# U(1) and $\mathbb{Z}_{2}$ spin liquids on the pyrochlore lattice

Chunxiao Liu  Department of Physics, University of California, Santa Barbara,
CA 93106-9530, USA  Gábor B. Halász  Materials Science and Technology
Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA  Quantum
Science Center, Oak Ridge, TN 37831, USA  Kavli Institute for Theoretical
Physics, University of California, Santa Barbara, CA 93106-4030, USA  Leon
Balents  Kavli Institute for Theoretical Physics, University of California,
Santa Barbara, CA 93106-4030, USA  Canadian Institute for Advanced Research,
661 University Ave., Toronto, ON M5G 1M1 Canada

###### Abstract

The geometrically frustrated 3D pyrochlore lattice has been long predicted to
host a quantum spin liquid, an intrinsic long-range entangled state with
fractionalized excitations. To date, most proposals for pyrochlore materials
have focused on quantum spin ice, a U(1) quantum spin liquid whose only low
energy excitations are emergent photons of Maxwell type. In this work, we
explore the possibility of finding pyrochlore quantum spin liquids whose low
energy theories go beyond this standard one. We give a complete classification
of symmetric $\mathrm{U(1)}$ and $\mathbb{Z}_{2}$ spin liquids on the
pyrochlore lattice within the projective symmetry group framework for
fermionic spinons. We find 18 $\mathrm{U(1)}$ spin liquids and 28
$\mathbb{Z}_{2}$ spin liquids that preserve pyrochlore space group symmetry
while, upon further imposing time reversal symmetry, the numbers of classes
become 16 and 48, respectively. For each class, the most general symmetry-
allowed spinon mean-field Hamiltonian is given. Interestingly, we find that
several U(1) spin liquid classes possess an unusual gapless multi-nodal-line
structure (“nodal star”) in the spinon bands, which is protected by the
projective actions of the three-fold rotation and screw symmetries of the
pyrochlore space group. Through a simple model, we study the effect of gauge
fluctuations on such a nodal star spin liquid and propose that the leading
terms in the low temperature specific heat have the scaling form
$C/T\sim\sqrt{T}+\sqrt{T}/\ln T$, in contrast to the form $C/T\sim T^{2}$ of
the standard U(1) pyrochlore spin liquid with gapped spinons.

###### pacs:

Valid PACS appear here

##  I Introduction

Quantum spin liquids (QSLs) are zero temperature phases of quantum magnets in
which localized spins evade magnetic long-range order due to strong quantum
fluctuations and form liquid-like states Savary and Balents (2016). Such
states are fundamentally characterized by intrinsic long-range entanglement
and support non-local excitations carrying fractionalized quantum numbers.
These non-local fractionalized excitations interact with each other via an
emergent gauge field. Therefore, QSLs are naturally described in terms of
gauge theories.

At a coarse level, different QSLs can be classified by their underlying low
energy effective theories. Depending on whether a mass exists, the
fractionalized spinon excitations (the matter fields) may or may not bear
relevance to the low energy description of QSLs. The gauge field may also be
gapless as is a U(1) gauge field of photons, or gapped as is a
$\mathbb{Z}_{2}$ gauge field which is of topological nature. All these types
have been extensively studied in two dimensions (2D). Perhaps the most well-
known $\mathbb{Z}_{2}$ QSL is the Kitaev model on the 2D honeycomb lattice:
this model supports either gapped or gapless Majorana fermions Kitaev (2006).
On the other hand, a compact U(1) gauge theory is confined in 2D Polyakov
(1977), and a QSL corresponding to the deconfined phase can appear only in the
presence of gapless matter fields. So far, the two most studied examples are
spinon Fermi surface and Dirac U(1) QSLs. On the experimental side, promising
Kitaev materials include a family of honeycomb iridates Singh and Gegenwart
(2010) and $\alpha$-RuCl3 Banerjee  _et al._ (2016), and new proposals are
still in progress Hermanns  _et al._ (2018). A spinon Fermi surface U(1) QSL
has been speculated to emerge in the 2D layered triangular materials YbMgGaO4
and NaYbSe2 Paddison  _et al._ (2017); Shen  _et al._ (2016); Dai  _et al._
(2020), while a U(1) Dirac QSL may be relevant for the 2D layered kagome
material Herbertsmithite Ran  _et al._ (2007); Yan  _et al._ (2011). The
recent material NaYbO2 may also realize a Dirac U(1) spin liquid Ding  _et
al._ (2019); Bordelon  _et al._ (2019). These spin liquid candidates provide a
natural ground for the experimental realization of exotic quantum phenomena
such as quantum electrodynamics in three dimensional spacetime (QED3) and 2D
topological order.

Moving to the three-dimensional (3D) world, arguably the most studied examples
are QSLs on the pyrochlore lattice. Consisting of corner-sharing tetrahedra,
the geometrically frustrated pyrochlore lattice has been proposed to host a
QSL phase since the birth of the concept of QSLs Anderson (1956). An important
theoretical advance occurred in 2004: through a rigorous mapping, Ref. Hermele
_et al._ (2004) showed that the pyrochlore Heisenberg model with local Ising
anisotropy has a U(1) QSL phase, commonly known as quantum spin ice, which is
described by the Maxwell theory at low energies. Since then, the properties of
such pyrochlore QSLs have been extensively studied Hermele  _et al._ (2004);
Savary and Balents (2012); Yao  _et al._ (2020); Yan  _et al._ (2020), and
numerous experiments have reported liquid-like behaviors in rare-earth
pyrochlore materials Sibille  _et al._ (2017, 2015); Gao  _et al._ (2019);
Gaudet  _et al._ (2019); Léger  _et al._ (2021).

Given the interest in quantum spin ice that realizes a prototypical low energy
theory, it is compelling to ask whether there are other spin liquids whose low
energy theory is of a new prototype. In a narrower sense, this amounts to
asking if gauge fields can interact with novel forms of gapless matter fields.
This has indeed been considered in other works. For example, Refs. Xu  _et
al._ (2012); Mishmash  _et al._ (2013) considered a class of QSLs with
symmetry protected quadratic spinon band touchings for the triangular spin
liquid candidates Ba3NiSb2O9, $\kappa$-BEDT, and DMIT. In three dimensions,
Refs. Huang  _et al._ (2017); Lawler  _et al._ (2008) studied possible
symmetric spin liquids on the hyperkagome lattice and found that certain
classes possess gapless nodal lines of spinons along high symmetry paths in
the Brillouin zone. A similar conclusion was drawn in Ref. Burnell  _et al._
(2009) for a QSL on the pyrochlore lattice. However, in these two 3D examples
it is not clear whether such gapless nodal structures are stable against
perturbations. Nodal lines of excitations also appear in other spin liquids,
which are either robust against gauge fluctuations Mandal and Surendran
(2009); Natori  _et al._ (2016) or symmetry protected Zhang and Senthil
(2017).

Another more systematic, yet formal, way of classifying QSLs is based on their
symmetry properties. Due to the absence of magnetic long-range order, a QSL
state usually preserves the full symmetry of the lattice and it may also
preserve time reversal symmetry. Crucially, due to the emergent gauge
structure, the fractionalized excitations of the system carry a _projective_
representation of the symmetry group – a representation of the group extension
of the original symmetry group by the gauge structure. The classification of
symmetric spin liquids can therefore be achieved by classifying all the
distinct projective representations for a given lattice symmetry and a given
gauge group type. In his seminal work Wen (2002), Wen coined this procedure
the classification of projective symmetry groups (PSGs). The PSG approach has
led to many fruitful results in our understanding of symmetric spin liquids.
For example, we now know that there are at most 20 different $\mathbb{Z}_{2}$
QSL classes on the kagome Lu  _et al._ (2011) and triangular Lu (2016)
lattices with distinct projective symmetries for fermionic spinons, while the
analogous numbers for the 3D hyperkagome Lawler  _et al._ (2008) and
hyperhoneycomb Huang  _et al._ (2018) lattices are 3 and 160, respectively.
The idea of the PSG has also been applied to the pyrochlore lattice, in the
hope of identifying experimental spin liquid candidates within the
classification Bergman  _et al._ (2006); Burnell  _et al._ (2009); Chen
(2016); Ern Chern and Kim (2018); Liu  _et al._ (2019).

In this work, we apply the PSG method for Abrikosov fermions to give a
complete classification of symmetric QSLs on the pyrochlore lattice with
either $\mathbb{Z}_{2}$ or U(1) gauge type. For each gauge type, we first
consider only space group symmetry, and later add time reversal symmetry. In
general, we allow spin-orbit coupling in the underlying spin system and do not
require SU(2) spin rotation symmetry. By following the general PSG principle
to solve the gauge-symmetry consistency equations, we find that there can be
at most 18 and 28 symmetric quantum spin liquids preserving the pyrochlore PSG
for the U(1) and $\mathbb{Z}_{2}$ gauge types, respectively. When time-
reversal symmetry is imposed, the number of possible symmetric spin liquids is
reduced to 16 for the U(1) type and is increased to 48 for the
$\mathbb{Z}_{2}$ type. For each class, the most general symmetry-allowed
spinon mean-field Hamiltonian is given. Importantly, we find that a large
family of spinon Hamiltonians possesses gapless nodal lines along the four
equivalent (111) directions of the Brillouin zone. We call this unusual nodal
structure a “nodal star” and show that it is stable at the mean-field level as
it is protected by the projective three-fold rotation and screw symmetries of
the system. We then go beyond the mean-field level and consider a full-fledged
low energy theory of the spinon nodal star coupled to a U(1) gauge field.
Specifically, we obtain thermodynamic properties of the system by computing
the photon contribution to the free energy. We find that the two most dominant
low temperature contributions to the specific heat are $C\sim T^{3/2}$ from
the bare spinons and $C\sim T^{3/2}/\ln T$ from the photon-spinon
interactions. This scaling of the low temperature specific heat may serve as a
clear evidence for the experimental discovery of a nodal star U(1) QSL.

The rest of this paper is organized as follows. In Sec. II, we discuss the
symmetry properties of the pyrochlore lattice, explain the main idea of the
PSG, and apply the PSG procedure to the classification of pyrochlore QSLs with
or without time reversal symmetry. In Sec. III, we construct mean-field
Hamiltonians for the fermionic spinons and analyze their symmetry properties.
We prove that several mean-field Hamiltonians obtained from the U(1) PSG
possess symmetry protected nodal lines. In Sec. IV, we construct a continuum
model for the spinon nodal lines coupled to a U(1) gauge field, and consider
the thermodynamic properties of the system. Finally, we summarize and discuss
our results in Sec. V.

##  II Projective symmetry group

Upon imposing symmetries, a spin liquid phase may split (or “fractionalize”)
into several distinct phases that all preserve the symmetry action. The phases
are distinguished by the distinct quantum numbers carried by the
fractionalized excitations under the symmetry action. These spin liquids
phases are called symmetric spin liquids. The classification of symmetric spin
liquids can be viewed as a generalized symmetry analysis which explores the
symmetry action on fractionalized excitations (in our case, the fermionic
spinons). The purpose of this section is to describe the procedure for this
classification.

###  II.1 Lattice and time reversal symmetries

In this subsection, we establish the convention and notation for this work and
give a brief introduction to the symmetry properties of the pyrochlore
lattice. A more detailed analysis of the pyrochlore space group can be found
in Ref. Liu  _et al._ (2019).

The pyrochlore lattice consists of four FCC-type sublattices which we label by
$\mu=0,1,2,3$. The lattice vectors $\bm{e}_{1}$, $\bm{e}_{2}$, and
$\bm{e}_{3}$ are defined as

$\displaystyle\bm{e}_{1}$
$\displaystyle=\frac{a}{2}(\hat{\bm{y}}+\hat{\bm{z}}),$ (1a)
$\displaystyle\bm{e}_{2}$
$\displaystyle=\frac{a}{2}(\hat{\bm{z}}+\hat{\bm{x}}),$ (1b)
$\displaystyle\bm{e}_{3}$
$\displaystyle=\frac{a}{2}(\hat{\bm{x}}+\hat{\bm{y}}),$ (1c)

where $a$ is the cubic lattice constant, the Cartesian coordinate has its
basis $\hat{\bm{x}}$, $\hat{\bm{y}}$, and $\hat{\bm{z}}$ aligned with the
cubic system of the pyrochlore lattice, and its origin sits on a $\mu=0$ site.
We define the following sublattice-dependent coordinate:

$\displaystyle(r_{1},r_{2},r_{3})_{\mu}$
$\displaystyle\equiv\bm{r}_{\mu}\equiv
r_{1}\bm{e}_{1}+r_{2}\bm{e}_{2}+r_{3}\bm{e}_{3}+\frac{1}{2}\bm{e}_{\mu},$ (2)

where it is implicitly understood that $\bm{e}_{0}=0$.

The space group of the pyrochlore lattice group is $Fd\overline{3}m$. It is
generated by the following five symmetry operations:

$\displaystyle T_{1}\colon$
$\displaystyle\bm{r}_{\mu}\rightarrow(r_{1}+1,r_{2},r_{3})_{\mu},$ (3a)
$\displaystyle T_{2}\colon$
$\displaystyle\bm{r}_{\mu}\rightarrow(r_{1},r_{2}+1,r_{3})_{\mu},$ (3b)
$\displaystyle T_{3}\colon$
$\displaystyle\bm{r}_{\mu}\rightarrow(r_{1},r_{2},r_{3}+1)_{\mu},$ (3c)
$\displaystyle\overline{C}_{6}\colon$
$\displaystyle\bm{r}_{\mu}\rightarrow(-r_{3}-\delta_{\mu,3},-r_{1}-\delta_{\mu,1},-r_{2}-\delta_{\mu,2})_{\overline{C}_{6}(\mu)},$
(3d) $\displaystyle S\colon$ $\displaystyle\bm{r}_{\mu}\rightarrow$
$\displaystyle(-r_{1}-\delta_{\mu,1},-r_{2}-\delta_{\mu,2},r_{1}+r_{2}+r_{3}+1-\delta_{\mu,0})_{S(\mu)},$

where $T_{1}$, $T_{2}$, and $T_{3}$ are translations along the lattice vectors
$\bm{e}_{1}$, $\bm{e}_{2}$, and $\bm{e}_{3}$, respectively, $\overline{C}_{6}$
is a sixfold rotoinversion around the $[111]$ axis, and $S$ is a nonsymmorphic
screw operation which is the composition of a twofold rotation around
$\bm{e}_{3}$ and a translation by $\bm{e}_{3}/2$. In the above equations, we
defined the symmetry action on the sublattice indices:
$\overline{C}_{6}(\mu)=0,2,3,1$ and $S(\mu)=3,1,2,0$ for $\mu=0,1,2,3$. By
definition, the rotoinversion can be written as the composition of an
inversion (with respect to the origin) and a threefold rotation around the
$[111]$ axis: $\overline{C}_{6}=I\circ{C}_{3}$, with $I=\overline{C}_{6}^{3}$
and ${C}_{3}=\overline{C}_{6}^{4}$. The point group of the pyrochlore lattice
is the cubic group $O_{\text{h}}$. It is generated by $\overline{C}_{6}$ and
$S^{\prime}$, where $S^{\prime}$ is a twofold rotation around $\bm{e}_{3}$.

In addition to the pyrochlore space group symmetries, time-reversal operation
$\mathcal{T}$ is an internal symmetry that commutes with all space-group
operations and satisfies $\mathcal{T}^{2}=-1$ when acting on a half-integer
spin state. The pyrochlore symmetry group is then completely characterized by
the following group relations:

$\displaystyle T_{i}T_{i+1}T_{i}^{-1}T_{i+1}^{-1}$ $\displaystyle=1,\quad
i=1,2,3,$ (4a) $\displaystyle\overline{C}^{6}_{6}$ $\displaystyle=1,$ (4b)
$\displaystyle S^{2}T^{-1}_{3}$ $\displaystyle=1,$ (4c)
$\displaystyle\overline{C}_{6}T_{i}\overline{C}^{-1}_{6}T_{i+1}$
$\displaystyle=1,\quad i=1,2,3,$ (4d) $\displaystyle
ST_{i}S^{-1}T^{-1}_{3}T_{i}$ $\displaystyle=1,\quad i=1,2,$ (4e)
$\displaystyle ST_{3}S^{-1}T_{3}^{-1}$ $\displaystyle=1,$ (4f)
$\displaystyle(\overline{C}_{6}S)^{4}$ $\displaystyle=1,$ (4g)
$\displaystyle(\overline{C}_{6}^{3}S)^{2}$ $\displaystyle=1,$ (4h)
$\displaystyle\mathcal{T}^{2}$ $\displaystyle=-1,$ (4i)
$\displaystyle\mathcal{T}\mathcal{O}\mathcal{T}^{-1}\mathcal{O}^{-1}$
$\displaystyle=1,\quad\mathcal{O}\in\\{T_{1},T_{2},T_{3},\overline{C}_{6},S\\},$
(4j)

where it is implicitly understood that $i+3\equiv i$.

###  II.2 Projective symmetry group

In this subsection, we describe the basic idea of the PSG and classify all
symmetric $\mathbb{Z}_{2}$ and U(1) spin liquids on the pyrochlore lattice. To
start with, one expresses spins in terms of the Abrikosov partons

$\hat{\bm{S}}_{\bm{r}_{\mu}}=\frac{1}{2}f^{\dagger}_{\bm{r}_{\mu}}\bm{\sigma}f^{\vphantom{\dagger}}_{\bm{r}_{\mu}},$
(5)

where $\bm{\sigma}=(\sigma^{1},\sigma^{2},\sigma^{3})$ are the Pauli matrices.
Formally, partons are mathematical representations of the fractionalized
excitations in a spin liquid phase, and the Abrikosov partons
$f_{\bm{r}_{\mu}}=\left(\begin{array}[]{c}f_{\bm{r}_{\mu},\uparrow}\\\
f_{\bm{r}_{\mu},\downarrow}\end{array}\right)$ are introduced here to describe
the fermionic spinon excitations that are of our interest. The transformation
in Eq. (5), however, is not a faithful one. The partons live in an enlarged
Hilbert space at each site $\bm{r}_{\mu}$, while the original Hilbert space is
recovered under the constraint

$\sum\limits_{\sigma=\uparrow,\downarrow}f^{\dagger}_{\bm{r}_{\mu},\sigma}f^{\vphantom{\dagger}}_{\bm{r}_{\mu},\sigma}=1,\quad\forall\,\bm{r}_{\mu}.$
(6)

As a consequence, the parton description contains redundant information: a
local U(1) gauge transformation $f_{\bm{r}_{\mu}}\rightarrow
e^{i\theta(\bm{r}_{\mu})}f_{\bm{r}_{\mu}}$ leaves
$\hat{\bm{S}}_{\bm{r}_{\mu}}$ invariant. In fact, such a gauge redundancy can
be enlarged to SU(2), which can be seen from the identity

$\hat{\bm{S}}_{\bm{r}_{\mu}}=\frac{1}{4}\mathrm{Tr}(\Psi^{\dagger}_{\bm{r}_{\mu}}\bm{\sigma}\Psi_{\bm{r}_{\mu}}),$
(7)

with
$\Psi_{\bm{r}_{\mu}}=\left(\begin{array}[]{cc}f_{\bm{r}_{\mu},\uparrow}&f^{\dagger}_{\bm{r}_{\mu},\downarrow}\\\
f_{\bm{r}_{\mu},\downarrow}&-f^{\dagger}_{r_{\mu},\uparrow}\end{array}\right)$,
and from the fact that any site-dependent SU(2) gauge transformation,

$G\colon\Psi_{\bm{r}_{\mu}}\rightarrow\Psi_{\bm{r}_{\mu}}W(\bm{r}_{\mu}),\quad
W(\bm{r}_{\mu})\in\text{SU(2)},$ (8)

leaves the spins $\hat{\bm{S}}_{\bm{r}_{\mu}}$ invariant. The enlargement of
the parton Hilbert space and the gauge redundancy must be properly treated to
validate the parton description.

The PSG method is a way to resolve this redundancy in the parton description
of a spin liquid with full lattice symmetries (a so-called symmetric spin
liquid). The crucial step is to realize that physical symmetries act
_projectively_ on the parton operators, and that seemingly different parton
Hamiltonians describe the same physics if they are related by gauge
transformations. Conversely, if two parton Hamiltonians cannot be related by
gauge transformations, they must carry different projective representations of
the physical symmetry – this suggests that the classification of projective
symmetry will lead to a full classification of symmetric spin liquids. We now
formulate this statement in a more concrete way. Consider a spin-orbit coupled
spin system on a pyrochlore lattice. Under a space group operation
$\mathcal{O}$ the spins transform as
$\mathcal{O}\colon\hat{\bm{S}}_{\bm{r}_{\mu}}\rightarrow
U_{\mathcal{O}}\hat{\bm{S}}_{\mathcal{O}(\bm{r}_{\mu})}U^{\dagger}_{\mathcal{O}}$,
where $U_{\mathcal{O}}$ is the SU(2) rotation matrix associated with the
operation $\mathcal{O}$. (When $\mathcal{O}$ is a pure translation, the SU(2)
matrix $U_{\mathcal{O}}$ is simply the identity matrix.) According to Eq. (7),
we naïvely expect that the partons transform as

$\mathcal{O}\colon\Psi_{\bm{r}_{\mu}}\rightarrow
U^{\dagger}_{\mathcal{O}}\Psi_{\mathcal{O}(\bm{r}_{\mu})}$ (9)

with nontrivial rotations

$U_{\overline{C}_{6}}=U_{C_{3}}=e^{-\frac{i}{2}\frac{2\pi}{3}\frac{(1,1,1)}{\sqrt{3}}\cdot\bm{\sigma}},\quad
U_{S}=e^{-\frac{i}{2}\pi\frac{(1,1,0)}{\sqrt{2}}\cdot\bm{\sigma}}.$ (10)

However, due to the SU(2) gauge redundancy, any operation $\mathcal{O}$ can be
accompanied by a site-dependent SU(2) gauge transformation of the form in Eq.
(8). The partons thus transform projectively as

$\widetilde{\mathcal{O}}=G_{\mathcal{O}}\circ\mathcal{O}\colon\Psi_{\bm{r}_{\mu}}\rightarrow
U^{\dagger}_{\mathcal{O}}\Psi_{O(\bm{r}_{\mu})}W_{\mathcal{O}}[\mathcal{O}(\bm{r}_{\mu})],$
(11)

where the symbol “$\circ$” indicates that the projective operation
$\widetilde{\mathcal{O}}$ is the composition of the physical symmetry
operation $\mathcal{O}$ and the gauge transformation $G_{\mathcal{O}}$.

The projective symmetry can be extended to include internal symmetries, and
here we consider time reversal operation $\mathcal{T}$ as an example. The
spins transform under $\mathcal{T}$ as
$\hat{\bm{S}}_{\bm{r}_{\mu}}\xrightarrow{\mathcal{T}}\mathcal{K}^{\dagger}U_{\mathcal{T}}\hat{\bm{S}}_{\bm{r}_{\mu}}U^{\dagger}_{\mathcal{T}}\mathcal{K}$,
where $U_{\mathcal{T}}=i\sigma^{2}$, and
$\mathcal{K}=\mathcal{K}^{\dagger}=\mathcal{K}^{-1}$ applies complex
conjugation to everything on its right. Using the special property of the
SU(2) algebra, one can design the projective action of ${\mathcal{T}}$ on
$\Psi$ to be unitary (see App. B for detailed derivation):

$\widetilde{\mathcal{T}}=G_{\mathcal{T}}\circ\mathcal{T}\colon\Psi_{\bm{r}_{\mu}}\rightarrow
U_{\mathcal{T}}\Psi_{\bm{r}_{\mu}}{W}_{\mathcal{T}}(\bm{r}_{\mu}),$ (12)

Note that this does not modify the anti-unitary nature of time reversal
symmetry.

For a symmetric spin liquid, the projective operations
$\widetilde{\mathcal{O}}$ and $\widetilde{\mathcal{T}}$ generate the symmetry
group of the parton Hamiltonian, commonly known as the projective symmetry
group (PSG). The classification of symmetric spin liquids amounts to the
classification of PSGs. To achieve this, one needs to find all the gauge-
inequivalent solutions for the gauge transformations $G_{\mathcal{O}}$ and
$G_{\mathcal{T}}$ that are consistent with the symmetry group of the system.
Any group relation of Eq. (4) can be written in the general form of

$\mathcal{O}_{1}\circ\mathcal{O}_{2}\circ\dots=1,$ (13)

which translates into the gauge-enriched group relation

$\widetilde{\mathcal{O}}_{1}\circ\widetilde{\mathcal{O}}_{2}\circ\dots=(G_{\mathcal{O}_{1}}\circ\mathcal{O}_{1})\circ(G_{\mathcal{O}_{2}}\circ\mathcal{O}_{2})\circ\dots=\mathcal{G},$
(14)

where $\mathcal{G}$ is a pure gauge transformation and corresponds to the
identity operation for the spins. We say that $\mathcal{G}$ is an element of
the invariant gauge group (IGG), the group of all pure gauge transformations
that leave the parton Hamiltonian invariant. The IGG transformation on each
site is a subgroup of SU(2), typically $\mathbb{Z}_{2}$ or U(1). In most
cases, there exists a gauge choice (the canonical gauge Wen (2002)) in which
the IGG transformation can be made “global” of the form
$\mathcal{G}=e^{i\sigma^{3}\chi}$ with a constant $\chi$. In this paper, we
will be classifying both $\mathbb{Z}_{2}$ and U(1) spin liquids, therefore we
consider both IGG $=\mathbb{Z}_{2}$ and U(1), for which $\chi=\\{0,\pi\\}$ and
$\chi\in[0,2\pi)$, respectively.

Making use of the general conjugation rule

$\mathcal{O}_{i}\circ
G_{\mathcal{O}_{j}}\circ\mathcal{O}^{-1}_{i}\colon\Psi_{\bm{r}_{\mu}}\rightarrow\Psi_{\bm{r}_{\mu}}W_{\mathcal{O}_{j}}[\mathcal{O}^{-1}_{i}(\bm{r}_{\mu})],$
(15)

which follows directly from Eqs. (9) and (11), Eq. (14) can be rewritten as

$\displaystyle G_{\mathcal{O}_{1}}$ $\displaystyle\circ(\mathcal{O}_{1}\circ
G_{\mathcal{O}_{2}}\circ\mathcal{O}^{-1}_{1})$ (16)
$\displaystyle\circ(\mathcal{O}_{1}\circ\mathcal{O}_{2}\circ
G_{\mathcal{O}_{3}}\circ\mathcal{O}^{-1}_{2}\circ\mathcal{O}^{-1}_{1})\circ\dots=\mathcal{G},$

which then becomes an SU(2) equation:

$\displaystyle
W_{\mathcal{O}_{1}}(\bm{r}_{\mu})W_{\mathcal{O}_{2}}[\mathcal{O}_{1}^{-1}(\bm{r}_{\mu})]W_{\mathcal{O}_{3}}\\{\mathcal{O}_{2}^{-1}[\mathcal{O}_{1}^{-1}(\bm{r}_{\mu})]\\}\dots=\mathcal{G}.$

The PSG classification is obtained by listing all group relations and finding
all solutions to the corresponding SU(2) equation (II.2). We emphasize that
solutions must be discriminated by the principle of gauge equivalence, rather
than resemblance. Indeed, by means of a general gauge transformation $G$ as in
Eq. (8), the gauge-enriched group relations in Eq. (14) can be rewritten as

$(G\circ G_{\mathcal{O}_{1}}\circ\mathcal{O}_{1}\circ G^{-1})\circ(G\circ
G_{\mathcal{O}_{2}}\circ\mathcal{O}_{2}\circ G^{-1})\circ\dots=\mathcal{G},$
(18)

which transforms $W_{\mathcal{O}_{i}}(\bm{r}_{\mu})$ according to

$W_{\mathcal{O}_{i}}(\bm{r}_{\mu})\rightarrow
W(\bm{r}_{\mu})W_{\mathcal{O}_{i}}(\bm{r}_{\mu})W^{-1}[\mathcal{O}_{i}^{-1}(\bm{r}_{\mu})].$
(19)

This indicates that two seemingly distinct solutions to the PSG equations can
in fact be equivalent.

###  II.3 Classification result

We first solve the PSG equations obtained from space group symmetries. The PSG
equations for U(1) and $\mathbb{Z}_{2}$ gauge groups are solved in App. A and
App. C, respectively. The results are presented in Table 1 for the U(1) gauge
group and in Table 2 for the $\mathbb{Z}_{2}$ gauge group. We find 18 gauge-
inequivalent PSG solutions for the U(1) gauge type and 28 gauge-inequivalent
PSG solutions for the $\mathbb{Z}_{2}$ gauge type. As a result, there can be
at most 18 U(1) and 28 $\mathbb{Z}_{2}$ symmetric spin liquids, ignoring
possible time reversal symmetry. Both the U(1) and the $\mathbb{Z}_{2}$
solutions have the following form:

$\displaystyle W_{T_{i}}(\bm{r}_{\mu})$
$\displaystyle=e^{i\sigma^{3}\phi_{T_{i}}(\bm{r}_{\mu})},\quad i=1,2,3,$ (20a)
$\displaystyle W_{\overline{C}_{6}}(\bm{r}_{\mu})$
$\displaystyle=W_{\overline{C}_{6},\mu}e^{i\sigma^{3}\phi_{\overline{C}_{6}}(\bm{r}_{\mu})},$
(20b) $\displaystyle W_{S}(\bm{r}_{\mu})$
$\displaystyle=W_{S,\mu}e^{i\sigma^{3}\phi_{S}(\bm{r}_{\mu})},$ (20c)

with

$\displaystyle\phi_{T_{1}}(\bm{r}_{\mu})$ $\displaystyle=0,$ (21a)
$\displaystyle\phi_{T_{2}}(\bm{r}_{\mu})$ $\displaystyle=-\chi_{1}r_{1},$
(21b) $\displaystyle\phi_{T_{3}}(\bm{r}_{\mu})$
$\displaystyle=\chi_{1}(r_{1}-r_{2}),$ (21c)
$\displaystyle\phi_{\overline{C}_{6}}(\bm{r}_{\mu})$
$\displaystyle=-\chi_{1}r_{1}(r_{2}-r_{3})$
$\displaystyle\quad-[{2\chi_{ST_{1}}+2\chi_{1}}+(\delta_{\mu,2}-\delta_{\mu,3})\chi_{1}]r_{1}$
$\displaystyle\quad+\delta_{\mu,2}\chi_{1}r_{3},$ (21d)
$\displaystyle\phi_{S}(\bm{r}_{\mu})$
$\displaystyle=\chi_{1}\left[\frac{(r_{1}+1)r_{1}}{2}-\frac{(r_{2}+1)r_{2}}{2}-r_{1}r_{2}\right]$
$\displaystyle\quad+[(\delta_{\mu,1}-\delta_{\mu,2})\chi_{1}+(2\chi_{1}-\chi_{ST_{1}})]r_{1}$
$\displaystyle\quad+[(2\delta_{\mu,1}-\delta_{\mu,2})\chi_{1}+3\chi_{ST_{1}}]r_{2}$
$\displaystyle\quad+[(\delta_{\mu,1}-\delta_{\mu,2})+2]\chi_{1}r_{3}.$ (21e)

The parameters $\chi_{1}$ and $\chi_{ST_{1}}$ are elements of the IGG defined
on the right-hand sides of the PSG equations obtained from Eq. (4).
Concretely:

  * •

The parameter $\chi_{1}$ is associated with
$T_{i}T_{i+1}T_{i}^{-1}T_{i+1}^{-1}=1$, and physically quantifies the
Aharonov-Bohm (AB) phase a spinon accumulates under such a sequence of
translations (see Fig. 1 for examples of the path). The AB phase is a gauge
invariant quantity. In U(1) PSG, such a phase is allowed to take on values
$0$, $\pi$ and $\pi/2$. They give rise to zero-flux, $\pi$-flux and
$\frac{\pi}{2}$-flux spin liquids, respectively. The $\frac{\pi}{2}$-flux spin
liquid explicitly breaks time reversal symmetry since a $\frac{\pi}{2}$ flux
changes sign under time reversal operation. In $\mathbb{Z}_{2}$ PSG, only 0-
and $\pi$-flux spin liquids are found.

  * •

The parameter $\chi_{ST_{1}}$ is associated with
$ST_{1}S^{-1}T_{3}^{-1}T_{1}=1$, and physically quantifies the AB phase a
spinon accumulates under the sequence of operations
$ST_{1}S^{-1}T_{3}^{-1}T_{1}$ (see Fig. 1 for examples of the path). Such a
phase is allowed to take on values $0$ and $\pi$ in both U(1) and
$\mathbb{Z}_{2}$ PSGs.

$W_{\overline{C}_{6},\mu}$ and $W_{S,\mu}$ in Eqs. (20b) and (20c) are the
SU(2) matrices at the origin, $\bm{r}_{\mu}=0$, which depend on additional
discrete parameters as given in Tables 1 and 2 for the two gauge types. The
parameters $\chi_{\overline{C}_{6}S}$, $\chi_{\overline{C}_{6}}$ and
$\chi_{S\overline{C}_{6}}$ are elements of the IGG associated with
$(\overline{C}_{6}S)^{4}=1$, $I^{2}=1$ and $(IS)^{3}=1$, respectively, and
have the same AB phase interpretation as explained above. We note that two
additional parameters $w_{\overline{C}_{6}}$ and $w_{S}$ appear in the U(1)
PSG classification: they are $\mathbb{Z}_{2}$-valued ($w=0,1$) and determine
whether or not the SU(2) matrices $W_{\overline{C}_{6},\mu}$ and $W_{S,\mu}$
belong to the IGG. It is necessary to introduce these two parameters in the
U(1) case a priori in order to simplify the SU(2) PSG equations to U(1) phase
equations for $\phi_{\mathcal{O}}(\bm{r}_{\mu})$. This is not required in the
$\mathbb{Z}_{2}$ case, since the phases $\phi_{\mathcal{O}}(\bm{r}_{\mu})$ are
$\mathbb{Z}_{2}$-valued and commute with $W_{\overline{C}_{6},\mu}$ and
$W_{S,\mu}$. The SU(2) equations for $W_{\overline{C}_{6},\mu}$ and
$W_{S,\mu}$, however, do rely on an additional discrete parameter $j$ in some
classes (see Table 2).

Figure 1: Examples of loops for (a) $T_{i}T_{i+1}T_{i}^{-1}T_{i+1}^{-1}=1$ and (b) $ST_{1}S^{-1}T_{3}^{-1}T_{1}=1$. Table 1: The 18 U(1) PSG classes of pyrochlore space group symmetry only and 16 U(1) PSG classes of pyrochlore space group symmetry combined with time reversal symmetry. The tabulation is such that the latter are embedded in the former; and that daggered square brackets “$\left[\phantom{ll}\right]^{\dagger}$” are used to enclose data that are specific to the former. The parameters $\chi_{1}$, $\chi_{ST_{1}}$, and $\chi_{\overline{C}_{6}S}$ are discrete elements of the IGG and $w_{\overline{C}_{6}}$ and $w_{S}$ are $\mathbb{Z}_{2}$-valued parameters introduced in the canonical gauge. Together they label the U(1) PSG classes. The gauge fixing of the SU(2) matrices $W_{\overline{C}_{6},\mu}$ and $W_{S,\mu}$ defined in Eq. (20) is given. $\chi_{1}$ | $\chi_{ST_{1}}$ | $\chi_{\overline{C}_{6}S}$ | $w_{\overline{C}_{6}}$ | $w_{S}$ | $(W_{\overline{C}_{6},0},W_{\overline{C}_{6},1},W_{\overline{C}_{6},2},W_{\overline{C}_{6},3})$ | $(W_{S,0},W_{S,1},W_{S,2},W_{S,3})$ | Number
---|---|---|---|---|---|---|---
of classes
$\left[\\#\right]^{\dagger}$ | $\\#$
$0$ or $\pi$ | $0$ |  $0$ or $\pi$ | 0 | 0 | $(1,1,1,e^{i\sigma^{3}(\chi_{1}-\chi_{ST_{1}})})$ | $(1,e^{i\sigma^{3}(\chi_{ST_{1}}+\chi_{\overline{C}_{6}S})},1,e^{i\sigma^{3}\chi_{1}})$ | $\left[4\right]^{\dagger}$ | 4
$0$ or $\pi$ | $0$ |  $0$ or $\pi$ | 0 | 1 | $(1,1,1,e^{i\sigma^{3}(\chi_{1}-\chi_{ST_{1}})})$ | $(i\sigma^{1},i\sigma^{1}e^{i\sigma^{3}(\chi_{ST_{1}}+\chi_{\overline{C}_{6}S})},i\sigma^{1},i\sigma^{1}e^{i\sigma^{3}\chi_{1}})$ | $\left[4\right]^{\dagger}$ | 4
$0$ or $\pi$ | $0$ |  $0$ or $\pi$ | 1 | 0 | $(i\sigma^{1},i\sigma^{1},i\sigma^{1},i\sigma^{1}e^{i\sigma^{3}(\chi_{1}-\chi_{ST_{1}})})$ | $(1,e^{i\sigma^{3}(\chi_{ST_{1}}+\chi_{\overline{C}_{6}S})},1,e^{i\sigma^{3}\chi_{1}})$ | $\left[4\right]^{\dagger}$ | 4
$0$ or $\pi$ | $0$ |  $0$ or $\pi$ | 1 | 1 | $(i\sigma^{1},i\sigma^{1},i\sigma^{1},i\sigma^{1}e^{i\sigma^{3}(\chi_{1}-\chi_{ST_{1}})})$ | $(i\sigma^{1},i\sigma^{1}e^{i\sigma^{3}(\chi_{ST_{1}}+\chi_{\overline{C}_{6}S})},i\sigma^{1},i\sigma^{1}e^{i\sigma^{3}\chi_{1}})$ | $\left[4\right]^{\dagger}$ | 4
$\left[\frac{\pi}{2}\right]^{\dagger}$ | $\left[\pi\right]^{\dagger}$ | $\left[0\text{ or }\pi\right]^{\dagger}$ | $\left[0\right]^{\dagger}$ | $\left[1\right]^{\dagger}$ | $\left[(1,1,1,e^{i\sigma^{3}(\chi_{1}-\chi_{ST_{1}})})\right]^{\dagger}$ | $\left[(i\sigma^{1},i\sigma^{1}e^{i\sigma^{3}(\chi_{ST_{1}}+\chi_{\overline{C}_{6}S})},i\sigma^{1},i\sigma^{1}e^{i\sigma^{3}\chi_{1}})\right]^{\dagger}$ | $\left[2\right]^{\dagger}$ | 0
Table 2: The 28 $\mathbb{Z}_{2}$ PSG classes of pyrochlore space group symmetry only and 48 $\mathbb{Z}_{2}$ PSG classes of pyrochlore space group symmetry combined with time reversal symmetry. The tabulation is such that if a class of the latter is derived from a class of the former then they are written in the same line; and that the double-daggered double-square brackets “$\left\llbracket\phantom{ll}\right\rrbracket^{\ddagger}$” are used to enclose data that are specific to the latter. The parameters $\chi_{1}$, $\chi_{ST_{1}}$, $\chi_{\overline{C}_{6}S}$, $\chi_{S\overline{C}_{6}}$, and $\chi_{\overline{C}_{6}}$ are discrete elements of the IGG and $j$ is an extra parameter introduced during solving the PSG equations. Together they label the $\mathbb{Z}_{2}$ PSG classes. The gauge fixing of the SU(2) matrices $W_{\overline{C}_{6},\mu}$ and $W_{S,\mu}$ defined in Eq. (20), and the time reversal SU(2) matrix defined in Eq. (23) for $\mu=0$ are given ($k=1,2,3$ corresponds to different gauge fixing). The time reversal parameters $\chi_{\mathcal{T}\overline{C}_{6}}$ and $\chi_{\mathcal{T}S}$ are related to $\eta_{\mu}$ in Eq. (23) by $(\eta_{0},\eta_{1},\eta_{2},\eta_{3})=(1,\eta_{\mathcal{T}\overline{C}_{6}}\eta_{\mathcal{T}S},\eta_{\mathcal{T}S},\eta_{\mathcal{T}S})$. We defined the shorthand notation $W^{\pi\pi 0}_{S,2}=e^{i\chi_{\overline{C}_{6}S}}i\sigma^{k+1}e^{i\frac{\pi}{6}\sigma^{k-1}}$. $\chi_{1}$ | $\chi_{ST_{1}}$ | $\chi_{\overline{C}_{6}S}$ | $\chi_{S\overline{C}_{6}}$ | $\chi_{\overline{C}_{6}}$ | $j$ | $\left\llbracket\begin{array}[]{{>{$}p{0.6cm}<{$}>{\raggedleft$}p{0.5cm}<{$}>{\raggedleft$}p{0.6cm}<{$}}}\chi_{\mathcal{T}\overline{C}_{6}}&\chi_{\mathcal{T}S}$\@add@raggedleft&W_{\mathcal{T},0}$\@add@raggedleft\end{array}\right\rrbracket^{\ddagger}$ | $(W_{\overline{C}_{6},0},W_{\overline{C}_{6},1},W_{\overline{C}_{6},2},W_{\overline{C}_{6},3})$ | $(W_{S,0},W_{S,1},W_{S,2},W_{S,3})$ | Number
---|---|---|---|---|---|---|---|---|---
of classes
$\\#$ | $\left\llbracket\\#\right\rrbracket^{\ddagger}$
$0$ or $\pi$ | $\chi_{1}$ |  $0$ or $\pi$ | $0$ | $0$ |  – | $\left\llbracket\begin{array}[]{{l>{\centering$}p{0.6cm}<{$}l}}{0}&{0}$\@add@centering&{i\sigma^{k\hphantom{+1}}}\end{array}\right\rrbracket^{\ddagger}$ | $(1,1,1,1)$ | $(1,1,e^{i\chi_{\overline{C}_{6}S}},1)$ | 4 | $\left\llbracket 4\right\rrbracket^{\ddagger}$
$0$ or $\pi$ | $\chi_{1}$ | $0$ or $\pi$ | $\pi$ | $\pi$ |  – | $\left\llbracket\begin{array}[]{{l>{\centering$}p{0.6cm}<{$}l}}{0}&{0}$\@add@centering&{i\sigma^{k}}\\\ {\pi}&{0}$\@add@centering&{i\sigma^{k-1}}\end{array}\right\rrbracket^{\ddagger}$ | $(-i\sigma^{k},1,1,i\sigma^{k})$ | $(1,1,e^{i\chi_{\overline{C}_{6}S}},1)$ | 4 | $\left\llbracket 8\right\rrbracket^{\ddagger}$
$0$ or $\pi$ | $\neg\chi_{1}$ | $0$ | $\pi$ | $0$ | $1$ | $\left\llbracket\begin{array}[]{{l>{\centering$}p{0.6cm}<{$}l}}{0}&{\pi}$\@add@centering&{i\sigma^{k-1}}\end{array}\right\rrbracket^{\ddagger}$ | $(-e^{i\chi_{\overline{C}_{6}S}}e^{i\frac{\pi j}{3}\sigma^{k-1}},1,1,1)$ | $(1,i\sigma^{k},i\sigma^{k}e^{i\frac{\pi j}{3}\sigma^{k-1}},1)$ | 2 | $\left\llbracket 2\right\rrbracket^{\ddagger}$
$0$ or $\pi$ | $\neg\chi_{1}$ | $0$ | $\pi$ | $0$ | $3$ | $\left\llbracket\begin{array}[]{{l>{\centering$}p{0.6cm}<{$}l}}{0}&{0}$\@add@centering&{i\sigma^{k}}\\\ {0}&{\pi}$\@add@centering&{i\sigma^{k-1}}\end{array}\right\rrbracket^{\ddagger}$ | $(-e^{i\chi_{\overline{C}_{6}S}}e^{i\frac{\pi j}{3}\sigma^{k-1}},1,1,1)$ | $(1,i\sigma^{k},i\sigma^{k}e^{i\frac{\pi j}{3}\sigma^{k-1}},1)$ | 2 | $\left\llbracket 4\right\rrbracket^{\ddagger}$
$0$ or $\pi$ | $\neg\chi_{1}$ | $\pi$ | $\pi$ | $0$ | $0$ | $\left\llbracket\begin{array}[]{{l>{\centering$}p{0.6cm}<{$}l}}{0}&{0}$\@add@centering&{i\sigma^{k}}\\\ {0}&{\pi}$\@add@centering&{i\sigma^{k-1}}\end{array}\right\rrbracket^{\ddagger}$ | $(-e^{i\chi_{\overline{C}_{6}S}}e^{i\frac{\pi j}{3}\sigma^{k-1}},1,1,1)$ | $(1,i\sigma^{k},i\sigma^{k}e^{i\frac{\pi j}{3}\sigma^{k-1}},1)$ | 2 | $\left\llbracket 4\right\rrbracket^{\ddagger}$
$0$ or $\pi$ | $\neg\chi_{1}$ | $\pi$ | $\pi$ | $0$ | $2$ | $\left\llbracket\begin{array}[]{{l>{\centering$}p{0.6cm}<{$}l}}{0}&{\pi}$\@add@centering&{i\sigma^{k-1}}\end{array}\right\rrbracket^{\ddagger}$ | $(-e^{i\chi_{\overline{C}_{6}S}}e^{i\frac{\pi j}{3}\sigma^{k-1}},1,1,1)$ | $(1,i\sigma^{k},i\sigma^{k}e^{i\frac{\pi j}{3}\sigma^{k-1}},1)$ | 2 | $\left\llbracket 2\right\rrbracket^{\ddagger}$
$0$ or $\pi$ | $\neg\chi_{1}$ | $0$ or $\pi$ | $0$ | $\pi$ |  – | $\left\llbracket\begin{array}[]{{l>{\centering$}p{0.6cm}<{$}l}}{0}&{0}$\@add@centering&{i\sigma^{k}}\\\ {\pi}&{\pi}$\@add@centering&{i\sigma^{k-1}}\end{array}\right\rrbracket^{\ddagger}$ | $(i\sigma^{k},1,1,i\sigma^{k})$ | $(1,i\sigma^{k},e^{i\chi_{\overline{C}_{6}S}}i\sigma^{k},1)$ | 4 | $\left\llbracket 8\right\rrbracket^{\ddagger}$
$0$ or $\pi$ | $\neg\chi_{1}$ |  $0$ or $\pi$ | $\pi$ | $\pi$ | $0$ | $\left\llbracket\begin{array}[]{{l>{\centering$}p{0.6cm}<{$}l}}{0}&{\pi}$\@add@centering&{i\sigma^{k-1}}\end{array}\right\rrbracket^{\ddagger}$ | $(e^{i\frac{\pi}{6}\sigma^{k-1}},1,1,i\sigma^{k-1})$ | $(1,i\sigma^{k},W^{\pi\pi 0}_{S,2},1)$ | 4 | $\left\llbracket 4\right\rrbracket^{\ddagger}$
$0$ or $\pi$ | $\neg\chi_{1}$ | $0$ or $\pi$ | $\pi$ | $\pi$ | $1$ | $\left\llbracket\begin{array}[]{{l>{\centering$}p{0.6cm}<{$}l}}{0}&{\pi}$\@add@centering&{i\sigma^{k-1}}\\\ {\pi}&{0}$\@add@centering&{i\sigma^{k}}\\\ {\pi}&{\pi}$\@add@centering&{i\sigma^{k+1}}\end{array}\right\rrbracket^{\ddagger}$ | $(-i\sigma^{k-1},1,1,i\sigma^{k-1})$ | $(1,i\sigma^{k},e^{i\chi_{\overline{C}_{6}S}}i\sigma^{k},1)$ | 4 | $\left\llbracket 12\right\rrbracket^{\ddagger}$

In the second step, we include the time reversal operation $\mathcal{T}$ in
the symmetry group. In an appropriate gauge, the solution for time reversal
operation can be chosen as

${W}_{\mathcal{T}}(\bm{r}_{\mu})=i\sigma^{1}$ (22)

for U(1) gauge type and

${W}_{\mathcal{T}}(\bm{r}_{\mu})=i\eta_{\mu}\sigma^{k}$ (23)

for $\mathbb{Z}_{2}$ gauge type, where $k=1,2,3$ depends on the PSG class and
$\eta_{\mu}=\pm$ is a sublattice-dependent sign factor. Applying this gauge
choice and the space group PSG solutions in Eq. (20) to the time reversal PSG
equations associated with Eqs. (4i) and (4j), we obtain the PSG solutions for
time reversal invariant symmetric spin liquids. We find that there are 16
classes for the U(1) gauge type and 48 classes for the $\mathbb{Z}_{2}$ gauge
type. We list these classes again in Tables 1 and 2 for U(1) and
$\mathbb{Z}_{2}$ gauge groups, respectively and explicitly mark the data that
are specific to the time reversal symmetric classes.

It is worth pointing out that including time reversal symmetry to the PSG has
opposite effects on the U(1) and the $\mathbb{Z}_{2}$ classes. In U(1) PSG,
the presence of time reversal symmetry forbids the two $\frac{\pi}{2}$-flux
PSG classes (last line of Table 1), thereby reducing the total number of PSG
classes from 18 to 16. On the other hand, in $\mathbb{Z}_{2}$ PSG, adding time
reversal symmetry introduces two additional discrete parameters
$\chi_{\mathcal{T}\overline{C}_{6}}$ and $\chi_{\mathcal{T}S}$ in the labeling
of the time reversal invariant classes (see columns 7 and 8 of Table 2). These
two parameters are the IGG elements associated with Eq. (4j) for
$\mathcal{O}=\overline{C}_{6}$ and $S$, respectively, and characterize the
phases that a spinon acquires in completing the corresponding spacetime
processes. We find that 20 classes obtained from the pure space group PSG are
further “fractionalized” as a result of these additional parameters, thereby
increasing the total number of PSG classes from 28 to 48. In U(1) PSG,
$\chi_{\overline{C}_{6}\mathcal{T}}$ and $\chi_{S\mathcal{T}}$ do not increase
the number of classes since they are fully determined by the $\mathbb{Z}_{2}$
parameters $w_{\overline{C}_{6}}$ and $w_{S}$ that are already introduced for
the pure space group PSG. The phenomenon described here in fact also happens
in the classification of other projective lattice symmetries: it is generally
true that adding time reversal symmetry will increase the number of
$\mathbb{Z}_{2}$ PSG classes and reduce the number of U(1) PSG classes.

##  III Analysis of the mean-field ansätze

The PSG classification in the last section provides the symmetry constraints
on constructing Hamiltonians that describe fractionalized spinons in symmetric
U(1) and $\mathbb{Z}_{2}$ spin liquids. Since the gauge fields are deconfined
in a spin liquid phase, a good description for the spinons is already achieved
at the mean-field level. In this section, we present a complete list of parton
mean-field Hamiltonians for symmetric spin liquids on the pyrochlore lattice.
These Hamiltonians can either be analyzed in their own right, or serve as a
first step towards a more realistic description of spin liquids upon adding
spinon interactions or fluctuating gauge fields.

###  III.1 Construction of the mean-field ansätze

We are now in the position to construct the mean-field ansatz for each PSG
class. The most general mean-field ansatz for fermionic spinons can be written
as

$\displaystyle H=\sum\limits_{\alpha={0,x,y,z}}H^{\alpha},\qquad
H^{\alpha}=\sum\limits_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}H^{\alpha}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}},$
(24) $\displaystyle
H^{\alpha}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}=\mathrm{Tr}\left[\sigma^{\alpha}\Psi_{\bm{r}_{\mu}}u^{(\alpha)}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\Psi^{\dagger}_{\bm{r}^{\prime}_{\nu}}\right],$

where $u^{(\alpha)}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}$ with
$\alpha=0,x,y,z$ contain all the sixteen real parameters for the bond
$\bm{r}_{\mu}\rightarrow\bm{r}^{\prime}_{\nu}$,

$\displaystyle u^{(0)}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}$
$\displaystyle=ia^{0}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}1-(b^{0}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{1}+c^{0}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{2}+d^{0}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{3}),$
(25) $\displaystyle u^{(x)}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}$
$\displaystyle=a^{x}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}1+i(b^{x}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{1}+c^{x}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{2}+d^{x}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{3}),$
$\displaystyle u^{(y)}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}$
$\displaystyle=a^{y}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}1+i(b^{y}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{1}+c^{y}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{2}+d^{y}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{3}),$
$\displaystyle u^{(z)}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}$
$\displaystyle=a^{z}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}1+i(b^{z}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{1}+c^{z}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{2}+d^{z}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\sigma^{3}).$

Note that $1$ denotes the $2\times 2$ identity matrix.

The bond parameters are subject to constraints provided by the PSG. The PSG
operators $\widetilde{\mathcal{O}}$ and $\widetilde{\mathcal{T}}$ are the
symmetry operators of the Hamiltonian $H$, meaning
$\widetilde{\mathcal{O}}\colon H\rightarrow H$ and
$\widetilde{\mathcal{T}}\colon H\rightarrow H$. Since the spinons transform
under $\widetilde{\mathcal{O}}$ and $\widetilde{\mathcal{T}}$ according to
Eqs. (11) and (12), we have the following rules:

  * •

For a general translation
$\bm{t}=t_{1}\bm{e}_{1}+t_{2}\bm{e}_{2}+t_{3}\bm{e}_{3}$, we have

$u^{(\alpha)}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}=u^{(\alpha)}_{(\bm{r}-\bm{t})_{\mu},(\bm{r}^{\prime}-\bm{t})_{\nu}}e^{i\sigma^{3}[t_{2}(r^{\prime}_{1}-r_{1})-t_{3}(r^{\prime}_{1}-r_{1}-r^{\prime}_{2}+r_{2})]\chi_{1}};$
(26)

  * •

For space-group elements $\mathcal{O}\in\\{\overline{C}_{6},S\\}$, the singlet
and the triplet parts transform as

$\displaystyle
W_{\mathcal{O}}[\mathcal{O}(\bm{r}_{\mu})]u^{(0)}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}W^{\dagger}_{\mathcal{O}}[\mathcal{O}(\bm{r}^{\prime}_{\nu})]$
$\displaystyle=u^{(0)}_{\mathcal{O}(\bm{r}_{\mu}),\mathcal{O}(\bm{r}^{\prime}_{\nu})},$
(27) $\displaystyle
W_{\mathcal{O}}[\mathcal{O}(\bm{r}_{\mu})]u^{(i)}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}\mathcal{R}^{\mathcal{O}}_{ij}W^{\dagger}_{\mathcal{O}}[\mathcal{O}(\bm{r}^{\prime}_{\nu})]$
$\displaystyle=u^{(j)}_{\mathcal{O}(\bm{r}_{\mu}),\mathcal{O}(\bm{r}^{\prime}_{\nu})},$

with

$\mathcal{R}^{\overline{C}_{6}}=\left(\begin{array}[]{ccc}&&1\\\ 1&&\\\
&1&\end{array}\right),\quad\mathcal{R}^{S}=\left(\begin{array}[]{ccc}&1&\\\
1&&\\\ &&-1\end{array}\right);$ (28)

  * •

For time reversal $\mathcal{T}$,

$u^{\alpha}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}=-{W}_{\mathcal{T}}(\bm{r}_{\mu})u^{\alpha}_{\bm{r}_{\mu},\bm{r}^{\prime}_{\nu}}{W}_{\mathcal{T}}^{\dagger}(\bm{r}^{\prime}_{\nu}).$
(29)

By solving Eqs. (26)-(29) for the sixteen real parameters at each bond, we
obtain the mean-field ansätze for the PSG classes.

The pyrochlore bonds can be categorized into equivalence classes (or orbits)
of the space group, where the bonds within each class are related by space
group transformations, while the bonds in different classes are unrelated. In
order to obtain the complete mean-field ansatz, it suffices to obtain the
mean-field solution for one representative bond of each equivalence class. We
choose and express these representative bonds as follows:

  * •

We use the Greek letters $\alpha$,$\beta$,$\gamma$,$\delta$ to parameterize
the representative onsite bond $(0,0,0)_{0}\rightarrow(0,0,0)_{0}$, the Latin
small letters $a$,$b$,$c$,$d$ to parameterize the representative nearest-
neighbor bond $(0,0,0)_{0}\rightarrow(0,0,0)_{1}$, and the Latin capital
letters $A$,$B$,$C$,$D$ to parameterize the representative next-nearest-
neighbor (NNN) bond $(0,0,0)_{1}\rightarrow(0,-1,0)_{2}$. The choice of
representative $i$th-nearest neighbor bonds and their parameterization for
$3\leq i\leq 8$ can be found in App. F.

  * •

Take the representative NN bond for example. In an ansatz for the
$\mathbb{Z}_{2}$ PSG without time reversal symmetry, all the 16 terms in Eq.
(25) may be nonvanishing. We then use eight complex numbers
$a_{h},b_{h},c_{h},d_{h},a_{p},b_{p},c_{p},d_{p}$ to paramaterize the 16 real
parameters in this bond: explicitly, we write (omitting the bond label
$(0,0,0)_{0}\rightarrow(0,0,0)_{1}$)

$\displaystyle u^{(0)}$
$\displaystyle=i\mathrm{Re}a_{h}1-\mathrm{Re}a_{p}\sigma^{1}-\mathrm{Im}a_{p}\sigma^{2}-\mathrm{Im}a_{h}\sigma^{3},$
(30) $\displaystyle u^{(x)}$
$\displaystyle=\mathrm{Re}b_{h}1+i(\mathrm{Re}b_{p}\sigma^{1}+\mathrm{Im}b_{p}\sigma^{2}+\mathrm{Im}b_{h}\sigma^{3}),$
$\displaystyle u^{(y)}$
$\displaystyle=\mathrm{Re}c_{h}1+i(\mathrm{Re}c_{p}\sigma^{1}+\mathrm{Im}c_{p}\sigma^{2}+\mathrm{Im}c_{h}\sigma^{3}),$
$\displaystyle u^{(z)}$
$\displaystyle=\mathrm{Re}d_{h}1+i(\mathrm{Re}d_{p}\sigma^{1}+\mathrm{Im}d_{p}\sigma^{2}+\mathrm{Im}d_{h}\sigma^{3}).$

In the U(1) PSG, we only have hopping bilinears, therefore the $\sigma^{1}$
and $\sigma^{2}$ terms in Eq. (25) vanish, and we parameterize Eq. (25) as

$\displaystyle u^{(0)}=i\mathrm{Re}a1-\mathrm{Im}a\sigma^{3},\quad
u^{(x)}=\mathrm{Re}b1+i\mathrm{Im}b\sigma^{3},$ (31) $\displaystyle
u^{(y)}=\mathrm{Re}c1+i\mathrm{Im}c\sigma^{3},\quad
u^{(z)}=\mathrm{Re}d1+i\mathrm{Im}d\sigma^{3}.$

The symmetry relations (26), (27), and (29) impose constraints on these
parameters, and the numbers of independent real bond parameters are usually
smaller than 16 or 8 in the $\mathbb{Z}_{2}$ and U(1) ansätze, respectively.
To determine the independent bond parameters, one needs to find all the
symmetry operations (the so called “stabilizers” of the symmetry group) that
leave the representative bonds invariant. All such space group symmetry
operations for $i$th-nearest neighbor bonds ($i\leq 8$) have been listed in
App. F. As an example, time reversal symmetry leaves any bond invariant. In
the U(1) case, applying the time reversal PSG in Eq. (22) to Eq. (29) reduces
the bond parameterization in Eq. (31) to

$\displaystyle u^{(0)}=-\mathrm{Im}a\sigma^{3},\ \ \
u^{(x)}=i\mathrm{Im}b\sigma^{3},$ (32) $\displaystyle
u^{(y)}=i\mathrm{Im}c\sigma^{3},\ \ \ u^{(z)}=i\mathrm{Im}d\sigma^{3}.$

In the $\mathbb{Z}_{2}$ case, applying the time reversal PSG in Eq. (23) to
Eq. (29) reduces the bond parameterization in Eq. (30) depending on the value
of $\eta_{\mu}$ for $\mu=0,1$:

$\displaystyle
u^{(0)}=-\mathrm{Re}a_{p}\sigma^{1}-\mathrm{Im}a_{p}\sigma^{2},\quad
u^{(x)}=i(\mathrm{Re}b_{p}\sigma^{1}+\mathrm{Im}b_{p}\sigma^{2}),$ (33)
$\displaystyle
u^{(y)}=i(\mathrm{Re}c_{p}\sigma^{1}+\mathrm{Im}c_{p}\sigma^{2}),\quad
u^{(z)}=i(\mathrm{Re}d_{p}\sigma^{1}+\mathrm{Im}d_{p}\sigma^{2})$

for $\eta_{0}=\eta_{1}$ and

$\displaystyle u^{(0)}=i\mathrm{Re}a_{h}1-\mathrm{Im}a_{h}\sigma^{3},\quad
u^{(x)}=\mathrm{Re}b_{h}1+i\mathrm{Im}b_{h}\sigma^{3},$ (34) $\displaystyle
u^{(y)}=\mathrm{Re}c_{h}1+i\mathrm{Im}c_{h}\sigma^{3},\quad
u^{(z)}=\mathrm{Re}d_{h}1+i\mathrm{Im}d_{h}\sigma^{3}$

for $\eta_{0}=-\eta_{1}$. Note that the form of Eq. (34) coincides with that
of a U(1) ansatz [see Eq. (31)]. However, pairing terms (in which parameters
have a subscript “$p$”) do appear for other bonds (e.g., the representative
NNN bond), and this is generally not a U(1) ansatz.

The final result of the mean-field parameters for representative bonds up to
NNN are presented in Table 3 for the U(1) PSG and Table 4 for the
$\mathbb{Z}_{2}$ PSG. The effect of time reversal symmetry has also been
addressed therein.

Table 3: Independent mean-field parameters and constraints for the 18 U(1) PSG classes of pyrochlore space group symmetry only and 16 U(1) PSG classes of pyrochlore space group symmetry combined with time reversal symmetry. The tabulation is such that the latter are embedded in the former; and that daggered square brackets “$\left[\phantom{ll}\right]^{\dagger}$” are used to enclose data that are specific to the former. The parameters not referenced are enforced to be zero. Note the zero- and $\pi$-flux PSG have identical free mean-field parameters up to NNN bonds, but this is no longer true when considering 3rd-nearest-neighbor bonds. Class | Independent nonzero parameters | Constraints
---|---|---
$\chi_{1}$–$(w_{\overline{C}_{6}}w_{S})$–$(\chi_{ST_{1}}\chi_{\overline{C}_{6}S})$ | Onsite | NN | NNN | NN | NNN
0– or $\pi$–$(0\;0)$–$(0\;0)$ | $\mathrm{Im}\alpha$ |  $\mathrm{Im}a$, $\mathrm{Im}c$ | $\mathrm{Im}A,\,{\left[\mathrm{Re}B\right]^{\dagger}},\,\mathrm{Im}B,\,\mathrm{Im}D$ | $\mathrm{Im}c=-\mathrm{Im}d$ | $B=-C^{*}$
0– or $\pi$–$(0\;0)$–$(0\;\pi)$ | $\mathrm{Im}\alpha$ |  $\mathrm{Im}b$, $[\mathrm{Re}c]^{\dagger}$ | $\mathrm{Im}A,\,{\left[\mathrm{Re}B\right]^{\dagger}},\,\mathrm{Im}B,\,\mathrm{Im}D$ | $\left[\mathrm{Re}c=\mathrm{Re}d\right]^{\dagger}$ | $B=-C^{*}$
0– or $\pi$–$(0\;1)$–$(0\;0)$ | – | $\mathrm{Im}c$ |  ${\left[\mathrm{Re}B\right]^{\dagger}}$, $\mathrm{Im}B$ | $\mathrm{Im}c=\mathrm{Im}d$ | $B=-C$
0– or $\pi$–$(0\;1)$–$(0\;\pi)$ | – | ${\left[\mathrm{Re}c\right]^{\dagger}}$ |  ${\left[\mathrm{Re}B\right]^{\dagger}}$, $\mathrm{Im}B$ | ${\left[\mathrm{Re}c=\mathrm{Re}d\right]^{\dagger}}$ | $B=-C$
0– or $\pi$–$(1\;0)$–$(0\;0)$ | – | $\mathrm{Im}c$ | $\mathrm{Im}A,\,\left[\mathrm{Re}B\right]^{\dagger},\,\mathrm{Im}B,\,\mathrm{Im}D$ | $\mathrm{Im}c=\mathrm{Im}d$ | $B=-C^{*}$
0– or $\pi$–$(1\;0)$–$(0\;\pi)$ | – | $\left[\mathrm{Re}c\right]^{\dagger}$ | $\mathrm{Im}A,\,{\left[\mathrm{Re}B\right]^{\dagger}},\,\mathrm{Im}B,\,\mathrm{Im}D$ | ${\left[\mathrm{Re}c=\mathrm{Re}d\right]^{\dagger}}$ | $B=-C^{*}$
0– or $\pi$–$(1\;1)$–$(0\;0)$ | – | $\mathrm{Im}a,\mathrm{Im}c$ | $\left[\mathrm{Re}B\right]^{\dagger},\,\mathrm{Im}B$ | $\mathrm{Im}c=-\mathrm{Im}d$ | $B=-C$
0– or $\pi$–$(1\;1)$–$(0\;\pi)$ | – | $\mathrm{Im}b,\,{\left[\mathrm{Re}c\right]^{\dagger}}$ | ${\left[\mathrm{Re}B\right]^{\dagger}},\,\mathrm{Im}B$ | ${\left[\mathrm{Re}c=\mathrm{Re}d\right]^{\dagger}}$ | $B=-C$
$\left[\frac{\pi}{2}\text{\\---}(0\;1)\text{\\---}(\pi\;0)\right]^{\dagger}$ | $\left[-\right]^{\dagger}$ | $\left[\mathrm{Im}c\right]^{\dagger}$ | $\left[\mathrm{Re}B,\,\mathrm{Im}B\right]^{\dagger}$ | $\left[\mathrm{Im}c=\mathrm{Im}d\right]^{\dagger}$ | $\left[B=-C\right]^{\dagger}$
$\left[\frac{\pi}{2}\text{\\---}(0\;1)\text{\\---}(\pi\;\pi)\right]^{\dagger}$ | $\left[-\right]^{\dagger}$ | $\left[\mathrm{Re}c\right]^{\dagger}$ | $\left[\mathrm{Re}B,\,\mathrm{Im}B\right]^{\dagger}$ | $\left[\mathrm{Re}c=\mathrm{Re}d\right]^{\dagger}$ | $\left[B=-C\right]^{\dagger}$
Table 4: Independent mean-field parameters and constraints for the 28 $\mathbb{Z}_{2}$ PSG classes of pyrochlore space group symmetry only and 48 $\mathbb{Z}_{2}$ PSG classes of pyrochlore space group symmetry combined with time reversal symmetry. The tabulation uses the double-daggered double-square brackets “$\left\llbracket\phantom{ll}\right\rrbracket^{\ddagger}$” to enclose data that are specific to the latter, and whenever a class of the latter is derived from a class of the former then a symbol “” is used to specify their relation. Also we set $k=1$ (see Table 2 for the definition of $k$). The parameters not referenced are enforced to be zero. We denoted $F_{2}=\\{\mathrm{Im}\alpha_{h},\mathrm{Im}\alpha_{p}\\}$, $F_{4}=\\{\mathrm{Im}a_{h},\mathrm{Im}c_{h},\mathrm{Im}a_{p},\mathrm{Im}c_{p}\\}$, $F_{5}=\\{{\mathrm{Re}B_{h},\mathrm{Im}A_{p},\mathrm{Im}B_{p},\mathrm{Im}D_{p}}\\}$, $F_{6}=\\{\mathrm{Im}A_{h},\mathrm{Im}B_{h},\mathrm{Im}D_{h},\mathrm{Im}A_{p},\,\mathrm{Im}B_{p},\,\mathrm{Im}D_{p}\\}$. Class ($\chi_{1}\chi_{ST_{1}}$)– | Independent nonzero parameters | Constraints (NN and NNN)
---|---|---
$(\chi_{\overline{C}_{6}S}\chi_{S\overline{C}_{6}}\chi_{\overline{C}_{6}})_{j}$ | Onsite | NN | NNN
$(00)$– or $(\pi\pi)$–$(000)$ | $\mathrm{Im}\alpha_{h},\alpha_{p}$ | $\mathrm{Im}a_{h},\mathrm{Im}c_{h},a_{p},c_{p}$ |  $\mathrm{Im}A_{h},B_{h},\mathrm{Im}D_{h}$, $A_{p},B_{p},D_{p}$ | $\mathrm{Im}d_{h}=-\mathrm{Im}c_{h},c_{p}=-d_{p},C_{h}=-B^{*}_{h},B_{p}=C_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(00)\right\rrbracket^{\ddagger}$ | ${\left\llbracket F_{2}\right\rrbracket^{\ddagger}}$ | ${\left\llbracket F_{4}\right\rrbracket^{\ddagger}}$ | ${\left\llbracket F_{6}\right\rrbracket^{\ddagger}}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(00)$– or $(\pi\pi)$–$(\pi 00)$ | $\mathrm{Im}\alpha_{h},\alpha_{p}$ | $\mathrm{Im}b_{h},\mathrm{Re}c_{h},b_{p}$ |  $\mathrm{Im}A_{h},B_{h},\mathrm{Im}D_{h}$, $A_{p},B_{p},D_{p}$ | $\mathrm{Re}d_{h}=\mathrm{Re}c_{h},C_{h}=-B^{*}_{h},B_{p}=C_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(00)\right\rrbracket^{\ddagger}$ | ${\left\llbracket F_{2}\right\rrbracket^{\ddagger}}$ | ${\left\llbracket\mathrm{Im}b_{h},\mathrm{Im}b_{p}\right\rrbracket^{\ddagger}}$ | ${\left\llbracket F_{6}\right\rrbracket^{\ddagger}}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(00)$– or $(\pi\pi)$–$(0\pi\pi)$ | $\mathrm{Re}\alpha_{p}$ | $\mathrm{Re}a_{h},c_{h},\mathrm{Im}c_{p}$ |  $A_{h},B_{h},D_{h}$, $\mathrm{Im}A_{p},B_{p},\mathrm{Im}D_{p}$ | $d_{h}=-c^{*}_{h},\mathrm{Im}d_{p}=\mathrm{Im}c_{p},C_{h}=B_{h},C_{p}=-B^{*}_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(00)\right\rrbracket^{\ddagger}$ | ${\left\llbracket-\right\rrbracket^{\ddagger}}$ | ${\left\llbracket\mathrm{Im}c_{h},\mathrm{Im}c_{p}\right\rrbracket^{\ddagger}}$ | ${\left\llbracket F_{6}\right\rrbracket^{\ddagger}}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(\pi 0)\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}\alpha_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}a_{h},\,c_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket A_{h},\,B_{h},\,D_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(00)$– or $(\pi\pi)$–$(\pi\pi\pi)$ | $\mathrm{Re}\alpha_{p}$ | $\mathrm{Re}b_{h},\mathrm{Re}c_{p}$ |  $A_{h},B_{h},D_{h}$, $\mathrm{Im}A_{p},B_{p},\mathrm{Im}D_{p}$ | $\mathrm{Re}d_{p}=\mathrm{Re}c_{p},rC_{h}=B_{h},C_{p}=-B^{*}_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(00)\right\rrbracket^{\ddagger}$ | ${\left\llbracket-\right\rrbracket^{\ddagger}}$ | ${\left\llbracket-\right\rrbracket^{\ddagger}}$ | ${\left\llbracket F_{6}\right\rrbracket^{\ddagger}}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(\pi 0)\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}\alpha_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}b_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket A_{h},\,B_{h},\,D_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(0\pi)$– or $(\pi 0)$–$(0\pi 0)_{1}$ |  – | $\mathrm{Re}b_{h},\mathrm{Re}c_{p}$ | $B_{h},\mathrm{Re}A_{p},B_{p},\mathrm{Re}D_{p}$ |  $\mathrm{Im}b_{h}=\sqrt{3}\mathrm{Re}b_{h},\mathrm{Im}c_{p}=\sqrt{3}\mathrm{Re}c_{p},c_{p}=d_{p}$,
$\mathrm{Im}A_{p}=-\sqrt{3}\mathrm{Re}A_{p},C_{h}=-B_{h}$,
$\mathrm{Im}D_{p}=-\sqrt{3}\mathrm{Re}D_{p},C_{p}=e^{-\frac{2\pi}{3}i}B_{p}^{*}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(0\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket-\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}b_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}A_{p},\,B_{p},\,\mathrm{Re}D_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(0\pi)$– or $(\pi 0)$–$(0\pi 0)_{3}$ | $\mathrm{Re}\alpha_{p}$ | $\mathrm{Re}b_{h},\mathrm{Re}c_{p}$ | $B_{h},\mathrm{Re}A_{p},B_{p},\mathrm{Re}D_{p}$ | $\mathrm{Re}d_{p}=\mathrm{Re}c_{p},C_{h}=-B_{h},C_{p}=B^{*}_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(00)\right\rrbracket^{\ddagger}$ | ${\left\llbracket-\right\rrbracket^{\ddagger}}$ | ${\left\llbracket-\right\rrbracket^{\ddagger}}$ | ${\left\llbracket\mathrm{Im}B_{h},\,\mathrm{Im}B_{p}\right\rrbracket^{\ddagger}}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(0\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}\alpha_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}b_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}A_{p},\,B_{p},\,\mathrm{Re}D_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(0\pi)$– or $(\pi 0)$–$(\pi\pi 0)_{0}$ | $\mathrm{Re}\alpha_{p}$ | $\mathrm{Re}a_{h},c_{h},\mathrm{Im}c_{p}$ | $B_{h},\mathrm{Re}A_{p},B_{p},\mathrm{Re}D_{p}$ | $d_{h}=-c^{*}_{h},\mathrm{Im}d_{p}=\mathrm{Im}c_{p},C_{h}=-B_{h},C_{p}=B^{*}_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(00)\right\rrbracket^{\ddagger}$ | ${\left\llbracket-\right\rrbracket^{\ddagger}}$ | ${\left\llbracket\mathrm{Im}c_{h},\mathrm{Im}c_{p}\right\rrbracket^{\ddagger}}$ | ${\left\llbracket\mathrm{Im}B_{h},\,\mathrm{Im}B_{p}\right\rrbracket^{\ddagger}}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(0\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}\alpha_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}a_{h},\,c_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}A_{p},\,B_{p},\,\mathrm{Re}D_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(0\pi)$– or $(\pi 0)$–$(\pi\pi 0)_{2}$ |  – | $\mathrm{Re}a_{h},c_{h},\mathrm{Re}c_{p}$ | $B_{h},\mathrm{Re}A_{p},B_{p},\mathrm{Re}D_{p}$ |  $\mathrm{Im}a_{h}=-\sqrt{3}\mathrm{Re}a_{h},d_{h}=e^{\frac{\pi}{3}i}c_{h}^{*},\mathrm{Im}c_{p}=\frac{1}{\sqrt{3}}\mathrm{Re}c_{p}$,
$d_{p}=c_{p},\mathrm{Im}A_{p}=\sqrt{3}\mathrm{Re}A_{p},C_{h}=-B_{h}$,
$C_{p}=e^{\frac{2\pi}{3}i}B_{p}^{*},\mathrm{Im}D_{p}=\sqrt{3}\mathrm{Re}D_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(0\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket-\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}a_{h},\,c_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}A_{p},\,B_{p},\,\mathrm{Re}D_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(0\pi)$– or $(\pi 0)$–$(00\pi)$ | $\mathrm{Re}\alpha_{p}$ | $\mathrm{Im}a_{h},\,\mathrm{Im}c_{h},\,a_{p},\,c_{p}$ | $\mathrm{Re}A_{h},B_{h},\mathrm{Re}D_{h},B_{p}$ | $\mathrm{Im}d_{h}=-\mathrm{Im}c_{h},d_{p}=-c_{p},C_{h}=B^{*}_{h},C_{p}=-B_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(00)\right\rrbracket^{\ddagger}$ | ${\left\llbracket-\right\rrbracket^{\ddagger}}$ | ${\left\llbracket F_{4}\right\rrbracket^{\ddagger}}$ | ${\left\llbracket\mathrm{Im}B_{h},\,\mathrm{Im}B_{p}\right\rrbracket^{\ddagger}}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(\pi\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}\alpha_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket a_{p},\,c_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}A_{h},\,B_{h},\,\mathrm{Re}D_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(0\pi)$– or $(\pi 0)$–$(\pi 0\pi)$ | $\mathrm{Re}\alpha_{p}$ | $\mathrm{Im}b_{h},\mathrm{Re}c_{h},b_{p}$ | $\mathrm{Re}A_{h},B_{h},\mathrm{Re}D_{h},B_{p}$ | $\mathrm{Re}d_{h}=\mathrm{Re}c_{h},C_{h}=B^{*}_{h},B_{p}=-C_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(00)\right\rrbracket^{\ddagger}$ | ${\left\llbracket-\right\rrbracket^{\ddagger}}$ | ${\left\llbracket\mathrm{Im}b_{h},\mathrm{Im}b_{p}\right\rrbracket^{\ddagger}}$ | ${\left\llbracket\mathrm{Im}B_{h},\,\mathrm{Im}B_{p}\right\rrbracket^{\ddagger}}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(\pi\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}\alpha_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket b_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}A_{h},\,B_{h},\,\mathrm{Re}D_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(0\pi)$– or $(\pi 0)$–$(0\pi\pi)_{0}$ |  – | $\mathrm{Re}b_{h},\mathrm{Re}c_{p}$ | $B_{h},\mathrm{Re}A_{p},B_{p},\mathrm{Re}D_{p}$ |  $\mathrm{Im}b_{h}=-\sqrt{3}\mathrm{Re}b_{h},\mathrm{Im}c_{p}=\frac{1}{\sqrt{3}}\mathrm{Re}c_{p},c_{p}=d_{p}$,
$\mathrm{Im}A_{p}=-\frac{1}{\sqrt{3}}\mathrm{Re}A_{p},C_{h}=-B_{h}$,
$C_{p}=e^{-\frac{\pi}{3}i}B_{p}^{*}$,
$\mathrm{Im}D_{p}=-\frac{1}{\sqrt{3}}\mathrm{Re}D_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(0\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket-\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}b_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}A_{p},\,B_{p},\,\mathrm{Re}D_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(0\pi)$– or $(\pi 0)$–$(\pi\pi\pi)_{0}$ |  – | $\mathrm{Re}a_{h},c_{h},\mathrm{Re}c_{p}$ | ${B_{h},\mathrm{Re}A_{p},B_{p},\mathrm{Re}D_{p}}$ |  $\mathrm{Im}a_{h}=-\sqrt{3}\mathrm{Re}a_{h},d_{h}=e^{i\frac{\pi}{3}}c_{h}^{*}$,
$\mathrm{Im}c_{p}=-\sqrt{3}\mathrm{Re}c_{p},c_{p}=d_{p},\mathrm{Im}A_{p}=-\frac{1}{\sqrt{3}}\mathrm{Re}A_{p}$,
$C_{h}=-B_{h},C_{p}=e^{-\frac{\pi}{3}i}B_{p}^{*}$,
$\mathrm{Im}D_{p}=-\frac{1}{\sqrt{3}}\mathrm{Re}D_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(0\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket-\right\rrbracket^{\ddagger}$ | ${\left\llbracket\mathrm{Re}a_{h},c_{h}\right\rrbracket^{\ddagger}}$ | $\left\llbracket{\,\mathrm{Re}A_{p},B_{p},\,\mathrm{Re}D_{p}}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(0\pi)$– or $(\pi 0)$–$(0\pi\pi)_{1}$ | – | $\mathrm{Re}b_{h},\mathrm{Im}c_{p}$ | $B_{h},\mathrm{Im}A_{p},B_{p},\mathrm{Im}D_{p}$ | $\mathrm{Im}d_{p}=\mathrm{Im}c_{p},C_{h}=-B_{h},C_{p}=-B^{*}_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(0\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket-\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}b_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Im}A_{p},\,B_{p},\,\mathrm{Im}D_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(\pi 0)\right\rrbracket^{\ddagger}$ | $\left\llbracket-\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}b_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket{\mathrm{Re}B_{h},\mathrm{Re}B_{p}}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(\pi\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket-\right\rrbracket^{\ddagger}$ | $\left\llbracket{-}\right\rrbracket^{\ddagger}$ | $\left\llbracket F_{5}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$(0\pi)$– or $(\pi 0)$–$(\pi\pi\pi)_{1}$ | – | $\mathrm{Re}a_{h},c_{h},\mathrm{Re}c_{p}$ | $B_{h},\mathrm{Im}A_{p},B_{p},\mathrm{Im}D_{p}$ |  $d_{h}=-c^{*}_{h},\mathrm{Re}d_{p}=\mathrm{Re}c_{p}$, $C_{h}=-B_{h},C_{p}=-B^{*}_{p}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(0\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket-\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}a_{h},\,c_{h}\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Im}A_{p},\,B_{p},\,\mathrm{Im}D_{p}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(\pi 0)\right\rrbracket^{\ddagger}$ | $\left\llbracket-\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}a_{h},{\mathrm{Re}c_{h},\mathrm{Re}c_{p}}\right\rrbracket^{\ddagger}$ | $\left\llbracket{\mathrm{Re}B_{h},\mathrm{Re}B_{p}}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$
$\drsh\left\llbracket(\chi_{\mathcal{T}\overline{C}_{6}}\chi_{\mathcal{T}S})=(\pi\pi)\right\rrbracket^{\ddagger}$ | $\left\llbracket-\right\rrbracket^{\ddagger}$ | $\left\llbracket\mathrm{Re}c_{p},{\mathrm{Im}c_{h}}\right\rrbracket^{\ddagger}$ | $\left\llbracket F_{5}\right\rrbracket^{\ddagger}$ | $\left\llbracket\text{Constraints inherited from above}\right\rrbracket^{\ddagger}$

###  III.2 Symmetry properties of the 0-flux ansätze

The symmetry constraints imposed by the PSG given in the last subsection are
formulated in real space. For analyzing the properties of the mean-field
ansätze, it is more helpful to see how the projective symmetry transformations
apply in momentum space. In the 0-flux case ($\chi_{1}=0$), the action of each
projective symmetry is simple and can be explicitly given. To start with, we
define the Bogoliubov-de Gennes (BdG) basis (where the spin indices are
suppressed):

$\Phi_{\bm{k}}=\left(f^{\phantom{{\dagger}}}_{\bm{k},0},f^{\phantom{{\dagger}}}_{\bm{k},1},f^{\phantom{{\dagger}}}_{\bm{k},2},f^{\phantom{{\dagger}}}_{\bm{k},3},f^{\dagger}_{-\bm{k},0},f^{\dagger}_{-\bm{k},1},f^{\dagger}_{-\bm{k},2},f^{\dagger}_{-\bm{k},3}\right)^{T}.$

The Hamiltonian is then written as

$H=\sum\limits_{\bm{k}\in\text{BZ}^{+}}\Phi^{\dagger}_{\bm{k}}\mathcal{H}_{\text{BdG}}(\bm{k})^{\vphantom{\dagger}}\Phi_{\bm{k}},$
(35)

where the momentum sum is over half of the Brillouin zone (BZ) with, say
$k_{3}>0$. This Hamiltonian has the standard Bogoliubov form

$\mathcal{H}_{\text{BdG}}(\bm{k})=\left(\begin{array}[]{cc}\mathcal{H}_{\text{U(1)}}(\bm{k})&\mathcal{H}_{p}(\bm{k})\\\
\mathcal{H}^{\dagger}_{p}&-\mathcal{H}_{\text{U(1)}}^{T}(-\bm{k})\end{array}\right),$
(36)

where $\mathcal{H}_{\text{U(1)}}$ and $\mathcal{H}_{p}$ correspond to the
hopping and pairing terms in Eq. (25) for the $\mathbb{Z}_{2}$ PSG. For the
U(1) PSG, the pairing terms vanish and the BdG form corresponds to two copies
of the U(1) Hamiltonian $\mathcal{H}_{\text{U(1)}}$.

In terms of the BdG Hamiltonian matrix $\mathcal{H}_{\text{BdG}}(\bm{k})$
describing each 0-flux ansatz, we have the following symmetry constraints:

$\displaystyle
W^{\dagger}_{\overline{C}_{6},w_{\overline{C}_{6}}}(\bm{k})\mathcal{H}_{\text{BdG}}(\bm{k})$
$\displaystyle W_{\overline{C}_{6},w_{\overline{C}_{6}}}(\bm{k})$ (37)
$\displaystyle=\mathcal{H}_{\text{BdG}}\left(\overline{C}_{6}(\bm{k})+\mu^{3}\phi_{ST_{1}}{\bm{b}}_{1}\right),$
$\displaystyle W^{\dagger}_{S,w_{S}}(\bm{k})\mathcal{H}_{\text{BdG}}(\bm{k})$
$\displaystyle W_{S,w_{S}}(\bm{k})$ (38)
$\displaystyle=\mathcal{H}_{\text{BdG}}\left(S(\bm{k})+\mu^{3}\phi_{ST_{1}}({\bm{b}}_{1}+3{\bm{b}}_{3})\right),$

where $\bm{b}_{i}$ (with $i=1,2,3$) are the three reciprocal lattice vectors
in units of $a^{-1}$, while $\mu^{i}$ (with $i=1,2,3$) are the Pauli matrices
acting in the particle-hole space $(f,f^{\dagger})^{T}$. The derivation of
these equations together with the forms of
$W_{\overline{C}_{6},w_{\overline{C}_{6}}}(\bm{k})$ and $W_{S,w_{S}}(\bm{k})$
can be found in App. E. We point out that, due to the projective nature of the
symmetry transformation, an annihilation operation $f$ can be mapped to either
an annihilation operator $f$ or a creation operator $f^{\dagger}$. Therefore,
the unitary matrices $W_{\mathcal{O},w_{\mathcal{O}}}(\bm{k})$ with
$\mathcal{O}=\overline{C}_{6},S$ are either off-diagonal (corresponding to
$w_{\mathcal{O}}=1$) or diagonal (corresponding to $w_{\mathcal{O}}=0$) in the
particle-hole space. Let us understand the implications of this property for a
U(1) 0-flux ansatz with $w_{\overline{C}_{6}}=1$ by considering the action of
$I=\overline{C}_{6}^{3}$:

$w_{\overline{C}_{6}}=1:\quad
W_{I}^{\dagger}\mathcal{H}_{\text{U(1)}}(\bm{k})W_{I}=-\mathcal{H}^{*}_{\text{U(1)}}(\bm{k})\quad\forall\bm{k}\in\text{BZ}.$
(39)

Therefore, projective inversion acts like the product of faithful inversion
and charge conjugation, which ensures that the energies come in $\pm
E(\bm{k})$ pairs for the entire BZ.

###  III.3 0-flux U(1) ansätze with $w_{S}=1$: projective symmetry protected
gapless nodal star

Figure 2: Illustration of the nodal star Fermi surface.

Unlike the $w_{\overline{C}_{6}}=1$ classes, a 0-flux U(1) ansatz with
$w_{\overline{C}_{6}}=0$ does not necessarily have energy levels coming in
$\pm E(\bm{k})$ pairs at each momentum $\bm{k}$. However, in the case of
$w_{S}=1$ (regardless of $w_{\overline{C}_{6}}$), there exists a one-
dimensional (1D) submanifold in the BZ along which this is true. In fact,
along this submanifold,

$\Lambda=\\{(\varsigma_{1},\varsigma_{2},\varsigma_{3})k|\varsigma_{1,2,3}=\pm\\},$
(40)

the energy levels not only come in $\pm E(\bm{k})$ pairs, but are always
_gapless_ at $E=0$, i.e., there are always two degenerate modes sitting at the
Fermi energy. We call this 1D submanifold $\Lambda$ the _star_ manifold, and
refer to the gapless modes as the _nodal star_ zero modes; see Fig. 2 for an
illustration. A direct check by adding up to 8th-nearest neighbor bonds (given
in App. F) with generic bond parameters shows that the zero modes are robust
as long as the projective symmetries are intact. This is strong evidence that
the nodal star zero modes are not accidental but are protected by the
projective symmetries.

Such a nodal star structure has been reported in other contexts. In Ref.
Burnell  _et al._ (2009), Burnell _et al._ studied an SU(2) invariant ansatz
with purely imaginary NN hoppings forming $\pi/2$ fluxes on all pyrochlore
faces: the so-called “Monopole Flux” state. Such an ansatz preserves charge
conjugation, the product of inversion and time reversal $I\circ\mathcal{T}$,
and the 24 “proper elements” of the point group $O_{h}$, but it breaks the
individual time reversal and inversion symmetries. The 24 “proper elements”
and the composed symmetry $I\circ\mathcal{T}$ allow the mapping of one NN bond
to all the other NN bonds, while charge conjugation and $I\circ\mathcal{T}$
ensure that the Hamiltonian is real and an odd function of $\bm{k}$. An
algebraic proof was then given to show that the matrix structure on the star
submanifold leads to nodal star zero modes and that the symmetries forbid a
chemical potential which could otherwise gap out the nodal star. However, the
proof relies on the restriction of the hopping to the nearest neighbors, and
it is not clear in the proof if the nodal star is truly symmetry protected,
i.e., whether further neighbor symmetry allowed hoppings can gap out the nodal
star. A similar nodal star state also appears in the bosonic description of a
$\mathbb{Z}_{2}$ spin liquid Liu  _et al._ (2019) at the NN level. In that
case, it was explicitly shown that an infinitesimal NNN bond amplitude is
enough to gap out the nodal line to discrete points.

Here, we provide a proof that the nodal star zero modes appearing in our
$w_{S}=1$ classes are indeed protected by the projective symmetries, see App.
H. The proof is algebraic, and can be viewed as a generalized version of that
given in Ref. Burnell  _et al._ (2009). The proof relies on the following
observation: while an unprojective screw symmetry relates the Hamiltonian at
momentum $\bm{k}=(k_{x},k_{y},k_{z})$ with that at momentum
$(k_{y},k_{x},-k_{z})$, a projective screw symmetry relates the Hamiltonian at
momentum $\bm{k}=(k_{x},k_{y},k_{z})$ with that at momentum
$(-k_{y},-k_{x},k_{z})$. Therefore, focusing on the $(1,1,1)$ nodal line
without loss of generality, the symmetry operation

$R\equiv S\circ C_{3}\circ C_{3}\circ S\circ C_{3}\circ S$ (41)

leaves the momenta along the nodal line unchanged in the case of a projective
$S$. This symmetry constrains the Hamiltonian along the nodal line [cf. Eq.
(39)],

$W_{R}^{\dagger}(k,k,k)\mathcal{H}_{\text{U(1)}}(k,k,k)W_{R}(k,k,k)=-\mathcal{H}^{*}_{\text{U(1)}}(k,k,k),$
(42)

and implies that the energy levels come in $\pm E(\bm{k})$ pairs. Considering
the analogous action of the threefold rotation $C_{3}$ [which also maps
$(k,k,k)$ to itself],

$W_{C_{3}}^{\dagger}(k,k,k)\mathcal{H}_{\text{U(1)}}(k,k,k)W_{C_{3}}(k,k,k)=\mathcal{H}_{\text{U(1)}}(k,k,k),$
(43)

and the specific forms of $W_{R}(k,k,k)$ and $W_{C_{3}}(k,k,k)$, it can then
be shown (see App. H) that the rank of the matrix
$\mathcal{H}_{\text{U(1)}}(k,k,k)$ is at most $6$, which implies that it has
at least two zero eigenvalues. Since our proof only relies on the symmetry
properties of the spinon Hamiltonian, the result universally applies to any
fully symmetric mean-field ansatz with a projective screw symmetry
($w_{S}=1$), even beyond the NN (or NNN) level. In turn, this suggests that a
pyrochlore spin liquid with a nodal star Fermi surface may be commonplace.

##  IV Nodal star U(1) spin liquid

In this section, we restrict ourselves to the study of U(1) spin liquids with
a gapless nodal star structure as was put forward in the last subsection. Our
goal is to develop a full-fledged low energy theory whose degrees of freedom
include both the nodal star spinons and the U(1) gauge field. The gauge field
has physical consequences and may lead to observable effects. Such effects
have been explored in U(1) spin liquids with a spinon Fermi surface where the
U(1) gauge fluctuations lead to $T^{2/3}$ and $T\ln(1/T)$ scaling in the
specific heat for two and three spatial dimensions, respectively Senthil  _et
al._ (2004); Motrunich (2005). These non-Fermi liquid behaviors have been
important experimental teststones for the discovery of spinon Fermi surface
U(1) spin liquids. In this regard, it is interesting to ask how gauge
fluctuations affect the thermodynamic properties of the nodal star U(1) spin
liquid. We hope that the answer to this question provided in this section can
serve as a primer for the more interesting properties of the nodal star U(1)
spin liquid.

###  IV.1 Low energy effective model for spinon nodal bands

In this subsection, we take a specific class of nodal star spin liquid and
study its low energy properties in detail. We choose the U(1) class
$0$–$(1\;1)$–$(0\;\pi)$ and only keep the NN mean-field parameters $b=ib_{i}$
and $c=c_{r}$. Although the energy at arbitrary momentum cannot be written in
a closed form, the energies along the star
$(\varsigma_{1}k,\varsigma_{2}k,\varsigma_{3}k)\in\Lambda$ have a simple
expression:

$\displaystyle E_{1,2}=0,\quad E_{3}=-E_{4}=4\sqrt{2}c_{r},$ (44)
$\displaystyle
E_{5,6}=-E_{7,8}=\sqrt{6b_{i}^{2}+20c_{r}^{2}-6(b_{i}^{2}-2c_{r}^{2})\cos k}.$

For simplicity, we set $b_{i}=\sqrt{2}c_{r}$; this specific ansatz should be
continuously connected to those at other parameter regions. The low energy
dispersion along the nodal lines and in vicinity of the $\Gamma$ point is then
well described by the following effective Hamiltonian:

$\displaystyle\mathcal{H}(\bm{k})$
$\displaystyle=\bm{d}_{\bm{k}}\cdot\bm{\sigma},$ (45a)
$\displaystyle\bm{d}_{\bm{k}}$ $\displaystyle=\left(\begin{array}[]{c}\cos
k_{3}-\cos k_{2}\\\ \cos k_{1}-\cos k_{3}\\\ \cos k_{2}-\cos
k_{1}\end{array}\right).$ (45e)

Along each momentum section perpendicular to the star lines, the spinon field
has the dispersion of a 2+1D Dirac field, where the Dirac velocity
$v=\sqrt{3}\sin k$ is a sinusoidal function of the star momentum
$(\varsigma_{1}k,\varsigma_{2}k,\varsigma_{3}k)$. In the vicinity of the
$\Gamma$ point, the model can be further simplified by expanding
$\bm{d}_{\bm{k}}$:

$\bm{d}_{\bm{k}}=(k_{2}^{2}-k_{3}^{2},k_{3}^{2}-k_{1}^{2},k_{1}^{2}-k_{2}^{2})+O(k^{4}).$
(46)

The spinon nodal star creates an interesting instance of U(1) gauge fields
interacting with gapless matter. This is in contrast with the quantum spin ice
model established in Ref. Hermele  _et al._ (2004) where the matter fields are
gapped and the low energy description is the Maxwell theory.

###  IV.2 Nodal star spinons with U(1) gauge field

We now assume that the nodal star spinons are coupled to a U(1) gauge field.
The low energy effective Hamiltonian describing the spinon nodal bands in Eq.
(45a) corresponds to a Lagrangian

$\mathcal{L}_{0}=\psi^{\dagger}_{k}(-ik_{0}+\mathcal{H}(\bm{k}))\psi_{k},$
(47)

where we use the imaginary time formulation and denote $k=(k_{0},\bm{k})$ as
the four-momentum in Euclidean spacetime. The U(1) gauge field has a Maxwell
term

$\mathcal{L}_{M}=\frac{1}{2g^{2}}A^{\mu}({k})(k^{2}\delta_{\mu\nu}-k_{\mu}k_{\nu})A^{\nu}(-{k}),$
(48)

which emerges in this low energy effective theory by integrating out the high
energy spinon bands. Finally, the gauge field couples to the spinon fields in
the form of

$\displaystyle\mathcal{L}_{1}=$
$\displaystyle\sum_{q}iA^{0}(-{q})\psi^{\dagger}_{{k}-{q}/2}\psi_{{k}+{q}/2}$
(49)
$\displaystyle\,~{}\quad+A^{i}(-{q})\psi^{\dagger}_{{k}-{q}/2}\frac{\partial\mathcal{H}(\bm{k})}{\partial
k_{i}}\psi_{{k}+{q}/2}$
$\displaystyle+\sum_{q,q^{\prime}}A^{i}({q})A^{j}({q}^{\prime})\psi^{\dagger}_{{k}+{q}^{\prime}}\frac{\partial^{2}\mathcal{H}(\bm{k})}{\partial
k_{j}\partial k_{i}}\psi_{{k}-{q}}+O(A^{3}),$

where the first two lines are the usual minimal coupling terms and the third
line is a diamagnetic coupling term. The complete theory describing the low
energy nodal star spinons and the U(1) gauge field is thus

$\mathcal{L}=\mathcal{L}_{0}+\mathcal{L}_{M}+\mathcal{L}_{1}+\mathcal{L}_{gf}+\mathcal{L}_{gh},$
(50)

where we have also included a gauge fixing term and a ghost term for later use
Peskin (2018):

$\displaystyle\mathcal{L}_{gf}$
$\displaystyle=\frac{1}{2\xi}k_{\mu}k_{\nu}A^{\mu}(k)A^{\nu}(-k),$ (51a)
$\displaystyle\mathcal{L}_{gh}$
$\displaystyle=\frac{1}{g^{2}}\bar{\eta}_{k}k^{2}\eta_{k}.$ (51b)

It is the goal of the next subsection to derive an effective theory for the
photon field.

###  IV.3 Vacuum polarization for the emergent photons

Figure 3: The two diagrams for the photon self-energy at one loop level: the
“vacuum polarization bubble” (left) and the “tadpole” (right). Solid (wavy)
lines denote spinon (photon) propagators.

We follow the usual perturbative approach to calculate the photon effective
action within the random phase approximation (RPA). The validity of this
calculation will be commented below.

At one-loop level, the spinon-gauge coupling $\mathcal{L}_{1}$ produces two
diagrams for the photon self-energy, as shown in Fig. 3. Due to gauge
invariance, the photon self-energy must (i) vanish when $q\rightarrow 0$ and
(ii) satisfy the Ward identity at small $q$. We prove these properties at one-
loop level in Appendix H. We show that, at $q=0$, the two diagrams of Fig. 3,
the “vacuum polarization bubble” and the “tadpole”, cancel each other.
Furthermore, at $q\neq 0$, the corrections to the tadpole are only $O(q^{2})$,
while, as we will show below, there are corrections to the vacuum polarization
bubble at a lower order of $q$. Therefore, at leading order in $q$, the photon
self-energy can be identified as the $q$-dependent part of the vacuum
polarization bubble. Ignoring a minus sign resulting from the fermion loop,
the vacuum polarization bubble reads

$\Pi^{\mu\nu}(q)=\int\frac{d^{4}k}{(2\pi)^{4}}\mathrm{Tr}\left[\Gamma^{\mu}(\bm{k})G_{0}(k+\frac{q}{2})\Gamma^{\nu}(\bm{k})G_{0}(k-\frac{q}{2})\right],$
(52)

where $G_{0}(k)=\frac{1}{ik_{0}-\mathcal{H}(\bm{k})}$ is the Green’s function
for the bare spinon Lagrangian $\mathcal{L}_{0}$, and the vertex
$\Gamma^{\mu}$ obtained from Eq. (49) has the following form:

$\displaystyle\Gamma^{0}(\bm{k})=i1_{2\times 2},$
$\displaystyle\quad\Gamma^{1}(\bm{k})=\sin k_{1}(\sigma^{2}-\sigma^{3}),$ (53)
$\displaystyle\Gamma^{2}(\bm{k})=\sin k_{2}(\sigma^{3}-\sigma^{1}),$
$\displaystyle\quad\Gamma^{3}(\bm{k})=\sin k_{3}(\sigma^{1}-\sigma^{2}).$

While the anisotropic forms of the Hamiltonian [Eq. (47)] and the vertices
[Eq. (53)] make it difficult to evaluate the polarization in Eq. (52) exactly,
it is physically clear that there are two distinct momentum regions in the BZ.
These two regions, the star region and the gapped region away from the star,
result in different scalings of the photon self energy. The contribution from
the gapped region away from the nodal lines, $\Pi_{1}$, is $O(q^{2})$, as can
be directly seen from expanding the polarization in Eq. (52). The final result
for $\Pi_{1}$ is constrained by the Ward identity (see Appendix H) to have the
form of the bare Maxwell term in Eq. (48), hence it simply renormalizes the
corresponding coupling constant $g$. In contrast, the contribution from the
star region, $\Pi_{0}$, is linear in $q$ _with logarithmic corrections_. This
means that $\Pi_{0}$ completely dominates over $\Pi_{1}$ at low energy and
small momentum. In the following, we focus on the calculation of $\Pi_{0}$.

We first provide an intuitive understanding for the linear scaling
$\Pi_{0}\sim|\bm{q}|$ and the existence of logarithmic corrections. In the BZ,
each plane perpendicular to the nodal direction has a Dirac dispersion
$E=v|\bm{k}_{\perp}|$ with a Dirac velocity $v=\sqrt{3}\sin k_{\parallel}$
that is a sinusoidal function of the nodal line momentum
$(\varsigma_{1},\varsigma_{2},\varsigma_{3})k_{\parallel}$. Restricted to such
a plane, the spinon-gauge coupled system can be viewed as a QED3. The vacuum
polarization diagram in QED3 scales linearly in
$q_{\perp}=(q_{0},v\bm{q}_{\perp})$ as

$\Pi^{ab}_{\text{QED3}}=\sqrt{q_{\perp}^{2}}\left(\delta^{ab}-\frac{q_{\perp}^{a}q_{\perp}^{b}}{q_{\perp}^{2}}\right),\quad
a=0,1,2.$ (54)

The polarization of one nodal line branch can therefore be obtained by
considering copies of QED3 interacting with each other. To the leading order
of $q$, these QED3 copies are decoupled, and we find that the polarization in
the star region, $\Pi_{0}$, summed over these QED3 copies and over different
line branches, scales linearly with $q$. Note, however, that this picture is
oversimplified as the vanishing Dirac velocity at the $\Gamma$ point would
lead to unphysical divergence in the limit of $q_{0}/|\bm{q}|\rightarrow 0$
when integrating over copies of QED3:

$\displaystyle\int^{\pi}_{0}dk_{\parallel}\Pi^{00}_{\text{QED3}}$
$\displaystyle\sim\int^{\pi}_{0}dk_{\parallel}\frac{1}{\sqrt{q_{0}^{2}+\bm{q}^{2}_{\perp}\sin^{2}k_{\parallel}}}$
(55) $\displaystyle\xrightarrow{q_{0}/|\bm{q}_{\perp}|\rightarrow
0}\mathrm{divergent!}$

In reality, the quadratic dispersion near the $\Gamma$ point takes over as the
Dirac dispersion flattens, which removes the unphysical divergence and
introduces a small momentum cutoff $\theta_{0}$ for the nodal line momentum
$k_{\parallel}$. The cutoff is determined by the criterion that the Dirac
dispersion becomes comparable to the quadratic dispersion around the $\Gamma$
point, $v|\bm{k}_{\perp}|\sim\bm{k}^{2}$, which gives $\theta_{0}\sim|\bm{k}|$
and thus changes the integration range in Eq. (55) as

$\int^{\pi}_{0}dk_{\parallel}\rightarrow\int^{\pi-|\bm{q}|}_{|\bm{q}|}dk_{\parallel}.$
(56)

This cutoff introduces a logarithmic correction to the $00$, $0i$ and $i0$
components of the polarization tensor.

To understand the scaling behavior of $\Pi_{0}$ in a more rigorous manner, we
provide in Appendix I the scaling analysis of the vacuum polarization tensor
in Eq. (52) for $q_{0}/|\bm{q}|\ll 1$. We find that

$\displaystyle\Pi_{0}^{00}$
$\displaystyle\sim-|\bm{q}|\ln\left(\frac{1}{\bm{q}^{2}+\omega^{2}/\bm{q}^{2}}\right)+|\bm{q}|f_{00}(q_{0}/\bm{q}^{2}),$
(57a) $\displaystyle\Pi_{0}^{0i}$ $\displaystyle\sim
iq_{0}\ln\left(\frac{1}{\bm{q}^{2}+\omega^{2}/\bm{q}^{2}}\right)+\bm{q}^{2}f_{0i}(q_{0}/\bm{q}^{2}),$
(57b) $\displaystyle\Pi_{0}^{ij}$
$\displaystyle\sim|\bm{q}|+|\bm{q}|^{3}f_{ij}(q_{0}/\bm{q}^{2}),$ (57c)

where, in each expression, the first and the second terms denote the
contributions from the nodal lines and the region near the $\Gamma$ point,
respectively, while $f_{00}(x)$, $f_{0i}(x)$, and $f_{ij}(x)$ are regular
functions for any $0<|x|<1$. We see that, in all components of the
polarization tensor, the contribution from the $\Gamma$ region is subdominant
compared to that from the nodal lines themselves.

The analysis in the preceding two paragraphs allows us to obtain the analytic
form of the dominant contribution to the photon self-energy by performing a
“QED3 type” calculation for the nodal lines. The detailed calculation can be
found in Appendix I. Here we stress that the “QED3 type” calculation performed
here must be understood with caution. In the usual perturbative calculation of
QED3, a large parameter $N$ (the number of fermion flavors) is introduced to
ensure the validity of the RPA and the convergence in the infrared (IR) limit.
While we introduce no explicit large parameter $N$ in our calculation, such a
large $N$ should be understood to be present whenever needed, and we hope that
the result can be analytically continued to the $N=1$ case.

With this caution in mind, we present the final result here: at the leading
order of $q$, the photon self-energy is

$\Pi(iq_{0},\bm{q})=\sum_{\varsigma_{1},\varsigma_{2},\varsigma_{3}=\pm
1}\Pi_{\varsigma_{1},\varsigma_{2},\varsigma_{3}}(iq_{0},\bm{q}),$ (58)

where $\Pi_{\varsigma_{1},\varsigma_{2},\varsigma_{3}}(iq_{0},\bm{q})$ is the
contribution from the nodal line branch
$(\varsigma_{1},\varsigma_{2},\varsigma_{3})$. The individual contributions
are

$\displaystyle\Pi_{\varsigma_{1},\varsigma_{2},\varsigma_{3}}(iq_{0},\bm{q})=\frac{\sqrt{q_{0}^{2}}}{16\sqrt{3}\pi}\times$
(59)
$\displaystyle\left(\begin{array}[]{cccc}\frac{Q^{2}}{q^{2}_{0}}F{}&-\varsigma_{1}\frac{Q_{1}}{q_{0}}F{}&-\varsigma_{2}\frac{Q_{2}}{q_{0}}F{}&-\varsigma_{3}\frac{Q_{3}}{q_{0}}F{}\\\
-\varsigma_{1}\frac{Q_{1}}{q_{0}}F{}&\quad\quad\;(1+\frac{Q^{2}_{11}}{Q^{2}})F{}+(1-\frac{Q^{2}_{11}}{Q^{2}})E{}&-\varsigma_{1}\varsigma_{2}\left[(\frac{1}{2}-\frac{Q^{2}_{33}}{Q^{2}})F{}+(\frac{1}{2}+\frac{Q^{2}_{33}}{Q^{2}})E{}\right]&-\varsigma_{1}\varsigma_{3}\left[(\frac{1}{2}-\frac{Q^{2}_{11}}{Q^{2}})F{}+(\frac{1}{2}+\frac{Q^{2}_{11}}{Q^{2}})E{}\right]\\\
-\varsigma_{2}\frac{Q_{2}}{q_{0}}F{}&-\varsigma_{1}\varsigma_{2}\left[(\frac{1}{2}-\frac{Q^{2}_{33}}{Q^{2}})F{}+(\frac{1}{2}+\frac{Q^{2}_{33}}{Q^{2}})E{}\right]&\quad\quad\;(1+\frac{Q^{2}_{22}}{Q^{2}})F{}+(1-\frac{Q^{2}_{22}}{Q^{2}})E{}&-\varsigma_{2}\varsigma_{2}\left[(\frac{1}{2}-\frac{Q^{2}_{11}}{Q^{2}})F{}+(\frac{1}{2}+\frac{Q^{2}_{11}}{Q^{2}})E{}\right]\\\
-\varsigma_{3}\frac{Q_{3}}{q_{0}}F{}&-\varsigma_{1}\varsigma_{3}\left[(\frac{1}{2}-\frac{Q^{2}_{22}}{Q^{2}})F{}+(\frac{1}{2}+\frac{Q^{2}_{22}}{Q^{2}})E{}\right]&-\varsigma_{2}\varsigma_{3}\left[(\frac{1}{2}-\frac{Q^{2}_{11}}{Q^{2}})F{}+(\frac{1}{2}+\frac{Q^{2}_{11}}{Q^{2}})E{}\right]&\quad\quad\;(1+\frac{Q^{2}_{33}}{Q^{2}})F{}+(1-\frac{Q^{2}_{33}}{Q^{2}})E{}\end{array}\right),$

where

$\displaystyle F$
$\displaystyle=F(\pi-|\bm{q}|,-Q^{2}/q_{0}^{2})-F(|\bm{q}|,-Q^{2}/q_{0}^{2}),$
(60a) $\displaystyle E$
$\displaystyle=E(\pi-|\bm{q}|,-Q^{2}/q_{0}^{2})-E(|\bm{q}|,-Q^{2}/q_{0}^{2})$
(60b)

are the incomplete elliptic integrals of the first and second kinds,
respectively, with elliptic modulus $Q/q_{0}$, and

$\displaystyle Q_{1}$
$\displaystyle=2\varsigma_{1}q_{1}-\varsigma_{2}q_{2}-\varsigma_{3}q_{3},$
(61a) $\displaystyle Q_{2}$
$\displaystyle=-\varsigma_{1}q_{1}+2\varsigma_{2}q_{2}-\varsigma_{3}q_{3},$
(61b) $\displaystyle Q_{3}$
$\displaystyle=-\varsigma_{1}q_{1}-\varsigma_{2}q_{2}+2\varsigma_{3}q_{3},$
(61c) $\displaystyle Q^{2}$
$\displaystyle=\frac{1}{3}(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}),$ (61d)
$\displaystyle Q_{11}^{2}$
$\displaystyle=Q^{2}-3(\varsigma_{2}q_{2}-\varsigma_{3}q_{3})^{2},$ (61e)
$\displaystyle Q_{22}^{2}$
$\displaystyle=Q^{2}-3(\varsigma_{3}q_{3}-\varsigma_{1}q_{1})^{2},$ (61f)
$\displaystyle Q_{33}^{2}$
$\displaystyle=Q^{2}-3(\varsigma_{1}q_{1}-\varsigma_{2}q_{2})^{2}.$ (61g)

For each branch $(\varsigma_{1},\varsigma_{2},\varsigma_{3})$,
$\Pi_{\varsigma_{1},\varsigma_{2},\varsigma_{3}}(iq_{0},\bm{q})$ has two zero
eigenvalues corresponding to eigenvectors $(q_{0},q_{1},q_{2},q_{3})$ and
$(0,\varsigma_{1},\varsigma_{2},\varsigma_{3})$; the former one is the
longitudinal four-momentum vector. The remaining two nonzero eigenvalues,
$\frac{\sqrt{3}}{32\pi}\sqrt{q_{0}^{2}}E$ and
$\frac{\sqrt{3}}{32\pi}\frac{q_{0}^{2}+\frac{1}{3}Q^{2}}{\sqrt{q_{0}^{2}}}F$,
are nondegenerate; they correspond to one “transverse” eigenvector
$\left(0,\varsigma_{1}(\varsigma_{2}q_{2}-\varsigma_{3}q_{3}),\varsigma_{2}(\varsigma_{3}q_{3}-\varsigma_{1}q_{1}),\varsigma_{3}(\varsigma_{1}q_{1}-\varsigma_{2}q_{2})\right)$
and one “longitudinal” eigenvector
$(-\frac{Q^{2}}{q_{0}},\zeta_{1}Q_{1},\zeta_{2}Q_{2},\zeta_{3}Q_{3})$, where
“transverse” and “longitudinal” are understood with respect to the spatial
three-momentum $(Q_{1},Q_{2},Q_{3})$. The nondegeneracy here implies that the
boost symmetry of the QED3 is broken in our theory, which can be traced back
to the “boost symmetry breaking” of the vertices in Eq. (53).

The photon self-energy $\Pi(iq_{0},\bm{q})$ contains the longitudinal four-
momentum vector as an eigenvector corresponding to eigenvalue zero, which
ensures that the Ward identity $q_{\mu}\Pi^{\mu\nu}=0$ is preserved. However,
since the four star branches have different three-momenta
$(Q_{1},Q_{2},Q_{3})$, $\Pi(iq_{0},\bm{q})$ can no longer be decomposed into
longitudinal and transverse modes.

Suppose $e_{i}(iq_{0},\bm{q})$ with $i=1,2,3$ are the three nonzero
eigenvalues of $\Pi(iq_{0},\bm{q})$ that correspond to the eigenvectors
$v_{i}(iq_{0},\bm{q})$. The dressed photon Green’s function is then

$D^{\mu\nu}(iq_{0},\bm{q})=\sum_{i=1}^{3}\frac{P^{\mu\nu}_{i}}{\frac{1}{g^{2}}q^{2}+e_{i}(q)}+\xi\frac{q^{\mu}q^{\nu}}{q^{4}},$
(62)

where $P^{\mu\nu}_{i}=v_{i}^{\mu}v_{i}^{\nu}$ are the projectors for the
$i=1,2,3$ modes. The last term results from the gauge fixing term
$\mathcal{L}_{gf}$ in Eq. (51a).

We remind the reader that the photon self-energy calculated here is for zero
temperature. A finite temperature calculation can also be considered following
Ref. Romatschke and Säppi (2019). We leave such a calculation to future work.

###  IV.4 Photon contribution to thermodynamics

We now proceed to calculate the photon contribution to the thermodynamics. The
photon free energy reads

$F=-\frac{1}{2\beta}\sum_{n}\int\frac{d^{3}\bm{q}}{(2\pi)^{3}}\left(\ln\det
D(i\omega_{n},\bm{q})+2\ln\beta^{2}(\omega^{2}_{n}+\bm{q}^{2})\right),$ (63)

where $\omega_{n}$ is the Matsubara frequency and $D$ is the photon Matsubara
Green’s function. The second term of $F$ comes from the fictitious free energy
for the ghost fields $\eta_{k}$ in Eq. (51b). Half of this fictitious term
will cancel the contribution from the gauge fixing term $\xi
q^{\mu}q^{\nu}/q^{4}$ in the photon Green’s function, while the other half
will contribute a positive term $\propto T^{4}$ to the free energy. Such a
term would cancel out the longitudinal mode in the free gauge theory, however,
as we will see, this is no longer the case in the full theory when the photons
are coupled to the spinons. Converting the Matsubara sum to a contour
integral, we then obtain

$\displaystyle F=$
$\displaystyle\frac{\pi^{2}}{90}T^{4}+\sum_{i=1}^{3}\int^{+\infty}_{-\infty}\frac{d\omega}{2\pi}\frac{1}{e^{\beta\omega}-1}\times$
(64)
$\displaystyle\,\,\int\frac{d^{3}q}{(2\pi)^{3}}\tan^{-1}\left(\frac{-\omega
0^{+}+\mathrm{Im}e^{R}_{i}(\omega,\bm{q})}{-\omega^{2}+\bm{q}^{2}+\mathrm{Re}e^{R}_{i}(\omega,\bm{q})}\right),$

where $e_{i}^{R}(\omega,\bm{q})$ are the eigenvalues of the retarded
polarization $\Pi^{R}(\omega,\bm{q})=\Pi(iq_{0}\rightarrow\omega^{+},\bm{q})$
with the notation $\omega^{+}=\omega+i0^{+}$. Note that the dressed photon
Green’s function corresponds to zero temperature and that the temperature
dependence of the free energy comes entirely from the Boltzmann function. The
momentum-frequency integral in the free energy can be separated into two
regions that give different scaling behaviors.
_The “dynamic” region: $|\omega|/|\bm{q}|\geq 2$_. In this region, we have
$\frac{Q^{2}}{\omega^{2}}\leq\frac{4\bm{q}^{2}}{\omega^{2}}\leq 1$, and the
elliptic functions $E$ and $F$ are both real. The eigenvalues $e^{R}_{1,2,3}$
are then purely imaginary due to the prefactor
$\sqrt{q_{0}^{2}}\rightarrow-i\omega$ on the upper half plane in Eq. (59).
Furthermore, the eigenvalues are of the same amplitude:

$e^{R}_{1,2,3}\rightarrow-\frac{i\mathrm{sgn}(\omega)}{4\sqrt{3}\pi}\text{\,\,
\,\,\,when\,\, }|\omega|/|\bm{q}|\rightarrow\infty,$ (65)

which is much larger than $-\omega^{2}+\bm{q}^{2}$ at small frequency. We then
have
$\tan^{-1}\left(\frac{\mathrm{Im}e^{R}_{i}(\omega,\bm{q})}{-\omega^{2}+\bm{q}^{2}}\right)\sim-\frac{\pi}{2}\mathrm{sgn}(\omega)$
and, hence, the free energy scales with temperature as

$F_{\text{dyn}}(T)\sim
3\int^{+\infty}_{-\infty}\frac{d\omega}{2\pi}\frac{1}{e^{\beta\omega}-1}\frac{\frac{4\pi}{3}}{(2\pi)^{3}}{\frac{\omega^{3}}{8}}\left(-\frac{\pi}{2}\right)=-\frac{\pi^{2}}{{480}}T^{4},$
(66)

where we have dropped an unphysical part $F_{\text{dyn}}(0)$ which is
divergent and temperature independent. One notices that the free energy (and
other thermodynamic properties, such as the entropy and the specific heat) of
each dressed photon is a fraction of that of a free photon, which can be
viewed as being contributed by a fractional degree of freedom. Such a
phenomenon also appears at infinite coupling of large $N$ QED3 Romatschke
(2019).
_The “static” region: $|\omega|/|\bm{q}|\ll 1$._ In this region, by using the
asymptotic forms of the elliptic functions, the polarization $\Pi^{\mu\nu}$ in
Eq. (59) agrees with the general scaling form in Eq. (57). The Ward identity
$\omega\Pi^{00}+q_{i}\Pi^{i0}=0$ suggests that
$\Pi^{0i}\sim\frac{|\omega|}{|\bm{q}|}\Pi^{00}\ll\Pi^{00}$. Therefore, the
$\Pi^{00}$ component is decoupled from the $3\times 3$ block of the
polarization tensor with spatial indices and is identified with one of the
eigenvalues, $e^{R}_{3}$. The remaining eigenvalues $e^{R}_{1,2}$ then
correspond to the two transverse polarization modes. All three eigenvalues
$e^{R}_{1,2,3}$ are generally complex, and we find the following scaling form
for them:

$\displaystyle e^{R}_{1,2}$
$\displaystyle\sim-i\omega\left[E\left(0,\frac{2\bm{q}^{2}}{(\omega^{+})^{2}}\right)-F\left(0,\frac{2\bm{q}^{2}}{(\omega^{+})^{2}}\right)\right]$
$\displaystyle\sim\ |\bm{q}|-i\frac{\omega^{2}}{|\bm{q}|}\text{sgn}(\omega),$
(67a) $\displaystyle e^{R}_{3}$ $\displaystyle\sim
i\frac{\bm{q}^{2}}{\omega}\left[F\left(\pi-\theta_{0},\frac{2\bm{q}^{2}}{(\omega^{+})^{2}}\right)-F\left(\theta_{0},\frac{2\bm{q}^{2}}{(\omega^{+})^{2}}\right)\right]$
$\displaystyle\xrightarrow{\theta_{0}\sim|\bm{q}|}|\bm{q}|\ln\left(\frac{1}{\max\\{\frac{|\omega|}{|\bm{q}|},|\bm{q}|\\}}\right)$
$\displaystyle\qquad\quad+i|\bm{q}|u\left(\frac{|\omega|}{|\bm{q}|}\right)\text{sgn}(\omega)\theta\left(\frac{|\omega|}{|\bm{q}|}-|\bm{q}|\right),$
(67b)

where $u(\frac{|\omega|}{|\bm{q}|})$ scales as
$u(x)\sim\mathrm{const.}+x^{2}$. The eigenvalues of the transverse modes,
$e^{R}_{1,2}$, do not diverge near $\Gamma$ or $\mathrm{L}$, therefore the
cutoff $\theta_{0}$ has been set to zero.

In writing these asymptotic expressions, we have neglected the angular
dependence in the momentum. The validity of this approximation has been
numerically verified. For $|\omega|>\bm{q}^{2}$, the physical modes $i=1,2,3$
lead to a dressed photon Green’s function of the form

$\displaystyle D^{\mu\nu}=$
$\displaystyle\frac{P^{\mu\nu}_{1}+P^{\mu\nu}_{2}}{\frac{1}{g^{2}}(\bm{q}^{2}c^{2}-\omega^{2})+\frac{1}{2\sqrt{3}\pi}(|\bm{q}|-i\text{sgn}(\omega)\frac{\omega^{2}}{|\bm{q}|})}$
(68)
$\displaystyle+\frac{P^{\mu\nu}_{3}}{\frac{1}{g^{2}}(\bm{q}^{2}c^{2}-\omega^{2})+\frac{1}{2\sqrt{3}\pi}(|\bm{q}|\ln\frac{|\bm{q}|}{|\omega|}+i\text{sgn}(\omega)|\bm{q}|)}.$

Note that the imaginary parts have opposite signs and that the
$P^{\mu\nu}_{3}$ term will contribute negatively to the specific heat. The
photon “bare” velocity $c$ and the coupling $g$ come from integrating out the
gapped spinon bands at higher energies. The “bare” velocity $c$ should be
comparable to the mean Dirac velocity of the nodal line spinons, $c\sim 1$,
and we take $g\lesssim 1$ following Ref. Gan and Wong (1993). Therefore, at
small $\bm{q}$, we have $|\bm{q}|>\bm{q}^{2}>\omega^{2}$, and we only keep
$|\bm{q}|$ in the real part. The free energy can then be written as
$F_{\text{sta}}=F^{1,2}_{\text{sta}}+F^{3}_{\text{sta}}$, where the
contributions $F^{1,2}_{\text{sta}}$ and $F^{3}_{\text{sta}}$ correspond to
the eigenvalues $e^{R}_{1,2}$ and $e^{R}_{3}$, respectively:

$\displaystyle F^{1,2}_{\text{sta}}$
$\displaystyle=-2\int_{0}^{\Lambda}\frac{q^{2}dq}{2\pi^{2}}\int^{|\bm{q}|}_{0}\frac{d\omega}{2\pi}\frac{1}{e^{\beta\omega}-1}2\tan^{-1}\frac{\omega^{2}}{\bm{q}^{2}},$
(69a) $\displaystyle F^{3}_{\text{sta}}$
$\displaystyle=-2\int_{0}^{\Lambda}\frac{q^{2}dq}{2\pi^{2}}\int^{<|\bm{q}|}_{\bm{q}^{2}}\frac{d\omega}{2\pi}\frac{1}{e^{\beta\omega}-1}\tan^{-1}\frac{1}{\ln\frac{|\omega|}{|\bm{q}|}},$
(69b)

where the dimensionless parameter $\Lambda$ is the upper momentum cutoff at
which the nodal line approximation becomes invalid. Setting
$x=\omega/|\bm{q}|$ in $F^{1,2}_{\text{sta}}$, we obtain

$F^{1,2}_{\text{sta}}\sim-\int^{\Lambda}_{0}q^{3}dq\int^{1}_{0}dx\frac{x^{2}}{e^{\beta|\bm{q}|x}-1}\sim-\int^{\Lambda}_{0}dqq^{3}F\left(\frac{|\bm{q}|}{T}\right),$
(70)

where $F(y)=\int^{1}_{0}dx\frac{x^{2}}{e^{yx}-1}$. For $y\rightarrow 0$, we
have $F(y)=\frac{1}{2y}+O(1)$, while for $y\rightarrow+\infty$, the upper
limit can be extended to $+\infty$, and we have $F(y)\rightarrow
2\zeta(3)/y^{3}$. Therefore, at low temperatures, $T<\Lambda$, we have
$F^{1,2}_{\text{sta}}\sim-2(\int^{T}_{0}dqq^{3}\frac{T}{q}+\int^{\Lambda}_{T}dqq^{3}\frac{T^{3}}{q^{3}})\sim-\Lambda
T^{3}+O(T^{4})$, and the leading contribution to free energy is $-\Lambda
T^{3}$.

For the remaining contribution $F^{3}_{\text{sta}}$, we perform the following
transformation:

$\displaystyle F^{3}_{\text{sta}}$ $\displaystyle\xrightarrow{\omega\equiv
q^{\alpha+1}}-2\int^{\Lambda}_{0}q^{2}dq\int^{\alpha_{0}>0}_{1}\frac{q^{\alpha+1}}{\alpha(e^{\beta
q^{\alpha+1}}-1)}d\alpha$ (71) $\displaystyle\xrightarrow{z\equiv\beta
q^{\alpha+1}}2\int^{1}_{\alpha_{0}>0}\frac{d\alpha}{\alpha(\alpha+1)}T^{\frac{\alpha+4}{\alpha+1}}\int^{\frac{\Lambda^{\alpha+1}}{T}}_{0}\frac{z^{\frac{3}{\alpha+1}}}{e^{z}-1}dz.$

Numerics show that $F^{3}_{\text{sta}}$ is independent of $\alpha_{0}$
whenever $\alpha_{0}<0.5$. Note also that, when $z\gg 1$, we can approximate
$e^{z}-1\sim e^{z}$, meaning that, at large $z$, the integral will contribute
to the free energy with an exponentially small term $e^{-\frac{1}{T}}$.
Therefore, we can safely extend the upper limit to infinity, and the integral
in $z$ then gives

$\int^{\infty}_{0}\frac{z^{\frac{3}{\alpha+1}}}{e^{z}-1}dz=\Gamma\left(\frac{4+\alpha}{1+\alpha}\right)\mathrm{Li}_{\frac{4+\alpha}{1+\alpha}}(1).$
(72)

Since it is approximately true for $0.5<\alpha<1$ that

$\frac{\Gamma\left(\frac{4+\alpha}{1+\alpha}\right)\mathrm{Li}_{\frac{4+\alpha}{1+\alpha}}(1)}{\alpha(\alpha+1)}\sim\frac{0.84}{\alpha^{2}},$
(73)

the contribution $F^{3}_{\text{sta}}$ takes the leading-order form

$F^{3}_{\text{sta}}\sim\int^{1}_{\alpha_{0}}\frac{1}{\alpha^{2}}T^{\frac{\alpha+4}{\alpha+1}}d\alpha\sim-\frac{T^{\frac{5}{2}}}{\ln
T}+O\left(\frac{T^{\frac{5}{2}}}{\ln^{2}T}\right).$ (74)

From all the analysis above, we conclude that the contribution
$F^{3}_{\text{sta}}$ dominates the dressed photon free energy at low
temperature. Note that this dominant term contributes negatively to the
specific heat.

###  IV.5 Final result for the specific heat

At the non-interacting level, the temperature scaling of the spinon free
energy can be obtained from the spinon density of states by a simple power
counting. The spinons near the nodal lines and the $\Gamma$ point have
densities of states $g(\epsilon)\propto\epsilon$ and
$g(\epsilon)\propto\sqrt{\epsilon}$, and contribute to the specific heat as
$c_{v}\propto T^{2}$ and $c_{v}\propto T^{\frac{3}{2}}$, respectively. The
final result for specific heat is then

$c_{v}\sim T^{\frac{3}{2}}+\frac{T^{\frac{3}{2}}}{\ln T}+\text{subleading
terms}.$ (75)

Compared to a U(1) QSL with gapped matter fields, the leading term in the
specific heat has a lower power law exponent, $c_{v}\propto T^{3/2}$, while
the subleading term has a negative contribution at low temperature. Since the
Dirac velocity $v$ is related to the pyrochlore spin exchange $J$ by $v\sim
Ja/\hbar$, the small value of $J$ (typically a few meV) indicates that this
$T^{3/2}$ scaling likely dominates at low temperature over non-magnetic
contributions and may serve as strong evidence for the observation of a nodal
star spin liquid.

##  V Discussion and outlook

###  V.1 Summary

In this paper, we obtain the complete classification of spin-orbit-coupled
spin liquids with either $\mathbb{Z}_{2}$ or U(1) gauge structure on the
pyrochlore lattice, within the PSG framework for Abrikosov fermions. We find
that there are at most 18 U(1) and 28 $\mathbb{Z}_{2}$ PSG classes with full
pyrochlore space group symmetry, and that the number of classes reduces to 16
for U(1) and increases to 48 for $\mathbb{Z}_{2}$ if time reversal symmetry is
further imposed. We present the explicit form of the mean-field Hamiltonian
for each PSG class upon gauge fixing. We also show that, in the U(1) case,
several classes of mean-field ansätze possess robust spinon zero modes along
high symmetry lines in the Brillouin zone and that these nodal lines are
protected by the projective screw symmetry. A low energy effective theory for
the nodal line spinons coupled to U(1) gauge fields is given. Finally, we
calculate the spinon contribution to the photon self energy at one loop level
and study the thermodynamics of the dressed photon within the RPA
approximation. We find that the most dominant contributions to the specific
heat are $T^{3/2}$ from the bare spinons and $T^{3/2}/\ln T$ from the dressed
photons.

###  V.2 $\mathbb{Z}_{2}$ PSGs from fermionic and bosonic partons

We now discuss the relationship between the $\mathbb{Z}_{2}$ PSGs obtained
from fermionic and bosonic partons. In a previous work Liu  _et al._ (2019),
we employed Schwinger bosons to classfy $\mathbb{Z}_{2}$ spin liquids on the
pyrochlore lattice with full lattice and time reversal symmetries. There, we
found 16 distinct PSG classes and labeled them by four $\mathbb{Z}_{2}$
parameters, $n_{1}$, $n_{ST_{1}}$, $n_{\overline{C}_{6}S}$, and
$n_{\overline{C}_{6}}$, that are the bosonic counterparts of the $\chi$’s in
the current work. The other IGG parameters are all related to these four
parameters, for example, we have
$n_{S\overline{C}_{6}}=n_{1}+n_{\overline{C}_{6}}+n_{ST_{1}}$,
$n_{\mathcal{T}\overline{C}_{6}}=n_{\overline{C}_{6}}$, and
$n_{\mathcal{T}S}=n_{1}+n_{ST_{1}}$.

By comparing these bosonic quantum numbers with the fermionic ones, we see
that each bosonic class corresponds to an appropriate class in the fermionic
PSG with lattice and time reversal symmetries. In fact, the bosonic classes
all have their counterparts already in the fermionic PSG with only pyrochlore
space group symmetry through the corresponding $\mathbb{Z}_{2}$ quantum
numbers
($\eta_{S\overline{C}_{6}}=\eta_{1}\eta_{ST_{1}}\eta_{\overline{C}_{6}}$). The
fermionic PSG, however, has a larger number of classes, some of which do not
have bosonic counterparts. From Appendix C, it is seen that the additional
fermionic classes exist due to the violation of the condition
$\eta_{S\overline{C}_{6}}=\eta_{1}\eta_{ST_{1}}\eta_{\overline{C}_{6}}$ or as
a result of multiple solutions to the SU(2) equation (which are distinguished
by an additional discrete parameter $j$). Upon imposing time reversal
symmetry, the number of fermionic classes increases from 28 to 48, and the 16
bosonic classes still have 16 counterparts among them.

Physically, the bosonic and fermionic PSGs are supposed to describe
fractionalized excitations with bosonic and fermionic statistics,
respectively: the bosonic and fermionic spinons. This has been well understood
for 2D $\mathbb{Z}_{2}$ QSLs with topological order. In the 2D case, the
elementary (fractionalized) excitations are bosonic spinons, fermionic
spinons, and visons. A fermionic spinon can be viewed as a bound state of a
bosonic spinon and a vison, which induces a corresponding product rule between
the vison, boson, and fermion PSGs. It was argued in Ref. Lu (2018) on general
grounds that the classes common in bosonic and fermionic PSGs realize gapped
$\mathbb{Z}_{2}$ symmetric spin liquids, while the additional classes in the
fermionic PSG realize symmetry protected gapless $\mathbb{Z}_{2}$ spin
liquids. However, it is not clear whether the claim directly applies to our
case. Concretely, one can compare the independent nonzero mean-field
parameters at a given bond level between the corresponding bosonic and
fermionic classes, and the numbers do not always match. The possible reasons
are that (i) the required assumption of U(1) spin symmetry for the proof of
Ref. Lu (2018) is absent in our case, and (ii) the dimensionality is different
in our case. Indeed, the dimensional augmentation to 3D may fundamentally
change the correspondence between the bosonic and fermionic PSGs since the
visons are now line-like objects and do not straightforwardly relate bosonic
and fermionic spinons to each other.

Understanding the relation between the fermionic and bosonic PSG
classifications is an important goal. In addition to extracting the statistics
of the fractionalized excitations discussed above, it can be used to map out
the phases proximate to a QSL and the possible transition types. Indeed, the
bosonic representation has the fundamental advantage that it can describe a
transition to a magnetically ordered state via the condensation of bosonic
spinons, while such a transition cannot be easily described in the fermionic
representation. Therefore, we hope to establish a clearer understanding of
this important relation in a future work.

###  V.3 $\mathbb{Z}_{2}$ and U(1) PSGs using fermionic partons

The $\mathbb{Z}_{2}$ mean-field ansätze have spinon pairing terms which
manifestly break the U(1) symmetry down to its subgroup $\mathbb{Z}_{2}$.
However, in special cases, when some of the mean-field parameters are switched
off, the $\mathbb{Z}_{2}$ ansätze may possess an enlarged symmetry. If this
enlarged symmetry is (or contains) U(1), the ansatz with parameters switched
off belongs to some “root” U(1) PSG class and the $\mathbb{Z}_{2}$ PSG can be
viewed as being derived from this U(1) PSG class by “gauge symmetry breaking”
via the Higgs mechanism. The simplest way one can enlarge $\mathbb{Z}_{2}$
symmetry to U(1) is by switching off all the pairing parameters in a
$\mathbb{Z}_{2}$ ansatz. If this can be consistently done without violating
the PSG, we obtain an explicit U(1) ansatz with only hopping terms. For
example, if we take the non-projective $\mathbb{Z}_{2}$ class $(00)$–$(000)$
and naively switch off the pairings, we get exactly the U(1) non-projective
mean-field state $0$–(00)–(00).

However, the correspondence between a $\mathbb{Z}_{2}$ and a U(1) ansatz may
not always be apparent and may be masked by the different gauge fixing
conventions used for the $\mathbb{Z}_{2}$ and the U(1) ansätze. For example, a
mean-field Hamiltonian with only singlet pairing terms ($a_{p}$ and its
equivalents at further neighbor bonds) also has a U(1) symmetry. This pairing
U(1) symmetry can be converted to the usual hopping U(1) symmetry by an
appropriate gauge transformation. For $\mathbb{Z}_{2}$ classes with time
reversal and
$(\chi_{\mathcal{T}\overline{C}_{6}},\chi_{\mathcal{T}S},k)=(0,0,3)$ (see
Table 2), such a gauge transformation can be chosen as

$W=e^{-i\frac{\pi}{4}\sigma^{2}},$ (76)

which transforms the time reversal PSG according to
$W_{\mathcal{T}}(\bm{r}_{\mu})=i\sigma^{3}\rightarrow
WW_{\mathcal{T}}(\bm{r}_{\mu})W^{\dagger}=i\sigma^{1}$. The mean-field
parameters therefore transform as

$i(\mathrm{Re}a\sigma^{1}+\mathrm{Im}b\sigma^{2})\rightarrow
i(-\mathrm{Re}a\sigma^{3}+\mathrm{Im}b\sigma^{2}),$

i.e., the real part of the pairing term is transformed into a hopping term,
which is consistent with Eq. (32). In this case, keeping only the real part of
the singlet pairing will recover a U(1) ansatz.

In closing this subsection, we point out that a general mapping between
$\mathbb{Z}_{2}$ and U(1) pyrochlore PSG classes is still lacking.
Understanding such a relation will be important in mapping out phase diagrams
containing various spin liquids and magnetic orders.

###  V.4 The non-projective U(1) class: topological insulator

Let us examine the topological properties of the U(1) PSG class
$0$–$(00)$–$(00)$: this is the “trivial” class in which symmetries are
realized linearly (i.e., nonprojectively). This symmetry structure also
applies to physical electrons instead of spinons: it can describe ordinary,
non-fractionalized itinerant electrons on the pyrochlore lattice. Such systems
have been intensely studied in the context of pyrochlore iridates Kurita  _et
al._ (2011); Guo and Franz (2009); Witczak-Krempa  _et al._ (2013). There, the
most striking prediction from theory is the existence of a topological
insulator phase, which occupies a finite volume in the phase space spanned by
spin-orbit couplings up to NNN Kurita  _et al._ (2011); Guo and Franz (2009);
Witczak-Krempa  _et al._ (2013).

The most complete of these prior works Kurita  _et al._ (2011); Guo and Franz
(2009); Witczak-Krempa  _et al._ (2013) is Ref. Witczak-Krempa  _et al._
(2013), which determined the general form of the Hamiltonian up to second
neighbor hoppings. Here, we provide an explicit mapping between the parameters
used there and those used in Table 3: there are in total two real independent
parameters $(t_{1},t_{2})$ for the NN bonds and three real independent
parameters $(t^{\prime}_{1},t^{\prime}_{2},t^{\prime}_{3})$ for the NNN bonds
Witczak-Krempa  _et al._ (2013), which are related to our PSG results by
$(t_{1},t_{2},t^{\prime}_{1},t^{\prime}_{2},t^{\prime}_{3})=(\mathrm{Im}a,\mathrm{Im}c,\mathrm{Im}A,\mathrm{Im}{B}+\mathrm{Im}{D},\mathrm{Im}{B}-\mathrm{Im}{D})$.
This serves as a partial check of our classification and allows the results in
Refs Kurita  _et al._ (2011); Guo and Franz (2009); Witczak-Krempa  _et al._
(2013) to directly apply in the PSG context.

Given the existence of a topological insulator phase in the “trivial” U(1) PSG
class, it is reasonable to believe that other classes may also support
nontrivial topological phases. Among them, it would be of specific interest to
identify those that are protected by the _projectiveness_ of the symmetry and
would appear only in systems with fractionalized degrees of freedom.

###  V.5 Future directions

The mean-field ansätze for the PSG classes listed in this work provide
abundant ground state candidates for model spin Hamiltonians on the pyrochlore
lattice. In the works reported so far, the monopole flux state Burnell  _et
al._ (2009) has the lowest energy as a variational mean-field ansatz for the
pyrochlore Heisenberg model. The monopole flux state does not belong to our
classification with the full pyrochlore lattice symmetry since it
spontaneously breaks lattice inversion. An interesting question is then
whether any of the fully symmetric states may be energetically favored by the
Heisenberg model, and if not, what is the physical reason for the energy being
lowered by spontaneous symmetry breaking. In this regard, it is interesting to
study the PSG classification of chiral spin liquids on the pyrochlore lattice,
in which certain space group symmetries are replaced by them composed with
time reversal symmetries. Furthermore, since rare-earth pyrochlore materials
are intrinsically spin-orbit coupled, it is also natural to take our PSG
ansätze as variational states for the full spin-orbit coupled pyrochlore
exchange model Ross  _et al._ (2011). These questions are addressed in an
ongoing work that will be reported elsewhere.

One outcome of this work is the realization, in several PSG classes, of the
nodal star U(1) spin liquid, which represents a new family of pyrochlore U(1)
spin liquids beyond the known prototypes, whose low energy nodal structure is
protected by the pyrochlore space group symmetries. We point out that the
proof of the symmetry protected nature of the nodal lines also applies to
several classes of chiral spin liquids, including the monopole flux state
Burnell  _et al._ (2009). In the present work, our main focus has been on the
spinon corrections to the gauge field. Subsequent questions – such as how the
gauge field feeds back into the spinons and how the vertices receive
corrections – have been outside the scope of this work and require a more
involved calculation. These calculations may reveal additional contributions
to the thermodynamic properties and will provide insights for another
important observable, the spin susceptibility. Even at the non-interacting
level, the nodal line spinons will lead to spectral features that should be
observable in, e.g., neutron scattering experiments. For example, a broad low
energy continuum should be seen along appropriate high-symmetry planes of the
Brillouin zone. The observation of such signatures may serve as direct
evidence for a pyrochlore spin liquid state. We leave the study of these
aspects of the nodal star U(1) spin liquid to a future work.

There are also several more broad directions to be explored. The pyrochlore
PSG may be studied from the perspective of symmetry protected crystalline
insulators and symmetry enriched topological orders. Apart from the case of
the non-projective U(1) ansätze mentioned in the preceding subsection, the
topological aspects of the spinon bands have not been investigated in this
work and deserve further study. Another future direction is the stability of
the gapless states beyond the mean field approximation. While at the latter
level, we have proven that the nodal line is symmetry protected, it is not
clear if it remains robust against symmetry allowed spinon interactions. In
this regard, it would be interesting to look for criteria that forbid a single
gapped many-body ground state of the spinon Hamiltonian from appearing in the
presence of spinon interactions and full pyrochlore symmetry, similar to the
proposals of Ref. Lu (2018). In the context of $\mathbb{Z}_{2}$ QSLs, a
related theme would be to generalize the Lieb-Schultz-Mattis theorem to the
pyrochlore lattice and other geometrically frustrated 3D lattice types. The
screw symmetry, which is crucial to the nodal star spin liquids, is an
instance of nonsymmorphic symmetry and, in this regard, it is compelling to
connect our results with recent works, such as Refs. Watanabe  _et al._
(2015); Else and Thorngren (2020) and especially Ref. Thorngren and Else
(2018) (considering that the classification result for the pyrochlore space
group there is missing).

###### Acknowledgements.

We acknowledge Yasir Iqbal, Federico Becca, Francesco Ferrari, and Mengxing Ye
for helpful discussions. C.L. and L.B. were supported by the DOE, Office of
Science, Basic Energy Sciences under Award No. DE-FG02-08ER46524. This
research benefitted from the facilities of the Kavli Institute for Theoretical
Physics, supported in part by the National Science Foundation under Grant No.
NSF PHY-1748958. The work of G.B.H. was supported by the U.S. Department of
Energy, Office of Science, National Quantum Information Science Research
Centers, Quantum Science Center.

##  Appendix A Solving U(1) PSG equations: space group part

The space group part of the PSG equations are

$\displaystyle(G_{T_{i}}T_{i})(G_{T_{i+1}}T_{i+1})(G_{T_{i}}T_{i})^{-1}(G_{T_{i+1}}T_{i+1})^{-1}\in\text{IGG},$
(77a) $\displaystyle(G_{\overline{C}_{6}}\overline{C}_{6})^{6}\in\text{IGG},$
(77b) $\displaystyle(G_{S}S)^{2}(G_{T_{3}}T_{3})^{-1}\in\text{IGG},$ (77c)
$\displaystyle(G_{\overline{C}_{6}}\overline{C}_{6})(G_{T_{i}}T_{i})(G_{\overline{C}_{6}}\overline{C}_{6})^{-1}(G_{T_{i+1}}T_{i+1})\in\text{IGG},$
(77d)
$\displaystyle(G_{S}S)(G_{T_{i}}T_{i})(G_{S}S)^{-1}(G_{T_{3}}T_{3})^{-1}(G_{T_{i}}T_{i})\in\text{IGG},$
(77e)
$\displaystyle(G_{S}S)(G_{T_{3}}T_{3})(G_{S}S)^{-1}(G_{T_{3}}T_{3})^{-1}\in\text{IGG},$
(77f)
$\displaystyle{}[(G_{\overline{C}_{6}}\overline{C}_{6})(G_{S}S)]^{4}\in\text{IGG},$
(77g)
$\displaystyle{}[(G_{\overline{C}_{6}}\overline{C}_{6})^{3}(G_{S}S)]^{2}\in\text{IGG}.$
(77h)

The corresponding SU(2) equations are

$\displaystyle
W_{T_{i}}(\bm{r}_{\mu})W_{T_{i+1}}[T^{-1}_{i}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W^{-1}_{T_{i}}[T^{-1}_{i+1}(\bm{r}_{\mu})]W^{-1}_{T_{i+1}}(\bm{r}_{\mu})$
$\displaystyle=e^{i\sigma^{3}\chi_{i}},$ (78a) $\displaystyle
W_{\overline{C}_{6}}(\bm{r}_{\mu})W_{\overline{C}_{6}}[\overline{C}_{6}^{-1}(\bm{r}_{\mu})]\cdot$
(78b) $\displaystyle
W_{\overline{C}_{6}}[\overline{C}_{6}^{-2}(\bm{r}_{\mu})]W_{\overline{C}_{6}}[\overline{C}_{6}^{-3}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W_{\overline{C}_{6}}[\overline{C}_{6}^{-4}(\bm{r}_{\mu})]W_{\overline{C}_{6}}[\overline{C}_{6}^{-5}(\bm{r}_{\mu})]$
$\displaystyle=e^{i\sigma^{3}\chi_{\overline{C}_{6}}}$ (78c) $\displaystyle
W_{S}(\bm{r}_{\mu})W_{S}[S^{-1}(\bm{r}_{\mu})]W^{-1}_{T_{3}}(\bm{r}_{\mu})$
$\displaystyle=e^{i\sigma^{3}\chi_{S}},$ (78d) $\displaystyle
W_{\overline{C}_{6}}(\bm{r}_{\mu})W_{T_{i}}[\overline{C}^{-1}_{6}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W^{-1}_{\overline{C}_{6}}[T_{i+1}(\bm{r}_{\mu})]W_{T_{i+1}}[T_{i+1}(\bm{r}_{\mu})]$
$\displaystyle=e^{i\sigma^{3}\chi_{\overline{C}_{6}T_{i}}},$ (78e)
$\displaystyle W_{S}(\bm{r}_{\mu})W_{T_{i}}[S^{-1}(\bm{r}_{\mu})]\cdot$
$\displaystyle W^{-1}_{S}[T^{-1}_{3}T_{i}(\bm{r}_{\mu})]\cdot$ $\displaystyle
W^{-1}_{T_{3}}[T_{i}(\bm{r}_{\mu})]W_{T_{i}}[T_{i}(\bm{r}_{\mu})]$
$\displaystyle=e^{i\sigma^{3}\chi_{ST_{i}}},$ (78f) $\displaystyle
W_{S}(\bm{r}_{\mu})W_{T_{3}}[S^{-1}(\bm{r}_{\mu})]\cdot$ $\displaystyle
W^{-1}_{S}[T^{-1}_{3}(\bm{r}_{\mu})]W^{-1}_{T_{3}}(\bm{r}_{\mu})$
$\displaystyle=e^{i\sigma^{3}\chi_{ST_{3}}},$ (78g) $\displaystyle
W_{\overline{C}_{6}}(\bm{r}_{\mu})W_{S}[\overline{C}_{6}^{-1}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W_{\overline{C}_{6}}[(\overline{C}_{6}S)^{-1}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W_{S}[(\overline{C}_{6}S\overline{C}_{6})^{-1}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W_{\overline{C}_{6}}[(\overline{C}_{6}S\overline{C}_{6}S)^{-1}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W_{S}[(\overline{C}_{6}S\overline{C}_{6}S\overline{C}_{6})^{-1}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W_{\overline{C}_{6}}[(\overline{C}_{6}S\overline{C}_{6}S\overline{C}_{6}S)^{-1}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W_{S}[(\overline{C}_{6}S\overline{C}_{6}S\overline{C}_{6}S\overline{C}_{6})^{-1}(\bm{r}_{\mu})]$
$\displaystyle=e^{i\sigma^{3}\chi_{\overline{C}_{6}S}},$ (78h) $\displaystyle
W_{\overline{C}_{6}}[\overline{C}_{6}^{-2}(\bm{r}_{\mu})]W_{S}[\overline{C}_{6}^{-3}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W_{\overline{C}_{6}}[(\overline{C}_{6}^{3}S)^{-1}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W_{\overline{C}_{6}}[(\overline{C}_{6}^{3}S\overline{C}_{6})^{-1}(\bm{r}_{\mu})]\cdot$
$\displaystyle
W_{\overline{C}_{6}}[(\overline{C}_{6}^{3}S\overline{C}^{2}_{6})^{-1}(\bm{r}_{\mu})]W_{S}[S(\bm{r}_{\mu})]$
$\displaystyle=e^{i\sigma^{3}\chi_{S\overline{C}_{6}}},$ (78i)

where all the $\chi\in[0,2\pi)$ for U(1) IGG, and $\chi\in\\{0,\pi\\}$ for
$\mathbb{Z}_{2}$ IGG.

The general form for $W_{\mathcal{O}}(\bm{r})$ is
$W_{\mathcal{O}}(\bm{r})=(i\sigma^{1})^{w_{\mathcal{O}}}e^{i\sigma^{3}\phi_{\mathcal{O}}(\bm{r})}$,
where $w_{\mathcal{O}}=0$ or $1$,
$\mathcal{O}\in\\{T_{1},T_{2},T_{3},\overline{C}_{6},S\\}$. From Eq. (78f) we
see we must have $w_{T_{3}}=0$, further from Eq. (78e) we have
$w_{T_{1}}=w_{T_{2}}=0$. Therefore Eq. (78a) becomes a pure phase equation

$\phi_{T_{i}}(\bm{r}_{\mu})+\phi_{T_{i+1}}[T_{i}^{-1}(\bm{r}_{\mu})]-\phi_{T_{i}}[T_{i+1}^{-1}(\bm{r}_{\mu})]-\phi_{T_{i+1}}(\bm{r}_{\mu})=\chi_{i},$
(79)

using gauge freedom to set $\phi_{T_{1}}(\bm{r}_{\mu})=0$,
$\phi_{T_{2}}(r_{1},0,0)_{\mu}=0$, and $\phi_{T_{3}}(r_{1},r_{2},0)=0$, we
have

$\phi_{T_{1}}(\bm{r}_{\mu})=0,\ \ \
\phi_{T_{2}}(\bm{r}_{\mu})=-\chi_{1}r_{1},\ \ \
\phi_{T_{3}}(\bm{r}_{\mu})=\chi_{3}r_{1}-\chi_{2}r_{2}.$ (80)

Eq. (78e) gives

$\displaystyle(-1)^{w_{\overline{C}_{6}}}\Big{(}\phi_{\overline{C}_{6}}(\bm{r}_{\mu})+\phi_{T_{i}}[\overline{C}^{-1}_{6}(\bm{r}_{\mu})]-\phi_{\overline{C}_{6}}[T_{i+1}$
$\displaystyle(\bm{r}_{\mu})]\Big{)}$ (81)
$\displaystyle+\phi_{T_{i+1}}[T_{i+1}(\bm{r}_{\mu})]$
$\displaystyle=\chi_{\overline{C}_{6}T_{i}},$

consistency condition

$\displaystyle\Delta_{i}\phi_{\overline{C}_{6}}(\bm{r}_{\mu})+\Delta_{i+1}\phi_{\overline{C}_{6}}[T_{i}^{-1}(\bm{r}_{\mu})]$
(82)
$\displaystyle=\Delta_{i+1}\phi_{\overline{C}_{6}}(\bm{r}_{\mu})+\Delta_{i}\phi_{\overline{C}_{6}}[T_{i+1}^{-1}(\bm{r}_{\mu})],\quad
i=1,2,3$

where we defined
$\Delta_{i}\phi(\bm{r}_{\mu})=\phi(\bm{r}_{\mu})-\phi[T_{i}^{-1}(\bm{r}_{\mu})]$,
requires

$\displaystyle(-1)^{w_{\overline{C}_{6}}}\chi_{1}=\chi_{3},$
$\displaystyle-\chi_{1}+(-1)^{w_{\overline{C}_{6}}}\chi_{2}=0,\ \ \ $ (83)
$\displaystyle\chi_{2}$ $\displaystyle=(-1)^{w_{\overline{C}_{6}}}\chi_{3},$

which means that, if $w_{\overline{C}_{6}}=0$, then
$\chi_{1}=\chi_{2}=\chi_{3}$ and no quantization condition is imposed on
$\chi_{1,2,3}$; if $w_{\overline{C}_{6}}=1$, then
$\chi_{1}=\chi_{2}=\chi_{3}=0$ or $\pi$. Combining the two cases we can write

$\displaystyle\phi_{\overline{C}_{6}}(\bm{r}_{\mu})=$ $\displaystyle-
r_{1}(r_{2}-r_{3})\chi_{1}$ (84)
$\displaystyle-(\delta_{\mu,2}-\delta_{\mu,3})\chi_{1}r_{1}+\delta_{\mu,2}\chi_{1}r_{3}$
$\displaystyle-(-1)^{w_{\overline{C}_{6}}}(\chi_{\overline{C}_{6}T_{3}}r_{1}+\chi_{\overline{C}_{6}T_{1}}r_{2}+\chi_{\overline{C}_{6}T_{2}}r_{3})$
$\displaystyle+\rho_{\mu},$

where we abbreviated $\rho_{\mu}\equiv\phi_{\overline{C}_{6}}(0,0,0)_{\mu}$.
Plug the expression (84) in Eq. (78c), we get the condition

  * •

When $w_{\overline{C}_{6}}=0$,

$\displaystyle 6\rho_{0}$
$\displaystyle=2(\rho_{1}+\rho_{2}+\rho_{3})+\chi_{\overline{C}_{6}T_{1}}+\chi_{\overline{C}_{6}T_{2}}+\chi_{\overline{C}_{6}T_{3}}$
(85) $\displaystyle=\chi_{\overline{C}_{6}};$

  * •

when $w_{\overline{C}_{6}}=1$,

$\chi_{\overline{C}_{6}}=\chi_{\overline{C}_{6}T_{1}}+\chi_{\overline{C}_{6}T_{2}}+\chi_{\overline{C}_{6}T_{3}}=0.$
(86)

Plug the result (80) in Eqs. (78f) and Eq. (78g), we get

$\displaystyle\Delta_{1}\phi_{S}(\bm{r}_{\mu})$
$\displaystyle=\chi_{1}(r_{1}-r_{2}+\delta_{\mu,1}-\delta_{\mu,2})$
$\displaystyle\quad+(-1)^{w_{S}}(-\chi_{ST_{1}}+\chi_{ST_{3}}),$ (87a)
$\displaystyle\Delta_{2}\phi_{S}(\bm{r}_{\mu})$
$\displaystyle=\chi_{1}(2r_{1}-(-1)^{w_{S}}r_{1}-r_{2}+2\delta_{\mu,1}-\delta_{\mu,2})$
$\displaystyle\quad+(-1)^{s_{W}}(-\chi_{ST_{2}}+\chi_{ST_{3}}),$ (87b)
$\displaystyle\Delta_{3}\phi_{S}(\bm{r}_{\mu})$
$\displaystyle=\chi_{1}((1+(-1)^{w_{S}})(r_{1}-r_{2})+\delta_{\mu,1}-\delta_{\mu,2})$
$\displaystyle\quad+(-1)^{w_{S}}\chi_{ST_{3}}.$ (87c)

Consistency conditions for $\phi_{S}$ similar to (82) give

$\displaystyle(-2+(-1)^{w_{S}})\chi_{1}=\chi_{1},\quad(1+(-1)^{w_{S}})\chi_{1}=0,$
(88) $\displaystyle\quad 0=-(1+(-1)^{w_{S}})\chi_{1},$

therefore when $w_{S}=0$, $\chi_{1}=0$ or $\pi$; when $w_{S}=1$, $\chi_{1}=0$,
$\frac{\pi}{2}$, $\pi$ or $\frac{3\pi}{2}$. And

$\displaystyle\phi_{S}(\vec{r}_{\mu})$
$\displaystyle=\left(\frac{(r_{1}+1)r_{1}}{2}-\frac{(r_{2}+1)r_{2}}{2}-r_{1}r_{2}\right)\chi_{1}$
(89)
$\displaystyle+\left[(\delta_{\mu,1}-\delta_{\mu,2})\chi_{1}+(-1)^{w_{S}}(\chi_{ST_{3}}-\chi_{ST_{1}})\right]r_{1}$
$\displaystyle+\left[(2\delta_{\mu,1}-\delta_{\mu,2})\chi_{1}+(-1)^{w_{S}}(\chi_{ST_{3}}-\chi_{ST_{2}})\right]r_{2}$
$\displaystyle+\left[(\delta_{\mu,1}-\delta_{\mu,2})\chi_{1}+(-1)^{w_{S}}\chi_{ST_{3}}\right]r_{3}$
$\displaystyle+\theta_{\mu},$

where we abbreviated $\theta_{\mu}=\phi_{S}(0,0,0)_{\mu}$. Plug the result
(89) and (80) into Eq. (78d), we get

  * •

When $w_{S}=0$,

$\displaystyle\chi_{ST_{3}}$ $\displaystyle=0,$ (90a)
$\displaystyle\theta_{0}+\theta_{3}$
$\displaystyle=2\theta_{1}-\chi_{1}+\chi_{ST_{1}}$
$\displaystyle=2\theta_{2}+\chi_{1}+\chi_{ST_{2}}$ $\displaystyle=\chi_{S};$
(90b)

  * •

When $w_{S}=1$,

$\displaystyle 2\chi_{1}+2\chi_{ST_{1}}-\chi_{ST_{3}}$
$\displaystyle=2\chi_{1}-2\chi_{ST_{2}}+\chi_{ST_{3}}$ $\displaystyle=0,$
(91a) $\displaystyle-\theta_{0}+\theta_{3}+\chi_{ST_{3}}$
$\displaystyle=-\chi_{1}-\chi_{ST_{1}}+\chi_{ST_{3}}$
$\displaystyle=\chi_{1}-\chi_{ST_{2}}+\chi_{ST_{3}}$
$\displaystyle=\theta_{0}-\theta_{3}$ $\displaystyle=\chi_{S}.$ (91b)

At this point we are left with two equations, (78h) and (78i). Depending on
the $\mathbb{Z}_{2}$ value of $w_{\overline{C}_{6}}$ and $w_{S}$ we have the
following four cases:

  * •

$(w_{\overline{C}_{6}},w_{S})=(0,0)$: Eq. (78h) gives

$\chi_{\overline{C}_{6}T_{1}}+\chi_{\overline{C}_{6}T_{2}}+\chi_{\overline{C}_{6}T_{3}}+\sum_{\mu=0}^{3}(\rho_{\mu}+\theta_{\mu})=\chi_{\overline{C}_{6}S},$
(92)

and Eq. (78i) gives

$\displaystyle
2\chi_{ST_{1}}-3\chi_{\overline{C}_{6}T_{1}}+3\chi_{\overline{C}_{6}T_{2}}-\chi_{\overline{C}_{6}T_{3}}$
$\displaystyle\quad=2\chi_{ST_{2}}-3\chi_{\overline{C}_{6}T_{1}}-\chi_{\overline{C}_{6}T_{2}}+3\chi_{\overline{C}_{6}T_{3}}$
$\displaystyle\quad=0,$ (93a)