# Observation of the anisotropic Dirac cone in the band dispersion of
112-structured iron-based superconductor Ca0.9La0.1FeAs2

Z. T. Liu  State Key Laboratory of Functional Materials for Informatics,
Shanghai Institute of Microsystem and Information Technology (SIMIT), Chinese
Academy of Sciences, Shanghai 200050, China  X. Z. Xing  Department of Physics
and Key Laboratory of MEMS of the Ministry of Education, Southeast University,
Nanjing 211189, China  M. Y. Li  State Key Laboratory of Functional Materials
for Informatics, Shanghai Institute of Microsystem and Information Technology
(SIMIT), Chinese Academy of Sciences, Shanghai 200050, China  W. Zhou
Department of Physics and Key Laboratory of MEMS of the Ministry of Education,
Southeast University, Nanjing 211189, China  Y. Sun  Department of Physics and
Key Laboratory of MEMS of the Ministry of Education, Southeast University,
Nanjing 211189, China  C. C. Fan  H. F. Yang  J. S. Liu  State Key Laboratory
of Functional Materials for Informatics, Shanghai Institute of Microsystem and
Information Technology (SIMIT), Chinese Academy of Sciences, Shanghai 200050,
China  Q. Yao  State Key Laboratory of Functional Materials for Informatics,
Shanghai Institute of Microsystem and Information Technology (SIMIT), Chinese
Academy of Sciences, Shanghai 200050, China  State Key Laboratory of Surface
Physics, Department of Physics, and Advanced Materials Laboratory, Fudan
University, Shanghai 200433, China  W. Li  State Key Laboratory of Functional
Materials for Informatics, Shanghai Institute of Microsystem and Information
Technology (SIMIT), Chinese Academy of Sciences, Shanghai 200050, China  State
Key Laboratory of Surface Physics, Department of Physics, and Advanced
Materials Laboratory, Fudan University, Shanghai 200433, China  CAS-Shanghai
Science Research Center, Shanghai 201203, China  Z. X. Shi  Department of
Physics and Key Laboratory of MEMS of the Ministry of Education, Southeast
University, Nanjing 211189, China  D. W. Shen<EMAIL_ADDRESS>State Key
Laboratory of Functional Materials for Informatics, Shanghai Institute of
Microsystem and Information Technology (SIMIT), Chinese Academy of Sciences,
Shanghai 200050, China  CAS-Shanghai Science Research Center, Shanghai 201203,
China  CAS Center for Excellence in Superconducting Electronics (CENSE),
Shanghai 200050, China  Z. Wang  State Key Laboratory of Functional Materials
for Informatics, Shanghai Institute of Microsystem and Information Technology
(SIMIT), Chinese Academy of Sciences, Shanghai 200050, China

###### Abstract

CaFeAs2 is a parent compound of recently discovered 112-type iron-based
superconductors. It is predicted to be a staggered intercalation compound that
naturally integrates both quantum spin Hall insulating and superconducting
layers and an ideal system for the realization of Majorana modes. We performed
a systematical angle-resolved photoemission spectroscopy and first-principle
calculation study of the slightly electron-doped CaFeAs2. We found that the
zigzag As chain of 112-type iron-based superconductors play a considerable
role in the low-energy electronic structure, resulting in the characteristic
Dirac-cone like band dispersion as the prediction. Our experimental results
further confirm that these Dirac cones only exists around the $X$ but not $Y$
points in the Brillouin zone, breaking the $S_{4}$ symmetry at iron sites. Our
findings present the compelling support to the theoretical prediction that the
112-type iron-based superconductors might host the topological nontrivial edge
states. The slightly electron doped CaFeAs2 would provide us a unique
opportunity to realize and explore Majorana fermion physics.

Topological superconductors have attracted great research interests since they
are proposed to host the unconventional Andreev surface states, which are a
condensed-matter realization of hypothetical Majorana fermions originating
from the field of particle physics Majorana1 ; Majorana2 ; Majorana3 ;
Majorana4 ; Majorana5 ; Majorana6 . Besides the novelty in the fundamental
research, the experimental realization of Majorana fermions in solids is as
well of significance for application since it is suggested to be a key to
achieving the fault-tolerant topological quantum computing computer1 ;
computer2 ; computer3 . In this context, a diversity of candidate materials
which are feasible to host Majorana fermions have been proposed in the past
several years, including quantum wires in proximity to superconductors
Majorana4 ; a , hybrid superconductor-semiconductor nanowires Majorana4 ; b ;
A ; B ; C  and topological insulators coupled to superconductors c ; d ; e ; f
. Interestingly, all these proposals are restricted to utilize the coupling
between topological insulators and ordinary superconductors. Recently, high-
temperature (high-$T_{c}$) superconductors such as Bi2Sr2CaCu2O8+x (Bi-2212)
have been also applied as the substrate materials to synthesize artificial
topological insulator/superconductor heterostructures, based on the
expectation that a larger superconducting gap could be resolved since more
robust cooper pairs might be introduced into the normal side through
superconducting proximity effect D ; E ; F . However, whether this proximity
system can really host Majorana fermions is still lack of further support, and
whether high-$T_{c}$ superconductors could work in realizing Majorana fermions
is a seductive but still open question JPHu1 ; JPHu2 .

Very recently, there is one fascinating proposal which suggests the newly
discovered 112-type iron-based high-temperature superconductor parent compound
CaFeAs2 be naturally a promising candidate to host Majorana fermions JPHu2 .
CaFeAs2 has a structure with alternating CaAs and FeAs layers [see Fig. 1(a)].
While FeAs layers have been proven to be responsible for the high-$T_{c}$
superconductivity similar to other iron-based superconductors FeSC1 , this
proposal has shown that each CaAs layer is actually a topologically nontrivial
two-dimensional quantum spin Hall insulator. Consequently, below $T_{c}$ the
proximity effect would drive the zigzag As chain of the CaAs layer to become a
one-dimensional topological superconductor. In this proposal, the key argument
is that the $p_{x}$ and $p_{y}$ orbitals of the As atoms in the unique arsenic
zigzag bond layers of CaFeAs2 would result in an anisotropic Dirac cone near
the Fermi level ($E_{F}$), which is gapped by the spin-orbit coupling (SOC)
and topological nontrivial JPHu1 ; JPHu2 . Once this Dirac-cone like band
dispersion could be confirmed experimentally, the staggered intercalation of
quantum spin Hall insulator and superconductor in this 112-type iron-based
superconductor could provide us a unique opportunity to realize Majorana modes
in solids. However, this theory is based on the 112-type iron-based
superconductor parent CaFeAs2, which actually has not been synthesized out
experimentally yet. Thus, whether this prediction as well applies to the
corresponding electron-doped superconductor Ca1-xLaxFeAs2 with a $T_{c}$ as
high as 40 K would be an ideal touchstone for this intriguing model. Besides,
so far there still have been no specific reports on this characteristic but
tenuous feature in the band dispersion measurements of 112-type iron-based
superconductors ARPES1 ; ARPES2 ; ARPES3 .

In this letter, using both first-principle calculations and angle-resolved
photoemission spectroscopy (ARPES), we resolved the characteristic Dirac-cone
like band dispersion of this material and then performed a comprehensive
investigation of this band feature. We find that the CaAs layers indeed play a
considerable role in its low-energy electronic structure, As $4p$ orbitals of
which form the unique Dirac cone-like band dispersion distinct from those of
other iron-based superconductors. Our experimental results further confirm
that these Dirac cones only exist around the $X$ but not $Y$ points in the
Brillioun zone, breaking the $S_{4}$ symmetry at iron sites. In addition,
these Dirac cones are rather two-dimensional nature, which show negligible
photon energy dependence. Our findings present the compelling support to the
theoretical prediction that the 112-type iron-based superconductors indeed
host the topological nontrivial bands originating from the unique zigzag As
chain.

Figure 1: (a) Crystal structure of Ca1-xLaxFeAs2.(b) Three-dimensional
Brillouin zone of Ca1-xLaxFeAs2. (c) Temperature dependence of normalized
resistance in ab-plane of Ca0.9La0.1FeAs2 single crystals.
Figure 2: (a) The calculated Fermi surface. (b) The calculated orbital-
projected band structure of Ca0.9La0.1FeAs2. The weight of individual As 4 _p
x_(_p y_) orbitals for different bands is presented by gradually varied
colors. The dashed boxes indicate the bands near $X$ and $Y$. (c) The zoom-in
band structure near the $X$ point of BZ in the vicinity of the Fermi level.

High-quality single crystals with nominal composition Ca0.9La0.1FeAs2 were
synthesized using the flux method as descried in previous report growth ;
growth1 ; growth2 ; growth3 . Fig. 1(a) illustrates the schematics of the
crystal structure of Ca1-xLaxFeAs2, containing alternately stacked FeAs and
CaAs layers as aforementioned. The three-dimensional reduced Brillouin zone
(BZ), corresponding to the unit cell containing two Fe atoms, is shown in Fig.
1(b). The normalized resistivity in the $ab$ plane gives a $Tc$ onset of 42.5
K, as illustrated in Fig. 1(c).

Our ARPES measurements were conducted at the beamline I05 of the Diamond Light
Source (DLS), and the beamline 13U of the National Synchrotron Radiation
Laboratory (NSRL). All the data were taken at 10 K in an ultrahigh vacuum
better than 8$\times$10-11 torr. This setup is equipped with a VG-Scienta
R4000 electron analyzer. The angular and the energy resolution was set to 0.2∘
and 10-20 meV, respectively. The electronic band calculations were performed
within the density-functional formalism as implemented in the VASP code VASP .
The plane-wave-basis method and the Perdew-Burke-Ernzerh of exchange-
correlation potential were applied.

Figure 3: (a) Photoemission integrated intensity map at $E_{F}$ ($\pm$10 meV)
of Ca0.9La0.1FeAs2 using 105 eV photons. The brown lines indicate the BZ. (b)
The indication of the cut direction in the projected two-dimensional BZ.
(c)-(e) The photoemission intensity plots and the corresponding momentum
distribution curves (MDCs) along Cut $\\#$1 taken with 105 eV photons. (f)
Lorentzian peaks fitting to a few representative MDCs (Nos. 1-5). (g)-(j)
Photoemission intensity plots along Cuts $\\#$2$\sim$5 taken with 105 eV
photons. The arrows are guides to the eye for the momenta where the Dirac
cone-like band dispersions can be detected. (k) Energy location of the Dirac
point with error bars determined from the second derivative plots and MDCs.

We calculated the low-energy electronic structure of Ca0.9La0.1FeAs2, in which
the chemical potential was determined by best fitting the experimental Fermi
surface (FS) [Fig. 3(a)] to the DFT calculation and the SOC was applied.
Similar to other iron-based superconductors, the calculated FS of this sample
[Fig. 2(a)] shows the common three hole-like pockets around BZ center and two
electron pockets around the corner attributed to the Fe $3d$ orbitals ARPES3 .
Besides, we note that there also exist the extra small red Fermi pockets
around $X$. Both the corresponding spaghetti drawing of the band structure
[Fig. 2(b)] and its close-up [Fig. 2(c)] around $X$ clearly show that these
small Fermi pockets are composed of Dirac-cone like bands around the $X$
point, which is in good agreement with the proposal for CaFeAs2 JPHu1 ; JPHu2
. Particularly, the gap isolating the upper and lower Dirac band still remains
though the carrier dosage has shifted the chemical potential away from the
Dirac node as in CaFeAs2. However, the gap size in Ca0.9La0.1FeAs2 is only
$\sim$50 meV, smaller than that predicted for CaFeAs2 (100 meV). Furthermore,
we have extracted the orbital characters of low-lying bands of
Ca0.9La0.1FeAs2, and the weight of individual As $4p_{x}$($4p_{y}$) orbitals
of the zigzag As chain for different bands is presented by the gradually
varied colors. It demonstrates that the Dirac-cone like bands near $X$ point
are mainly contributed by the two-dimensional 4$p_{x}$($p_{y}$) orbitals of
the zigzag As in Ca0.9La0.1FeAs2 or CaFeAs2, which are suggested to play vital
roles in determining the topological nontrivial properties of CaAs layers in
the theoretical proposal.

Fig. 3(a) shows the photoemission intensity map, which agrees well with the
DFT calculation result. However, it is difficult to resolve the predicted
small pockets around $X$ induced by the Dirac-cone like bands
straightforwardly from the map. This might be due to the overwhelming
intensity of some broad bands from Fe $3d$ orbitals. To seek for the tenuous
Dirac cones in the vicinity of $E_{F}$, we performed a detailed survey of the
photoemission spectra around both $X$ and $Y$ points in the BZ, as illustrated
in Fig. 3(b). In Fig. 3(c), the intensity plot shows the band dispersion taken
along the Cut$\\#1$, slightly off the $\Gamma-X$ direction but crossing the
Dirac cone around $X$. This data together with the corresponding second
derivative plot [Fig. 3(d)] and momentum distribution curves (MDCs) [Fig.
3(e)] clearly show fast dispersing bands around $X$ with the characteristic
linearity, which is quantitatively confirmed by the peaks fitting of MDCs as
shown in Fig. 3(f). Also, the Dirac point energy and Dirac Fermi velocity are
determined to be 90$\pm$10 meV below $E_{F}$ [Fig. 3(k)] and
(3.8$\pm$0.3)$\times$105 m/s, respectively, which are in remarkable agreement
with the theoretically predicted values 80 meV below $E_{F}$ and
4.18$\times$105 m/s.

Figure 4: Photo-energy dependence of Dirac cone near the X point of BZ.
(a)-(d) The photoemission intensity plots along $\Gamma-X^{\prime}$ taken with
different photo-energy. (e)-(h) The second derivative plots corresponding to
(a)-(d) The lines and arrows are guides to the eye for the Dirac cone-like
band dispersions. (i) The indication of the cut direction in the projected
two-dimensional BZ. (k) The photon-energy dependence of the MDCs near EF along
$\Gamma-X^{\prime}$.

For the Ca1-xLaxFeAs2, the As atoms in the CaAs layers form zigzag chains with
a distorted checkerboard lattice in which an As atom moves in the $x$
direction and is off the center. Thus, the $S_{4}$ symmetry for As $4p$
orbitals should be broken. The $X$ and $Y$ points would no longer be
equivalent in BZ, and thus the Dirac-cone like feature could only be observed
around one point JPHu1 ; JPHu2 . Actually, our DFT calculation [Fig. 2(b)] and
detailed spectral scans around both $X$ and $Y$ provide the compelling proof
of this prediction. While the photoemission data along Cut #1 can reveal Dirac
bands around both equivalent X points [Figs. 3(c-d)], the data along Cuts
#2$\sim$5 do not show any sign of photoemission spectral weight around the $Y$
point. Here, a detailed survey around $Y$ was performed to prevent missing the
small momentum region in which the Dirac cones could exist. Note that the
photoemission experimental geometry for Cuts #1 and #2$\sim$5 is exactly the
same except of the different mounting azimuthal angles of samples [Fig. 3(b)],
so the interference of matrix element effect should be minimized.
Consequently, this result clearly confirms the anisotropy of these Dirac cone-
like bands.

To comprehensively understand these Dirac cones in the three-dimensional BZ,
we have performed detailed $k_{z}$ dependent measurement of the bands around
the $X$ point taken along the $\Gamma(Z)-X^{\prime}(R^{\prime})$ direction
[Fig 4(i)]. Using the free-electron final-state model Kz , in which the inner
potential was set to 15 eV ARPES3 , we can estimate the $k_{z}$ value probed
with different photons. Figs. 4(a-d) show the photoemission intensity plots
taken with photon energies ranged from 105 eV to 120 eV. Together with the
corresponding second derivative plots [Figs. 4(e-h)], we find some hole-like
bands around the $\Gamma(Z)$ points demonstrate substantial photon energy
dependence, which is in consistent with our previous report on the
considerable three-dimensionality of bands attributed to Fe $3d$ orbitals in
Ca1-xLaxFeAs2 ARPES3 . However, for the Dirac cones around $X(R)$, their band
dispersions show negligible $k_{z}$ dependence, as summarized in the MDCs
[Fig. 4(k)], indicating their rather two-dimensional nature considering the
fact that the photon energies we applied have covered more than half of the BZ
along $k$$\perp$ [Fig. 4(j)]. This finding further confirms that the Dirac-
cone like bands and the in-gap states around $X$ are attributed to the more
two-dimensional As $4p_{x}$($p_{y}$) orbitals in the zigzag chain.

Our DFT calculation and experimental data confirm that the unique As zigzag
chain in 112-type iron-based superconductors would result in anisotropic Dirac
cones in band dispersion as the theoretical prediction. For such a material
like CaFeAs2, the calculation of the Wannier function centers of the occupied
has shown that the Wannier states would switch partners and there should exist
topologically nontrivial spin-polarized edge states in the gap JPHu2 . On the
other hand, although our photoemission experiments do not directly reveal the
superconducting gap of this compound below $T_{c}$ due to the sample quality,
previous ARPES result has implied the s-wave superconductivity in 112-type and
other iron-based superconductors ARPES3 ; FeSC2 . Thus, both essential
ingredients of hosting topological superconductivity should have been
fulfilled in lightly electron doped CaFeAs2, and this compound is an ideal
system for the realization of Majorana modes.

However, although we observe some weak spectral weight in between the upper
and lower Dirac bands, we can still not pin down these in-gap states as the
long-sought topological non-trivial edge states due to their extremely weak
and blurred feature. Particularly, the characteristic Dirac point of the edge
states inside the bulk valence band gap cannot be resolved. This might be
still limited by the single crystal quality in the current stage or the rather
weak spectral weight of the one-dimensional edge states. These edge states
might be checked by more local and surface sensitive probes such as scanning
tunneling microscopy in the future. Besides, to realize topological
superconductors, an appropriate amount of electron doping to as-grown CaFeAs2
is needed so as to induce necessary carriers to the FeAs layers and shift the
chemical potential to cross the down edge bands only.

In summary, we report a systematical DFT calculation and ARPES study of the
low-lying electronic structure of the the 112-type iron-based superconductor
Ca0.9La0.1FeAs2. Besides the commonly hole- and electron-like bands as
reported in other iron-based pnictides, there exists extra Dirac-cone like
bands around the $X$ point of the BZ, which are rather anisotropic and two-
dimensional. We figure out a rather promising candidate Ca0.9La0.1FeAs2 of the
long-sought topological superconductors, and provide compelling evidences. Our
work may provide a promising platform for researchers to explore Majorana mode
which is the key for the realization of topological quantum computer.

We gratefully acknowledge the helpful discussions with Dr. Meixiao Wang. This
work is supported by the National Key R&D Program of the MOST of China (Grant
No. 2016YFA0300204) and the National Science Foundation of China (Grant Nos.
11274332, 11574337, 11227902 and U1432135). D. W. S. is also supported by the
“Strategic Priority Research Program (B)” of the Chinese Academy of Sciences
(Grant No. XDB04040300) and the “Youth Innovation Promotion Association CAS”.

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# Deconfined Quantum Criticality among Grand Unified Theories

Yi-Zhuang You  Department of Physics, University of California, San Diego, CA
92093, USA  Juven Wang  Center of Mathematical Sciences and Applications,
Harvard University, Cambridge, MA 02138, USA

###### Abstract

This article overviews the recent developments in applying the idea of
deconfined quantum criticality in condensed matter physics to understand
quantum phase transitions among grand unified theories in high energy physics
in the 4-dimensional spacetime. In particular, dictated by a mod 2 class
nonperturbative global mixed gauge-gravitational anomaly, there can be a
gapless deconfined quantum critical region between Georgi-Glashow and Pati-
Salam models — not only the Standard Model occurs as a neighbor phase, but
also beyond the Standard Model phenomena (such as deconfined dark gauge force
and fractionalized/fragmentary fermionic partons) emerge near the critical
region. An invited contribution to Professor Chen-Ning Yang Centenary
Festschrift.

##  I Introduction

The Standard Model (SM) [1, 2, 3, 4] of particle physics summarizes our
current knowledge of the fundamental building blocks of our universe in the
4-dimensional spacetime (4d). It describes three generations of fermionic
matter particles interacting with gauge forces and an electroweak Higgs field.
In each generation, if we assume that there exists a right-handed neutrino,
there are sixteen Weyl fermion matter particles, containing left- and right-
handed quarks (three colors and two flavors) and leptons (two flavors), as
summarized in Fig. 1(a). One of the open questions about the SM is whether the
three fundamental gauge forces (electromagnetic, weak, and strong) are
actually manifestations of a single all-encompassing force. This is the dream
of the Grand Unified Theory (GUT). Throughout the history, many GUT models has
been proposed, including the Pati-Salam (PS) model (or the
$\mathfrak{so}(6)\times\mathfrak{so}(4)$ GUT) [5] , the Georgi-Glashow (GG)
model (or the $\mathfrak{su}(5)$ GUT) [6] , and the $\mathfrak{so}(10)$ GUT
[7]. The pattern that matter particles are unified under these GUTs are
illustrated in Fig. 1(b-d) respectively.

Figure 1: The first generation of matter particles in (a) the Standard Model
(SM), (b) the Pati-Salam (PS) model, i.e. the
$\mathfrak{so}(6)\oplus\mathfrak{so}(4)$ GUT, (c) the Georgi-Glashow (GG)
model, i.e. the $\mathfrak{u}(5)$ GUT, and (d) the $\mathfrak{so}(10)$ GUT.
$Y$ denotes the electroweak hypercharge, and $X=5(B-L)-\frac{2}{3}Y$ with
$(B-L)$ as the baryon minus lepton number. Figure 2: Two different
perspectives of how GUTs can be accessed from the SM: (a) high energy physics
perspective, (b) condensed matter perspective. Experiments like proton decay
only rules out the perspective (a), but not (b).

Despite the theoretical elegance of GUTs, experiments (such as proton decay
[8]) have ruled out many (non-supersymmetric) GUTs as the high-energy
completion of the SM, meaning that these GUTs might not be achieved only by
raising up to a higher energy scale (or higher temperature scale) from the SM
through thermal phase transitions, as shown in Fig. 2(a). However, experiments
did not rule out the possibilities that these GUTs can be accessed by tuning
model parameters away from the SM through quantum phase transitions, as shown
in Fig. 2(b). From the high energy physics perspective, different GUTs are
just different quantum field theories describing how gauge forces can be
unified at higher energy scales. However, in condensed matter physics,
different quantum field theories are often considered low-energy effective
descriptions of different quantum phases of matter. From this perspective, the
SM and various GUT vacua may be viewed as competing quantum phases of our
universe, such that it becomes natural to discuss quantum phase transitions
(and the associated quantum criticality) among different GUT vacua. By quantum
criticality, we mean to emphasize the near zero-temperature quantum physics of
gapless, scale or conformal invariant critical phenomena [9]. The quantum
vacua (phases) tuning parameters appear perturbatively irrelevant at the SM
fixed point in the renormalization group (RG) setup, hence their non-
perturbative effects are largely overlooked in the previous literature.
Exploring the effects of these tuning parameters and investigating quantum
phase transitions among the SM and GUTs has motivated a series of works
recently [10, 11, 12].

The major differences between these field theories are their gauge groups.
Therefore, the phase transition between GUTs (and the SM) are generally gauge-
Higgs transitions, which is driven by the condensation of certain GUT Higgs
field (distinct from the electroweak Higgs field in the SM) to Higgs down the
gauge group. (An important remark: The present work only focuses on the GUT
Higgs fields and their induced phase transitions; we do not consider the
electroweak Higgs field or electroweak symmetry breaking below the electroweak
energy scale within SM. So whenever Higgs fields are mentioned, they stand for
the GUT Higgs fields for GUTs instead of the electroweak Higgs field within
the SM.) If one conceptually suppresses the gauge fluctuation and treats the
gauge group $G$ to be a global internal symmetry in $G$, the gauge-Higgs
transitions will be closely analogous to spontaneous symmetry breaking
transitions. Therefore below we will switch between the gauge group $G$ and
the global internal symmetry group $G$ perspectives, based on the Weyl’s
principle of “gauging” or ”ungauging” via “summing over” or “fixing” the
$G$-bundle with $G$-connections in the field theory path integral, depending
on the context. Besides, in this article, we intentionally simplify the
discussion of the global symmetry by focusing only on the internal symmetry
$G$ but omitting the spacetime symmetry (see the full spacetime-internal
symmetry in [10, 11, 12]).

The study of symmetry-breaking transitions has a long history in condensed
matter physics [13]. The conventional symmetry breaking phases and their phase
transitions are successfully described by the Landau-Ginsburg-Wilson (LGW)
paradigm in terms of order parameters [14]. However, the condensed matter
literature has also seen much discussion [15, 16, 17, 18, 19, 20, 21, 22] on
exotic quantum phase transitions beyond the LGW paradigm, one notable example
is the deconfined quantum criticality (DQC) [15, 16, 17]. The DQC occurs in a
quantum system with a global symmetry $G$ that has two spontaneous symmetry
breaking phases which break $G$ to its distinct subgroups $G_{1}$ and $G_{2}$
respectively. When the low-energy effective theory has a ’t Hooft anomaly of
$G$, the two symmetry breaking phases can not share a trivially gapped
$G$-symmetric intermediate phase, resulting in a gapless critical point (or a
gapless critical region as an intermediate new phase) between the two symmetry
breaking phases. The theory for the quantum critical point/phase is usually
not expressed in terms of order parameters in $G/G_{1}$ or $G/G_{2}$, but in
terms of “fractionalized” degrees of freedom (such that the order parameters
are composite of deconfined fractionalized particles, i.e. partons), and hence
the terminology “deconfined”.

This article reviews the recent progress [10] in exploring the possibility of
DQC in quantum phase transitions between different GUT vacua. The study
reveals the SM as a neighboring phase of two competing GUT phases (i.e. PS and
GG). The competition could give rise to an intermediate DQC phase where the
GUT Higgs field fractionalizes to deconfined particles and emergent gauge
fields beyond the SM. Some of the fractionalized particles may persist in the
SM phase, which contributes to dark matter candidates of our universe.

##  II GUT Higgs Condensation Transitions among GUT Phases

The Yang-Mills gauge theory [23], or more generally the non-Abelian gauge
theory, provides the basis for the theoretical formulation of the SM and GUTs.
In this framework, the matter particles are excitations of Weyl fermion
fields, and the gauge forces are mediated by non-Abelian gauge fields. The
action of the Yang-Mills theory coupled to Weyl fermions in the 4d spacetime
generally takes the form of

$S_{\text{YM}}=\int_{\mathcal{M}^{4}}\big{(}\mathrm{d}^{4}x\;\psi^{\dagger}(\mathrm{i}\bar{\sigma}^{\mu}D_{\mu,A})\psi\big{)}+\operatorname{Tr}(F\wedge\star
F),$ (1)

where $\psi$ denotes the Lorentz spinor Weyl fermion field, the covariant
derivative $D_{\mu,A}=\partial_{\mu}-\mathrm{i}gA_{\mu}$ contains the Lie
algebra-valued gauge field $A_{\mu}$, and $F=\mathrm{d}A-\mathrm{i}gA\wedge A$
is the gauge curvature 2-form (given $A=A_{\mu}\mathrm{d}x^{\mu}$) with the
Yang-Mills gauge coupling $g$. All Weyl fermions in the theory can be taken to
have the left-handed helicity, with the choice of
$\bar{\sigma}^{\mu}=(\sigma^{0},-\sigma^{1},-\sigma^{2},-\sigma^{3})$, as
right-handed fermions can always be conjugated to their left-handed
antiparticles. Each generation of matter particles consists of sixteen or
fifteen Weyl fermions. In particular, the 16th Weyl fermion is the right-
handed neutrino [24, 25] which is sterile to the SM gauge force and has not
been observed in experiments so far. However, there is a good reason to
include the sterile neutrino, as it provides the simplest way to cancel a
$\mathbb{Z}_{16}$ global anomaly of the theory [26, 27, 28, 29, 30].
Otherwise, with only fifteen Weyl fermions, the anomaly must be canceled in
more exotic ways [31, 32, 33], either by additional 4d or 5d gapped
topological quantum field theories (TQFTs) or by 4d gapless interacting
conformal field theories (CFTs).

Assuming sixteen Weyl fermions in each generation, which couple to gauge
fields that mediate gauge interactions, the main difference among different
GUT models relies on their different choices of the gauge group $G$ (or the
gauge algebra $\mathfrak{g}$), summarized as follows:

  * •

SM:
$G_{\text{SM}}=(\mathrm{SU}(3)\times\mathrm{SU}(2)_{L})\times_{\mathbb{Z}_{6}}\mathrm{U}(1)_{Y}\times_{\mathbb{Z}_{6}}\mathrm{U}(1)_{X}$
(where $\mathrm{SU}(3)$ denotes the color group, $\mathrm{SU}(2)_{L}$ denotes
the weak isospin group, $\mathrm{U}(1)_{Y}$ denotes the electroweak
hypercharge group, and $\mathrm{U}(1)_{X}$ denotes the $X\equiv
5(B-L)-\frac{2}{3}Y$-charge [34] group which is a variant of baryon minus
lepton $(B-L)$-like symmetry), correspondingly,
$\mathfrak{g}_{\text{SM}}=\mathfrak{su}(3)\oplus\mathfrak{su}(2)_{L}\oplus\mathfrak{u}(1)_{Y}\oplus\mathfrak{u}(1)_{X}$.
Note that we introduce this extra $s\mathfrak{u}(1)_{X}$ into
$\mathfrak{g}_{\text{SM}}$.

  * •

PS: $G_{\text{PS}}=\mathrm{Spin}(6)\times_{\mathbb{Z}_{2}}\mathrm{Spin}(4)$,
correspondingly,
$\mathfrak{g}_{\text{PS}}=\mathfrak{so}(6)\oplus\mathfrak{so}(4)=\mathfrak{su}(4)\oplus\mathfrak{su}(2)_{L}\oplus\mathfrak{su}(2)_{R}$.

  * •

GG:
$G_{\text{GG}}=\mathrm{SU}(5)\times_{\mathbb{Z}_{5|2}}\mathrm{U}(1)_{X}=\mathrm{U}(5)_{2}$
(where the subscript $2$ indicates that the $\mathbb{Z}_{5}$ center of
$\mathrm{SU}(5)$ is identified with the $\mathbb{Z}_{5}$ subgroup of
$\mathrm{U}(1)_{X}$ by a double covering, or more precisely defined by the
short exact sequence
$1\to\mathrm{SU}(5)\to\mathrm{U}(5)_{m}\xrightarrow{\det}\mathrm{U}(1)/\mathbb{Z}_{5m}\to
1$ [11]), correspondingly,
$\mathfrak{g}_{\text{GG}}=\mathfrak{su}(5)\oplus\mathfrak{u}(1)_{X}$.

  * •

$\mathfrak{so}(10)$ GUT: $G_{\mathfrak{so}(10)}=\mathrm{Spin}(10)$,
correspondingly, $\mathfrak{g}_{\mathfrak{so}(10)}=\mathfrak{so}(10)$.

The notation $G_{1}\times_{N}G_{2}\equiv\frac{G_{1}\times G_{2}}{N}$ denotes
the product of two groups $G_{1}$ and $G_{2}$ quotient their common normal
subgroup $N$. One may as well treat these groups as global symmetry groups by
“un-gauging” the gauge theory, i.e. by considering the dynamical gauge field
as a static background probe. This allows us to understand the transitions
between these models from the perspective of spontaneous symmetry breaking.

Figure 3: Unification of gauge forces as Lie algebra embedding. Each dot
represents a gauge boson (corresponding to a Lie algebra generator).
$\mathfrak{u}(1)_{Y}\oplus\mathfrak{u}(1)_{X}$ and
$\mathfrak{u}(1)_{3,R}\oplus\mathfrak{u}(1)_{3(B-L)}$ generators are linearly
mixed to accommodate different unification schemes, see Eq. (4).

A phase with a larger symmetry group $G_{\text{large}}$ can undergo a symmetry
breaking transition to a phase with a smaller symmetry group
$G_{\text{small}}$, if $G_{\text{small}}$ is a subgroup of $G_{\text{large}}$,
or equivalently if $G_{\text{small}}$ can be embedded in $G_{\text{large}}$
(denoted by $G_{\text{small}}\hookrightarrow G_{\text{large}}$). The group of
the SM and various GUTs can be embedded as [35]

$\begin{matrix}G_{\text{GG}}&\hookrightarrow&G_{\mathfrak{so}(10)}\\\
\mathrel{\rotatebox[origin={c}]{90.0}{$\hookrightarrow$}}&&\mathrel{\rotatebox[origin={c}]{90.0}{$\hookrightarrow$}}\\\
G_{\text{SM}}&\hookrightarrow&G_{\text{PS}}\end{matrix}.$ (2)

The embedding structure indicates that the $\mathfrak{so}(10)$ GUT can go
through symmetry-breaking transitions from the $\mathfrak{so}(10)$ GUT phase
to the adjacent PS or GG phases. The PS and GG phases can further break
symmetry to their common adjacent SM phase. The Lie group embedding structure
in Eq. (2) also applies to their corresponding Lie algebras, as illustrated in
Fig. 3. It turns out that $\mathfrak{g}_{\text{SM}}$ is the unique common
subalgebra of $\mathfrak{g}_{\text{PS}}$ and $\mathfrak{g}_{\text{GG}}$, while
$\mathfrak{g}_{\mathfrak{so}(10)}$ is the minimal algebra that contains both
$\mathfrak{g}_{\text{PS}}$ and $\mathfrak{g}_{\text{GG}}$:

$\begin{split}\mathfrak{g}_{\text{PS}}\cap\mathfrak{g}_{\text{GG}}&=\mathfrak{g}_{\text{SM}},\\\
\mathfrak{g}_{\text{PS}}\cup\mathfrak{g}_{\text{GG}}&=\mathfrak{g}_{\mathfrak{so}(10)}.\\\
\end{split}$ (3)

More than just Lie algebra intersection/union, there are also corresponding
statements in Lie groups [35, 10, 11]. The stringent relations between these
symmetry groups (or algebras) suggest that there might be a unifying theory
that describes the phase transitions among the SM and different GUTs.

It is worth mentioning that the subalgebra
$\mathfrak{u}(1)_{Y}\oplus\mathfrak{u}(1)_{X}\cong\mathfrak{u}(1)_{3,R}\oplus\mathfrak{u}(1)_{3(B-L)}$
inside $\mathfrak{g}_{\text{SM}}$ is shared between $\mathfrak{g}_{\text{PS}}$
and $\mathfrak{g}_{\text{GG}}$, but it can have different choices of generator
basis. Let $Y\in\mathbb{Z}$ be the weak hypercharge (at triple-scale),
$X\in\mathbb{Z}$ be the $X$-charge, $Q_{3,R}=2I_{3,R}\in\mathbb{Z}$ be the
right-isospin (at double-scale), $(B-L)\in\mathbb{Z}/3$ be the baryon minus
lepton number. They are related by the linear relation

$\left[\begin{matrix}Y\\\ X\end{matrix}\right]=\left[\begin{matrix}3&1\\\
-2&1\end{matrix}\right]\left[\begin{matrix}Q_{3,R}\\\
3(B-L)\end{matrix}\right].$ (4)

The electromagnetic $\mathfrak{u}(1)_{\text{em}}$ charge is given by
$Q_{\text{em}}=I_{3,L}+\frac{1}{6}Y=I_{3,L}+I_{3,R}+\frac{1}{2}(B-L)$, where
$I_{3,L}\in\mathbb{Z}/2$ is the left-isospin (the weak isospin).

Within the Landau-Ginzburg-Wilson paradigm, the symmetry-breaking transition
from a larger symmetry group $G_{\text{large}}$ (or its algebra
$\mathfrak{g}_{\text{large}}$) to a smaller symmetry group $G_{\text{small}}$
(or its algebra $\mathfrak{g}_{\text{small}}$) can be formulated as the
condensation of the order parameter field, whose target manifold is
$G_{\text{large}}/G_{\text{small}}$. If the symmetries are dynamically gauged,
the symmetry-breaking transition can be reinterpreted as the Higgs transition,
in which the order parameter field is promoted to the Higgs field. The Higgs
field that drives the transition between GUT (and SM) phases will be called
the GUT Higgs fields to distinguish from the electroweak Higgs field in the
SM. In order for the Higgs condensation to break $\mathfrak{g}_{\text{large}}$
but not $\mathfrak{g}_{\text{small}}$, it must be in a non-trivial
representation of $\mathfrak{g}_{\text{large}}$ but in the trivial
representation of $\mathfrak{g}_{\text{small}}$. One can look up such cases
from Lie algebra branching rules.

For example, to break $\mathfrak{g}_{\mathfrak{so}(10)}=\mathfrak{so}(10)$ to
$\mathfrak{g}_{\text{PS}}=\mathfrak{su}(4)\oplus\mathfrak{su}(2)_{L}\oplus\mathfrak{su}(2)_{R}$,
the branching rule $\mathbf{54}\to(\mathbf{1},\mathbf{1},\mathbf{1})+\cdots$
suggests that one could condense a GUT Higgs field $\Phi_{\mathbf{54}}$ in the
$\mathbf{54}$ representation of $\mathfrak{g}_{\mathfrak{so}(10)}$ to the
$(\mathbf{1},\mathbf{1},\mathbf{1})$ representation of
$\mathfrak{g}_{\text{PS}}$ to achieve the desired symmetry breaking.
Similarly, to break $\mathfrak{g}_{\mathfrak{so}(10)}$ to
$\mathfrak{g}_{\text{GG}}=\mathfrak{su}(5)\oplus\mathfrak{u}(1)_{X}$, the
branching rule $\mathbf{45}\to\mathbf{1}_{0}+\cdots$ suggests that one could
condense a GUT Higgs field $\Phi_{\mathbf{45}}$ in the $\mathbf{45}$
representation of $\mathfrak{g}_{\mathfrak{so}(10)}$ to the $\mathbf{1}_{0}$
representation of $\mathfrak{g}_{\text{GG}}$ to achieve the desired symmetry
breaking. Furthermore, if both GUT Higgs fields $\Phi_{\mathbf{45}}$ and
$\Phi_{\mathbf{54}}$ are condensed, the internal symmetry will be broken down
to $\mathfrak{g}_{\text{SM}}$, as $\mathfrak{g}_{\text{SM}}$ is the
intersection of $\mathfrak{g}_{\text{PS}}$ and $\mathfrak{g}_{\text{GG}}$ that
survives both symmetry breaking. More concretely, starting from the PS phase
where the GUT Higgs field $\Phi_{\mathbf{54}}$ is already condensed, to
further break $\mathfrak{g}_{\text{PS}}$ to
$\mathfrak{g}_{\text{SM}}=\mathfrak{su}(3)\oplus\mathfrak{su}(2)_{L}\oplus\mathfrak{u}(1)_{Y}\oplus\mathfrak{u}(1)_{X}$,
the branching rules
$(\mathbf{1},\mathbf{1},\mathbf{3})\to(\mathbf{1},\mathbf{1})_{0,0}+\cdots$
and
$(\mathbf{15},\mathbf{1},\mathbf{1})\to(\mathbf{1},\mathbf{1})_{0,0}+\cdots$
in the $\mathbf{45}$ sector indicates that the GUT Higgs field
$\Phi_{\mathbf{45}}$ needs to be condensed as well (more precisely, a certain
combination of $(\mathbf{1},\mathbf{1},\mathbf{3})$ and
$(\mathbf{15},\mathbf{1},\mathbf{1})$ should be condensed, whose specific form
will be discussed later). On the other hand, starting from the GG phase where
the GUT Higgs field $\Phi_{\mathbf{45}}$ is already condensed, to further
break $G_{\text{GG}}$ to $G_{\text{SM}}$, the branching rule
$\mathbf{24}_{0}\to(\mathbf{1},\mathbf{1})_{0,0}+\cdots$ in the $\mathbf{54}$
sector indicates that the GUT Higgs field $\Phi_{\mathbf{54}}$ needs to be
condensed as well. Both analyses points to the same conclusion that the SM
phase can be achieved from the $\mathfrak{so}(10)$ GUT phase by simultaneously
condensing the two GUT Higgs fields $\Phi_{\mathbf{45}}$ and
$\Phi_{\mathbf{54}}$.

Figure 4: Schematic quantum phase diagram of GUT phases adjacent to the SM
phase. Phase labels stand for: SM - the Standard Model phase, PS - the Parti-
Salam $\mathfrak{so}(6)\oplus\mathfrak{so}(4)$ phase, GG - the Georgi-Glashow
$\mathfrak{u}(5)$ phase, and SMG - the symmetric mass generation phase.

To connect different GUT phases, one can start with the largest internal
symmetry (or gauge) group $G_{\mathfrak{so}(10)}=\mathrm{Spin}(10)$ and access
lower-symmetry phases by Higgs condensation. The Higgs condensation can be
described by

$S_{\text{H}}=\int_{\mathcal{M}^{4}}\mathrm{d}^{4}x\big{(}\sum_{\mathbf{R}}(D_{\mu,A}\Phi_{\mathbf{R}})^{2}+V(\Phi_{\mathbf{R}})\big{)},$
(5)

where $\Phi_{\mathbf{R}}$ is the Lorentz scalar GUT Higgs field in the
$\mathbf{R}$ representation of $\mathfrak{so}(10)$, and the Higgs potential
takes the form of
$V(\Phi_{\mathbf{R}})=r_{\mathbf{R}}\Phi_{\mathbf{R}}^{2}+\lambda_{\mathbf{R}}\Phi_{\mathbf{R}}^{4}+\cdots$.
The $\Phi^{4}$ coefficient $\lambda_{\mathbf{R}}>0$ is always assumed to be
positive. The $\Phi^{2}$ coefficients $r_{\mathbf{R}}$ is the tuning parameter
that drives the transition between GUT phases. At the mean-field level, when
$r_{\mathbf{R}}<0$, the corresponding Higgs field $\Phi_{\mathbf{R}}$ will
condense, i.e. $\langle\Phi_{\mathbf{R}}\rangle\neq 0$. Based on the above
discussion, a schematic phase diagram can be concluded as in Fig. 4 (on the
$r_{\mathbf{1}}>0$ side).

The GUT Higgs field in both the $\mathbf{45}$ and $\mathbf{54}$ representation
motivates us to include an additional GUT Higgs field $\Phi_{\mathbf{1}}$ in
the $\mathbf{1}$ (trivial) representation, such that
$\mathbf{1}+\mathbf{45}+\mathbf{54}$ can be unified as the $\mathfrak{so}(10)$
bivector representation $\mathbf{10}\times\mathbf{10}$, while the
$\mathfrak{so}(10)$ vector representation $\mathbf{10}$ can be further
composed by the product of spinor representations as
$\mathbf{16}\times\mathbf{16}\to\mathbf{10}+\mathbf{120}+\overline{\mathbf{126}}$.
This enables us to write down the most natural Yukawa-like coupling between
the GUT Higgs field and the matter fermion field (in the $\mathbf{16}$
representation),

$\begin{split}S_{\text{Y}}&=\int_{\mathcal{M}^{4}}\mathrm{d}^{4}x\frac{1}{2}\Big{(}\phi^{\intercal}\Phi\phi\\\
&+\sum_{a=1}^{5}\big{(}\psi^{\intercal}\mathrm{i}\sigma^{2}(\phi_{2a-1}\Gamma_{2a-1}-\mathrm{i}\phi_{2a}\Gamma_{2a})\psi+\text{h.c.}\big{)}\Big{)}.\end{split}$
(6)

Here the GUT Higgs field $\Phi$ is written in the $\mathfrak{so}(10)$ bivector
representation as a $10\times 10$ matrix field (with totally 100 real scalar
components). The $\mathfrak{so}(10)$ vector field $\phi$ has 10 real scalar
components, which corresponds to the 10 orthogonal Majorana masses (pairing
terms) among 16 Weyl fermions. $\Gamma_{a}$ (for $a=1,2,\cdots,10$) are ten
rank-16 real symmetric matrices satisfying
$\\{\Gamma_{2a-1},\Gamma_{2b-1}\\}=2\delta_{ab}$,
$\\{\Gamma_{2a},\Gamma_{2b}\\}=2\delta_{ab}$,
$\\{\Gamma_{2a-1},\Gamma_{2b}\\}=0$ (for $a,b=1,2,\cdots,5$). In Eq. (6),
$\mathrm{i}\sigma^{2}$ acts in the 2-dimensional spacetime spinor subspace and
$\Gamma_{a}$ acts in the 16-dimensional flavor subspace. Unlike the
electroweak Higgs field in the SM, which gives quadratic mass to the matter
fermions, the GUT Higgs field $\Phi$ does not generate fermion-bilinear mass,
but rather serves as a four-fermion interaction. This can be seen by treating
the $\mathfrak{so}(10)$ vector field $\phi$ as a Lagrangian multiplier.
Integrating out $\phi$ will leads to the quartic coupling of the form
$\Phi\psi\psi\psi\psi$. As four-fermion interactions are perturbatively
irrelevant for 4d Weyl fermions, perturbative vacuum expectation value of the
GUT Higgs field (i.e. $\langle\Phi\rangle\neq 0$ but small) will not affect
the matter fermion spectrum.

As an $\mathfrak{so}(10)$ bivector representation
$\mathbf{10}\times\mathbf{10}$, the GUT Higgs field $\Phi$ can be decomposed
into $\mathbf{1}+\mathbf{45}+\mathbf{54}$ sectors, which defines

$\begin{split}\Phi_{\mathbf{1}}&:\operatorname{Tr}\Phi=\sum_{a}\Phi_{aa},\\\
\Phi_{\mathbf{45}}&:\Phi_{[a,b]}=\tfrac{1}{2}(\Phi_{ab}-\Phi_{ba}),\\\
\Phi_{\mathbf{54}}&:\Phi_{\\{a,b\\}}=\tfrac{1}{2}(\Phi_{ab}+\Phi_{ba}).\\\
\end{split}$ (7)

Starting from the $\mathfrak{so}(10)$ GUT, the condensation of
$\Phi_{\mathbf{45}}$ or/and $\Phi_{\mathbf{54}}$ will drive symmetry breaking
transitions to various lower-symmetry phases, as summarized in Fig. 4. In
particular, if $\Phi$ condenses to a specific configuration
$\langle\Phi\rangle$, the original algebra $\mathfrak{g}_{\text{large}}$ will
be broken to its subalgebra
$\mathfrak{g}_{\text{small}}=\\{x\in\mathfrak{g}_{\text{large}}|[x,\langle\Phi\rangle]=0\\}$.
It can be verified that the following condensation configurations

$\begin{split}\langle\Phi_{\mathbf{54}}\rangle&\propto\left[\begin{smallmatrix}-3&&&&\\\
&-3&&&\\\ &&2&&\\\ &&&2&\\\
&&&&2\end{smallmatrix}\right]\otimes\left[\begin{smallmatrix}1&\\\
&1\end{smallmatrix}\right],\\\
\langle\Phi_{\mathbf{45}}\rangle&\propto\left[\begin{smallmatrix}1&&&&\\\
&1&&&\\\ &&1&&\\\ &&&1&\\\
&&&&1\end{smallmatrix}\right]\otimes\left[\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right],\end{split}$ (8)

induce the following symmetry breaking

${\mathfrak{g}_{so(10)}}$${\mathfrak{g}_{\text{PS}}}$${\mathfrak{g}_{\text{GG}}}$${\mathfrak{g}_{\text{SM}}}$$\scriptstyle{\langle\Phi_{\mathbf{54}}\rangle\neq
0}$$\scriptstyle{\langle\Phi_{\mathbf{45}}\rangle\neq
0}$$\scriptstyle{\langle\Phi_{\mathbf{45}}\rangle\neq
0}$$\scriptstyle{\langle\Phi_{\mathbf{54}}\rangle\neq 0}$ (9)

$\langle\Phi_{\mathbf{54}}\rangle$ explicitly distinguishes the first four-
dimensional subspace from the last six-dimensional subspace of the
$\mathfrak{so}(10)$ vector, therefore breaking
$\mathfrak{g}_{\mathfrak{so}(10)}=\mathfrak{so}(10)$ down to
$\mathfrak{g}_{\text{PS}}=\mathfrak{so}(6)\oplus\mathfrak{so}(4)$.
$\langle\Phi_{\mathbf{45}}\rangle$ is proportional to the
$\mathfrak{u}(1)_{X}$ generator, which effectively requires the unbroken
generators to commute with $\mathfrak{u}(1)_{X}$ generator, which singles out
$\mathfrak{g}_{\text{GG}}=\mathfrak{su}(5)\oplus\mathfrak{u}(1)_{X}$. When
both of them condense, the symmetry is broken to
$\mathfrak{g}_{\text{PS}}\cap\mathfrak{g}_{\text{GG}}=\mathfrak{g}_{\text{SM}}$.

The condensation of $\Phi_{\mathbf{1}}$ in the trivial sector will not break
the $G_{\mathfrak{so}(10)}$ symmetry, and a perturbative vacuum expectation
value of $\langle\Phi_{\mathbf{1}}\rangle<\Phi_{\mathbf{1},c}$ below a
critical value also has no effect on the fermion spectrum, so the theory
remains in the $\mathfrak{so}(10)$ GUT phase. However, when the condensation
$\langle\Phi_{\mathbf{1}}\rangle>\Phi_{\mathbf{1},c}$ exceeds a critical
value, the quartic interaction induced by $\langle\Phi_{\mathbf{1}}\rangle$
can have non-perturbative effects: it can generate an interacting gap for all
fermions without breaking the $G_{\mathfrak{so}(10)}$ symmetry. The mechanism
is known as the symmetric mass generation (SMG), and has been explored in
various situations, as some selective examples, for fermion zero modes in 1d
[36], for chiral fermions in 2d [37, 38], for Dirac fermions in 3d [39, 40],
and most notable for Weyl fermions in 4d [41, 26, 27, 28, 42, 43, 44, 45, 46].
To consider the possibility of SMG, one can investigate the strongly
interacting limit of $\langle\Phi_{\mathbf{1}}\rangle\to\infty$, where the
matter fermion dynamics is dominated by the interaction $S_{\text{Y}}$ in Eq.
(6). In this limit, the field theory decouples in each spatial location and
can be solved exactly. The solution shows that the many-body ground state at
each location is unique with a finite excitation gap [27], indicating all
fermions are gapped by the interaction without spontaneous symmetry breaking.

##  III Deconfined Quantum Criticality

The phase diagram in Fig. 4 resembles the phase diagram of deconfined quantum
criticality in (2+1)D quantum magnet [15, 16, 22], where two distinct symmetry
breaking phases – the antiferromagnetic Néel phase and the valence bond solid
(VBS) phase – are connected by a direct continuous quantum phase transition
[19]. The Néel phase spontaneously breaks the spin $\mathrm{SO}(3)$ rotation
symmetry and the VBS phase spontaneously breaks the lattice rotation
$\mathbb{Z}_{4}$ symmetry (or $\mathrm{SO}(2)$ symmetry in the continuum). The
direct transition between these two quantum phases is exotic, which involves
fractionalized excitations and emergent gauge fields. The deconfinement of the
fractionalized excitations happens at and only at the Néel-VBS critical point,
therefore the critical point is also called the deconfined criterial point
(DQCP). In the phase diagram Fig. 4, the PS and GG phases also breaks the
$G_{\mathfrak{so}(10)}=\mathrm{Spin}(10)$ symmetry differently, which
motivates the investigation of the possible DQCP physics along the phase
boundary between the PS and GG phases.

The key mechanism of DQCP is that the topological defect of one order
parameter carries the charge of the other order parameter. In this case, the
two order parameters are said to be intertwined orders. For example, the
vortex of VBS order parameter carries a spin-1/2 degree of freedom that is
charged under the spin $\mathrm{SO}(3)$ symmetry in a fractionalized and
projective representation [18]. To explore similar scenarios in the GUTs, one
should first understand the possible topological defects of the order
parameters in the PS and GG phases. When the GUT Higgs fields
$\Phi_{\mathbf{45}}$ and $\Phi_{\mathbf{54}}$ condense to a finite amplitude,
their orientational fluctuations belongs to the following target manifolds

$\Phi_{\mathbf{45}}\in\frac{\mathrm{SO}(10)}{\mathrm{U}(5)},\quad\Phi_{\mathbf{54}}\in\frac{\mathrm{SO}(10)}{\mathrm{SO}(6)\times\mathrm{SO}(4)}.$
(10)

These manifolds have non-trivial second homotopy groups

$\pi_{2}\Big{(}\frac{\mathrm{SO}(10)}{\mathrm{U}(5)}\Big{)}=\mathbb{Z},\quad\pi_{2}\Big{(}\frac{\mathrm{SO}(10)}{\mathrm{SO}(6)\times\mathrm{SO}(4)}\Big{)}=\mathbb{Z}_{2},$
(11)

which indicates that the $\Phi_{\mathbf{45}}$ order parameter admits
$\mathbb{Z}$-classified topological point defects in the 3D space, and the
$\Phi_{\mathbf{54}}$ order parameter admits $\mathbb{Z}_{2}$-classified
topological point defects in the 3D space. The intertwinement of these two
symmetry-breaking orders is equivalently characterized by the non-trivial
mutual braiding statistics of their topological defects, i.e. the
$\Phi_{\mathbf{45}}$ defect and the $\Phi_{\mathbf{54}}$ defect will see each
other as a monopole emitting totally $2\pi$ statistical Berry phase flux.

Figure 5: Illustration of defect worldsheets in extended 5d spacetime. The two
colors indicate the $\Phi_{\mathbf{45}}$ or the $\Phi_{\mathbf{54}}$
topological defects. Each pair of mutually linked 2-spheres of the two types
of topological defects will contribute a $(-1)$ sign to the path integral
weight.

The braiding statistics of topological defects can be more precisely defined
by extending the 4d spacetime to 5d with an auxiliary dimension. As shown in
Fig. 5, the topological defect world-lines in the 4d spacetime will extended
to pseudo world-sheets in 5d. The defect braiding statistics corresponds to
the effect that each pair of linked 2-spheres of the world-sheets of two
different types of defects will contribute a $(-1)$ sign to the partition
function weight [47, 48]. On the field theory level, this effect can be
described by a Wess-Zumino-Witten (WZW) term, defined in the 5d extension,

$e^{\mathrm{i}S_{\text{WZW}}[\Phi]}=\exp\Big{(}\mathrm{i}\pi\int_{\mathcal{M}_{5}}B(\Phi_{\mathbf{54}})\smile\delta
B^{\prime}(\Phi_{\mathbf{45}})\Big{)}\Big{|}_{\mathcal{M}_{4}=\partial\mathcal{M}_{5}},$
(12)

where $B$ and $B^{\prime}$ are two-cocycles also two-cohomology class of the
second cohomology group that evaluate the defect topological number when they
are integrated over a 2-sphere enclosing the point defect in the 3D space,

$\begin{split}B&\in{\rm
H}^{2}\Big{(}\frac{\mathrm{SO}(10)}{\mathrm{SO}(6)\times\mathrm{SO}(4)},\mathbb{Z}_{2}\Big{)}\\\
&=\text{Hom}\Big{(}\pi_{2}\Big{(}\frac{\mathrm{SO}(10)}{\mathrm{SO}(6)\times\mathrm{SO}(4)}\Big{)},\mathbb{Z}_{2}\Big{)},\\\
B^{\prime}&\in{\rm
H}^{2}\Big{(}\frac{\mathrm{SO}(10)}{\mathrm{U}(5)},\mathbb{Z}\Big{)}=\text{Hom}\Big{(}\pi_{2}\Big{(}\frac{\mathrm{SO}(10)}{\mathrm{U}(5)}\Big{)},\mathbb{Z}\Big{)}.\\\
\end{split}$ (13)

Here we use the fact that if $K$ is a $(n-1)$-connected topological space,
then ${\rm H}^{n}(K,{\rm A})=\text{Hom}(\pi_{d}(K),{\rm A})$ for some abelian
${\rm A}$, by the Hurewicz theorem and universal coefficient theorem.

With the additional WZW term, the field theory (which has the spacetime-
internal symmetry $\mathrm{Spin}\times_{\mathbb{Z}_{2}}\mathrm{Spin}(10)$
structure) takes the form of

$S=S_{\text{YM}}[\psi,A]+S_{\text{H}}[\Phi,A]+S_{\text{Y}}[\psi,\Phi]+\mathrm{i}S_{\text{WZW}}[\Phi].$
(14)

This field theory has a $w_{2}w_{3}$ $\mathbb{Z}_{2}$-class mixed gauge-
gravitational global anomaly [49, 50, 51] captured by a 5d bulk invertible
topological quantum field theory (invertible TQFT)

$\begin{split}&\exp\Big{(}\mathrm{i}\pi\int_{\mathcal{M}_{5}}w_{2}(V_{\mathrm{SO}(10)})w_{3}(V_{\mathrm{SO}(10)})\Big{)}\\\
=&\exp\Big{(}\mathrm{i}\pi\int_{\mathcal{M}_{5}}w_{2}(TM)w_{3}(TM)\Big{)},\end{split}$
(15)

where $w_{2},w_{3}$ denotes the Stiefel-Whitney classes of either the
associated $V_{\mathrm{SO}(10)}$ vector bundle or the $TM$ tangent bundle.
This anomaly originated purely from the GUT Higgs sector $\Phi$. The matter
fermions $\psi$ does not contribute to it. In the presence of this anomaly,
the GUT Higgs field should be viewed as the 4d boundary modes of a 5d
invertible TQFT. Due to the anomaly, the $\mathrm{Spin}(10)$ symmetry can not
be dynamically gauged on the 4d boundary alone. The $\mathrm{Spin}(10)$ gauge
field must propagates into the 5d bulk.

The $w_{2}w_{3}$ anomaly only exists in the presence of the
$G_{\mathfrak{so}(10)}=\mathrm{Spin}(10)$ symmetry. It is lifted when the
$\mathrm{Spin}(10)$ symmetry is broken to any subgroups of interest:
$G_{\text{PS}}$, $G_{\text{GG}}$ or $G_{\text{SM}}$. Thus condensing either
$\Phi_{\mathbf{45}}$ or $\Phi_{\mathbf{54}}$ GUT Higgs fields will break the
$\mathrm{Spin}(10)$ symmetry and cancels the $w_{2}w_{3}$ anomaly. The
resulting PS, GG, and SM phases are just ordinary spontaneous symmetry
breaking phases (or Higgs phases if symmetries are gauged). What becomes non-
trivial is the $\mathrm{Spin}(10)$ symmetric $\mathfrak{so}(10)$ GUT phase.
Due to the anomaly introduced by the WZW term, the GUT Higgs fields can not be
trivially disordered (namely, trivially and featurelessly gapped without
ground state degeneracy associated with topological order or symmetry breaking
order). One possibility is that the GUT Higgs field enters a “spin liquid”
fractionalized phase when the $\mathrm{Spin}(10)$ symmetry is restored, which
happens either at the PS-GG transition point or in the $\mathfrak{so}(10)$ GUT
phase.

Figure 6: Phase diagram in the presence of the WZW term. The dashed circle
denotes the confine-deconfine transition of the emergent
$\mathrm{U}(1)^{\prime}$ gauge field. The solid-line phase boundaries between
two neighbor phases all are described by Higgs-condensation continuous quantum
phase transitions.

The fractionalized phase of the GUT Higgs field can be understood from a
parton construction for the WZW field theory. The WZW term can be replaced by
quantum electrodynamics (QED) theory in 4d spacetime, in terms of
fractionalized fermionic partons $\xi$ coupled to an emergent
$\mathrm{U}(1)^{\prime}$ gauge field (this $\mathrm{U}(1)^{\prime}$ is beyond
the SM, not any of the $\mathrm{U}(1)$ gauge group in the SM or GUTs),

$S_{\text{QED}^{\prime}_{4}}=\int_{\mathcal{M}^{4}}\mathrm{d}^{4}x\;\bar{\xi}(\mathrm{i}\gamma^{\mu}D_{\mu,a}-\Phi_{\mathbf{54}}-\mathrm{i}\gamma^{5}\Phi_{\mathbf{45}})\xi.$
(16)

The fermionic parton field $\xi$ contains 10 Dirac fermions forming the
representation $\mathbf{10}$ of $\mathrm{SO}(10)$, which also carries 1 unit
of an emergent $\mathrm{U}(1)^{\prime}$ gauge charge. The gauge field $a$ is
neural under $\mathrm{SO}(10)$. The emergent $\mathrm{U}(1)^{\prime}$ gauge
bosons can be called “dark” photons as they do not couple to any of the SM
particles. The dark photons mediate the gauge interaction between fermionic
partons and bind them back into the GUT Higgs field $\Phi$. The GUT Higgs
field $\Phi$ is in the $\mathbf{10}\times\mathbf{10}$ bivector representation
of $\mathrm{SO}(10)$, which is fractionalized as the fermionic parton bilinear
mass terms

$\Phi_{\mathbf{54}}\sim\bar{\xi}\xi,\quad\Phi_{\mathbf{45}}\sim\bar{\xi}\mathrm{i}\gamma^{5}\xi.$
(17)

The phase described by the QED${}^{\prime}_{4}$ parton theory is similar to
the $\mathrm{U}(1)$ Dirac spin liquid [52, 53] discussed in the condensed
matter physics context.

To check that the proposed parton theory Eq. (16) reproduces the same
$w_{2}w_{3}$ anomaly as the WZW term, one can temporally turn off the GUT
Higgs coupling to partons, which allows the gauge group to be enlarged to
$\mathrm{SU}(2)^{\prime}$ (with $\mathrm{U}(1)^{\prime}$ being its subgroup).
Now the theory
$\int_{\mathcal{M}^{4}}\mathrm{d}^{4}x\;\bar{\xi}(\mathrm{i}\gamma^{\mu}D_{\mu,a})\xi$
contains $2\times 10=20$ Weyl fermions, transforming as
$(\mathbf{2},\mathbf{10})$ in $\mathrm{SU}(2)^{\prime}\times\mathrm{SO}(10)$.
The anomaly can be matched by tracking the fermion representations

$\begin{array}[]{ccccc}\mathrm{U}(1)^{\prime}\times\mathrm{SO}(10)&\hookrightarrow&\mathrm{SU}(2)^{\prime}\times\mathrm{SO}(10)&\hookrightarrow&\mathrm{Sp}(10)\\\
{\bf 10}_{1}&&({\bf 2},{\bf 10})&\sim&{\bf 20}\\\
&&&&\rotatebox[origin={c}]{-90.0}{$\cong$}\\\
\mathrm{SU}(2)^{\prime\prime}\times\mathrm{Sp}(8)&\hookrightarrow&\mathrm{Sp}(2)\times\mathrm{Sp}(8)&\hookrightarrow&\mathrm{Sp}(10)\\\
({\bf 4},{\bf 1})\oplus({\bf 1},{\bf 16})&\sim&({\bf 4},{\bf 1})\oplus({\bf
1},{\bf 16})&\sim&{\bf 20}\end{array}.$ (18)

The twenty Weyl fermions can be further embedded in the $\mathbf{20}$-dim
representation of $\mathrm{Sp}(10)$, which transforms the fermion fields as
ten quaternion Grassmann numbers. On the other hand, it can be split to
$\mathrm{Sp}(2)\times\mathrm{Sp}(8)$ that separately rotate the first two and
the last eight quaternion components $({\bf 4},{\bf 1})\oplus({\bf 1},{\bf
16})$. The $\mathrm{Sp}(2)$ group has an $\mathrm{SU}(2)^{\prime\prime}$
subgroup as its maximal special embedding. It is known that odd number of
fermions in the $\mathbf{4}$-dim (spin-3/2) representation of $\mathrm{SU}(2)$
can generate the new $\mathrm{SU}(2)$ anomaly [51], described by the
$w_{2}w_{3}$ anomaly term. Using this fact, the
$\mathrm{SU}(2)^{\prime\prime}$ sector will give rise to the $w_{2}w_{3}$
anomaly and the $\mathrm{Sp}(8)$ sector will not (as it contains an even
number of spin-3/2 fermions when broken down to its $\mathrm{SU}(2)$ subgroup
similarly). By anomaly matching, one can conclude that the fermions will
generate the $w_{2}w_{3}$ anomaly of the $\mathrm{Sp}(10)$ bundle. However,
the emergent $\mathrm{U}(1)^{\prime}$ sector of fermionic partons alone does
not give rise to the $w_{2}w_{3}$ anomaly, so the anomaly must be contained in
the $\mathrm{SO}(10)$ sector of fermionic partons, which reproduces the
anomaly of the WZW term.

The anomaly matching argument indicates that the gapless fermionic partons can
induce the desired braiding statistics between the topological defects of
$\Phi_{\mathbf{45}}$ and $\Phi_{\mathbf{54}}$, which provides a possible state
that saturates the anomaly. In this state, the Higgs field $\Phi$ deconfines
into fermionic partons coupled by the emergent $\mathrm{U}(1)^{\prime}$ gauge
field governed by the QED${}^{\prime}_{4}$ theory in Eq. (16). Naïvely and
intuitively, the GG and PS phases can then be connected by a continuous
transition with a deconfined quantum critical (DQC) point, as the white dot in
Fig. 6. However, since the deconfined $\mathrm{U}(1)^{\prime}$ gauge theory is
stable in 4d, the critical point will extend into a critical phase in the
$\mathfrak{so}(10)$ GUT phase (in the presence of the WZW term), as indicated
by the DQC region in Fig. 6. The DQC region (including the DQC point) is
described by the QED${}^{\prime}_{4}$ theory.

Starting from the DQC phase, condensing the $\Phi_{\mathbf{45}}$ or
$\Phi_{\mathbf{54}}$ Higgs fields will generate bilinear masses for the
fermionic partons and break the $\mathrm{Spin}(10)$ symmetry at the same time.
When the fermionic partons become fully gapped, the emergent
$\mathrm{U}(1)^{\prime}$ dark photons can remain deconfined and gapless, as
the Maxwell theory has a stable deconfined phase in 4d. Therefore, the
resulting phases are PS, GG, or SM theories coexisting with a decoupled pure
$\mathrm{U}(1)^{\prime}$ gauge theory. They are denoted as PS∗, GG∗ and SM∗ in
Fig. 6 (where the superscript $*$ indicates the coexistence of dark photons in
the low-energy spectrum).

However if the $\mathrm{U}(1)^{\prime}$ gauge coupling is strong enough, the
$\mathrm{U}(1)^{\prime}$ gauge theory can also be driven to its confined
phase. The confinement transition is indicated by the dashed circle in Fig. 6.
After $\mathrm{U}(1)^{\prime}$ confinement, the DQC physics will disappear.
The PS∗, GG∗ and SM∗ phases will fall back to the conventional PS, GG, and SM
phases. But the $\mathfrak{so}(10)$ GUT phase with the WZW term (even after
the $\mathrm{U}(1)^{\prime}$ confinement) will remain non-trivial due to the
$w_{2}w_{3}$ anomaly. The anomaly may also be saturated by other CFT (differed
from the QED${}^{\prime}_{4}$ CFT proposed here), or by some TQFT.

Figure 7: Quantum numbers of fermionic partons in the DQC phase.

The quantum numbers of the ten Dirac fermionic partons are summarized in Fig.
7. In terms of the $\mathfrak{g}_{\text{SM}}$ representations, the fermionic
partons have charge assignments that are very different from those of the
matter fermions in the SM. Some fermionic partons (call colorons
$c,c^{\prime}$) carries the $\mathfrak{su}(3)$ color charge but not the weak
$\mathfrak{su}(2)_{L}$ flavor charge, while other fermionic partons (called
flavorons $f,f^{\prime}$) carries the weak $\mathfrak{su}(2)_{L}$ flavor
charge but not the color $\mathfrak{su}(3)$ charge. This color-flavor
separation on the fermionic partons is one important feature of these
additional fermions that are fractionalized from the GUT Higgs field. This is
analogous to the spin-charge separation [54, 55, 56] in condensed matter
physics.

Suppose the DQC phase once existed in the early history of our universe, now
we have entered the SM∗ phase, where the GUT Higgs field $\Phi$ has condensed.
All these fermionic partons will be gapped by the Higgs condensation at the
GUT scale. The $\mathrm{U}(1)^{\prime}$ dark photons are decoupled from the
fermionic matter particles in the SM, and remain as dark cosmic background
radiation, which could provide a potential dark matter candidate.

##  IV Summary

Standard lore ritualizes our quantum vacuum in the 4-dimensional spacetime
governed by the Standard Models, while lifting towards one of Grand
Unifications (GUTs) at higher energy scales. In contrast, in recent studies
[10, 11, 12], we introduce an alternative view that the SM is a low-energy
quantum vacuum arising from various neighbor GUT vacua competition in an
immense quantum phase diagram. We study the quantum phase transition between
the SM and various GUTs, including the Georgi-Glashow model, the Pati-Salam
model, and the $\mathfrak{so}(10)$ GUT. We found that the competition between
the GG and PS phases could lead to a gapless quantum critical phase, similar
to the deconfined quantum critical point in quantum magnets of condensed
matter, where the GUT Higgs field fractionalizes into fermionic partons
coupled by an emergent gauge field. This fractionalization is entailed by a
$w_{2}w_{3}(TM)=w_{2}w_{3}(V_{\mathrm{SO}(10)})$ nonperturbative global
anomaly in the presence of a Wess-Zumino-Witten-like term for the GUT Higgs
field. The quantum critical phase features fragmentary low-energy excitations
of Color-Flavor separation of fermionic partons and emergent Dark Gauge force,
which may leave observable consequences even away from the critical phase.

###### Acknowledgements.

The authors are honored to contribute to the Professor Chen-Ning Yang
Centenary Festschrift. A related presentation is given by Yi-Zhuang You at the
Ultra-Quantum Matter Annual Meeting on January 21, 2022 [57].

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