# Mimicking Negative Mass Properties

S. D. Campos<EMAIL_ADDRESS>Applied Mathematics Laboratory - CCTS/DFQM,
Federal University of São Carlos, Sorocaba, CEP 18052780, Brazil

###### Abstract

In the present work, one analyzes two systems trying to obtain physical
conditions where some properties attributed to negative mass can be mimicked
by positive mass particles. The first one is the well-known 1/2-spin system
described by the Dirac equation in the presence of an external electromagnetic
field. Assuming some physical restrictions, one obtains that the use of
$e\rightarrow-e$ can lead to the same results as using $m\rightarrow-m$. In
particular, for a null dielectric function, it is possible to obtain a
negative mass behavior from a positive mass system composed of negatively
charged particles. The second system is based on the de Broglie matter wave.
The dispersion relation of such a wave can be negative (real or imaginary
valued) if one assumes an imaginary wavenumber. The consequence is the
emergence of a negative refractive index for positive mass particles. However,
this behavior is generally attributed to a negative mass system.

negative mass; Dirac equation; dispersion relation; spin-orbit

###### pacs:

11.55.Fv, 05.30.Fk, 71.70.Ej

##  I Introduction

For some intriguing reason, unicorns do not exist, even though the laws of
Nature do not explicitly prohibit their existence. The general belief is that
these laws allow for any phenomenon unless there is a specific constraint or
limitation explicitly preventing it. The laws of Nature are inclusive rather
than restrictive unless stated otherwise. Hence, it is theoretically possible
to create one of these fantastic creatures. Of course, it is not a true-born
unicorn since it was built following some recipe (not necessarily the best
one) and obeying some (not necessarily all) rules displayed on the table.

In physics, there is no fundamental law of classical or quantum mechanics
prohibiting the existence of negative mass (see Ref. SDC.2023  and references
therein). However, the lack of experimental evidence in the present day
unfortunately suggests that negative mass does not exist. Indeed, the main
assumption in SDC.2023  is that negative mass particles have imaginary
velocity, which prohibits their direct experimental determination. However,
the transformation $v\rightarrow iv$ provides positive kinetic energy,
certainly contributing to the total energy of the system.

One of the first alleged experimental evidence of a system presenting physical
properties related to negative effective mass is from 1959
G.C.Dousmanis.R.C.DuncanJr.J.J.Thomas.R.C.Williams.PRL.1(11)404.1958 . Using
cyclotron resonance, Dousmanis and collaborators suggested the detection of
negative mass carriers in Ge crystals at 4 K. In 2017, more than 60 years
later, some physical effects expected for negative effective mass were
generated in a spin-orbit coupled Bose-Einstein Condensate
M.A.Khamehchi.K.Hossain.M.E.Mossman.Y.Zhang.Th.Busch.M.M.Forbes.P.Engels.PRL118.155301.2017
. Specifically, Khamehchi and collaborators suggest that a modified dispersion
relation and a negative effective mass may explain their results as well as
other related
J.P.Ronzheimer.M.Schreiber.S.Braun.S.S.Hodgman.S.Langer.I.P.McCulloch.F.Heidrich.Meisner.I.Bloch.U.Schneider.PRL110.205301.2013
;
A.Reinhard.J.Riou.L.A.Zundel.D.S.Weiss.S.Li.A.M.Rey.R.Hipolito.PRL110.033001.2013
; K.Henderson.H.Kelkar.B.Gutierrez.Medina.T.C.Li.M.G.Raizen.PRL96.150401.2006
;
Th.Anker.M.Albiez.R.Gati.S.Hunsmann.B.Eiermann.A.Trombettoni.M.K.Oberthaler.PRL94.020403.2005
;
B.Eiermann.P.Treutlein.Th.Anker.M.Albiez.M.Taglieber.K.P.Marzlin.M.K.Oberthaler.Phys.Rev.Lett91.060402.2003
.

Here, in the first step, the system studied describes the Dirac equation in
the presence of an external electromagnetic field, showing that
$m\rightarrow-m$ is not everywhere equivalent to $e\rightarrow-e$. Indeed,
both transformations describe the same system only when some physical
conditions, involving the scalar electric field or the dielectric function,
are assumed to be valid. An interesting case emerges when the dielectric
function is null for some frequencies. In this case, a positively charged
system with negative mass has the same physical behavior as a negatively
charged system with positive mass. Similar behavior can be viewed in some
metamaterials where a negative mass behavior emerges from the plasma
oscillations of a free electron gas
E.Bormashenko.I.Legchenkova.Materials.Basel.13.8.1890.2020 ;
E.Bormashenko.I.Legchenkova.M.Frenkel.Materials.Basel.13.16.3512.2020 . There
are several applications for metamaterials with properties not found in usual
materials. In particular, some metamaterials allow the emergence of a negative
refractive index
M.A.Khamehchi.K.Hossain.M.E.Mossman.Y.Zhang.Th.Busch.M.M.Forbes.P.Engels.PRL118.155301.2017
; J.A.Dionne.Optics.Express.16.23.19001.2008 ; J.Valentine.Nature.455.376.2008
, connected with the next case studied here.

The second step presents the de Broglie matter wave dispersion relation.
Rather than resorting to a negative mass for achieving a negative dispersion
relation, it is obtained by employing an imaginary wavenumber, indicating the
wave decaying within the medium. The resulting group velocity is imaginary,
expressing the presence of a non-conservative force. Depending on the
frequency, a negative imaginary dispersion relation can also arise, with the
corresponding imaginary group velocity pointing out a non-restorative force
affecting the system. In any case, the negative real or negative imaginary
dispersion relation may be viewed as being due to a negative mass
Y.Liu.G.P.Wang.J.B.Pendry.S.Zhang.Light.Sci.Appl.11.276.2022 ;
T.Suzuki.M.Sekiya.T.Sato.Y.Takebayashi.Opt.Express.26.8314.2018 ;
M.A.Khamehchi.K.Hossain.M.E.Mossman.Y.Zhang.Th.Busch.M.M.Forbes.P.Engels.PRL118.155301.2017
; D.Forcella.C.Prada.R.Carminati.Phys.Rev.Lett.118.134301.2017 ;
S.A.Ramakrishna.Rep.Prog.Phys.68.449.2005 ;
K.Y.Bliokh.Y.P.Bliokh.Sov.Phys.Uspekhi.47.393.2004 ;
Y.Ben.Aryeh.J.Mod.Opt.52.1871.2005 ; J.B.Pendry.Phys.Rev.Lett.85.3966.2000 .

This paper is organized as follows. In section II, one shows that the use of a
negative mass in the Dirac equation (free-particle) is equivalent to a time-
reversal procedure. However, in the presence of an electromagnetic 4-vector,
it is not generally true as shall be seen in Section III. Section IV presents
some considerations about the negative dispersion relation. Discussion and
remarks are left for the final section V.

##  II Negative Mass and Charge in the Dirac Equation

The use of $-m$ instead of $m$ in the Dirac equation (DE) seems to be a simple
mathematical replacement. However, this alteration could carry significant
physical ramifications since the interaction properties of negative mass are
quite different from positive ones (see Ref. SDC.2023  and references
therein).

In this first case, the DE is used to describe $1/2$-spin particles in the
absence of a potential, being written as

$\displaystyle\mathcal{H}\psi(t,\mathbf{r})=i\hbar\frac{\partial\psi(\mathbf{r},t)}{\partial
t},$ (1)

where $\mathbf{r}=(x^{1},x^{2},x^{3})$ and

$\displaystyle\mathcal{H}=\mathbf{\alpha}\cdot\bigl{(}-i\hbar\mathbf{\nabla}\bigr{)}-\alpha_{4}mc^{2}$
(2)

for $\mathbf{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3})$. For a matter of
internal consistency, $\alpha_{i}$’s obey the following known relations
($i,j=1,2,3$)

$\displaystyle\alpha_{4}^{2}=\mathcal{I},~{}~{}\\{\alpha_{4},\alpha_{i}\\}=0,~{}~{}\\{\alpha_{i},\alpha_{j}\\}=2\delta_{ij}\mathcal{I},$
(3)

where $\\{a,b\\}\equiv ab+ba$ is the anticommmutator and $\mathcal{I}$ is the
$4\times 4$ identity matrix. Using $\gamma$ matrices, one can rewrite DE (1)
in the compact form mandl.shawn

$\displaystyle(i\hbar\gamma^{\mu}\partial_{\mu}-mc)\psi(\mathbf{r},t)=0.$ (4)

From the mathematical point of view, replacing $m$ with $-m$ does not alter
the solutions of DE: it still yields positive energy solutions for electrons
and negative energy solutions for positrons. Then, the equation

$\displaystyle(i\hbar\gamma^{\mu}\partial_{\mu}\pm mc)\psi(\mathbf{r},t)=0,$
(5)

where $+$ and $-$ signs stand for positive and negative mass, respectively,
has the same set of solutions (interpreted according to the mass sign). For
the sake of simplicity, one looks here just for the stationary solutions of
(5), written as

$\displaystyle\psi(\mathbf{r},t)=\chi(\mathbf{r})e^{-i(\mathcal{E}/\hbar)t},$
(6)

where $\mathcal{E}$ is the energy that can be positive (particles) or negative
(antiparticles). The two-component of the time-independent spinor field
$\chi(\mathbf{r})$ is given by

$\displaystyle\chi(\mathbf{r})=\begin{pmatrix}\phi(\mathbf{r})\\\
\eta(\mathbf{r})\\\ \end{pmatrix}.$ (7)

Nevertheless, is there another parameter within the DE that, when replaced,
could yield equivalent physical interpretations for the solutions obtained
when using $m\rightarrow-m$? The answer is yes.

It is straightforward to observe that the transformation $t\rightarrow-t$
leads to the same set of solutions if one assumes $+\mathcal{E}$ for the
particle and $-\mathcal{E}$ for the antiparticle. Consequently, from a
dynamics standpoint, replacing $m\rightarrow-m$ in DE is equivalent to using
$t\rightarrow-t$ in the same differential equation, taking into account the
convenient definition of particles and antiparticles.

As shall be seen, the presence of an electromagnetic field introduces some
interesting physical results when one uses $m\rightarrow-m$ or
$e\rightarrow-e$.

##  III Dirac Equation in the Presence of an External Electromagnetic Field

###  III.1 Basic Picture

For the sake of simplicity, let us assume that the vector
$\mathbf{A}=\mathbf{A}(\mathbf{r})$ and the scalar $\phi=\phi(\mathbf{r})$
potentials are time-independent, and associated with the magnetic and electric
components of the external electromagnetic field by the relations jackson

$\displaystyle\mathbf{B}=\nabla\times\mathbf{A},~{}~{}\mathbf{E}=-\nabla\phi,$

where $\mathbf{B}=\mathbf{B}(\mathbf{r})$ and
$\mathbf{E}=\mathbf{E}(\mathbf{r})$.

The Dirac equation for a particle with charge $q$ subject to an external
electromagnetic field has the following Hamiltonian bransden.joachin

$\displaystyle\mathcal{H}=\frac{1}{2m}\bigl{[}\mathbf{p}-q\mathbf{A}\bigr{]}^{2}+q\phi-\frac{q\hbar}{2m}\bigl{[}\mathbf{\sigma}\cdot\mathbf{B}\bigr{]},$
(8)

where the linear momentum $\mathbf{p}$ is defined as
$\mathbf{p}=-i\hbar\mathbf{\nabla}$ and the term
$e\hbar/2m\bigl{(}\mathbf{\sigma}\cdot\mathbf{B}\bigr{)}$ corresponds to the
interaction term relating the spin and the magnetic field $\mathbf{B}$, and,
for the sake of simplicity, hereafter $m=m_{e}=m_{p}$. The spin and magnetic
moment operators are related by

$\displaystyle\mathbf{\mu}_{q}=\frac{q\hbar}{2m}\mathbf{S},$ (9)

and for the electron, it is usual to represent the above equality as

$\displaystyle\mathbf{\mu}_{e}=-\mu_{B}\mathbf{\sigma}=-\frac{e\hbar}{2m}\mathbf{S}$
(10)

to denote that $\mathbf{\mu}_{e}$ and $\mathbf{S}=(\hbar/2)\mathbf{\sigma}$
are antiparallel vectors M.Vogel.W.Quint.Phys.Rep.490.1.2010 . Moreover,
$\mu_{B}$ is the Bohr magneton, and
$\sigma=(\sigma_{x},\sigma_{y},\sigma_{z})$ are the usual Pauli spin matrices.

In the case of electrons $-e$ and positrons $+e$, replacing (8) in (4) should
be followed by the Dirac prescription
P.A.M.Dirac.Proc.Roy.Soc.Lond.Ser.A.133.80.1931 ; Bethe.Salpeter_book :
transformation $\mathcal{E}\rightarrow-\mathcal{E}$ is also followed by
changing the sign of $\mathbf{p}$, $e\mathbf{A}$, $e\phi$, and the interaction
term. This procedure ensures the solutions for negative $\mathcal{E}$ are only
the opposite of the solutions for positive $\mathcal{E}$. The results for
positive and negative $\mathcal{E}$ are the well-known Pauli equations (for
electrons and positrons)

$\displaystyle\mathcal{E}_{\pm}\chi_{\pm}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\pm\left[\frac{1}{2m}\bigl{(}\mathbf{p}+e\mathbf{A}\bigr{)}^{2}-e\phi+\frac{e\hbar}{2m}\bigl{(}\mathbf{\sigma}\cdot\mathbf{B}\bigr{)}\right]\chi_{\pm}(\mathbf{r}).$
(11)

Notice, as expected, that $\mathcal{E}_{-}=-\mathcal{E}_{+}$. On the other
hand, the transformation $t\rightarrow-t$ furnishes the same result as above
only if the Dirac prescription is used. It seems reasonable since the
transformation $t\rightarrow-t$ is equivalent to
$\mathcal{E}\rightarrow-\mathcal{E}$ in the stationary solutions. One writes
for electrons and positrons (note that $t\rightarrow-t$ implies
$dt\rightarrow-dt$)

$\displaystyle\mathcal{E}_{\pm}\chi_{\pm}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\pm\left[\frac{1}{2m}\bigr{(}\mathbf{p}+e\mathbf{A}\bigl{)}^{2}-e\phi+\frac{e\hbar}{2m}\bigl{(}\mathbf{\sigma}\cdot\mathbf{B}\bigr{)}\right]\chi_{\pm}(\mathbf{r}).$
(12)

Comparing (11) and (12), one observes the well-known result that the time and
charge inversion represent the same physical situation for the Dirac equation
in the presence of an external electromagnetic field.

The interest here is to study the effects of the transformation
$m\rightarrow-m$ in the Dirac equation using the above Hamiltonian and looking
for stationary solutions (6). Let us start by using the stationary solution
(6) and replacing $m\rightarrow-m$ into DE. The results can be exhibited in a
single equation for electrons and positrons

$\displaystyle\mathcal{E}_{\pm}\chi_{\pm}(\mathbf{r})=\mp\left[\frac{1}{2m}\bigl{(}\mathbf{p}+e\mathbf{A}\bigr{)}^{2}+e\phi+\frac{e\hbar}{2m}\bigl{(}\mathbf{\sigma}\cdot\mathbf{B}\bigr{)}\right]\chi_{\pm}(\mathbf{r}).$
(13)

Comparing (13) and (11), one observes that in the DE equation for the
Hamiltonian (8), the transformations $e\rightarrow-e$ and $m\rightarrow-m$ do
not lead to the same equations. However, these transformations represent the
same physical situation if the following condition is satisfied

$\displaystyle e\phi\chi(\mathbf{r})=0.$ (14)

The simplest way to accomplish the above condition is assuming the static
charge distribution $\rho=\rho(\mathbf{r})$ in a given volume $V$ is null

$\displaystyle\phi(\mathbf{r})=\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho}{|\mathbf{r}-\mathbf{r}^{\prime}|}dV=0.$
(15)

which occurs for a zero charge density within the homogeneous volume $V$ and a
transverse electric field. In general, result (14) implies

$\displaystyle\nabla\cdot\mathbf{E}=-\nabla^{2}\phi=0.$ (16)

However, there is another way to obtain (14) without assuming
$\nabla\cdot\mathbf{E}=0$. Notice the Gauss law can be written by using the
displacement field
$\mathbf{D}=\mathbf{D}(\mathbf{r})=\epsilon(\omega)\mathbf{E}+4\pi\mathbf{P}$
as jackson

$\displaystyle\nabla\cdot\mathbf{D}=\rho,$ (17)

where $\epsilon(\omega)$ is the dielectric function that depends on the
frequency $\omega$, and the polarization field $\mathbf{P}$ is associated with
the induced charge density. For a linear homogeneous isotropic material, the
polarization is null. Hence, one writes

$\displaystyle\epsilon(\omega)\bigr{[}\nabla\cdot\mathbf{E}\bigl{]}=0.$ (18)

The product shown in result (18) is null $\nabla\cdot\mathbf{E}=0$ or for
$\epsilon(\omega)=0$, for some frequencies. Assuming $\epsilon(\omega)=0$, for
some frequencies, it implies the existence of a non-null longitudinal
contribution for the electric field allowing the plasma to have a charge
fluctuation, where the resulting plasma oscillation is the so-called plasmon
Stefan.A.Maier.Plasmonics.Fundamentals.and.Applications.Springer.2007 .

The above assumption on the dielectric function implies that the plasmon
emerging in a plasma composed of negatively charged particles (with positive
mass) behaves as a plasmon emerging in a positively charged plasma (with
negative mass). An effect similar to this one was already observed in
metamaterials E.Bormashenko.I.Legchenkova.Materials.Basel.13.8.1890.2020 ;
E.Bormashenko.I.Legchenkova.M.Frenkel.Materials.Basel.13.16.3512.2020 , where
plasma oscillations of a free electron gas induces a negative effective mass
in the system due to the vibrations of a metallic particle with frequency
$\omega$, close to the plasma oscillation frequency
E.Bormashenko.I.Legchenkova.Materials.Basel.13.8.1890.2020 .

##  IV Dispersion Relation

Considering massive particles, the de Broglie matter waves can be obtained by
considering the usual relativistic energy relation

$\displaystyle(\hbar\omega)^{2}=(pc)^{2}+\bigl{(}mc^{2}\bigr{)}^{2},$ (19)

where $\mathbf{p}=\hbar\mathbf{k}$, $\mathbf{k}$ is the wavevector
($|\mathbf{k}|=k$), and $\omega=\omega(k)$ is the angular frequency (also
known as dispersion relation). Solving the above equation for $\omega$, one
obtains the well-known relativistic dispersion relation for matter waves

$\displaystyle\omega=\frac{mc^{2}}{\hbar}\sqrt{1+\left(\frac{\hbar
k}{mc}\right)^{2}}.$ (20)

Observe that (20) can assume negative values by considering $m<0$ and a
convenient range of frequency. However, the main purpose here is to present
physical conditions where $\omega$ can be negative without assuming a priori
negative mass since one seeks physical situations where a positive mass can
mimic the expected behavior of a negative mass.

Indeed, a similar result can be achieved by assuming that $k$ is an imaginary
number representing an exponential decay away from the source. Thus, the
wavenumber is (almost) locally defined near the source. Indeed, this condition
occurs in the so-called evanescent wave and indicates it does not propagate,
representing a local vibration with fast decay away from the source
V.Giurgiutiu.M.F.Haider.Materials.Basel.12.2.269.2019 . Then, one writes the
imaginary wavenumber as

$\displaystyle k=i\delta,$ (21)

where $\delta$ corresponds to the decaying of the wave energy through the
medium. Then, inserting the assumption (21) into equation (20), one obtains

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle-\frac{m_{0}c^{2}}{\hbar}\sqrt{1-\left(\frac{\hbar\delta}{m_{0}c}\right)^{2}}.$
(22)

Notice the emergence of the negative sign within the square root imposes some
care in the study of the dispersion relation (22). Indeed, the above
dispersion relation is negative if

$\displaystyle-\frac{m_{0}c}{\hbar}<\delta<\frac{m_{0}c}{\hbar},$ (23)

while it is imaginary for $\delta>|m_{0}c/\hbar|$.

It is important to stress that replacing $m\rightarrow-m$ into (20) is
equivalent to using the imaginary wavenumber only if the bound (23) is
satisfied. Thus, the dispersion relation (22), in this simple case, is not due
to the mass sign but to the introduction of an imaginary wavenumber, that
represents the introduction of a non-conservative force. Therefore, the
introduction of the imaginary wavenumber allows a positive mass to have a
negative dispersion relation.

A similar behavior is expected for light in some metamaterials, where the
medium bends light to a negative angle relative to the surface normal, i.e.
the incident and refracted waves lie on the same side of the surface normal
Y.Liu.G.P.Wang.J.B.Pendry.S.Zhang.Light.Sci.Appl.11.276.2022 . This behavior
is exposed in
M.A.Khamehchi.K.Hossain.M.E.Mossman.Y.Zhang.Th.Busch.M.M.Forbes.P.Engels.PRL118.155301.2017
, where the negative dispersion is argued to indicate a negative refraction
index. There is a growing interest in materials with a negative refraction
index Y.Liu.G.P.Wang.J.B.Pendry.S.Zhang.Light.Sci.Appl.11.276.2022 ;
T.Suzuki.M.Sekiya.T.Sato.Y.Takebayashi.Opt.Express.26.8314.2018 ;
M.A.Khamehchi.K.Hossain.M.E.Mossman.Y.Zhang.Th.Busch.M.M.Forbes.P.Engels.PRL118.155301.2017
; D.Forcella.C.Prada.R.Carminati.Phys.Rev.Lett.118.134301.2017 ;
S.A.Ramakrishna.Rep.Prog.Phys.68.449.2005 ;
K.Y.Bliokh.Y.P.Bliokh.Sov.Phys.Uspekhi.47.393.2004 ;
Y.Ben.Aryeh.J.Mod.Opt.52.1871.2005 ; J.B.Pendry.Phys.Rev.Lett.85.3966.2000 .
Causality is invoked to argue that dissipation is a necessary condition for a
negative refraction index M.I.Stockman.Phys.Rev.Lett.98.177404.2007 . In the
case of matter waves presented here, the introduction of an imaginary
wavenumber means the introduction of a non-restorative force (respecting
causality).

On the other hand, some values of $\delta$ can also allow an imaginary
dispersion relation, representing the absorption of the wave by the medium.
Here, the imaginary dispersion relation (22) is negative, indicating the
decaying amplitude in the direction of the wave motion. This result agrees
with the introduction of a dissipative force.

For the case of a negative real dispersion relation, the group velocity
obtained from (22) is imaginary ($dk=id\delta$) and opposite to the dispersion
relation, being given by

$\displaystyle\frac{d\omega}{d\delta}=\frac{i\hbar\delta}{m_{0}\sqrt{1-\left(\frac{\hbar\delta}{m_{0}}\right)^{2}}}.$
(24)

The above result is typical of evanescent modes
J.W.Ryu.N.Myoung.H.C.Park.Scientific.Reports.7.8746.2017 , indicating values
of frequency for which the wave does not propagate. Notice the energy related
to this velocity is negative $E\sim(id\omega(\delta)/d\delta)^{2}$ since the
mass is positive.

On the other hand, considering a negative imaginary dispersion relation, the
group velocity is negative real,

$\displaystyle\frac{d\omega}{d\delta}=-\frac{\hbar\delta}{m_{0}\sqrt{1-\left(\frac{\hbar\delta}{m_{0}}\right)^{2}}},$
(25)

which can be interpreted as a negative refraction index
V.Veselago.L.Braginsky.V.Shklover.C.Hafner.J.Comput.Theor.Nanosci.3.1.2006 .

It is important to stress that the second derivative of the frequency (22)
should be positive to ensure the minimum value of the potential is achieved.
This condition holds if

$\displaystyle-\frac{m_{0}}{\hbar}<\delta<\frac{m_{0}}{\hbar},$ (26)

implying the dispersion relation is negative real while the group velocity is
imaginary since $(-m_{0}/\hbar,m_{0}/\hbar)\in(-m_{0}c/\hbar,m_{0}c/\hbar)$.
Nevertheless, if this condition (26) is released, then the dispersion
relations can also be negative imaginary, whereas the group velocity is
negative real.

Therefore, if one assumes $k=i\delta$, one obtains two distinct behaviors for
the dispersion relation: (a) a negative real dispersion relation and an
imaginary group velocity or (b) a negative imaginary dispersion relation and a
negative real group velocity. Both cases depend on the range of $\delta$, i.e.
the frequency of the system. However, whether the dispersion relation real or
imaginary valued function, the negative refraction index seems to be a natural
consequence of that.

##  V Discussion and Remarks

The present work focuses on studying the physical conditions where a positive
mass can mimic the behavior of a negative one. In the first case of study, one
uses $m\rightarrow-m$ in the Dirac equation in the presence of an external
electromagnetic field.

The equivalence between $m\rightarrow-m$ and $e\rightarrow-e$ can be achieved
under the assumptions that (a) the scalar field $\phi$ is null or (b) the
dielectric function is null. The latter furnishes the possibility of a
negative mass behavior emerging, for example, from a plasma negatively charged
with a positive mass
E.Bormashenko.I.Legchenkova.Materials.Basel.13.8.1890.2020 ;
E.Bormashenko.I.Legchenkova.M.Frenkel.Materials.Basel.13.16.3512.2020 . This
possibility may allow the construction of metamaterials with specific
properties and, in particular, with a negative refraction index, which is the
subject of the second system studied here.

In the de Broglie matter wave the emergence of a negative dispersion relation
can be achieved without assuming a negative mass. Indeed, it can be done by
assuming an imaginary wavenumber representing the decaying of the wave energy
through the medium. This assumption, for some range of frequencies, leads to
(a) a negative dispersion relation or (b) an imaginary negative dispersion
relation. The condition (a) results in an imaginary negative group velocity
while (b) the group velocity is negative real. In both cases, the consequence
is the emergence of a negative refraction index, which may be attributed to a
negative mass Y.Liu.G.P.Wang.J.B.Pendry.S.Zhang.Light.Sci.Appl.11.276.2022 ;
T.Suzuki.M.Sekiya.T.Sato.Y.Takebayashi.Opt.Express.26.8314.2018 ;
M.A.Khamehchi.K.Hossain.M.E.Mossman.Y.Zhang.Th.Busch.M.M.Forbes.P.Engels.PRL118.155301.2017
; D.Forcella.C.Prada.R.Carminati.Phys.Rev.Lett.118.134301.2017 ;
S.A.Ramakrishna.Rep.Prog.Phys.68.449.2005 ;
K.Y.Bliokh.Y.P.Bliokh.Sov.Phys.Uspekhi.47.393.2004 ;
Y.Ben.Aryeh.J.Mod.Opt.52.1871.2005 ; J.B.Pendry.Phys.Rev.Lett.85.3966.2000 .

As a last comment, it should be pointed out that the Klein-Gordon equation is
a relativistic second-order differential equation that correctly describes
spinless composite particles, written as

$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\psi(\mathbf{r},t)}{\partial
t^{2}}=\nabla^{2}\psi(\mathbf{r},t)-\frac{m^{2}c^{2}}{\hbar^{2}}\psi(\mathbf{r},t).$
(27)

The use of $m\rightarrow-m$ or $t\rightarrow-t$ is indifferent, leading to the
same set of solutions for $\psi(\mathbf{r},t)$. It indicates that in the
Klein-Gordon equation, there is no physical difference between particles with
positive or negative mass. It corroborates to the usual understanding that the
spinless particles are also their antiparticles.

## Acknowledgments

The author thanks to UFSCar.

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# Edge physics at the deconfined transition
between a quantum spin Hall insulator and a superconductor

Ruochen Ma  Perimeter Institute for Theoretical Physics, Waterloo, Ontario,
Canada N2L 2Y5  Department of Physics and Astronomy, University of Waterloo,
Waterloo, Ontario, Canada N2L 3G1  Liujun Zou  Perimeter Institute for
Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5  Chong Wang  Perimeter
Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5

###### Abstract

We study the edge physics of the deconfined quantum phase transition (DQCP)
between a spontaneous quantum spin Hall (QSH) insulator and a spin-singlet
superconductor (SC). Although the bulk of this transition is in the same
universality class as the paradigmatic deconfined Neel to valence-bond-solid
transition, the boundary physics has a richer structure due to proximity to a
quantum spin Hall state. We use the parton trick to write down an effective
field theory for the QSH-SC transition in the presence of a boundary. We
calculate various edge properties in an $N\to\infty$ limit. We show that the
boundary Luttinger liquid in the QSH state survives at the phase transition,
but only as “fractional” degrees of freedom that carry charge but not spin.
The physical fermion remains gapless on the edge at the critical point, with a
universal jump in the fermion scaling dimension as the system approaches the
transition from the QSH side. The critical point could be viewed as a gapless
analogue of the quantum spin Hall state but with the full $SU(2)$ spin
rotation symmetry, which cannot be realized if the bulk is gapped.

##  I Introduction

The deconfined quantum critical point (DQCP) is a prototypical example of
quantum phase transitions beyond Landau’s paradigm [1, 2]. While DQCP has been
often discussed in the context of quantum magnetism, as a transition between a
Neel antiferromagnet and a valence-bond-solid (VBS), the same bulk
universality class describes the transition, in a fermion system, from a
spontaneous quantum spin Hall (QSH) insulator to a spin-singlet s-wave
superconductor (SC) [3, 4]. In the fermionic realization, the QSH insulator
spontaneously breaks the spin $SU(2)$ symmetry down to a $U(1)_{z}$ while
preserving the charge $U(1)_{c}$ conservation; the superconductor in contrary
breaks the charge $U(1)_{c}$ while preserving the full $SU(2)$.

The fermionic realization of DQCP, as a QSH-SC transition, has some features
not present in the Neel-VBS transition. The QSH insulator [5, 6], being an
early example of symmetry-protected topological (SPT) phase [7, 8], has
gapless fermion edge modes protected jointly by the unbroken charge $U(1)$ and
spin $U(1)_{z}$ (or time-reversal $\mathbb{Z}_{2}^{T}$) symmetries. A natural
question is: how would this aspect of SPT physics affect the phase transition,
especially when the system has a boundary?

The broader topic of the interplay between SPT physics and bulk criticality
has attracted some attention in recent years [9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 21, 22]. In particular, for the QSH-SC deconfined
transition, Ref. [16] argued, based on the notion of emergent anomaly, that
the physical fermion must be gapless on the edge even though it is gapped in
the bulk – recall that all the gapless degrees of freedom in the bulk are
bosonic, carrying integer spins (the QSH order parameter) and even electric
charges (the Cooper pair). Heuristically the gaplessness of the edge fermions
can be understood as a consequence of the gapless edge states in the QSH
phase, together with the continuity of the transition. The emergent anomaly,
however, further predicts a specific form of topological response that is an
analogue of the quantum spin Hall effect, but with the full $SU(2)$ spin
rotation symmetry – this is well known to be impossible for a gapped bulk
system.

In this work, we explicitly write down an effective field theory for the QSH-
SC deconfined transition, in the presence of a boundary. The field theory is a
gauge theory constructed using the parton trick [23]. Our field theory takes
the usual form of $CP^{1}$ gauge theory in the bulk, containing a dynamical
$U(1)$ gauge field coupled with a critical Higgs field that carries
spin-$1/2$. On the boundary, however, there is an additional pair of helical
fermions which couples to the dynamical $U(1)$ gauge field. These helical
fermions, carrying electric charge but not spin, are the remnants of the QSH
edge states at the critical point. We discuss various kinematic aspects of
this theory, including details of the parton construction and the choice of
boundary conditions, in Sec. II.

In Sec. III, using the large-$N$ technique (with the spin rotation symmetry
generalized to $SU(N)$), the boundary critical behaviours of this quantum
phase transition are analyzed. As we will demonstrate, the boundary helical
fermions dynamically decouple from the critical bulk at large $N$, making the
problem tractable. Since the symmetry “fractionalizes” at the DQCP, the
expressions of various symmetry order parameters at the edge not only involve
the helical edge fermions, but are also dressed by the critical degrees of
freedom from the bulk. As a result, these order parameters show rather
distinct scaling behaviours at the critical point from those in the familiar
single-component Luttinger liquid theory [24], which describes the boundary of
the neighbouring QSH insulator. We also predict a universal jump in the
scaling dimension of the boundary fermions as the system approaches the
transition from the QSH side. For $N$ small ($N=2$ being the directly relevant
one), we argue in Sec. III that our edge theory is still stable in certain
parameter regimes – in particular the helical fermions still decouple with
other degrees of freedom (DOFs) as long as the scaling dimension of the Cooper
pair (decided by the Luttinger parameter) is sufficiently large.

In Sec. IV we demonstrate explicitly how the edge states result in a
nontrivial bulk response (also known as emergent anomaly). In particular, a
$\pi$-flux of the $U(1)_{c}$ symmetry will trap a spin-$1/2$ moment in the
bulk – this is analogous to the famous quantum spin Hall response, but now
with the full spin $SU(2)$ symmetry. Such response is impossible if the bulk
is gapped – this is the familiar statement that topological insulators require
spin-orbit coupling. The bulk criticality, with the emergent anomaly, provides
a route to realize this peculiar response. Another notable consequence of the
emergent anomaly is that the enlarged $SO(5)$ symmetry, which emerges in the
critical bulk system [25, 26], no longer exists on the boundary as long as
time-reversal symmetry is unbroken.

We end with some discussions on open problems and future directions in Sec. V.

##  II Kinematics

###  II.1 Parton gauge theory

The DQCP between a superconductor and spontaneous quantum spin Hall insulator
is usually formulated in terms of a Wess-Zumino-Witten sigma model [3] due to
its natural connection with Dirac fermions. For our purpose of studying the
edge physics, it is more convenient to adapt a gauge theoretic formulation.
Below we construct the gauge theory using the parton trick [23], from which
the form of the edge theory will also become apparent.

The fundamental degree of freedom (DOF) in the system is the electron
$c_{\alpha}$ (with $\alpha=1,2$ the spin index), which carries unit charge
under the $U(1)$ charge conservation symmetry and spin-$1/2$ under $SU(2)$
spin rotation symmetry. To access the DQCP between the SC and QSH, we employ
the following parton decomposition of the electron:

$c=\begin{pmatrix}c_{1}\\\
c_{2}\end{pmatrix}=Bf=\begin{pmatrix}b_{1}&-b_{2}^{\dagger}\\\
b_{2}&b_{1}^{\dagger}\end{pmatrix}\begin{pmatrix}f_{1}\\\
f_{2}^{\dagger}\end{pmatrix}.$ (1)

This decomposition has an $SU(2)$ gauge redundancy, under which $B\to
BU^{\dagger}$ and $f\to Uf$ with $U$ an $SU(2)$ matrix. We will arrange a
parton mean field that breaks the $SU(2)$ gauge structure to its diagonal
$U(1)$ subgroup. Consequently, the low-energy effective theory takes the
following form

$S=S_{[B,a]}+S_{[f,a]}+S_{[B,f]}+S_{[a]},$ (2)

where $S_{[B,a]}$($S_{[f,a]}$) denotes the action of $B$($f$) that is
minimally coupled to $a$, an emergent $U(1)$ gauge field. $b$ and $f$ carry
opposite gauge charges, i.e., under the $U(1)$ gauge transformation,

$\displaystyle b_{1,2}\rightarrow e^{-i\theta}b_{1,2}\quad f_{1,2}\rightarrow
e^{i\theta}f_{1,2}$ (3)

$S_{[B,f]}$ represents the coupling between $B$ and $f$, and $S_{[a]}$ is the
action of the $U(1)$ gauge field $a$.

The global $U(1)_{c}$ charge conservation symmetry acts on the partons, up to
a gauge choice, as

$B\to B,\quad f\to e^{i\theta}f.$ (4)

Under the $SU(2)$ spin rotational symmetry,

$B\to VB,\quad f\to f,$ (5)

where $V$ is an $SU(2)$ matrix. Under time reversal symmetry $\mathcal{T}$,
one has

$B\to B^{*},\quad f\to i\sigma^{2}f,\quad i\to-i.$ (6)

Note that $\mathcal{T}$ is an anti-unitary symmetry. One can check that
$\mathcal{T}$ takes $c\to i\sigma^{2}c$, as $B$ has a structure of an
$SU(2)\cong USp(2)$ matrix. We choose the boson $B$ to be a Kramers singlet,
which will be important in the following discussion.

Next, we put the $f_{1}$ fermion into a Chern band with Chern number $C=+1$,
and $f_{2}$ into a Chern band with $C=-1$. One can see from the above symmetry
implementation that $f_{1}$ couples to the gauge field $-(a+A)$ and $f_{2}$
couples to $-(a-A)$, where $A$ is the probe gauge field of the global $U(1)$
charge conservation symmetry. Integrating out the fermions converts
$S_{[f,a]}$ into a Chern-Simons response

$\frac{1}{4\pi}(a+A)d(a+A)-\frac{1}{4\pi}(a-A)d(a-A)=\frac{1}{\pi}Ada.$ (7)

The coefficient of the Chern-Simons response is such that a strength-$1$
monopole operator $\mathcal{M}_{a}$, which creates $2\pi$ flux of $a$, carries
physical electric charge $Q=2$ \- it will be identified with a spin-singlet
Cooper pair of electrons 111The existence of local fermionic degrees of
freedom with spin-$1/2$ and $Q=1$ (the electrons), which carry half-charge and
half-spin of a strength-$1$ monopole, guarantees the global symmetries can be
realized non-anomalously. This is in contrast to the standard DQCP in the
context of quantum magnets. See Sec. IV and Appendix B for more details..

Now we are ready to apply this gauge theory to describe different quantum
phases. The bosons, $B$, can be condensed or gapped out by tuning the boson
mass in $S_{[b_{\alpha},a]}$. If the $b_{\alpha}$ fields are gapped, we can
simply integrate them out, which induces a Maxwell dynamics for the gauge
field $a$. Then the monopoles proliferate, i.e.,
$\langle\mathcal{M}_{a}\rangle\neq 0$, leading to a short-range entangled
phase [27], which should be identified as a spin-singlet SC, due to the
quantum numbers of the monopole operator.

On the other hand, a QSH results if the $b_{\alpha}$ fields are condensed. To
see it, suppose $b_{1}$ is condensed with a real-valued expectation $\langle
b_{1}\rangle\neq 0$ and $\langle b_{2}\rangle=0$. 222One can always bring the
condensation pattern into this form by a spin rotation in Eq. (5). In this
case the $SU(2)$ spin rotational symmetry is broken to $U(1)_{z}$, the
rotation along $z$-axis, while the time reversal symmetry is intact (which is
due to the fact that $B$ is a Kramers singlet). The gauge field $a_{\mu}$ is
then Higgsed and the monopole is no longer relevant. Moreover, the $f$
fermions can be identified as an electronic quasiparticle with a renormalized
quasiparticle residue $Z\sim|\langle b_{\uparrow}\rangle|^{2}$, according to
the relation in Eq. (1). Due to the Chern numbers of the $f$ fermions, the
resulting state is simply a QSH, with a gapped bulk and a gapless left-moving
(right-moving) spin-up (spin-down) mode on the edge.

The story gets more interesting right at the SC-QSH critical point, where the
$b_{\alpha}$ bosons are critical. The bulk physics of this transition has
already been investigated [3, 4], with a critical theory given by the
following CP1 model

$\displaystyle S_{\rm bulk}=\int
d^{3}x\left[\sum_{\alpha=1,2}|D_{\mu}b_{\alpha}|^{2}+\frac{\lambda}{4}(|b_{1}|^{2}+|b_{2}|^{2})^{2}\right]$
(8)

with $D_{\mu}=\partial_{\mu}-ia_{\mu}$ the covariant derivative. This is also
the critical theory proposed for the DQCP of spin-1/2 antiferromagnet on a
square lattice. The bulk critical properties of this theory has been studied
extensively over the past two decades. A partial sampling of numerical works
include Ref. [28, 29, 30, 31, 25]. One subtlety that emerged recently is the
possibility that the transition may be very weakly first order due to fixed
points collision [31, 26, 32, 33, 34, 35] (some more direct numerical
evidences also emerged recently [36, 37, 38]). This subtlety does not arise in
local correlation functions as long as the temperature is not too low, so in
this work we will ignore this issue and treat the above gauge theory as
describing a truly continuous transition.

We now proceed to describe the edge physics, which has been largely unexplored
so far. The bulk $+$ boundary system at the critical point is described by

$\begin{split}S=&S_{\rm bulk}+\int_{\rm
edge}d^{2}x\Sigma_{p=L,R}\psi^{\dagger}_{p}(\partial_{\tau}+ia_{\tau})\psi_{p}+\mathcal{H}_{\rm
edge}\\\ &+\mathcal{H}_{\rm b-e}\end{split}$ (9)

where $\psi_{L,R}$ are the left- and right-moving edge modes resulted from the
$f$ fermions, with the edge Hamiltonian

$\begin{split}\mathcal{H}_{edge}=&\psi_{L}^{\dagger}(-i\partial_{x}+(a+A)_{x})\psi_{L}\\\
&-\psi_{R}^{\dagger}(-i\partial_{x}+(a-A)_{x})\psi_{R}+...\end{split}$ (10)

where $...$ includes other edge couplings (such as four-fermion interactions)
allowed by symmetry and gauge invariance. Note although $\psi_{L,R}$ lives on
the edge, $a$ in this equation lives on the edge and also the bulk.
$\mathcal{H}_{\rm b-e}$ denotes other coupling between the bulk and the edge –
we shall discuss these additional couplings in more detail in Sec. III.

###  II.2 Boundary conditions

We now consider the appropriate boundary conditions for the bulk fields
$b_{\alpha}$ and $a_{\mu}$. We denote the boundary as $z=0$ separating the
bulk $z<0$ from the vacuum $z>0$, and the two other coordinates parrallel to
the edge spacetime are denoted $x,\tau$.

For the scalar field $b_{\alpha}$ we will only consider the Dirichlet boundary
condition $b_{\alpha}|_{z=0}=0$, which is also known as the “ordinary”
boundary condition, where the scalar fields on the boundary and in the bulk
condense simultaneously as the bulk mass changes sign. This boundary condition
is particularly appropriate in the large-$N$ generalizations to be considered
in Sec. III. Other more exotic possibilities may exist for relatively small
$N$ [39], but are beyond the scope of this work.

For the gauge field $a_{\mu}$, it turns out that we should also adopt the
Dirichlet boundary condition $a|_{z=0}=0$ (up to a gauge transformation). The
reason is again due to the large-$N$ generalization of the theory to be
considered in Sec. III. In this generalization, when the number of flavors of
the matter fields $b_{\alpha}$ is large, for all spacetime dimensions $d<4$,
the effective photon dynamics at low energies is entirely induced by the
bosonic matter $b_{\alpha}$ [40, 27]. The Dirichlet boundary condition
$b_{\alpha}|_{z=0}=0$ then immediately leads to the Dirichlet boundary
condition of the gauge field $a_{\mu}$. Physically, the matter fields
$b_{\alpha}$ and $\bm{e}$, the electric field of $a$, change continuously at
the system boundary when the Dirichlet boundary conditions are imposed. We
will come back to this choice of boundary conditions in Sec. III. An
additional remark is that this discussion applies to the case where the system
is a half infinite plane, and it should be slightly modified if the system is,
say, a disc (see Appendix A).

The vanishing of electric field $\bm{e}|_{z=0}=0$ at the edge also means that
when we put test gauge charges on the boundary, the electric field line
emitted from the gauge charge cannot be restricted to the boundary (since the
field strength must vanish there) and must penetrate into the bulk. So the
usual mechanism of $1d$ confinement, due to the electric field line restricted
to $1d$ with a finite tension, does not apply on the boundary, and we expect
the boundary gauge charges to be deconfined. This is just the boundary
manifestation of the familiar deconfinement of $U(1)$ gauge theory with a
conserved flux-current.

##  III Edge dynamics

Let us first make some general remarks on the edge physics. In the SC, there
is no nontrivial edge mode. In the QSH, the edge has a $U(1)_{c}\times
U(1)_{z}$ symmetry, which cannot be broken spontaneously due to the Mermin-
Wagner theorem, and which carries the standard axial ’t Hooft anomaly. This
implies that the edge state in the QSH cannot be gapped. Quite generically, it
can be described by a Luttinger liquid.

The bosonized description of a Luttinger liquid reads [24],

$S_{\rm
LL}=\frac{1}{2\pi}\int_{x,\tau}[\frac{(\partial_{x}\phi)^{2}}{K}+K(\partial_{x}\theta)^{2}]+\frac{i}{\pi}\int_{x,\tau}\partial_{x}\theta\partial_{\tau}\phi,$
(11)

where $K$ is the effective Luttinger liquid parameter, and $\phi$ and $\theta$
are two $\pi$-periodic compact bosons, satisfying

$[\partial_{x}\phi(x),\theta(x^{\prime})]=[\partial_{x}\theta(x),\phi(x^{\prime})]=i\pi\delta(x-x^{\prime}).$
(12)

Various microscopic operators are expressed as vertex operators, e.g.,
$\Psi_{R}\sim e^{-i\theta+i\phi}$, $\Psi_{L}\sim e^{-i\theta-i\phi}$ and
$S^{+}\sim\Psi_{L}^{\dagger}\Psi_{R}\sim e^{i2\phi}$, where $\Psi_{L,R}$ are
the left- and right-moving edge electronic operators (which carry no charge
under $a$), and $S^{+}$ is the $XY$ spin order parameter333The boundary
conditions on $\theta$ and $\phi$ are coupled, such that the expressions of
the chiral fermions are compatible with the periodicity of $\theta$ and
$\phi$. Also do not confuse the physical edge modes $\Psi_{L,R}$ with the edge
modes of the partons $\psi_{L,R}$. In the quantum spin Hall phase, we have
$\Psi_{L}\propto\psi_{L}$ and $\Psi_{R}\propto\psi_{R}^{\dagger}$.. Another
interesting operator is the Cooper pair, $\Psi_{L}\Psi_{R}\sim e^{-i2\theta}$.
The charge density and the spin density are expressed in terms of the bosons
as $\rho_{c}=\frac{1}{\pi}\partial_{x}\phi$ and
$\rho_{s}=\frac{1}{\pi}\partial_{x}\theta$, respectively. The time reversal
symmetry $\mathcal{T}$ acts as

$\theta\to-\theta,\quad\phi\to\phi+\pi/2,$ (13)

which is chosen such that the electric charge is even under $\mathcal{T}$
while the spin is odd under $\mathcal{T}$.

The scaling dimensions of the vertex operators depend on the Luttinger
parameter $K$ [24, 41]. The most important operators for us are the electrons
$e^{-i\theta\pm i\phi}$ with $\Delta_{e}=(K+1/K)/4$, the $XY$ spin operator
$e^{i2\phi}$ with $\Delta_{s}=K$, and the Cooper pair $e^{-i2\theta}$ with
$\Delta_{c}=\frac{1}{K}$. Certain combinations of these dimensions are
universal (independent of $K$), for example

$\displaystyle\Delta_{s}\Delta_{c}=1.$ (14)

Now we tune the bulk to the DQCP between the SC and QSH. From considerations
based on emergent anomalies, nontrivial edge states still exist at the DQCP
[15]. In particular, the fermions, which are gapped in the bulk throughout the
transition, are required to be gapless on the boundary. However, the detailed
edge physics remains unknown. For example, one may ask how the scaling
dimensions of various operators will differ from a simple Luttinger liquid.

The edge behaviour of the DQCP can be systematically studied in certain
large-$N$ generalization of the action in Eq. (9). We start by ignoring the
gapless fermions and focusing on the sector of $b_{\alpha}$ and $a$. Letting
the bulk gauge field $a_{\mu}$ be coupled to $N$ species of critical bosons,
$S_{\rm bulk}$ becomes a $\mathrm{CP}^{N-1}$ model

$S_{\rm
bulk}^{(N)}=\int_{bulk}d^{3}x\sum_{j=1}^{N/2}\sum_{\alpha=1,2}|D_{\mu}b_{j,\alpha}|^{2}+\frac{\lambda}{2N}(\sum_{j,\alpha}|b_{j,\alpha}|^{2})^{2}.$
(15)

The explicit $N$-dependence in the generalized action has been chosen to lead
to a consistent large-$N$ limit with $\lambda\sim\mathcal{O}(1)$. The global
symmetry of the model is

$(\mathrm{PSU}(N)\times U(1)_{top})\rtimes Z_{2}^{T},$ (16)

where $U(1)_{top}$ denotes the topological $U(1)$ symmetry, whose conserved
current is the flux of $a_{\mu}$,
$J^{\mu}=\frac{1}{2\pi}\varepsilon^{\mu\nu\rho}\partial_{\nu}a_{\rho}$. As
discussed in the previous section, the $U(1)_{top}$ corresponds to the
microscopic $U(1)$ charge conservation symmetry. $Z_{2}^{T}$ is the time
reversal, which acts as

$\mathcal{T}:\,b_{i}\to b_{i}^{*},\quad a_{\mu}\to a_{\mu}.$ (17)

The $\mathcal{T}$ action above is indicated for the spatial components. The
time component will transform with opposite sign. The $PSU(N)$ symmetry
reduces to the spin rotational symmetry at $N=2$. We will use symmetry
representations to organize the operators in the large-$N$ discussion below.

The standard technique to study the theory in Eq. (15) at large-$N$ is to
introduce a Hubbard-Stratonovich auxiliary field $\sigma$, such that

$S_{\rm
bulk}^{(N)}=\int_{bulk}d^{d}x\sum_{i=1}^{N}|D_{\mu}b_{i}|^{2}+\sigma|b_{i}|^{2}-\frac{N}{2\lambda}\sigma^{2},$
(18)

where the indices $(j,\,\alpha)$ are collected into a single index $i$.
Integrating out the $\sigma$ field, one reproduces the original action. Eq.
(18) is now Gaussian in $b_{i}$, and the integral over the field $b_{i}$ can
be performed to yield:

$S_{[\sigma,a_{\mu}]}=N\mathrm{Tr}\,\mathrm{log}(-D_{\mu}^{2}+\sigma)-\frac{N}{2\lambda}\sigma^{2}.$
(19)

This effective action is proportional to $N$, so the fluctuations of $\sigma$
and $a_{\mu}$ are suppressed in the large-$N$ limit, and the behaviour of
these fields are dominated by the saddle point equations. In particular, the
saddle point equation for the gauge field reads

$\mathrm{Tr}[\frac{(\partial_{\mu}-ia_{\mu})}{-D_{\nu}D^{\nu}+\sigma}]=0.$
(20)

By translational and rotational symmetry in the $x\tau$ plane, $a_{\mu}=0$ is
always a solution for $\mu=x,\,\tau$. The equation for $a_{z}$ is much more
non-trivial due to the presence of a boundary. However, one can utilize the
gauge freedom to set $a_{z}=0$. We thus conclude that, similar as the case of
infinite systems, the gauge field plays no role in the saddle point equations,
which then reduce to the saddle point equations of the $O(2N)$ non-linear
sigma model (NLSM) in the semi-infinite plane [40]. As we expand the
functional determinant in Eq. (19) around the saddle point, one can see that
the effective photon propagator is proportional to $\frac{1}{N}$, so the gauge
fluctuations are negligible at large $N$ [27]. Therefore, the scaling
behaviour of the model should be that of the $O(2N)$ NLSM in the $N\to\infty$
limit.

Let us now turn our attention to the boundary critical behaviour of the
$O(2N)$ NLSM. For bulk space-time dimension $d>3$, there are three possible
universality classes: the ordinary, the extraordinary and the special [42].
The ordinary universality class denotes the case where the boundary remains
disordered and gapped throughout the bulk disordered phase. For the
extraordinary fixed point, the bulk undergoes a symmetry breaking phase
transition in the presence of an ordered boundary. The multi-critical point
between the ordinary universality class and the extraordinary universality
class is known as the special transition. In the physical dimension $d=3$, a
modified version of the extraordinary universality class could survive for $N$
smaller than some critical value [39]. However at $N=\infty$ one finds only an
ordinary universality class [43]. This can be understood physically by noting
that for spacetime dimension $d=3$, the boundary does not order when the bulk
is disordered, due to the Mermin-Wagner constraint. The appropriate boundary
condition for the ordinary transition is the Dirichlet boundary condition
$b_{\alpha}|_{z=0}=0$, which justifies the choice made in Sec. II.2.

At the critical point, the form of the correlation function on a semi-infinite
plane is fixed by conformal invariance. We have [44]

$\langle\phi^{i}(x,z)\phi^{j}(x^{\prime},z^{\prime})\rangle=\frac{\delta^{ij}}{(zz^{\prime})^{\Delta_{\phi}}}\Phi(v),\quad
v=\frac{\rho^{2}+z^{2}+z^{\prime 2}}{2zz^{\prime}},$ (21)

with $\rho=|x-x^{\prime}|$ the separation along the $x\tau$ plane and $\Phi$ a
universal function. $\Delta_{\phi}$ is the bulk scaling dimension of the
$\phi$ field, the $O(2N)$ vector, which is related to our original
$\mathrm{CP}^{N-1}$ field via $b_{j}=\phi_{2j-1}+i\phi_{2j}$. RG calculations
and exact results suggest that for $\rho\to\infty$ with $z$, $z^{\prime}$
fixed, $\Phi$ decays as $\rho^{-2\Delta_{\phi}^{s}}$ [42, 43], where
$\Delta_{\phi}^{s}$ can be defined as a surface critical exponent. We thus
expect an operator product expansion near the boundary

$\phi_{i}(x,z)\sim\mu_{\phi}z^{\Delta_{\phi}^{s}-\Delta_{\phi}}\hat{\phi}_{i}(x)+...,$
(22)

where $\mu_{\phi}$ is a universal constant only depending on the normalization
of the surface operator $\hat{\phi}_{i}$. (We denote boundary version of the
scaling operators with a hat.) The operator $\hat{\phi}_{i}$ can be viewed as
the leading representation of the scaling operators with a certain symmetry
quantum number at the surface.

For the ordinary universality class in bulk dimension $d$, the exact result of
the correlation function at $N\to\infty$ is obtained by Bray and Moore [43]:

$\begin{split}&\langle\phi^{i}(x,z)\phi^{j}(x^{\prime},z^{\prime})\rangle\\\
&\sim\delta^{ij}(\frac{zz^{\prime}}{[\rho^{2}+(z-z^{\prime})^{2}][\rho^{2}+(z+z^{\prime})^{2}]})^{(d-2)/2}.\end{split}$
(23)

Therefore, at $d=3$ the scaling dimension of the bulk field $b_{i}$ is
$[b_{i}]=\Delta_{\phi}=1/2$, and the scaling dimension of the boundary field
$\hat{b}_{i}$ is $[\hat{b}_{i}]=\Delta_{\phi}^{s}=1$. By the equation of
motion of the Hubbard-Stratonovich field $\sigma$, we see that $\sigma$ plays
the role of the $PSU(N)$ singlet operator in any correlation functions. The
induced propagator of $\sigma$ is obtained from summing the geometric series
of bubbles in Fig. (1), which gives [45]

$\langle\sigma(x,z)\sigma(x^{\prime},z^{\prime})\rangle\propto\frac{(zz^{\prime})^{-2}}{N}\frac{v}{(v^{2}-1)^{2}}.$
(24)

We thus read off the scaling dimensions444This agrees with the expectation
that $\hat{\sigma}$ should also be the boundary descendent of the bulk stress
tensor [42], which always has dimension $d=3$. $\Delta_{\sigma}=2$ and
$\Delta_{\hat{\sigma}}=3$, using the behaviour of the above correlation
function in the limit $\rho\to\infty$ with $z$, $z^{\prime}$ fixed. Since the
edge singlet operator $\hat{\sigma}$ has scaling dimension
$\Delta_{\hat{\sigma}}>2$, there is no relevant perturbations at the boundary
that respect the $PSU(N)$ symmetry. Another important surface operator is the
boundary descendent of the gauge field $\hat{a}_{\mu}$. The effective photon
propagator at the leading order, namely $O(1/N)$, is obtained by summing
similar geometric series as that in Fig. (1) [46]. By a simple dimensional
analysis, one can obtain the boundary scaling dimension of the gauge field is
$\Delta_{\hat{a}}=2$.

Figure 1: The induced propagator of the $\sigma$ field. The dashed line stands
for the $\sigma$ propagator at tree level. The solid line stands for the exact
propagator of $b_{i}$ at the $N\to\infty$ limit.

Now we include the gapless boundary fermions. Again we bosonize the fermion
modes and write it as a Luttinger liquid Eq. (11), with $\psi_{R}\sim
e^{i\theta-i\phi}$ and $\psi_{L}\sim e^{-i\theta-i\phi}$. The Luttinger liquid
has a minimal coupling to the boundary gauge field, which can be written as
$\frac{1}{2\pi}\hat{a}_{\mu}\epsilon^{\mu\nu}\partial_{\nu}\theta$. Since the
gauge field has a boundary scaling dimension $\Delta_{\hat{a}}>1$, this
coupling is irrelevant and has no consequence in terms of dynamics (the gauge
field still imposes a global constraint of vanishing gauge charge). There are
additional couplings between the Luttinger liquid and other degrees of
freedom, but they all turn out to have scaling dimension larger than $d-1=2$
and are therefore irrelevant (at least when generalized to large $N$): (a) the
operator $e^{2i\phi}$ carries gauge charge $2$ and is time-reversal odd, so it
couples to $\epsilon^{\alpha\beta}b_{\alpha}\partial_{t}b_{\beta}$ which is
irrelevant on the edge; (b) the operator $e^{2i\theta}$ is the physical Cooper
pair and therefore couples to the monopole operator $\mathcal{M}_{a}$ of the
bulk $U(1)$ gauge field. At large $N$ the monopole operator has scaling
dimension $\sim O(N)$ so this coupling is again irrelevant [47, 48]; (c) the
operator $\partial_{x}\theta$ is odd under time-reversal, and could couple
with the gauge field $e_{z}=\partial_{0}a_{z}-\partial_{z}a_{0}$, but since
the gauge flux is a conserved current, $[e_{z}]=2$ in the bulk so the boundary
coupling is irrelevant; (d) the operator $\partial_{x}\phi$ transforms
trivially under all microscopic symmetries (this is just a chemical potential
term), so the leading coupling involving this operator is with either
$\hat{\sigma}$ or the boundary descendent of the gauge magnetic field
$\partial_{x}a_{y}-\partial_{y}a_{x}$, and in either case the coupling will be
irrelevant.

Notice that all the above couplings are already irrelevant in the $N\to\infty$
limit, so including further $O(1/N)$ corrections will not immediately change
the picture. We therefore conclude that, at least when $N$ is sufficiently
large, the DOF in the $\mathrm{CP}^{N-1}$ model (either living in the bulk or
edge) and the edge $\psi_{L/R}$ fermions decouple in the RG sense. So the edge
$\psi$ fermions behave the same as ordinary Luttinger liquids, with a
Luttinger liquid parameter $K$ that is exactly marginal. For smaller values of
$N$, most of the above edge couplings will still be irrelevant. The only
exception is the coupling $e^{2i\theta}\mathcal{M}_{a}$, since
$[\mathcal{M}_{a}]$ can become quite small for small $N$. In this case the
coupling is irrelevant only if $[e^{2i\theta}]=1/K$ is sufficiently large,
i.e. if $K$ is smaller than some critical value $K_{c}$. Practically, this
requires the Cooper pair to be “heavy” on the edge, and can be achieved, if
necessary, by turning on repulsive interactions on the edge. Therefore for
small $N$, what we have found is a boundary fixed line that exists in certain
parameter regimes. If $K>K_{c}$, the Luttinger liquid no longer decouples, and
the theory flows to strong coupling over which we do not have analytical
control. In this case a natural possibility is that the Cooper pair
$e^{2i\theta}$ may spontaneously condense (or logarithmically so [39]) due to
the long-range couplings induced by the bulk fluctuations. The edge fermions
will effectively be gapped in this scenario.

We now compute the boundary scaling dimensions of the electron, Cooper pair
and spin operators in the $N\to\infty$ limit. The electron operator
$\Psi_{\alpha}\sim b_{\alpha}f$, and since the Luttinger liquid of the $f$
fermions decouple with the rest of DOFs on the edge,

$\displaystyle\Delta_{\Psi}=[b]+[f]=\frac{K}{4}+\frac{1}{4K}+1.$ (25)

It is interesting to compare $\Delta_{\Psi}$ at the transition point and in
the QSH phase but close to the transition. Since the Luttinger liquid sector
decouples with other DOFs on the edge, the Luttinger parameter $K$ should
change smoothly as the system approaches the phase transition from the QSH
phase. We therefore conclude that there is a universal jump in the fermion
boundary scaling dimension as the system reaches the phase transition from the
QSH side:

$\displaystyle\delta\Delta_{\Psi}\equiv\Delta^{DQCP}_{\Psi}-\Delta^{QSH}_{\Psi}=1.$
(26)

For finite $N$, we expect $\delta\Delta_{\Psi}$ to deviate from $1$, but
remains universal as long as the Luttinger liquid decouples with the other
DOFs, which requires a sufficiently small $K$.

The above universal jump in the fermion boundary scaling dimension can be
viewed as a hallmark of electron fractionalization at the transition. For
other more conventional transitions, for example a Landau transition from the
QSH to a superconductor (without restoring the full $SU(2)$ symmetry), such a
universal jump is not expected even if the Luttinger liquid (from the QSH
boundary modes) decouples with the bulk critical fields.

The scaling dimension of the spin operator can be obtained as follows. A
large-$N$ generalization of the spin operator is
$O_{s}\sim\sum_{j=1}^{N/2}[b_{j,1}^{\dagger}b_{j,2}(f_{1}^{\dagger}f_{1}+f_{2}^{\dagger}f_{2})+(b_{j,1}^{\dagger})^{2}f_{1}^{\dagger}f_{2}^{\dagger}-b_{j,2}^{2}f_{2}f_{1}]$,555$O_{s}$
can be written in a more compact form in terms of the matrix notation in Eq.
(1),
$O_{s}\sim\frac{1}{2}\sum_{j=1}^{N/2}f^{\dagger}B_{j}^{\dagger}\sigma^{+}B_{j}f$.
which transforms in the vector representation of the $SO(3)$ spin rotational
symmetry, and carries no electric charge. Under $\mathcal{T}$, $O_{s}$ has the
expected transformation of the spin operator $S^{+}$,
$O_{s}\to-(O_{s})^{\dagger}$. The boundary scaling dimension of the conserved
charge
$(f_{1}^{\dagger}f_{1}+f_{2}^{\dagger}f_{2})_{\mathrm{edge}}\sim(\psi_{L}^{\dagger}\psi_{L}+\psi_{R}^{\dagger}\psi_{R})$
is $1$, while
$(f_{1}^{\dagger}f_{2}^{\dagger})_{\mathrm{edge}}\sim\psi_{L}^{\dagger}\psi_{R}^{\dagger}$
has dimension $K$ (similarly for $f_{1}f_{2}$). The operator
$\sum_{j}b_{j,1}^{\dagger}b_{j,2}$ transforms in the adjoint representation of
the $PSU(N)$ symmetry, whose boundary scaling dimension is

$\Delta_{\hat{\mathrm{adj}}}=2[\hat{b}_{i}]=2,$ (27)

due to the fact that the vertex correction to the scaling dimension of
$PSU(N)$ adjoint operators is $O(\frac{1}{N})$. The operator
$\sum_{j}(b_{j,1}^{\dagger})^{2}=\sum_{j}(b_{j,1}^{\dagger 2}+b_{j,2}^{\dagger
2})+(b_{j,1}^{\dagger 2}-b_{j,2}^{\dagger 2})$ contains both $PSU(N)$ singlet
and adjoint components (similarly for $\sum_{j}b_{j,2}^{2}$). Because
$\Delta_{\hat{\mathrm{adj}}}<\Delta_{\hat{\sigma}}$, the adjoint component
dominates at long distances. We conclude that the boundary scaling dimension
of the $XY$ order parameter is

$\Delta_{s}=2+\mathrm{min}(1,\,K)$ (28)

at the $N\to\infty$ limit.

The physical Cooper pair can be expressed in terms of the edge fermions as
$c_{\uparrow}c_{\downarrow}\sim\psi_{R}^{\dagger}\psi_{L}$. One may check it
carries global electric charge $2$, and is invariant under both $\mathcal{T}$
and spin rotational symmetry. In the large-$N$ limit, the edge scaling
dimension of the Cooper pair is determined simply by the Luttinger parameter

$\Delta_{c}=1/K.$ (29)

Unlike the Luttinger liquid on the QSH boundary, the universal relation Eq.
(14) is no longer true at the critical point.

##  IV Edge states and anomaly

We now show how our edge theory, in particular the existence of the gapless
$\psi$ fermion modes, satisfies the constraints imposed by the emergent
anomaly of DQCP. We remind the reader that our system has a vanishing bulk
gap, while the bulk gap for single fermionic excitations remains open.

Let us first review the argument in Ref. [16], in a language that will be
convenient for our purpose (we give a more detailed and formal account in
Appendix B). Consider the system in a disk geometry, and turn on a global
$U(1)$ flux $\int dA=\pi$. This flux is supposed to be turned on
adiabatically, so that no bulk fermion can be excited, and only the gapless
part of the system can respond to it. In the bulk the minimally charged
gapless DOF is the Cooper pair, which sees the $\pi$-flux as effectively
$2\pi$-flux. It is well known that for DQCP such a flux will trap a spin-$1/2$
moment of the $SO(3)$ symmetry [49, 50]. If the fermions are gapped everywhere
(bulk and boundary), then below the fermion gap the $\pi$-flux is not
observable from far away, and therefore should only induce a local excitation
– but there is no local excitation with spin-$1/2$ below the fermion gap. This
is a contradiction, which in turn requires the fermion gap to vanish on the
boundary.

We can now check the above reasoning explicitly using our edge theory. The
$U(1)$ flux is seen as a $\pm\pi$ flux by $\psi_{L/R}$ (see Eq. (10)).
Following a standard integer quantum Hall (or chiral anomaly) calculation, we
see that the flux should carry a unit gauge charge (under $a$) but no global
$U(1)$ charge – in the bosonized language the flux corresponds to the operator
$e^{i\phi}$. This is just the boundary manifestation of the QSH physics of the
$f$ sector. Accordingly, when the $\pi$-flux is inserted, it will trap an
excitation with gauge charge 1 from the $f$ sector, which combines with $b$
boson to yield the spin-1/2 property of the flux anticipated from our
understanding of the DQCP. 666A nontrivial question is how to observe this
spin-$1/2$ physically, given the existence of critically fluctuating spin
DOFs. The general expectation is that the spin-$1/2$ moment cannot be fully
screened by local gapless excitations that only carry integer spins. However
the details may be complicated and deserve further investigations (for a
recent work in this direction see Ref. [17]).

The above topological response (also known as emergent anomaly) is a close
analogue of the celebrated quantum spin Hall effect, in which a $\pi$-flux
traps an $S_{z}=1/2$ (or Kramers degeneracy if the state is protected by time-
reversal symmetry). This is the fully $SU(2)$ symmetric version, which is
known to be impossible for gapped SPT states – this can be understood simply
by noticing that if the bulk is fully gapped, an adiabatic insertion of a
$\pi$-flux cannot turn the system from a spin singlet state into a doubly-
degenerate spin-$1/2$ state. The bulk criticality is what makes this response
possible. For this reason Ref. [16] calls such states “intrinsically gapless”
topological phases.

Another consequence of the emergent anomaly is that while the bulk is known to
have an emergent $SO(5)$ symmetry at the critical point [25, 26], at the
boundary the symmetry is reduced back to the microscopic $SO(3)\times U(1)$
since the two order parameters (QSH and SC) will in general have different
boundary scaling dimensions. At the anomaly level this is because the emergent
$SO(5)$ anomaly [26] remains a nontrivial $\theta$-term even if fermions (in
the $SO(5)$ spinor representation) are introduced, and the $\theta$-term
cannot be trivialized as long as time-reversal symmetry is preserved (similar
to the standard topological insulator [51]) – see Appendix B for more details.
So the boundary gapless fermions can only trivialize the bulk $SO(3)\times
U(1)$ anomaly, but not the larger $SO(5)$ anomaly, unless time-reversal
symmetry is broken with nonzero $SO(5)$ and thermal Hall conductance. Because
the bulk $SO(5)$ anomaly remains nontrivial in the presence of the time
reversal symmetry, while the vacuum hosts no anomaly, the anomaly should be
ill defined on the boundary since this is the interface between the bulk and
the vacuum, which means that the $SO(5)$ and time reversal symmetries cannot
be simultaneously preserved on the boundary. An example where $SO(5)$ is
preserved but time reversal is broken on the boundary is given in Ref. [52],
in which the fermions transform as $SO(5)$ spinors and the time-reversal
symmetry is realized through the (anti-unitary) particle-hole symmetry in a
half-filled Landau level. The boundary of this system can have the full
$SO(5)$ (or more precisely $Spin(5)$), but the particle-hole symmetry of the
Landau level is inevitably broken on the boundary. The edge state in this case
can be formulated starting from bulk descriptions that are explicitly $SO(5)$
invariant, such as the $SU(2)$ QCD3 theory in Ref. [26]. We leave the details
of such edge theory to future work.

##  V Discussion

In this work we constructed the phase transition between an SPT phase, namely,
a quantum spin Hall insulator, and a superconductor which breaks part of the
protecting symmetries of the SPT phase. In contrast to the ordering
transitions of SPT phases discussed in the literature [53, 39], the transition
in this work separates two phases with different symmetry breaking patterns,
one of which supports an SPT protected by the unbroken symmetry. The critical
theory is therefore a DQCP in the bulk, instead of a conventional Ginzburg-
Landau type theory. Due to the existence of the symmetry-protected edge modes,
the phase transition exhibits unconventional boundary critical behaviours,
which were studied using the large-$N$ technique. We calculated the edge
scaling dimensions of the electron, the spin, and the Cooper pair operators at
the $N\to\infty$ limit. The expressions of these order parameters may involve
both the helical edge modes and the critical bosons from the bulk, as a result
of symmetry fractionalization at the critical point. We discussed several
measurable consequences. In particular, we predict a universal jump in the
scaling dimension of the boundary fermions as the system approaches the phase
transition from the QSH phase. The scaling dimensions at the transition also
obey different relations than a conventional Luttinger liquid, which in
principle can help distinguishing the critical point from the neighbouring
quantum spin Hall state. Our results should be helpful for future numerical
studies on the boundary criticality of the QSH-SC deconfined transition.

There are many potential future directions following our work. An obvious one
is to compute the leading $1/N$ corrections on edge critical behaviour. Beyond
the zeroth order, the emergent gauge field can mediate couplings between the
edge modes and the critical bulk, which may lead to interesting new physics. A
conceptually more intriguing question is whether entirely new boundary fixed
point, in which the boundary helical modes no longer decouple from the other
DOFs, can appear at small $N$, including the most relevant case of $N=2$.

The insight of this paper can also be generalized to other systems with
different symmetries. The key idea is to have two phases with distinct
symmetry breaking patterns, one (or even both) of which can support SPT phases
protected by the unbroken symmetry. The critical point between these two, if
exists, should exhibit unconventional behaviours both in the bulk and at the
boundary.

###### Acknowledgements.

We thank F. Assaad for illuminating discussions. RM and CW acknowledge support
from the Natural Sciences and Engineering Research Council of Canada (NSERC)
through a Discovery Grant. Research at Perimeter Institute is supported in
part by the Government of Canada through the Department of Innovation, Science
and Industry Canada and by the Province of Ontario through the Ministry of
Colleges and Universities.

##  Appendix A Boundary condition of the gauge field for a disc

In the main text, we have shown that the gauge field $a_{\mu}$ should be taken
to have a Dirichlet boundary condition (up to a gauge transformation). This
means that the field strength vanishes and the gauge field does not contain
any physical degree of freedom at the boundary. However, though there is
always a well defined gauge transformation that sets $a|_{z=0}=0$ if the
system is a half infinite plane, this is generically not true if the system
is, say, a disc.

In fact, there is a physical difference between a half infinite plane and a
disc. For the former, the only gauge invariant objects that can be constructed
from the gauge field $a_{\mu}$ on the boundary are just the field strengths
(and their composites), which vanish. So in this case the boundary gauge field
contains no dynamical degree of freedom and can be gauged away, which is the
physical reason behind the Dirichlet boundary condition. However, for a disc,
besides the field strengths, there is another gauge invariant operator made of
$a_{\mu}$, which is the Wilson loop. In fact, the Wilson loop $\int
d\bm{l}\cdot\bm{a}$ along the edge counts the physical $U(1)$ charge in the
bulk, which cannot be gauged away. It is useful to separate $a_{\mu}$ into two
parts, the part that depends on the spacetime coordinates and contributes to
the field strengths, and the part that is independent of the space time
coordinates and contributes to the Wilson loop. On a disc, although the former
part can still be gauged away since the field strengths vanish, the latter
part is a dynamical degree of freedom, so the boundary condition is not really
a Dirichlet one.

So how should we think of this boundary condition? Because the spacetime
independent part of $a_{\mu}$ couples to the total gauge charge and current on
the boundary, integrating it out in the path integral yields a constraint that
the total gauge charge and current vanish on the boundary. Therefore, this
boundary condition can be viewed as a Dirichlet boundary condition of
$a_{\mu}$ together with the constraint that the boundary hosts a vanishing
gauge charge and current. In contrast, this constraint is absent on a half
infinite plane.

##  Appendix B Details on emergent anomaly

In this Appendix we review some of the formal aspects of the emergent
anomalies in DQCP. A more detailed review of the mathematical backgrounds can
be found in, for example, Ref. [26].

The maximum symmetry of the DQCP is an $O(5)^{T}$ symmetry, where the
superscript “T” means that the improper $\mathbb{Z}_{2}$ rotation of $O(5)$
should always be accompanied by a time reversal. The $SO(3)\times
SO(2)\times\mathbb{Z}_{2}^{T}$ of our interest is a subgroup of $O(5)^{T}$. If
we couple the DQCP to a background $O(5)$ gauge field, the partition function
is not gauge invariant unless we introduce a bulk topological term living in
one extra dimension [54]:

$S^{O(5)^{T}}_{bulk}=i\pi\int d^{4}xw_{4}^{O(5)},$ (30)

where $w_{i}^{O(5)}$ is the $i$-th Stiefel-Whitney class of the $O(5)$ gauge
bundle, with a constraint that $w_{1}^{O(5)}=w_{1}^{TM}\ ({\rm mod\ }2)$ to
capture the locking between improper rotation of $O(5)$ and time reversal,
where $w_{i}^{TM}$ is the $i$-th Stiefel-Whitney class of the tangent bundle
of the spacetime manifold where the bulk lives. This is a nontrivial t’Hooft
anomaly. One consequence of this anomaly is that the system cannot have a
boundary if the symmetry is strictly an on-site $O(5)^{T}$, because the theory
can only be defined on the boundary of a higher dimensional bulk and it cannot
further have its own boundary. Breaking the symmetry to $SO(3)\times O(2)$
does not help as the anomaly remains nontrivial:

$S^{SO(3)\times O(2)}_{bulk}=i\pi\int w_{2}^{SO(3)}\cup w_{2}^{O(2)},$ (31)

where $\cup$ is the cup product.

The anomaly can be trivialized if we enlarge the symmetry by introducing
physical fields that transform as fractional (projective) representations of
the original symmetry group. In our case, we are introducing physical fermions
that carry half-integer $SO(3)$ spins, odd electric charges and Kramers
degeneracy (recall that in our example the superconductor order parameter
carries even electric charges). At the level of gauge bundles, the existence
of such physical fermions puts the following constraint (cocycle condition) on
the Stiefel-Whitney classes, when integrated over any $2$-cycle $C_{2}$:

$\int_{C_{2}}[w_{2}^{SO(3)}+w_{2}^{TM}+w_{2}^{O(2)}]=0\hskip
10.0pt{\rm{mod}}\hskip 5.0pt2,$ (32)

The faithful global symmetry is sometimes called $U(2)_{c}^{T}$ (the subscript
$c$ denotes the fermionic nature of the fundamental operators and the
superscript $T$ implies that the fermions are Kramers doublets). Substituting
this constraint into Eq. (31) we get

$\displaystyle\int(w_{1}^{TM})^{2}\cup w_{2}^{O(2)}=\int
Sq^{1}(Sq^{1}w_{2}^{O(2)})=0,$ (33)

where the first equality follows from the Wu formula. The anomaly therefore
vanishes with the physical fermions we introduce and the system is then
allowed to have a consistent edge theory. However, as the fermions remain
gapped in the bulk, the effective symmetry at low energies in the bulk is
still $SO(3)\times SO(2)\times\mathbb{Z}_{2}^{T}$ and the emergent anomaly Eq.
(31) still governs the low-energy response in the bulk, in the sense that a
flux quantum in the $SO(2)$ gauge field still traps a Kramers doublet half-
integer spin of $SO(3)$, as discussed in the main text.

The anomaly can also be trivialized by other ways of extending the symmetry
group. For example, if we introduce physical bosons that carry half-integer
SO(3) spins, even electric charge and Kramers degeneracy, the global symmetry
becomes $SU(2)\times U(1)\times\mathbb{Z}_{2}^{T}$. The cocycle condition is
now $w_{2}^{SO(3)}+(w_{1}^{TM})^{2}=0$ (mod $2$) and the anomaly vanishes. In
this case the edge theory is almost identical to the one discussed in the main
text, with a boundary Luttinger liquid $(\theta,\phi)$ coupled to the bulk CP1
theory with ordinary boundary condition. The only difference is that the
fundamental operators in the edge Luttinger liquid are now bosonic, including
$e^{i\theta}$ and $e^{2i\phi}$ (instead of $e^{i(\theta\pm\phi)}$ in the
fermionic case).

Note that not all symmetry extensions trivialize the anomaly. For example, if
we use $U(2)$ with bosonic Kramers singlet (doublet) fundamental particles,
the anomaly becomes $i\pi\int w_{2}^{SO(2)}\cup w_{2}^{SO(2)}$
($i\pi\int(w_{2}^{SO(2)}+(w_{1}^{TM})^{2})\cup w_{2}^{SO(2)}$) and is not
trivial.

We now return to the fully $O(5)^{T}$ symmetric anomaly Eq. (30) and show that
it cannot be trivialized. For this purpose, it suffices to consider orientable
spacetime manifold for the bulk, such that the anomaly in Eq. (30) reduces to
$i\pi\int d^{4}xw_{4}^{SO(5)}$. It is known that if we introduce physical
fermions in the spinor representation of the $SO(5)$ (so that the faithful
symmetry is $Spin(5)_{c}$), the DQCP can be realized if we put the fermions in
half-filled Landau levels [52]. At the anomaly level the existence of the
physical $Spin(5)$ fermions imposes the cocycle constraint

$w_{2}^{SO(5)}=w_{2}^{TM}\hskip 10.0pt{\rm{mod}}\hskip 5.0pt2.$ (34)

We then use the relation (true for $SO(N)$ bundles on a $4d$ orientable
manifolds)

$w_{4}^{SO(5)}=\frac{1}{2}p_{1}-\frac{1}{2}\mathcal{P}(w_{2}^{SO(5)})\hskip
10.0pt{\rm{mod}}\hskip 5.0pt2,$ (35)

where $\mathcal{P}$ is the Pontryagin square and $p_{1}$ is the first
Pontryagin number

$p_{1}=\frac{1}{8\pi^{2}}\int{\rm{tr}}_{SO(5)}F\wedge F.$ (36)

We can now substitute the cocycle condition Eq. (34) into the above equations
and use the relation for the signature $\sigma$ of a $4d$ manifold (with
Riemann curvature tensor $R$):

$\frac{\sigma}{2}=\frac{1}{2}\mathcal{P}(w_{2}^{TM})=-\frac{1}{48\pi^{2}}\int
R\wedge R\hskip 10.0pt{\rm{mod}}\hskip 5.0pt2.$ (37)

We conclude that the anomaly becomes the following gauge and gravity theta
terms:

$S_{bulk}^{Spin(5)_{c}}=\frac{1}{16\pi}\int{\rm{tr}}_{SO(5)}F\wedge
F+\frac{1}{48\pi}\int R\wedge R.$ (38)

The above theta term is nontrivial for $Spin(5)_{c}$ symmetry as long as time-
reversal (or any orientation-reversing) symmetry is preserved. This means that
for the $Spin(5)_{c}$ system to have a physical boundary, either the $Spin(5)$
or time-reversal symmetry must be explicitly broken on the boundary in order
to have a consistent boundary theory. In our main example the $Spin(5)$
symmetry is broken to $(U(1)\times SU(2))/Z_{2}=U(2)$ while time-reversal
symmetry is preserved. In the half-filled Landau level [52] the $Spin(5)$
symmetry is kept but the anti-unitary particle hole symmetry (playing the role
of time-reversal) is broken on the boundary due to the nature of Landau
levels.

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