to a
charge-free Gauss’s law, which is derived from the Taylor expansion of
$\mathbf{T}(\mathbf{q})$ around the band touching point. Specifically, if for
the $a^{\text{th}}$ spin in the unit cell the lowest order term in the
expansion is of order $m_{a}\geq 1$

$T_{a}(\mathbf{q})=\sum_{j=0}^{m_{a}}c_{aj}(-ik_{x})^{j}(-ik_{y})^{m_{a}-j},\quad
a=1,\dots,N,$ (9)

then the ground states described by the spin configurations orthogonal to
$\mathbf{T}(\mathbf{q})$, i.e.,

$\mathbf{T}(\mathbf{q})\cdot\tilde{\mathbf{S}}(\mathbf{q})=0,$ (10)

in momentum space. Reverse-Fourier transforming this back to real space, we
obtain the Gauss’s law

$\sum_{a=1}^{N}\left(\sum_{j=0}^{m_{a}}c^{\ast}_{aj}(\partial_{x})^{j}(\partial_{y})^{m_{a}-j}S_{a}\right)\equiv\sum_{a=1}^{N}D_{a}^{(m_{a})}S_{a}=0,$
(11)

where $D_{a}^{(m_{a})}$ denotes the differential operator of order $m_{a}\geq
1$ from Fourier transforming the $(-ik_{x})^{j}(-ik_{y})^{m_{a}-j}$ terms.
This principle also applies to models with multiple constraints per unit cell
(see Refs. [35, 36] for detailed discussions).

2\. Fragile Topological CSL: When the bottom flat band is entirely gapped from
the higher dispersive band, $\mathbf{T}(\mathbf{q})$ becomes a non-zero,
smoothly defined vector field in the target manifold $\mathbb{C}P^{N-1}$ (if
complex) or $\mathbb{R}P^{N-1}$ (if real) across the entire BZ. It can be
classified by how it winds around the BZ, which is a $d$-torus, ${T}^{d}$. The
winding is encoded by the relative homotopy group $[T^{d},\mathbb{C}P^{N-1}]$
(or $[T^{d},\mathbb{R}P^{N-1}]$) of the map

$\hat{\mathbf{T}}(\mathbf{q}):{T}^{d}\rightarrow\mathbb{C}P^{N-1}(\text{or
}\mathbb{R}P^{N-1})$ (12)

The homotopy class is invariant under smooth changes to the Hamiltonian as
long as it maintains the constrainer form and the gap between the bottom flat
and upper dispersive bands. If the map $\hat{\mathbf{T}}(\mathbf{q})$ belongs
to a nontrivial homotopy class, the corresponding gapped phase is (fragile)
topological. Otherwise, the CSL is topologically trivial. The fragility of the
classification stems from the fact that adding (say, $P$) spins to the unit
cell without closing the spectral gap changes the target manifold to
$\mathbb{C}P^{N+P-1}$ (or $\mathbb{R}P^{N+P-1}$), whose relative homotopy
group may be trivial. The homotopy class may also change by closing the
spectral gap, at which point a band touching characterizes an algebraic CSL.
Thus, the boundaries of fragile topological CSLs are algebraic CSLs. The
homotopy classification generalizes to systems with multiple degenerate flat
bands or multiple higher bands (also see Refs. [35, 36] for detailed
discussions).

##  III Crystalline symmetry analysis

Table 1: Common lattices and their band representations ($\mathcal{BR}$) [37, 43, 44, 45]; notation defined below Eq. (13).  Lattice | Space group $\cal G$ |  $\mathcal{BR}$ of scalar/spin | Representation at high symmetry points
---|---|---|---
Lieb | $P4/mmm$ | $({\mathrm{A}}_{\mathrm{g}})_{1{\mathrm{a}}}\uparrow{\cal G}\mathbin{\ooalign{$\vbox{\hbox{$+$}}$\cr$\boxempty$\cr}}({\mathrm{A}}_{\mathrm{g}})_{2{\mathrm{f}}}\uparrow{\cal G}$ | $(2\Gamma_{1}^{+}\oplus\Gamma_{2}^{+})+(2X_{1}^{+}\oplus X_{4}^{-})+(M_{1}^{+}\oplus M_{5}^{-})$
square | $P4/mmm$ | $({\mathrm{A}}_{\mathrm{g}})_{1{\mathrm{c}}}\uparrow{\cal G}$ | $(\Gamma_{1}^{+})+(X_{4}^{-})+(M_{4}^{-})$
kagome | $P6/mmm$ | $({\mathrm{A}}_{\mathrm{g}})_{3{\mathrm{f}}}\uparrow{\cal G}$ | ($\Gamma_{1}^{+}\oplus\Gamma_{5}^{+})+(K_{1}\oplus K_{5})+(M_{1}^{+}\oplus M_{3}^{-}\oplus M_{4}^{-})$
hexagonal | $P6/mmm$ | $({\mathrm{A}}_{1\mathrm{g}})_{1{\mathrm{a}}}\uparrow{\cal G}$ | $(\Gamma_{1}^{+})+(K_{1})+(M_{1}^{+})$
honeycomb | $P6/mmm$ | $({\mathrm{A}}_{1}^{\prime})_{2{\mathrm{c}}}\uparrow{\cal G}$ | $(\Gamma_{1}^{+}\oplus\Gamma_{4}^{-})+(K_{5})+(M_{1}^{+}\oplus M_{4}^{-})$
pyrochlore | $Fd\bar{3}m$ | $({\mathrm{A}}_{2\mathrm{g}})_{16{\mathrm{c}}}\uparrow{\cal G}$ | $(\Gamma_{2}^{+}\oplus\Gamma_{4}^{+})+(X_{2}\oplus X_{4})+(W_{1}\oplus W_{2})$

In previous studies [35, 36], several models utilizing the constrainer
formalism have been reviewed and proposed (see Tables II and III in Ref.
[36]). To classify a Hamiltonian written in the constrainer form, it is
typically necessary to Fourier transform it into momentum space, diagonalizing
the Hamiltonian.

An essential inquiry at this juncture concerns the ability to determine the
class of the CSL solely based on the crystalline symmetry information of the
model, without knowing the exact form of the constrainer. The question is
motivated by analogy to TQC [37], which uses symmetry to constrain the
topology and connectivity of band structures without knowledge of the exact
Hamiltonian. The answer to this question provides significant insights into
the physics of any lattice model, such as discerning whether an algebraic CSL
is protected by crystalline symmetry, or merely accidental.

In this section, we outline the symmetry analysis employed to identify band
touchings and determine the topology of the flat bands within the constrainer
Hamiltonian. As one may anticipate, this analysis has a close relation to the
band irreducible representations (irreps) of the crystalline symmetry group
used in TQC. However, this approach by itself is insufficient to produce the
CSL classification, as we will now explain.

One notable example that will be extensively examined later in Sec. IV is the
comparison between the kagome antiferromagnetic (AFM) model and the kagome
hexagon model. Although both models feature spins arranged on the kagome
lattice, the former exhibits symmetry-protected gapless points on the bottom
flat band, whereas the latter does not. While the band symmetry analysis does
predict gap-closures at specific high symmetry points within the Brillouin
Zone (BZ), it fails to discern whether these closures occur on the bottom flat
band(s) (which is crucial for the CSL classification) or among the higher
dispersive bands (which is irrelevant for the CSL classification).

From this shortcoming, we discover that, apart from the irrep analysis of the
microscopic spins and their lattice symmetries, the constrainers, whose
centers define an “auxiliary lattice” that is generically distinct from the
lattice of spins, play a critical role in determining the physics of the CSL.
This second aspect of physics goes beyond the irrep analysis of the local
spins and encapsulates crucial information regarding the properties of the
flat bands.

We now describe the symmetry classification scheme. Consider a CSL consisting
of spins on a lattice $S$, which is invariant under a space group
$\mathcal{G}$. Every local spin at a particular lattice site transforms as a
representation of the site-symmetry group at that site. The local
representation induces a band representation $\mathcal{BR}_{S}$, which
describes the symmetry of the entire spectrum, i.e., of both the dispersive
bands and the flat bands [37, 43, 44, 45, 46]. While $\mathcal{BR}_{S}$
contains information about band touching points in the spectrum, it does not
distinguish band touching points between the dispersive bands themselves from
band touchings between the dispersive and flat bands. Thus, to derive symmetry
constraints specific to the latter, we need more information.

The extra information lies in the constrainers. Specifically, the dispersive
bands live in the Hilbert subspace spanned by the constrainers. Thus, to
distinguish properties of the flat and dispersive bands, we must apply TQC to
the constrainers. Let $C$ denote the lattice comprised by the centers
$\mathbf{R}$ of each constrainer ${\cal C}_{I}(\mathbf{R})$ in Eq. (1). Note
that this lattice is virtual, in a sense that it is generically distinct from
the lattice $C$ inhabited by physical spins. Since the constrainers must also
satisfy the space group symmetry, each constrainer transforms as a
representation of the site-symmetry group of the corresponding site in the
lattice $C$, which induces a band representation ${\cal BR}_{C}$ describing
the symmetry of the dispersive bands.

Since the flat bands and dispersive bands together comprise the entire
spectrum, the band representation of the flat bands is determined by

$\mathcal{B}_{\text{FB}}=\mathcal{BR}_{S}\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}{\cal
BR}_{C}.$ (13)

where $\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}$ denotes the
“difference” of two band representations, i.e., for each point in the BZ, the
set of representations in ${\cal BR}_{S}$ not contained in ${\cal BR}_{C}$.
Examples of band representations are listed in Table 1, where $+$ indicates
appending irreps at different momenta, $\oplus$ indicates the addition of
irreps at the same momentum, and
$\mathbin{\ooalign{$\vbox{\hbox{$+$}}$\cr$\boxempty$\cr}}$ indicates the union
of two band representations, defined analogously to
$\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}$.

The symmetry data determines the symmetry enforced band touching points and
the topology of the gapped flat bands, as follows:

  * •

If ${\cal BR}_{C}\nsubseteq{\cal BR}_{S}$, then the spectrum has symmetry
protected band touching points on the bottom flat band and the model is an
algebraic CSL.

The degeneracy of symmetry protected band touching points between the flat
bands and the dispersive bands are determined by the subduced representations
of ${\cal BR}_{C}$ and ${\cal BR}_{S}$ at each momentum ${\mathbf{q}}$.
Specifically, the number of dispersive bands is dim$({\cal
BR}_{C}\downarrow{\mathbf{q}}\,\cap\,{\cal BR}_{S}\downarrow{\mathbf{q}})$ and
similarly, the number of zero energy states $n_{0}$ can be deduced from the
band representations (see Appendix A).

  * •

If ${\cal BR}_{C}\subseteq{\cal BR}_{S}$, then the spectrum has either no band
touching points on the bottom flat band, or the band touching points are not
protected by symmetry. The system belongs to FT-CSL class.

In this case, at each $\mathbf{q}$ the irrep of the constrainer band is
contained in ${\cal BR}_{S}$, and the flat bands are fully gapped from the
dispersive bands. The topology of the flat bands is determined by its symmetry
data vector $\mathcal{B}_{\text{FB}}$ and can be either atomic (trivial) or
non-trivial but fragile, classified using the symmetry indicators developed
for electron band structures [47, 48, 49, 50].

The classification of flat bands in our constrainer Hamiltonians is
mathematically the same as the classification of flat bands in the bipartite
single-particle band structures studied in Ref [51]. In Appendix B we prove
their equivalence by constructing a map between the two systems, which relies
on the introduction of the auxiliary lattice defined by the constrainers.

##  IV Application: two Kagome models

We now demonstrate the symmetry analysis with a concrete example of the kagome
AFM model versus the kagome hexagon model (cf. Fig. 2(a) for the kagome
lattice). The two models have the same band representation ${\cal BR}_{S}$
because they both feature spins on the kagome lattice. However, their CSL
properties are different. The former is an algebraic CSL whose spectrum has a
gapless point on the bottom flat bands, and the ground state fluctuations obey
the emergent Maxwell Gauss’s law $\bm{\nabla}\cdot\mathbf{E}=0$. The latter is
a topological CSL, with two degenerate bottom flat bands that are fully gapped
away from the higher dispersive band. Its higher dispersive band eigenvector
$\mathbf{T}(\mathbf{q})$ is a three-component real vector on $S^{2}$, which
has homotopy classes known as skyrmion number. The distinction between the two
models can only be discerned by considering the symmetry of the constrainers.

###  IV.1 Kagome AFM model

Figure 2:  (a) Kagome lattice for the Kagome AFM model (Eq. (14)). (b) The two
constrainers of the kagome model involve sites in the shaded regions. The
ground states are defined by the constraint that the sum of spins on each
triangle must vanish (Eq. (14)). (c) Spectrum $\omega({\bf q})$ that arises
from diagonalizing the Hamiltonian Eq. (14). There is one flat band at the
bottom of the spectrum and two higher dispersive bands with gap-closing points
on the flat band. (d) Spin structure factor showing pinch points at the
position of gap-closing points.

The Hamiltonian of the kagome AFM model is

$\begin{split}\mathcal{H}_{\mathsf{KAFM}}&=\sum_{\langle
i,j\rangle}S_{i}S_{j}+2\sum_{i}S_{i}^{2}\\\
&=\sum_{\bigtriangleup}\left(\sum_{i\in\bigtriangleup}S_{i}\right)^{2}+\sum_{\bigtriangledown}\left(\sum_{i\in\bigtriangledown}S_{i}\right)^{2}\\\
&\equiv\sum_{\mathbf{R}\text{ for
}\bigtriangleup}[\mathcal{C}_{\mathsf{KAFM1}}(\mathbf{R})]^{2}+\sum_{\mathbf{R}\text{
for
}\bigtriangledown}[\mathcal{C}_{\mathsf{KAFM2}}(\mathbf{R})]^{2},\end{split}$
(14)

The Hamiltonian consists of two constrainers
$\mathcal{C}_{\mathsf{KAFM1,2}}(\mathbf{R})$, defined by sums over the spins
on an up- or down-pointing triangle centered at $\mathbf{R}$, respectively
(see Fig. 2(b)); the sums in Eq. (14) are over the constrainer centers. Ground
states are hence defined by the two constraints:

$\mathcal{C}_{\mathsf{KAFM1,KAFM2}}(\mathbf{R})=0\quad\forall\ \mathbf{R}\ ,$
(15)

on every triangular plaquette.

Diagonlizing the Hamiltonian in momentum space, we obtain the spectrum shown
in Fig. 2(c). There is one bottom flat band with gapless points where the
dispersive bands touch. Using the techniques developed in Refs. [35, 36],
expanding the dispersive band eigenvector at the gapless point yields the
Gauss’s law, which turns out to be that of Maxwell’s theory:

$\begin{split}&3\partial_{x}(-S_{2}+S_{3})+\sqrt{3}\partial_{y}(2S_{1}-S_{2}-S_{3})\\\
&\equiv\partial_{x}E_{x}+\partial_{y}E_{y}=0.\end{split}$ (16)

This is manifested in the pinch points in the equal-time spin correlation
function shown in Fig. 2(d).

We now apply the symmetry analysis introduced in the previous section to prove
that the band touching between the flat and dispersive bands is symmetry
required. This model is in space group ${\cal G}=P6/mmm$. The spins are
located at the 3f Wyckoff position, which forms a kagome lattice. Each
classical spin transforms as the scalar irrep ${\mathrm{A}}_{\mathrm{g}}$ of
the site symmetry group. Thus,

${\cal BR}_{S}=({\mathrm{A}}_{\mathrm{g}})_{3{\mathrm{f}}}\uparrow{\cal G}.$
(17)

Its irreps at high symmetry points are listed in Table 1.

The constrainers are centered around the 2c Wyckoff position, forming a
honeycomb lattice where the honeycomb sites are located at the center of each
triangle in the kagome lattice. Each constrainer transforms as the scalar
irrep ${\mathrm{A}}_{1}^{\prime}$ of the site symmetry group. Thus,

${\cal
BR}_{C,\mathsf{KAFM}}=({\mathrm{A}}_{1}^{\prime})_{2{\mathrm{c}}}\uparrow{\cal
G}.$ (18)

Applying Eq. (13), we get the irreps of the flat band (see Table 1)

$\begin{split}\mathcal{B}_{\mathrm{FB},\mathsf{KAFM}}&={\cal
BR}_{S}\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}{\cal
BR}_{C,\mathsf{KAFM}}\\\
&=\left(\mathrm{A}_{\mathrm{g}}\right)_{3\mathrm{f}}\uparrow\mathcal{G}\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}\left(\mathrm{A}_{1}^{\prime}\right)_{2\mathrm{c}}\uparrow\mathcal{G}\\\
&=\left(\Gamma_{5}^{+}\ominus\Gamma_{4}^{-}\right)+\left(\mathrm{K}_{1}\right)+\left(\mathrm{M}_{3}^{-}\right).\end{split}$
(19)

At the $\Gamma$ point, there is an irrep difference
$\Gamma_{5}^{+}\ominus\Gamma_{4}^{-}$. The fact that an irrep difference
appears rather than a sum indicates a band touching point at $\Gamma$ enforced
by symmetry.

###  IV.2 Kagome hexagon model

Figure 3:  (a) Kagome lattice for the Kagome hexagon model (Eq. (20)). (b)
Constrainer of the Kagome-Hexagon model. Classical spins are arranged on a
kagome lattice, with ground states defined by the constraint that the sum of
spins on each hexagonal plaquette must vanish (Eqs. (21)). (c) Spectrum
$\omega({\bf q})$ that arises from diagonalizing the Hamiltonian (Eq. (20)) in
momentum space. There are two degenerate flat bands at the bottom of the
spectrum and a dispersive upper band with no band touchings between the upper
and lower bands. (d) Spin structure factor showing an absence of
singularities.

We now discuss the kagome-hexagon model [13] as an example of a fragile
topological CSL with short-ranged correlations. Its Hamiltonian is defined as

$\mathcal{H}_{\mathsf{KH}}=\sum_{\mathbf{R}\in\text{all
hexagons}}[\mathcal{C}_{\mathsf{KH}}(\mathbf{R})]^{2}\ ,$ (20)

where the sum of $\mathbf{R}$ runs over hexagon centers on the kagome lattice
(indicated in Fig. 3(a,b)), or equivalently the centers of all unit cells. The
constrainer $\mathcal{C}_{\mathsf{KH}}(\mathbf{R})$ is the sum of the six
spins around each hexagon as labeled in Fig. 3(a,b):

$\displaystyle\mathcal{C}_{\mathsf{KH}}(\mathbf{R})=\sum_{i\in\text{hex. at
}\mathbf{R}}S_{i}\ .$ (21)

The ground states are hence defined by the constraint
$\mathcal{C}_{\mathsf{KH}}(\mathbf{R})=0$ on every hexagonal plaquette.

Diagonalizing $\mathcal{H}_{\sf KH}$ in momentum space yields a spectrum with
three bands, of which the lowest two are flat and degenerate (Fig. 3(c)). This
is the case of one constrainer per unit cell discussed in Sec. II, and the
eigenvector of the top band can be found by Fourier-transforming the
constrainer (as done in Refs. [35, 36]),

${\mathbf{T}}({\mathbf{q}})=\begin{pmatrix}\cos(\sqrt{3}q_{x})\\\
\cos\left(-\frac{\sqrt{3}}{2}q_{x}+\frac{3}{2}q_{y}\right)\\\
\cos\left(-\frac{\sqrt{3}}{2}q_{x}-\frac{3}{2}q_{y}\right)\end{pmatrix},$ (22)

and its dispersion is
$\omega(\mathbf{q})=\left|{\mathbf{T}}({\mathbf{q}})\right|^{2}$. One can then
explicitly see that there are no band touchings between the upper band and the
two flat bands at any point in the BZ, and consequently no pinch points in the
correlation function (Fig. 3(d)). Accordingly, the real space correlations
remain short ranged with a correlation length on the order of the nearest-
neighbor distance at $T=0$. The ground state fluctuations are not described by
any effective Gauss’s law due to the absence of gapless points.

We now use symmetry to explain why the flat band is gapped. As in the previous
example, this model is in space group ${\cal G}=P6/mmm$, and with spins are
located at the 3f Wyckoff position, forming a kagome lattice. The band
representation is thus the same as in Eq. (17):

${\cal BR}_{S}=({\mathrm{A}}_{\mathrm{g}})_{3{\mathrm{f}}}\uparrow{\cal G}.$
(23)

The symmetry of the constrainers are different, however. They are located at
the 1a Wyckoff positions, which form a triangular lattice with each site at
the center of a hexagon in the kagome lattice. Each constrainer transforms as
an irrep ${\mathrm{A}}_{1{\mathrm{g}}}$ of the site symmetry group. Thus,
their lattice irreps are

${\cal
BR}_{C,\mathsf{KH}}=({\mathrm{A}}_{1{\mathrm{g}}})_{1{\mathrm{a}}}\uparrow{\cal
G},$ (24)

which is distinct from the previous example of the kagome AFM (contrast with
the Eq. (18)).

Applying Eq. (13), we obtain (see Table 1)

$\begin{split}\mathcal{B}_{\mathrm{FB},\mathsf{KH}}&={\cal
BR}_{S}\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}{\cal
BR}_{C,\mathsf{KH}}\\\
&=\left(\mathrm{A}_{\mathrm{g}}\right)_{3\mathrm{f}}\uparrow\mathcal{G}\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}\left(\mathrm{A}_{1_{\mathrm{g}}}\right)_{1\mathrm{a}}\uparrow\mathcal{G}\\\
&=\left(\Gamma_{5}^{+}\right)+\left(\mathrm{K}_{5}\right)+\left(\mathrm{M}_{3}^{-}\oplus\mathrm{M}_{4}^{-}\right).\end{split}$
(25)

In contrast to Eq. (19), no
$\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}$ signs appear in the
last line of Eq. (25). Thus, all irreps of ${\cal BR}_{C,\mathsf{KH}}$ are
included in ${\cal BR}_{S}$, which indicates that the flat band is fully
gapped.

However, the irreps that appear in $\mathcal{B}_{\mathrm{FB},\mathsf{KH}}$ do
not correspond to any sum of elementary band representations [47, 49, 50].
Thus, the flat bands have fragile topology, i.e., there is no way to
understand them as coming from localized degrees of freedom.

Comparing these two examples on the kagome lattice proves that the irreps of
the spins on the lattice, which determine ${\cal BR}_{S}$, do not provide
enough information to determine the class of the CSL. Specifically, ${\cal
BR}_{S}$ requires a gap-closing at the $\Gamma$ point because of the two-
dimensional irrep $\Gamma_{5}^{+}$, but it does not specify whether the band
touching is between the flat band and a dispersive band (kagome AFM), between
the two degenerate flat bands (kagome-hexagonal) or even between the two
dispersive bands. To distinguish between these possibilities from symmetry
requires the irreps of the constrainers, which are contained in ${\cal
BR}_{C}$.

We note that the two constrainer models of spins on the kagome lattice
discussed above can be mapped onto the two bipartite models of electrons
discussed in the main text and Fig. 1 in Ref. [51], with the distinction that
the physical spins are situated on only one sublattice – the other, virtual
sublattice corresponds to the centers of the constrainers. The band
representation analysis is however identical to that in Ref. [51], as follows
from the equivalence we prove in Appendix B.

##  V Construction of new classical spin liquid models

Figure 4:  (a) The cubic lattice. It has three sites per unit cell sitting on the face centers, forming three sublattices indicated in red, blue and green. (b) Constrainers of the cubic model. (c) Spectrum $\omega({\bf q})$ from diagonalizing the Hamiltonian (Eq. (30)) in momentum space at $q_{y}=0$. There is one flat band at the bottom of the spectrum and two higher dispersive bands, with band touchings between the flat and dispersive bands along the lines $q_{x,z}=0$. (d) Spin structure factor at $q_{y}=0$. (e,f) Spectrum and spin structure factor at $q_{y}=\pi/4$.  Table 2: The band representations ($\mathcal{BR}$) of the three-dimensional pinch line model [37, 43, 44, 45].  | Spin on face center($S$) | Constrainer($C$) | Flat bands
---|---|---|---
$\mathcal{BR}$ | ${\cal BR}_{S}=({\mathrm{B}}_{\mathrm{1}})_{3{\mathrm{d}}}\uparrow{\cal G}$ | ${\cal BR}_{C}=(^{1}{\mathrm{E}}_{\mathrm{g}})_{1{\mathrm{a}}}\uparrow{\cal G}\mathbin{\ooalign{$\vbox{\hbox{$+$}}$\cr$\boxempty$\cr}}(^{2}{\mathrm{E}})_{1{\mathrm{a}}}\uparrow{\cal G}$ | ${\cal BR}_{S}\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}{\cal BR}_{C}$
$\Gamma(0,0,0)$ | $\Gamma_{4}(3)$ | $\Gamma_{2}(1)\oplus\Gamma_{3}(1)$ | $\Gamma_{4}(3)\ominus\Gamma_{2}(1)\ominus\Gamma_{3}(1)$
$M(\pi,\pi,0)$ | $2M_{1}(1)\oplus M_{2}(1)$ | $2M_{1}(1)$ | $M_{2}(1)$
$R(\pi,\pi,\pi)$ | $R_{1}(1)\oplus R_{2}(1)\oplus R_{3}(1)$ | $R_{2}(1)\oplus R_{3}(1)$ | $R_{1}(1)$
$X(\pi,0,0)$ | $X_{1}(1)\oplus X_{2}X_{3}(2)$ | $2X_{1}(1)$ | $X_{2}X_{3}(2)\ominus X_{1}(1)$
$\Delta(u,0,0)$ | $\Delta_{1}(1)\oplus 2\Delta_{2}(1)$ | $2\Delta_{1}(1)$ | $2\Delta_{2}(1)\ominus\Delta_{1}(1)$
$Z(u,\pi,0)$ | $2Z_{1}(1)\oplus Z_{2}(1)$ | $2Z_{1}(1)$ | $Z_{2}(1)$
$T(u,\pi,\pi)$ | $3T_{1}(1)$ | $2T_{1}(1)$ | $T_{1}(1)$
$\Sigma(u,u,0)$ | $3\Sigma_{1}(1)$ | $2\Sigma_{1}(1)$ | $\Sigma_{1}(1)$
$\Lambda(u,u,u)$ | $\Lambda_{1}(1)\oplus\Lambda_{2}(1)\oplus\Lambda_{3}(1)$ | $\Lambda_{2}(1)\oplus\Lambda_{3}(1)$ | $\Lambda_{1}(1)$

Let us now showcase the usefulness of the crystalline symmetry analysis by
designing a new CSL model. It has symmetry protected nodal lines on the bottom
flat band, i.e. the gapless points form a line. In spin structure factor, the
nodal lines host pinch points around each point on the line, hence called
pinch lines in spin liquid literature [14]. Although also in the algebraic CSL
category, the pinch-line spin liquids exhibit very different physics than the
more common algebraic CSLs with a gapless point. It is thus interesting to
construct robust pinch-line models protected by symmetry for future study.
More specifically, we require the degeneracy nodal lines to be guaranteed to
exist when both the number of constrainers and their symmetry property is
fixed, but not the exact form of constrainers. Similar examples can also be
found in Refs. [14, 36].

We consider a model with a cubic unit cell and spins located at the face
centers (see Fig. 4(a)). In the large-$\mathcal{N}$ limit, these spins are
effectively treated as scalars or soft spins. The orientation of each spin is
pointing out of the face, in the positive
$\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}$ directions respectively.
For example, the spin on the $x-$normal face in points along $x-$direction.
The model is symmetric under the symmetries of the magnetic space group ${\cal
G}=P4^{\prime}32^{\prime}$ [52, 53] (No. 207.42 in the standard notation of
Ref. [54]), which is generated by $C_{4,[001]}{\cal T}$, $C_{3,[111]}$ and
$C_{2,[110]}$, where the first subscript denotes the rotation order and the
second three the rotation axis; $\mathcal{T}$ denotes time-reversal. Let us
denote the spin on $x/y/z$-normal face as $S_{x/y/z}$ (note that they are not
spin components but labels of spins on different sites). The symmetries act on
the spins by:

$\displaystyle C_{3,[111]}:~{}$
$\displaystyle(S_{x},S_{y},S_{z})\mapsto(S_{z},S_{x},S_{y})$ (26)
$\displaystyle C_{2,[110]}:$
$\displaystyle(S_{x},S_{y},S_{z})\mapsto(S_{y},S_{x},S_{z})$ (27)
$\displaystyle C_{4,[001]}{\cal T}:$
$\displaystyle(S_{x},S_{y},S_{z})\mapsto(S_{y}(-\mathbf{a}_{x}),-S_{x},S_{z}),$
(28)

where $S_{y}(-\mathbf{a}_{x})$ indicates that the transformed spin on the
$x$-normal face center of the current unit cell comes from the $S_{y}$ spin of
the $-\mathbf{a}_{x}$ unit cell.

We use the constrainers centered at the center of each cube (see Fig. 4(b)):

$\displaystyle{\cal C}_{1}(\mathbf{R})$
$\displaystyle=S_{x}(\mathbf{R})-S_{y}(\mathbf{R})-S_{x}(\mathbf{R}-\mathbf{a}_{x})+S_{y}(\mathbf{R}-\mathbf{a}_{y})$
$\displaystyle{\cal C}_{2}(\mathbf{R})$
$\displaystyle=S_{y}(\mathbf{R})-S_{z}(\mathbf{R})-S_{y}(\mathbf{R}-\mathbf{a}_{y})+S_{z}(\mathbf{R}-\mathbf{a}_{z})$
$\displaystyle{\cal C}_{3}(\mathbf{R})$
$\displaystyle=S_{z}(\mathbf{R})-S_{x}(\mathbf{R})-S_{z}(\mathbf{R}-\mathbf{a}_{z})+S_{x}(\mathbf{R}-\mathbf{a}_{x}),$
(29)

which obey the crystalline symmetries described in the preceding paragraph;
the minus signs come from the transformation under $C_{4}{\cal T}$. The
Hamiltonian is then

$\mathcal{H}_{\mathsf{PL}}=\sum_{\mathbf{R}}\left[{\cal
C}^{2}_{1}(\mathbf{R})+{\cal C}^{2}_{2}(\mathbf{R})+{\cal
C}^{2}_{3}(\mathbf{R})\right]$ (30)

The Fourier transformation of the constrainers are

$\displaystyle\mathbf{T}_{1}(\mathbf{q})$
$\displaystyle=2i\left(\sin\frac{q_{x}}{2},~{}-\sin\frac{q_{y}}{2},~{}0\right)^{T}$
$\displaystyle\mathbf{T}_{2}(\mathbf{q})$
$\displaystyle=2i\left(0,~{}\sin\frac{q_{y}}{2},~{}-\sin\frac{q_{z}}{2}\right)^{T}$
$\displaystyle\mathbf{T}_{3}(\mathbf{q})$
$\displaystyle=2i\left(-\sin\frac{q_{x}}{2},~{}0,~{}\sin\frac{q_{z}}{2}\right)^{T}$
(31)

The three vectors span the subspace of the higher dispersive bands. Note that
there are two dispersive bands and one flat band (see Figs. 4(c,d)) since the
identity
$\mathbf{T}_{1}(\mathbf{q})+\mathbf{T}_{2}(\mathbf{q})+\mathbf{T}_{3}(\mathbf{q})=0$
implies that the rank of the three vectors at a generic $\mathbf{q}$ is $2$.
The line nodes are along the high-symmetry line $\Delta=(u,0,0)$ and its
symmetry related partners in momentum space.

The Gauss’s law can be obtained by examining the eigenvector configuration
around a gapless point. At the $\Gamma$ point ($\mathbf{q}=\mathbf{0}$), the
charge-free Gauss’s law is

$\begin{split}\partial_{x}E_{x}-\partial_{y}E_{y}=0,\\\
\partial_{y}E_{y}-\partial_{z}E_{z}=0,\\\
\partial_{z}E_{z}-\partial_{x}E_{x}=0.\end{split}$ (32)

The two linearly independent constraints reflect the fact that both dispersive
bands touch the bottom flat band at $\Gamma$. Along the line $\Delta=(0,u,0)$,
the Gauss’s law is

$\partial_{x}E_{x}-\partial_{z}E_{z}=0,$ (33)

while the degree of freedom $E_{y}$ becomes “gapped” and is not involved in
the Gauss’s law. This is reflected in the spectrum in Fig. 4(c) where there is
one higher dispersive band closes that touches the bottom flat band, and
another stays gapped away from the origin.

We now apply the symmetry analysis to determine the symmetry protected
touchings between the dispersive bands and the flat bands. The spins are
located at the $3{\mathrm{d}}$ Wyckoff position of the aforementioned group
$P4^{\prime}32^{\prime}$, corresponding to the face centers of the cubic unit
cell. Each spin transforms as the irrep ${\mathrm{B}}_{1}$ of the site
symmetry group $4^{\prime}22^{\prime}$ [52, 53]. Thus,

${\cal BR}_{S}=\left({\mathrm{B}}_{1}\right)_{3{\mathrm{d}}}\uparrow{\cal G}.$
(34)

Its irreps at high symmetry points are listed in Table 2.

The constrainers are centered at the $1{\mathrm{a}}$ Wyckoff position which is
at the center of a cube. The constrainers transform as the representation
${}^{1}{\mathrm{E}}\oplus{}^{2}{\mathrm{E}}$ of the site symmetry group
$4^{\prime}32^{\prime}$. Thus,

${\cal BR}_{C}=\left({}^{1}{\mathrm{E}}\right)\uparrow{\cal
G}\mathbin{\ooalign{$\vbox{\hbox{$+$}}$\cr$\boxempty$\cr}}\left({}^{2}{\mathrm{E}}\right)\uparrow{\cal
G}.$ (35)

Applying Eq. (13) and using the relevant band representations listed in Table
2, yields the representations in the flat band

$\begin{split}\mathcal{B}_{\mathrm{FB}}&={\cal
BR}_{S}\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}{\cal
BR}_{C}\\\ &=\left({\mathrm{B}}_{1}\right)_{3{\mathrm{d}}}\uparrow{\cal
G}\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}\left({}^{1}{\mathrm{E}}\right)_{1{\mathrm{a}}}\uparrow{\cal
G}\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}\left({}^{2}{\mathrm{E}}\right)_{1{\mathrm{a}}}\uparrow{\cal
G}.\end{split}$ (36)

The representations at high symmetry points and along pertinent high symmetry
lines are listed in Table 2. Importantly, Table 2 shows that along the
$\Delta$ line, there are irreps in $\mathcal{BR}_{C}$ which do not appear in
${\cal BR}_{S}$, guaranteeing that the dispersive band touches the flat band
along the $\Delta$ line.

##  VI Summary

In summary, we have developed the mathematical formalism for analysing the
effects of crystalline symmetries on band touchings and topology of CSLs. In
short, comparing the band representations generated by the constrainers to
those generated by the spins determines whether the CSL is gapless (algebraic)
or gapped. In the former case, the band representations also determine the
degeneracy and location of gapless points in the BZ, while in the latter case,
the symmetry data encodes the topology of the gapped band.

Our symmetry analysis goes beyond applying TQC to the spins on the lattice: on
the contrary, as we have shown with explicit examples, understanding the
symmetry of the constrainer terms in the Hamiltonian is imperative to deduce
the band crossings. This is different than the TQC classification of electron
band structures, which holds independent of the form of the one-body
Hamiltonian. We note that while the mathematical formalism of the ‘difference’
of band representations in Eq. (13) is identical to that developed for
electron band structures on bipartite lattices in Ref. [51], there are
important distinctions. First, the physical spins only inhabit one sublattice
(S, which need not be bipartite), and there are no intersublattice ‘hopping’
terms, which are central to the construction in Ref. [51]. Second, the dual,
virtual lattice (C) is determined solely by the centers of the constrainer
clusters in Eq. (1) and depends crucially on the form of the spin
interactions. In fact, the constrainer spin Hamiltonian can be thought of, in
a sense, as a square of a one-particle hopping Hamiltonian (see Appendix B).

The formalism we have derived is a powerful tool for both understanding the
robustness of spectral features in known models as well as for reverse-
engineering CSLs with desired properties. We have demonstrated the latter by
introducing a new CSL with symmetry-protected nodal line degeneracies. The
symmetry formalism will be an essential part of the comprehensive
classification of the CLSs in the large-$\mathcal{N}$ limit going forward. Our
study paves way to a high-throughout examination of all possible lattice
models that have specific spectral features of interest.

## Acknowledgements

H.Y. thank Owen Benton for helpful discussions. H.Y. and A.H.N. were supported
by the U.S. National Science Foundation Division of Materials Research under
the Award DMR-1917511. H.Y. and J.C. gratefully acknowledge support from the
Simons Center for Geometry and Physics at Stony Brook University at which this
project was initiated. Y.F. and J.C. acknowledge support from the National
Science Foundation under Grant No. DMR-1942447. J.C. also acknowledges the
support of the Flatiron Institute, a division of the Simons Foundation, and
the Alfred P. Sloan Foundation through a Sloan Research Fellowship.

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##  Appendix A The dimension of symmetry protected band touching

From topological quantum chemistry, given the orbitals on the lattice, the
representation of bands in momentum space can be determined by the induced
representation. In our case, the spin degrees on the lattice site form a
representation $\rho_{S}$ of the site symmetry group, which is defined by the
symmetries that leave the site $S$ invariant. Then the band representation of
the full spectrum (flat bands and the dispersive bands) is given by ${\cal
BR}_{S}=\rho_{S}\uparrow{\cal G}$. The corresponding representation of the
little group at momentum $\mathbf{q}$ is ${\cal
BR}_{S}\downarrow{\mathbf{q}}$. The explicit construction of these induced
representations can be found in Refs. [37, 46, 55].

The representation of the constrainers can be determined in the same way. The
constrainer $\mathcal{C}(\mathbf{R})$ is a sum over a local cluster of spins
as we have defined in Eq. (1). Denote its Fourier transform by
${\mathbf{T}}(\mathbf{q})$. If $\mathcal{C}(\mathbf{R})$ transforms as the
representation $\rho_{C}$ of its site symmetry group, then
${\mathbf{T}}(\mathbf{q})$ transforms as a representation of the little group
at $\mathbf{q}$ given by $\rho_{C}^{\mathbf{q}}=\rho_{C}\uparrow{\cal
G}\downarrow{\mathbf{q}}$. The band representation of $\mathbf{T}(\mathbf{q})$
is ${\cal BR}_{C}=\rho_{C}\uparrow{\cal G}$. Notice this is identical to the
way to determine ${\cal BR}_{C}$ as constructed in the topological quantum
chemistry [37, 46, 55]. However, the new feature that appears in this work is
that the basis of the representation, ${\mathbf{T}}(\mathbf{q})$, can vanish
at a particular $\mathbf{q}$. When this happens, the transformation of
$\mathbf{T}(\mathbf{q})$ under the little group symmetries is ill-defined,
despite the fact that $\rho_{C}^{\mathbf{q}}$ is perfectly well-defined, since
it is obtained from a Fourier transform of $\rho_{C}\uparrow\mathcal{G}$.
Since $\mathbf{T}(\mathbf{q})$ vanishes precisely where the dispersive bands
and flat bands touch, the symmetry data at this point gives information about
the band touching, as we now describe.

Since the Hilbert space of the dispersive bands are spanned by
${\mathbf{T}}(\mathbf{q})$, the dispersive bands transform as the induced
representation ${\cal BR}_{C}\downarrow{\mathbf{q}}$. On the other hand, the
representation of the dispersive bands must be part of the complete
representation induced from all spins in the model, ${\cal
BR}_{S}\downarrow{\mathbf{q}}$. Let us now compare ${\cal
BR}_{S}\downarrow{\mathbf{q}}$ to ${\cal BR}_{C}\downarrow{\mathbf{q}}$. There
are two cases: (i) ${\cal BR}_{C}\downarrow{\mathbf{q}}\subseteq{\cal
BR}_{S}\downarrow{\mathbf{q}}$. In this case there is no symmetry protected
band touching between the dispersive bands and the flat bands since they
transform as two representations ${\cal BR}_{C}\downarrow{\mathbf{q}}$ and
${\cal BR}_{S}\downarrow{\mathbf{q}}\ominus{\cal
BR}_{C}\downarrow{\mathbf{q}}$. Note there may be accidental band touchings
which are not protected by symmetry and hence not detected by this analysis.
(ii) ${\cal BR}_{C}\downarrow{\mathbf{q}}\nsubseteq{\cal
BR}_{S}\downarrow{\mathbf{q}}$. Such cases exist when
${\mathbf{T}}(\mathbf{q})=0$, where the induced representation ${\cal
BR}_{C}=\rho_{C}\uparrow{\cal G}$ fails to describe the representation of the
dispersive bands. Therefore, when an irrep in ${\cal
BR}_{C}\downarrow{\mathbf{q}}$ does not appear in ${\cal
BR}_{S}\downarrow{\mathbf{q}}$, it indicates a band touching.

Specifically, the dimension of the dispersive bands above zero energy is

$n_{\text{dispersive}}=\text{dim}({\cal
BR}_{C}\downarrow{\mathbf{q}}\,\cap\,{\cal BR}_{S}\downarrow{\mathbf{q}})$
(37)

and the number of zero energy states $n_{0}$ is

$\displaystyle n_{0}$ $\displaystyle=\text{dim}\big{(}{\cal
BR}_{S}\downarrow{\mathbf{q}}\ominus({\cal
BR}_{C}\downarrow{\mathbf{q}}\,\cap\,{\cal
BR}_{S}\downarrow{\mathbf{q}})\big{)}$ (38)
$\displaystyle=\text{dim}\big{(}{\cal
BR}_{C}\downarrow{\mathbf{q}}\,\cup\,{\cal
BR}_{S}\downarrow{\mathbf{q}})\ominus{\cal
BR}_{C}\downarrow{\mathbf{q}}\big{)},$ (39)

where $\ominus$ denotes taking the difference of irreps at each momentum. Away
from the band touching point, $n_{0}$ gives the number of flat bands, while at
the band touching point, $n_{0}$ gives the dimension of the band touching.

##  Appendix B  Mapping between the bipartite Hamiltonian and the constrainer
Hamiltonian

Ref. [51] classifies flat electronic bands on bipartite lattices. In this
Appendix, we show there is a bijective mapping between the constrainer
Hamiltonians and the bipartite electronic Hamiltonians.

The constrainer Hamiltonian of CSLs in Eq. (1) can be rewritten in terms of
the $N\times 1$ constrainer vector ${\mathbf{T}}_{I}({\mathbf{q}})$, $I=1\dots
M$, which is the Fourier transform of the constrainer
$\mathcal{C}_{I}(\mathbf{R})$ defined as the sum over a local cluster of
spins, as follows

${\cal
H}=\sum_{\mathbf{q}\in\text{BZ}}\sum_{i,j=1}^{N}s^{\dagger}_{\mathbf{q},i}\big{[}H_{\text{constrainer}}(\mathbf{q})\big{]}_{ij}s_{\mathbf{q},j}$
(40)

where the constrainer Hamiltonian matrix $H_{\text{constrainer}}$ is

$H_{\text{constrainer}}(\mathbf{q})=\sum_{i=1}^{M}{\mathbf{T}}_{i}({\mathbf{q}}){\mathbf{T}}_{i}^{\dagger}({\mathbf{q}})=\left(S^{\dagger}S\right)_{N\times
N}$ (41)

by taking
$S^{\dagger}=[{\mathbf{T}}_{1}({\mathbf{q}}),\dots,{\mathbf{T}}_{M}({\mathbf{q}})]$,
and $s^{\dagger}_{\mathbf{q},i}$ $s_{\mathbf{q},i}$ are the
creation/annihilation operators of our spin degrees at site $i$ and momentum
$\mathbf{q}$.

Now we construct an abstract block-off-diagonal matrix based on the matrix $S$
by

$Q=\begin{pmatrix}&S^{\dagger}_{N\times M}\\\ S_{M\times N}&\end{pmatrix}.$
(42)

Our constrainer Hamiltonian matrix appears in the first diagonal block of
$Q^{2}$

$\displaystyle Q^{2}$
$\displaystyle=\begin{pmatrix}H_{\text{constrainer}}(\mathbf{q})&\\\
&\left(SS^{\dagger}\right)_{M\times M}\end{pmatrix}$ (43)

We now define a set of auxiliary orbitals on the lattice sites formed by the
constrainers, on which the creation/annihilation operators
$c^{\dagger}_{\mathbf{q},I}$ $c_{\mathbf{q},I}$ are defined, where
$I=1,\dots,M$. Then the matrix $Q^{2}$ represents a Hamiltonian over the
Hilbert space spanned by the basis
$(s^{\dagger}_{\mathbf{q},i},c^{\dagger}_{\mathbf{q},I})$, where $i=1,\dots,N$
and $I=1,\dots,M$.

To connect with the bipartite Hamiltonian [51], we interpret $Q$ as a
Hamiltonian on a bipartite lattice where $S$ and $S^{\dagger}$ represent
hopping between the two sublattices, which are denoted $L$ and $\bar{L}$ in
Ref. [51]. However, in our CSL Hamiltonian, the bipartite lattice is virtual,
i.e., it is a mathematical construction but does not correspond to a physical
hopping between spins and constrainers. This construction shows that every
constrainer Hamiltonian can be mapped to an electronic Hamiltonian on a
bipartite lattice, and vice versa.

We now explain how to use the matrices $Q$ and $Q^{2}$ to re-derive the
symmetry analysis we obtained in Appendix A. First, the matrix $Q$ defined in
Eq. (42) has an anti-commuting chiral symmetry $\cal C$, satisfying
$\\{Q,{\cal C}\\}=0$, given by

${\cal C}=\begin{pmatrix}1_{N\times N}&\\\ &-1_{M\times M}\end{pmatrix}.$ (44)

Thus, an eigenstate $\psi$ of the matrix $Q$ at energy $E$ has a chiral
symmetry related partner $\cal C\psi$, which is an eigenstate of $Q$ at energy
$-E$. Both $\psi$ and ${\cal C}\psi$ are eigenstates of $Q^{2}$ with
eigenvalue $E^{2}$. Since $[{\cal C},Q^{2}]=0$, one can construct linear
combinations of these eigenstates that only live on the Hilbert subspace
described by either the upper or lower block of $Q^{2}$, i.e.,
$\psi_{+}=\psi+{\cal C}\psi$ and $\psi_{-}=\psi-{\cal C}\psi$. These
wavefunctions are defined in the basis of
$(s^{\dagger}_{\mathbf{q},i},c^{\dagger}_{\mathbf{q},I})$; projecting
$\psi_{\pm}$ onto the upper/lower block yields an eigenstate of either
$S^{\dagger}S$ or $SS^{\dagger}$. Thus, we have shown that the eigenvalue
spectrum of $S^{\dagger}S$ and $SS^{\dagger}$ are identical, with one
exception, which is where $\psi_{\pm}$ is ill-defined. Assuming $N>M$ and
$\operatorname{rank}(S)=M$, this happens precisely when
$\psi=\mathcal{C}\psi$, which implies $E=0$ and $\psi_{-}=0$. In other words,
the constrainer Hamiltonian $S^{\dagger}S$ has zero energy eigenstates that
are not eigenstates of $SS^{\dagger}$.

Now consider the symmetry properties of the bands. In the constrainer models
we consider, every symmetry $g$ commutes with $\mathcal{C}$, since symmetries
do not mix spins with constrainers. Thus, if $\psi$ is simultaneously an
eigenstate of $Q$ with energy $E$ and an eigenstate of a symmetry $g$ with
eigenvalue $\xi$, then $\cal C\psi$, $\psi_{+}$ and $\psi_{-}$ are also
eigenstates of $g$ with eigenvalue $\xi$ because $[{\cal C},g]=0$. It follows
that the representation of the dispersive bands in $S^{\dagger}S$ and the
bands of $SS^{\dagger}$ are also identical. However, the flat bands in
$S^{\dagger}S$ do not have partners in $SS^{\dagger}$. Therefore the
representation of the flat bands are given by the difference between the band
representations of $S$ and $C$, i.e., ${\cal
BR}_{S}\mathbin{\ooalign{$\vbox{\hbox{$-$}}$\cr$\boxempty$\cr}}{\cal BR}_{C}$.
If the flat bands are gapped from the dispersive bands, then every irrep that
appears in ${\cal BR}_{C}$ must also appear in ${\cal BR}_{S}$ since the
dispersive bands of each have the same irreps. Thus, if an irrep appears in
${\cal BR}_{C}$ that is not in ${\cal BR}_{S}$, it must be that the flat bands
are not gapped from the dispersive bands. In this case, the representation of
the dispersive bands at the band touching point is not given by
${\cal{BR}}_{C}$ since the symmetry analysis only works for nonzero energy
states. The results here are identical to what we have shown in Appendix A.