# Emergent $U(1)$ Symmetries and $\tau$-$\sigma$ Duality in Gapless
Superfluids or Superconductors

Fei Zhou  Department of Physics and Astronomy, University of British Columbia,
6224 Agricultural Road, Vancouver, BC, V6T 1Z1, Canada

(August 28, 2024)

###### Abstract

A superfluid spontaneously breaks the usual $U(1)$ symmetry because of
condensation. In this article, we illustrate six independent families emergent
$U(1)$ symmetries that naturally appear in infrared limits in a broad class of
gapless topological superfluids (that either belong to a stable phase or are
quantum critical). In gapless states we have considered, emergent $U(1)$
symmetry groups are embedded in an $Spin(4)=SU(2)\otimes SU(2)$ group that
double covers (and algebraically is isomorphic to) an $SO(4)$ group. All
$U(1)$ charges associated with symmetries are further invariant under an
$SU(2)$ spin group or an equivalent of it but always break pre-existing higher
space-time Lorentz symmetry of $SO(3,1)$ group. Emergent $U(1)$ symmetries can
be further spontaneously broken only if interactions are strong enough and
resultant strong coupling states become fully gapped. However if states remain
gapless, emergent $U(1)$ symmetries are always present, despite that these
states may exhibit much lower space-time symmetries compared to their weakly
interacting gapless Lorentz symmetric counter parts. In the limit of our
interests, we have identified all possible gapless real fermions with or
without Lorentz symmetries and find that they all display emergent $U(1)$
symmetries in the infrared limit. We argue emergent $U(1)$ symmetries are
intrinsic in a broad class of interacting gapless superfluid or
superconducting states and are typically well defined in high dimensions where
there are infrared stable fixed points dictating emergent properties.

##  I Introduction

It is well known that the $U(1)$ symmetry is always spontaneously broken in
superfluids (or superconductors). So in those states, charges are not
conserved because of condensates in ground states and low energy dynamics are
generically characterized by emergent real fermions rather than conventional
complex fermions. These emergent particles are not only crucial in discussions
of topological states, but also play critical roles in studies of topological
quantum criticality as well as in applications to quantum technonolgiesFu08 ;
Schnyder08 ; Kitaev09 ; Qi10 ; Volovik88 ; Volovik03 ; Read00 ; Qi11 ;
Bernevig13 ; Hasan10 .

Naturally related to emergent fermions are the emergent symmetries of these
fields or particles. In a broad class of topological states where not only
$U(1)$ symmetry but also other continuous symmetries such as rotation one are
broken, there can be surprising emergent symmetries of very large groups in
the low energy subspace. These symmetries can be totally unexpected as one
usually does not anticipate their appearance in low energy scales solely based
on microscopic considerations. Their existence at first sight even appears to
be inconsistent with underlying symmetries. Nevertheless, they do emerge in
many physical systems.

Consider a topologically non-trivial 3D $p$-wave superfluid or superconductor
with time reversal symmetry. A well-known concrete example for this can be a
BW phase of $p$-wave superfluidLeggett75  that breaks $U(1)$ symmetry and
spatial rotation symmetry (but with spin-spatial rotation symmetry intact). In
strong coupling limits where pairing is very strong or the one-particle band
is flat so that Fermi surface effects are completely suppressed, the low
energy dynamics can be fully described by real fermions with an emergent
Lorentz symmetry while underlying fermions are entirely non-
relativisticVolovik03 ; Qi11 . In this case, strong interactions lead to
highly surprising dynamics with very high symmetry.

These emergent symmetries have played very important roles in previous studies
of topological quantum criticality in superfluids. In fact, they are part of
symmetry groups that have been used to identify universality classes, apart
from discrete global symmetriesYang19 ; Yang21 ; Zhou22 . While continuous
emergent symmetries usually lead to concrete scaling symmetries, discrete
global symmetries define the number of relevant low energy degrees in fermion
fields or central charges. Both are crucial in studies of thermodynamics and
dynamics near topological critical points.

The emergent symmetries in gapless limits are also usually further higher than
adjacent gapped phases. This can be understood in terms of mass operators in
gapped phases. Presence of these additional mass operators always lowers pre-
existing symmetries of their gapless counterparts . From this point of view,
gapless states, either critical or belonging to stable phases usually have the
highest emergent symmetries when compared with surrounding gapped phases. This
article is mainly focused on those highly symmetric gapless states.
Topological and dynamic stability of various gapless states had been focuses
in many previous studiesSato17 ; Wen02 ; Sato06 ; Beri10 ; Kobayashi14 ;
Zhao16 ; Schnyder11 ; Wan11 ; Burkov11 ; Burkov18 ; Armitage18 ; Meng12 ;
Cho12 ; Grover14  and we will refer them to those original research.

Below we will mainly emphasize two important aspects of translationally
invariant gapless superfluids or superconductors. One is about connections
between physically different gapless states that exhibit the Lorentz invariant
dynamics with the same partition functions, and relations between emergent
$U(1)$ symmetries in different systems. This is effectively to classify and
establish equivalent families of apparently different gapless states with
different emergent $U(1)$ symmetries unique to gapless states.

More specially, each unique emergent symmetry in a specific gapless state is
associated with invariance under a particular transformation induced by a
generator. In the limit of our interests, all these generators turn out to be
purely imaginary or anti-symmetric Hermitian operators. The complete set of
such operators can be classified into ones of an $SU(2)$ spinor rotation
subgroup of Lorentz transformation generated by $S^{ij}$, $i\neq j=x,y,z$, and
ones of its dual $SU(2)$ group induced by $S_{D}^{ij}$, $i\neq j=x,y,z$ that
are mutually commuting with $S^{ij}$. The combination of these two $SU(2)$
groups, $SU(2)\otimes SU(2)$, forms an $Spin(4)$ group of real fermions that
double covers an $SO(4)$ group with isomorphic algebras.

Emergent $U(1)$ symmetry groups are thus embedded in this
$Spin(4)=SU(2)\otimes SU(2)$ group. The conserved charge associated with the
emergent symmetry is always rotationally invariant under the action of
$S^{ij}$ and has to be represented by one of the generators in the dual group,
$S_{D}^{ij}$. However, if gapless states become gapped, we find the unique
emergent $U(1)$ symmetry associated with gapless liquids is always broken
although the Lorentz symmetry always remains.

The second objective is to explore relations between gapless states with
higher symmetries including $SO(3,1)$ Lorentz symmetry and more general
gapless states with much lower symmetries i.e. without the Lorentz symmetry.
In the limit we have considered, the emergent $U(1)$ symmetry associated to
the dual group generated by $S^{ij}_{D}$, $i,j=x,y,z$ and its conserved
charges are always present as far as states are gapless, disregarding the
Lorentz symmetry. Quantum dynamics of gapless liquids strongly depend on
whether they display a full $SO(3,1)$ Lorentz symmetry, or there is only a
lower $SO(3)$ or $SO(2)$ rotation symmetry. However, it appears that they all
share the same emergent $U(1)$ symmetries associated with the dual group of
$S_{D}^{ij}$, disregarding differences in dynamics. We conjecture such an
emergent symmetry to be fundamental to many gapless states in superfluids or
superconductors.

The article is organized as follows. In section II, we introduce effective
field theories for discussions of gapless states in superfluids or
superconductors with a Lorentz symmetry. We show a simple relation between
different physical systems with the Lorentz symmetry and relations between
different emergent $U(1)$ symmetries. In section III and IV, we illustrate the
structure of theories of real fermions in a specific limit that we focus on in
this article and unitary rotations of real fields. In section V, we show
explicitly connections between physically different gapless states. In section
VI and VII, we show that an emergent $U(1)$ symmetry as a robust feature in
gapless states near a stable infrared fixed point. In section VIII, we
illustrate that the emergent infrared $U(1)$ symmetry can be an asymptotic
symmetry in more generic interacting gapless states. Moving into higher
energies, $U(1)$ can naturally evolve into $Z_{2}$ symmetries in gapless
liquids. We further show that by condensing one of mass operators so to break
the Lorentz symmetry, the gapless liquids can be further converted into other
gapless states without $SO(3,1)$ Lorentz symmetry. Nevertheless, the $U(1)$
symmetry is still intact in all gapless liquids we have examined. It can be
broken only when the Lorentz symmetry remains unbroken but states become fully
gapped. In section IX, we conclude our studies and point out a few open
questions.

##  II Effective Field Theories (EFTs) for Relativistic Real Fermions

###  II.1 Effective Field Theories

To understand interaction dynamics in superfluids and superconductors
especially topological aspects, it is often very convenient to employ the real
fermion representation to faithfully represent the intrinsic charge
conjugation symmetry. Below we are going to use the real fermion
representation to explore simple relations between physically very different
superconducting states or superfluids. The purpose is to show that dynamics in
many different systems are entirely equivalent and are universal and therefore
studying one is equivalent to exploring the whole equivalent class.

Without loosing generality, we can cast an interacting Hamiltonian of gapless
real fermions in the following infrared form,

$\displaystyle H_{eff}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int d{\bf
r}[\chi^{T}({\bf r}){\bf\alpha}\cdot{i\nabla}\chi({\bf r})$ (1)
$\displaystyle+$ $\displaystyle
g_{1}\chi^{T}\beta_{1}\chi\chi^{T}\beta_{1}\chi+g_{2}\chi^{T}\beta_{2}\chi\chi^{T}\beta_{2}\chi+...]$

where ${\bf\alpha}=(\alpha_{x},\alpha_{y},\alpha_{z})$. And $\alpha_{i}$,
$i=x,y,z$ and $\beta_{1,2}$ are mutually anti-commuting Hermitian matrices.
That is,

$\displaystyle\\{\alpha_{i},\alpha_{j}\\}=2\delta_{i,j},\\{\beta_{m},\beta_{n}\\}=2\delta_{m,n},\\{\alpha_{i},\beta_{m}\\}=0;$
$\displaystyle\alpha^{\dagger}_{i}=\alpha_{i},\beta^{\dagger}_{m}=\beta_{m},i=x,y,z;m=1,2.$
(2)

We will restrict to real fermions with four components, which turns out to be
a minimum number of degrees of freedom for our discussions;

$\displaystyle\chi^{T}=(\chi_{1},\chi_{2},\chi_{3},\chi_{4}),\chi^{\dagger}_{i}({\bf
r})=\chi_{i}({\bf r}),\\{\chi_{i},\chi_{j}\\}=\delta_{ij}.$ (3)

We have also chosen to introduce two interactions, $g_{1,2}$ for later
discussions on emergent symmetries, although for the four component real
fermions, these two operators turn out to be always equivalent. So for the
convenience of the rest of discussions in this section, we first set
$g_{1}=g_{0}$ and $g_{2}=0$, and $\beta_{1}=\beta_{0}$; this is equivalent to
a procedure of gauge-fixing in the case of emergent global $U(1)$ symmetries
or dropping a less relevant channel in the case of emergent $Z_{2}$
symmetries. One can also verify that to be fully consistent with real fermion
representations, $\beta_{0}$ also has to be purely imaginary and anti-
symmetric; otherwise the four fermion operator becomes nullified. And $\alpha$
matrices are real and symmetric ones. That is,

$\displaystyle\alpha_{i}^{T}=\alpha_{i}=\alpha_{i}^{*},\beta^{T}_{0}=-\beta_{0}=-\beta_{0}^{*};i=x,y,z.$
(4)

In presenting Eq.(1), we have only kept most relevant kinetic and interaction
terms.

1) We have muted the terms bilinear in $\chi$ but higher order in $\nabla$,
i.e. $\chi\nabla^{2}\chi,\chi\nabla^{3}\chi$ (as $...$) as they are less
relevant in the infrared limit. We have also re-adjusted the velocity of
fermions along $x,y,z$ direction to be equal by a trivial rescaling of $x,y,z$
and keep the terms linear in $\nabla$. In very special cases when one of the
velocities is exactly zero and the leading terms involve $\nabla^{2}$, the
effective theory shall be of quantum Lifshitz majorana fields. The physics of
those was discussed in Ref.Yang21  and will not be the focus here. However,
most of the conclusions derived here shall also be applicable to those models
in the infrared limit as generally, $\nabla^{2}$ terms are less relevant than
terms linear in $\nabla$ we have kept here. Eq.(1) is a generic infrared
theory for a broad class of gapless superfluids or superconductors with
intrinsic (relativistic) particle-hole symmetries and with dynamics captured
by leading linear-in-$\nabla$ terms.

From the point of view of scaling dimensions, Lifshitz majorana fields are
less generic as they require fine tuning of fermion velocity to zero so that
more relevant kinetic terms vanish identically. Eq.(1) shall be considered to
be a more generic form of low energy interacting real fermions.

2) We also have muted the four fermion terms involving additional $\nabla$ (as
$...$ in line 2) as they are also less relevant compared to the four-fermion
terms kept.

This generic form of gapless real fermions naturally have a very high space-
time Lorentz symmetry. We will take this as a starting point of discussions on
gapless fermions. For the purpose of emergent $U(1)$ symmetries to be
discussed later, it turns out that this naturally emergent high space-time
symmetry is un-essential at all and the emergent $U(1)$ symmetry can appear in
all other Lorentz non-invariant gapless fermion systems including the ones
with Fermi surfaces. Nevertheless, this highly symmetric limit is the most
convenient focal point where other gapless fermions can be easily related to
and for that reason, we will spend quite bit efforts to examine this limit
first before extending to other less symmetric but equally interesting cases.

For the same reason, we also first restrict ourselves to Lorentz symmetric
interactions $g_{1,2}$ only and do not consider less relevant interactions
that break the Lorentz symmetry. Later, we will see algebras in Eq.(2) are
sufficient for emergent Lorentz symmetry, even when interactions are strong.

###  II.2 Emergent Lorentz Symmetry

The generic model for infrared physics defined by ${\bf\alpha}$, $\beta_{0}$
has an emergent Lorentz symmetry as in the standard relativistic theories. For
instance, one can then construct the $4\times 4$ gamma matrices and Lorentz
group generators in a standard way.

$\displaystyle\Gamma_{0}=\beta_{0},\Gamma_{i}=\beta_{0}{\bf\alpha}_{i},i=x,y,z.$
$\displaystyle\\{\Gamma_{\mu},\Gamma_{\nu}\\}=g_{\mu\nu},\mu,\nu=0,x,y,z.$ (5)

The Lorentz boost are generated by $S^{0i}$, $i=x,y,z$ and rotations are
$S^{ij}$, $i\neq j=x,y,z$. Together they generate the $SO(3,1)$ Lorentz group.
They can be explicitly defined in terms of $\alpha_{i},\beta_{0}$, $i=x,y,z$;

$\displaystyle
S^{0i}=\frac{i}{2}\alpha_{i},S^{ij}=-S^{ji}=-\frac{i}{4}[\alpha_{i},\alpha_{j}];i,j=x,y,z.$
(6)

Furthermore, the boost operators are anti-hermitian, symmetric and purely
imaginary; the rotation ones as usually are hermitian and antisymmetric and
purely imaginary, i.e.,

$\displaystyle{S^{0i}}^{\dagger}$ $\displaystyle=$
$\displaystyle-S^{0i}=-{S^{0i}}^{T}={S^{0j}}^{*};$
$\displaystyle{S^{ij}}^{\dagger}$ $\displaystyle=$ $\displaystyle
S^{ij}=-{S^{ij}}^{T}=-{S^{ij}}^{*},i\neq j=x,y,z.$ (7)

It is important to notice that the Lorentz $SO(3,1)$ group can be fully
generated by $S^{0i}$ and $S^{ij}$ which are independent of the choices of
$\beta$ as far as Eq.(2) are satisfied. So the Lorentz symmetry can emerge
even when $g_{2}$ is nonzero and interactions of $\beta_{2}$-type are present.
From now on, we can relax the condition of $g_{2}=0$ and again keep both
$g_{1,2}$.

We can further define $s_{i}=\frac{1}{2}\epsilon_{ijk}S^{jk}$. Following the
general anti-commuting relations between $\alpha_{i}$ in Eq.(2) , we verify
that $s_{i}$, $i=x,y,z$, being anti-symmetric and purely imaginary, satisfy
the simple $SU(2)$ algebra, i.e. $S^{ij}$ form the algebras of an $SU(2)$
spinor subgroup in the Lorentz group of $SO(3,1)$. Together with
$t_{i}=S^{0i}$, $\\{s_{i},t_{i}\\}$, $i=x,y,z$ in Eq.(6) indeed lead to the
$SO(3,1)$ algebras of the Lorentz group,

$\displaystyle[s_{i},s_{j}]$ $\displaystyle=$ $\displaystyle
i\epsilon_{ijk}s_{k},[t_{i},t_{j}]=-i\epsilon_{ijk}s_{k},[s_{i},t_{j}]=i\epsilon_{ijk}t_{k},$

where $i,j.k=x,y,z$.

For an operator to be Lorentz invariant, the operator has to commute with the
rotation subgroup generators $S^{ij}$, $i\neq j=x,y,z$ but anti-commute with
the boost generators of $S^{0i}$. Following Eq.(6), any operator anti-
commuting with $\alpha_{i}$ or the booster operators $S^{0i}$ is also
rotationally invariant; so it is further Lorentz invariant. For this reason,
$\beta_{0}$ is obviously Lorentz invariant by this definition,

$\displaystyle\\{\beta_{0},S^{0i}\\}=[\beta_{0},S^{ij}]=0,i\neq j=x,y,z.$ (9)

Figure 1:  Equivalence between real fermions dynamics in gapless superfluids
or superconductors. $\phi$ is the rotation angle of an $SU(2)$ rotation
defined in Eq.(22) where ${\bf n}=1/\sqrt{3}(1,1,1)$. a). P1 refers to a time
reversal invariant $p$-wave state; P2 as state P1 but with an additional
$\pi/2$-phase shift to further break the time reversal symmetry; W is related
to a real fermion representation of complex Weyl fermions. b). The
$\tau-\sigma$ duals of the dynamics in a). DP1(2) is dual to P1(2) and the
dual of state W in a) is N, that can be representing dynamics of a nodal
phase.

##  III Hermitian matrices in EFTs of real fermions

To construct effective field theories (EFT) of real fermions and find a
specific representation for $\alpha_{i},\beta_{0}$, $i=x,y,z$, we shall group
all hermitian matrices into two big categories:

1) real and symmetric ones that can only appear along with gradient operators
or odd powers of gradient operators;

2) imaginary and antisymmetric ones that can only appear along with mass terms
or even powers of gradient operators.

In this article, we are mainly interested in four component real fermion field
theories, which is equivalent to $N_{f}=1/2$ of standard three dimensional
four-component Dirac fermions. The $4\times 4$ hermitian matrices form a
representation of $SU(4)$ algebras that has three independent $SU(2)$
subgroups; these subgrounps play important roles in later discussions. We
construct these operators via a tensor product of two $SU(2)$ group algebras,
one acting on the particle-anti-particle Nambu space by $\tau_{1,2,3}$ and the
other acting on the standard spin space by $\sigma_{x,y,z}$. The 15 hermitian
matrices constructed in this way are

Six Imaginary and Asymmetric Operators $\displaystyle
K_{1}=\tau_{y}\otimes\sigma_{x},K_{2}=\tau_{y}\otimes\sigma_{z},K_{3}=1\otimes\sigma_{y},$
$\displaystyle
F_{1}=\tau_{x}\otimes\sigma_{y},F_{2}=\tau_{z}\otimes\sigma_{y},F_{3}=\tau_{y}\otimes
1;$ Nine Real and Symmetric Operators
$\displaystyle\tau_{x}\otimes\sigma_{x},\tau_{x}\otimes\sigma_{z},\tau_{z}\otimes
1,$
$\displaystyle\tau_{z}\otimes\sigma_{x},\tau_{z}\otimes\sigma_{z},\tau_{x}\otimes
1,$ $\displaystyle
1\otimes\sigma_{x},1\otimes\sigma_{z},\tau_{y}\otimes\sigma_{y}.$ (10)

Note that six asymmetric operators form two independent $SU(2)$ subgroup
algebras isomorphically and respectively;

$\displaystyle[K_{i},K_{j}]=i2\epsilon_{ijk}K_{k},[F_{i},F_{j}]=i2\epsilon_{ijk}F_{k},$
$\displaystyle[K_{i},F_{j}]=0,i,j=1,2,3.$ (11)

Unlike the standard Pauli matrices, these purely imaginary hermitian matrices
form a real representation of $SU(2)$ rotations and can be employed to perform
$SU(2)$ rotations of real fermion fields. They play paramount roles in
emergent symmetries and spontaneously symmetry breaking of them. We will name
the symmetries associated with real $SU(2)$ rotations generated by
$K_{i},i=1,2,3$ as the $\sigma$-symmetry, and the ones associated with $SU(2)$
rotations generated by $F_{i},i=1,2,3$ as the $\tau$-symmetry.

Two $SU(2)$ groups also double cover an $SO(4)$ rotation group. In fact, they
naturally form the standard $Spin(4)$ group for real fermions under
considerations. Their algebras are isomorphic to those of an $SO(4)$ group;

$Spin(4)=SU(2)\otimes SU(2)\rightarrow SO(4).$ (12)

So when acting on real fermion fields, they naturally induce $SO(4)$ rotations
in the Hamiltonian of real fermion fields. In the later discussions, we will
mainly focus on the algebraic aspect of this group rather than the topological
ones and sometimes use $Spin(4)$ and $SO(4)$ interchangeably without further
distinguishing them. $Spin(n),n=4$ group naturally emerges in our current
discussions of symmetry as spin groups are generic to real or charge neutral
fermions. They are also special limits of more general Clifford algebras
defined for general Riemannian manifolds. We refer to more general discussions
on relations between spin groups and Clifford algebras to Ref.CliffordA . From
physics point of view, spin groups $Spin(n)$ can be thought as quantized
version of classical groups $SO(n)$ as groups of $Spin(n)$ take into account
essential quantum aspects of real fermion spin dynamics.

##  IV Unitary Transformation of Real Fermions and $\tau-\sigma$ duality

###  IV.1 Unitary Transformation

Readers can skip this section and move to Sec.V if they are not interested in
details of algebraical structures of the symmetry groups.

Unitary transformation of real fermions needs to leave fermions in a real
representation. That requires that all unitary rotations be implemented in a
real representation and so effectively become orthogonal transformations. For
that purpose, we can only utilize pure imaginary, antisymmetry generators. The
only generators fall into this class are the ones specified as $K_{i}$ and
$F_{i}$, $i=x,y,z$ in Eq.(III), (11) which are indeed isomorphic to $SO(4)$
group algebras and generate the entire $Spin(4)$ group.

To highlight the structure of $Spin(4)$ group, we study the general rotations
of real fermion fields induced by $SU_{K}(2)$ or K-rotations and $SU_{F}(2)$
or F-rotations in $Spin(4)$ group. By construction, each group element in
$Spin(4)$ can be specified as $[U_{K},U_{F}]$, a pair of $K$-rotation and
$F$-rotation. Just like in standard constructions of $SU(2)$ rotations, we
specify each of these $SU(2)$ rotations by Euler angles $({\bf n},{\phi})$,
i.e. a rotation axis ${\bf n}=(n_{x},n_{y},n_{z})$ (${\bf n}$ is a unit
vector) and a rotation angle $\phi\in[0,2\pi]$. Each $Spin(4)$ group element
is thus specified by two sets of Euler angles, one for $K$-group, i.e.$({\bf
n}_{K},\phi_{K})$; the other for $F$-group, i.e. $({\bf n}_{F},\phi_{F})$.

Under the action of $Spin(4)$ group, real fermions can transform accordingly.
In our case, transformations of real fermions are by multiplication of the
pair of $SU(2)$ rotations in $[U_{K},U_{F}]$ of $Spin(4)$ group. That is

$\displaystyle\chi\rightarrow U_{T}({\bf n}_{K},\phi_{K};{\bf
n}_{F},\phi_{F})\chi$ $\displaystyle U_{T}({\bf n}_{K},\phi_{k};{\bf
n}_{F},\phi_{F})=U_{K}({\bf n}_{K},\phi_{K})\cdot U_{F}({\bf n}_{F},\phi_{F})$
(13)

where the subindices $K,F$ refer to $SU(2)$ rotations of $K$-type and $F$-type
respectively. In presenting this result, we have taken into account that each
of three generators $K_{i}$,$i=x,y,z$ in the $SU(2)$ group of $K$ rotations
commutes with each of $F_{i}$, $i=x,y,z$ in the $SU(2)$ group of $F$
rotations. So, $U_{T}$ is simply a product of two $SU(2)$ transformations,
$U_{K}$ and $U_{F}$ in $Spin(4)$ group. Explicit structures of $U_{F,K}$ will
be presented in the next section.

It is important to notice that $SU_{K}(2)\otimes SU_{F}(2)$, the product of
$SU(2)$ groups of $K$-type $U_{K}$ and $F$-type $U_{F}$ are isomorphic to
$S^{3}\otimes S^{3}$. Each $S^{3}$ can be conveniently defined by a set of
hyperspherical coordinates,

$\displaystyle(\cos\frac{\phi}{2},\sin\frac{\phi}{2}{\bf n})$ (14)

where $\phi\in[0,2\pi]$ and ${\bf n}$ again is a three dimensional unit vector
projecting out an $S^{2}$.

An inversion in a single $S^{3}$ corresponds to $({\bf
n},\phi)\rightarrow(-{\bf n},2\pi-\phi)$. Applying the standard $SU(2)$
algebras, one can verify explicitly that under such an inversion, indeed

$\displaystyle U_{F}(-{\bf n}_{F},2\pi-\phi_{F})$ $\displaystyle=$
$\displaystyle-U_{F}({\bf n}_{F},\phi_{F});$ $\displaystyle U_{K}(-{\bf
n}_{K},2\pi-\phi_{K})$ $\displaystyle=$ $\displaystyle-U_{K}({\bf
n}_{K},\phi_{K}).$ (15)

Thus, under an inversion in an $S^{3}\otimes S^{3}$, both $U_{F}$ and $U_{K}$
acquire minus signs. That is each element of $Spin(4)$ group acquires a minus
sign under inversion; and is mapped into minus of itself, i.e.
$[U_{K},U_{F}]\rightarrow[-U_{K},-U_{F}]$. This however leaves the bilinear
structure in $U_{T}$ invariant under such an inversion, although each of
$U_{F,K}$ is not. That is

$\displaystyle U_{T}(-{\bf n}_{K},2\pi-\phi_{K};-{\bf
n}_{F},2\pi-\phi_{F})=U_{T}({\bf n}_{K},\phi_{K};{\bf n}_{F},\phi_{F}).$ (16)

A pair of inverted points in $S^{3}\otimes S^{3}$ therefore lead to the same
orthogonal rotation of real fermions. Eq.(16) indicates a double covering of
$SO(4)$ by $S^{3}\otimes S^{3}$ and hence by the spin group
$Spin(4)=SU(2)\otimes SU(2)$ that is isomorphic to $S^{3}\otimes S^{3}$.

In addition, spin rotations of $S^{ij}$ can be covered by an $SU(2)$ group;
and $S^{ij}$ are also antisymmetry and hermitian. This indicates that they
shall also be in one of the two $SU(2)$ groups, either $K$-group or $F$-group
in Eqs.III,11. Whatever it is, it remains invariant under the actions of the
other group.

For instance, if $S^{ij}$, $i\neq j=x,y,z$ are represented by $K$-group
generators, they are invariant under any action of $F$-group, and vice versa.
However, under the actions of the same group of $S^{ij}$ or $K$-group actions,
$S^{ij}$ can be further rotated and are not invariant. On the other hand,
$S^{0i}$, $i=x,y,z$ are not invariant under actions of either group, $K$\- or
$F$-group.

###  IV.2 $\tau-\sigma$ Duality

One shall notice that $K$\- and $F$-rotations are related by a $\tau-\sigma$
duality,

$\displaystyle F_{i}\leftrightarrow K_{i}\mbox{ when
$\tau_{i}\leftrightarrow\sigma_{i}$.}$ (17)

Therefore, generally speaking, $Spin(4)=SU(2)_{F}\otimes SU(2)_{K}$ is simply
generated by $S^{ij}$ and its $\tau-\sigma$ dual, whichever $S^{ij}$ are. We
name the $\tau-\sigma$ dual of $S^{ij}$ defined in Eq.(6) as $S^{ij}_{D}$.
$S^{ij}$ are also purely imaginary and antisymmetric and hermitian.

$\displaystyle S^{ij}\leftrightarrow S^{ij}_{D}\mbox{ when
$\tau_{i}\leftrightarrow\sigma_{i}$.}$ (18)

This simple observation, the invariance of $S^{ij}$ under its dual
$S^{ij}_{D}$, either $K$-group or $F$-group, suggests a way to classify EFTs
and identify them with different physics reality depending on the structure of
$S^{ij}$.

$\tau$-symmetry below is assigned to EFTs where their Lorentz rotation
subgroup of $S^{ij}$ is identified as $K$-rotations and remains invariant
under the actions of its dual $S^{ij}_{D}$ or $F$-group. $\sigma$-symmetry is
assigned to EFTs where their Lorentz rotation subgroup of $S^{ij}$ is
identified as $F$-rotations and remains invariant under the actions of its
dual $S^{ij}_{D}$, which now is $K$-rotations. $S^{ij}$ together with its
$\tau-\sigma$ dual, $S^{ij}_{D}$, form the algebra group of $Spin(4)$ and
generate the $Spin(4)$ group.

For each class, we can identify a parent Hamiltonian and generate the rest
members in the class by the dual of $S^{ij}$, $S^{ij}_{D}$ which can be either
$K$-group or $F$-group, whichever is the dual of $S^{ij}$. Within each class,
EFTs have the same representation for the Lorentz subgroup group $S^{ij}$ up
to a rotation induced by $S^{ij}$ itself; and we do not discuss such a trivial
spin rotations generated by $S^{ij}$ in this article.

The action of $S^{i,j}_{D}$, the dual of $S^{ij}$, on $S^{ij}$ is trivial as
$S^{ij}$ is invariant under its dual $S^{ij}_{D}$; it leaves $S^{ij}$
unchanged. However, $S^{ij}_{D}$ acts on $S^{0i}$ or $\alpha_{i}$ non-
trivially. It can generate other members of EFTs with the same $S^{ij}$
rotation group and they correspond to different physical systems. On the other
hand, because different real fermion representations, or Hamiltonians are
related by simple rotations, dynamics and the partition functions shall be
obviously identical. This is the focus of this article, to explore the classes
of EFTs that form a group representation of the dual of the rotation group
$S^{ij}$, i.e. the $SU(2)$ subgroup in $Spin(4)$ generated by $S^{ij}_{D}$.

###  IV.3 Masses

As all the bilinear terms or mass terms for real fermions are represented by
anti-symmetric hermitian matrices, they have to be one of $K$\- or
$F$-rotation generators. Also as $K$-rotation and $F$-rotation generators are
mutually commuting, all mass terms associated $K$ generators are invariant
under $F$-rotations and vice versa.

As we will be mainly interested in spatially rotationally invariant states and
three rotation generators defined by $S^{ij}$ are either associated with $K$\-
or $F$-generators, the mass operators have to be identified as the dual of
$S^{ij}$, that is $S^{ij}_{D}$ that can be either $F$\- or $K$-generators.

The maximum number of mass terms allowed by a given gapless state with
$\alpha_{i},i=x,y,z$ in EFTs already identified is therefore three.
Furthermore, we also find only two of these three rotationally invariant mass
operators anti-commute with Lorentz boost operators $S^{0i},i=1,2,3$, and are
Lorentz invariant.

The third one that commutes with $S^{0i}$ or $\alpha_{i}$, $i=x,y,z$
explicitly breaks Lorentz symmetry. On the other hand, it represents an
emergent or hidden $U(1)$ symmetry in gapless states we will focus on.

The emergent $U(1)$ symmetry can be broken whenever one of the other two mass
operators in $S^{ij}_{D}$, or both, condense and develop finite expectation
values in the ground states while the Lorentz symmetry is preserved.

At last, if the third mass operator which is Lorentz non-invariant condenses
with non-zero expectation values, the Lorentz symmetry is broken; typically,
there shall be an emergent Fermi surface in gapless states.

Finally, one can also have mass terms that further break $SO(3)$ rotation
symmetries, in addition to breaking the Lorentz symmetry. This is actually a
path leading to nodal phases. However, in this case, a new Lorentz symmetry
can re-emerge replacing the original Lorentz group; the mechanism is similar
to the emergence of relativistic physics in $(1+1)$D due to emergence of Fermi
points at fermion surfaces.

###  IV.4 Identifying an EFT with a Physical System

Real fermions are emergent particles in physical systems instead of
fundamental ones. The relations between physical complex fermions and real
fermions are usually simple in one particular representation than other ones.
So although dynamically different systems can be equivalent, when it comes to
physical interpretations or predictions, we always prefer one with the
simplest relation between physical complex fermions and real fermions. If we
specify real fermions $\chi$ directly in terms of complex ones $\psi$ for a
given physical system and only work with a particular choice of real fermion
fields, then EFT Hamiltonian in term of those real fermion is entirely fixed.
Further rotated EFTs involve $\chi$ fields defined in different ways in terms
of complex fermions. A popular choice is to identify

$\displaystyle\chi_{1}({\bf r})=\frac{1}{\sqrt{2}}[\psi_{\uparrow}({\bf
r})+\psi^{\dagger}_{\uparrow}({\bf r})],$ $\displaystyle\chi_{2}({\bf
r})=\frac{1}{\sqrt{2}}[\psi_{\downarrow}({\bf
r})+\psi^{\dagger}_{\downarrow}({\bf r})],$ $\displaystyle\chi_{3}({\bf
r})=\frac{i}{\sqrt{2}}[\psi_{\uparrow}({\bf r})-\psi^{\dagger}_{\uparrow}({\bf
r})],$ $\displaystyle\chi_{4}({\bf
r})=\frac{i}{\sqrt{2}}[\psi_{\downarrow}({\bf
r})-\psi^{\dagger}_{\downarrow}({\bf r})].$ (19)

Here ${1,2,3,4}$ are indices for the real fermions, and $\uparrow,\downarrow$
are ones for spins or pseudo spins depending on microscopic starting points.

For this reason if we only work with this particular choice of $\chi$ fields
or effectively fix an $SU(2)$ gauge, every Hamiltonian in the two classes
discussed above corresponds to one single physical reality. This is the point
we will be taking in this article. Every physical system has a specific EFT.
As each can be further related to other Hamiltonians in the same family if one
performs a purely real unitary rotation of $\chi$, dynamically speaking,
different physical systems can be mapped into other ones by redefining $\chi$
fields. And we will be exploring the relations between emergent symmetries in
different states using the $K$-group and $F$-group.

##  V Mapping between real fermions

###  V.1 $F$-rotations and mapping

We first focus on a class of EFTs where $SU(2)$ rotation subgroup of $S^{ij}$
in Lorentz group $SO(3,1)$ is given by $K$-rotations in $Spin(4)$ group. We
will study the members generated by the dual group $S^{ij}_{D}$ which in this
case is $SU_{F}(2)$ subgroup in $Spin(4)$. All $F$-rotations leave $S^{ij}$
and their rotation group $SU_{K}(2)$ or $K$-rotations invariant; all EFTs here
have the same structures of $S^{ij}$, $i\neq j=x,y,z$.

We start with the most well known model for $p$-wave superconductors or
superfluids in a strong coupling limit, an EFT for topological quantum
critical points (TQCPs) with time reversal-symmetry.

$\displaystyle H_{TS}=\frac{1}{2}\int d{\bf r}[\chi^{T}({\bf
r})(\tau_{z}\otimes(\sigma_{x}{i\nabla_{z}}-\sigma_{z}{i\nabla_{x}})+\tau_{x}\otimes\mathbb{I}{i\nabla_{y}})\chi({\bf
r})+g_{1}\chi^{T}\tau_{y}\chi\chi^{T}\tau_{y}\chi+g_{2}\chi^{T}\tau_{z}\otimes\sigma_{y}\chi\chi^{T}\tau_{z}\otimes\sigma_{y}\chi]$
(20)

where
$\chi^{T}=(\chi_{1\uparrow},\chi_{1\downarrow},\chi_{2\uparrow},\chi_{2\downarrow})$.
$\chi$ are real fermions defined by the following standard algebra,

$\displaystyle\chi^{\dagger}_{\alpha}({\bf r})=\chi_{\alpha}({\bf
r}),\\{\chi_{\alpha}({\bf r}),\chi_{\beta}({\bf r})\\}=\delta_{\alpha,\beta},$
$\displaystyle\alpha,\beta=1\uparrow,1\downarrow,2\uparrow,2\downarrow.$ (21)

[ Here we have renamed $\chi_{1,2,3,4}$ in terms of
$\chi_{1\uparrow,1\downarrow,2\uparrow,2\downarrow}$ to illustrate the spin
structures explicitly ].

The EFT in Eq.(20) is a minimum representation for TQCPsYang21 ; Zhou22  with
spinful time reversal symmetry and emergent Lorentz symmetry. It can further
have emergent supersymmetry in strong coupling limits. We now exam the
effective field theories or EFT that we can further obtain via $SU(2)$
rotations generated by $F$-group in Eq.(11) and the $\tau$-symmetry associated
with it. The general structure for a rotation along the direction of ${\bf
n}=(n_{x},n_{y},n_{z})$ with an angle $\phi$ (following the right hand rule)
is

$\displaystyle U_{F}({\bf n},\phi)$ $\displaystyle=$
$\displaystyle\cos\frac{\phi}{2}-i\sin\frac{\phi}{2}(n_{x}\tau_{x}\otimes\sigma_{y}+n_{y}\tau_{y}+n_{z}\tau_{z}\otimes\sigma_{y});$
$\displaystyle U_{F}({\bf n},\phi)$ $\displaystyle=$ $\displaystyle
U_{F}^{*}({\bf n},\phi),U_{F}^{-1}({\bf n},\phi)=U^{T}_{F}({\bf n},\phi).$
(22)

Note for $F$ rotations, the range of $\phi$ shall be set as

$2\pi\geq\phi\geq 0.$ (23)

As expected, $U_{F}({\bf n},\phi)$ defines a simply connected three-sphere
manifold $S^{3}$. Specially,

$\displaystyle U_{F}(-{\bf n},2\pi-\phi)=-U_{F}({\bf n},\phi)$ (24)

which effective defines a pair of diagonally opposite points in an $S^{3}$. On
the other hand, $U_{F}({\bf n},\phi)$ and $-U_{F}({\bf n},\phi)$ obviously
results in the identical rotations in the Hamiltonian manifold. Inclusion of
the other three $S^{3}$ defined by $U_{K}$ that is the dual of $U_{F}$
effectively allows a double coverage of an $SO(4)$ group by an $Spin(4)$
group, similar to a universal coverage of $SO(3)$ by a quantum spin group
$Spin(3)=SU(2)$.

Below are two examples where ${\tilde{H}}$ in EFTs after $F$-rotations are
transformed into the Hamiltonians in other superconducting states or
superfluids when we apply the same identification in Eq.(19) to the rotated
real fermions. Hence we show that interaction dynamics are completely
identical as they are given by the same partition functions.

Under such a purely real rotation,

$\displaystyle\chi({\bf r})\rightarrow{\tilde{\chi}}({\bf r})=U_{F}\chi({\bf
r}),H[\\{\chi({\bf r})\\}]\rightarrow{H}={\tilde{H}}[\\{{\tilde{\chi}}({\bf
r})\\}]$

where fermions remain to be real, i.e.${\tilde{\chi}}^{T}({\bf
r})={\tilde{\chi}}^{\dagger}({\bf r})$.

A1: TQCP model with time-reversal-symmetry broken

First we consider a rotation where

$\displaystyle{\bf n}=\frac{1}{\sqrt{3}}(1,1,1),\phi=-\frac{2\pi}{3}$ (26)

One can easily verify that this is also equivalent to a rotation along ${\bf
n}=(0,1,0)$ axis and $\phi=\frac{\pi}{2}$ therefore the resultant EFT
describes a $p-wave$ state with a $\pi/2$ phase shift. The state therefore
breaks the time reversal symmetry, physically distinct from a $T$-invariant
state. The EFT for ${\tilde{\chi}}$ (here we rename them as $\chi$) is

$\displaystyle H_{TS1}=\frac{1}{2}\int d{\bf r}[\chi^{T}({\bf
r})(\tau_{x}\otimes(\sigma_{x}{i\nabla_{z}}-\sigma_{z}{i\nabla_{x}})-\tau_{z}\otimes\mathbb{I}{i\nabla_{y}})\chi({\bf
r})+g_{1}\chi^{T}\tau_{y}\chi\chi^{T}\tau_{y}\chi+g_{2}\chi^{T}\tau_{x}\otimes\sigma_{y}\chi\chi^{T}\tau_{x}\otimes\sigma_{y}\chi]$
(27)

where
$\chi^{T}=(\chi_{1\uparrow},\chi_{1\downarrow},\chi_{2\uparrow},\chi_{2\downarrow})$.

B: Weyl Fermions

We then consider a rotation,

$\displaystyle{\bf n}=\frac{1}{\sqrt{3}}(1,1,1),\phi=\frac{2\pi}{3}.$ (28)

This transformation leads a more surprising EFT, one that is related to a real
fermion representation of interacting Weyl fermions but with single chirality;

$\displaystyle H_{Weyl}=\frac{1}{2}\int d{\bf r}[\chi^{T}({\bf
r})(\mathbb{I}\otimes(\sigma_{x}{i\nabla_{x}}+\sigma_{z}{i\nabla_{z}})+\tau_{y}\otimes\sigma_{y}{i\nabla_{y}})\chi({\bf
r})+g_{1}\chi^{T}\tau_{x}\otimes\sigma_{y}\chi\chi^{T}\tau_{x}\sigma_{y}\chi+g_{2}\chi^{T}\tau_{z}\otimes\sigma_{y}\chi\chi^{T}\tau_{z}\otimes\sigma_{y}\chi].$
(29)

This mapping was also applied to understand topological quantum criticality in
previous studiesZhou22 . Phenomenologically, the emergent symmetries we are
going to discuss below can be easily related to a chiral symmetry in Weyl
fermions but with a very distinct feature. The standard $U(1)$ chiral
anomalies in quantum field theories are absent and Weyl fermions as emergent
particles implied in Eq.(29) have only one fixed chirality, either left or
right but not both. Weyl fermions with one single chirality usually do not
appear in three dimensional bulk Weyl metals so their appearance here in an
EFT is a big surprise.

###  V.2 $K$-rotations and mapping and $\tau-\sigma$ duality

We now turn to a class of EFTs where $SU(2)$ rotation subgroup of $S^{ij}$ in
Lorentz group $SO(3,1)$ is given by $F$-rotations instead of $K$-rotations in
$Spin(4)$ group. We will study the members generated by the dual group
$S^{ij}_{D}$ which in this case is $SU_{K}(2)$ subgroup in $Spin(4)$. All
$K$-rotations leave $S^{ij}$ and their rotation group $SU_{F}(2)$ or
$F$-rotations invariant; all EFTs here again are given by the same $S^{ij}$.

We can further apply $K$ rotations and generate mapping between EFTs for
different physics systems. The discussions are very similar to those in the
previous subsection and we simply list the results below. They can also be
related to $F$ rotations by a simple $\tau-\sigma$ duality transformation,

$\displaystyle\tau_{i}\rightarrow\sigma_{i},\sigma_{i}\rightarrow\tau_{i},i=x,y,z.$
(30)

C: Dual of Model A

$\displaystyle H_{DTS}=\frac{1}{2}\int d{\bf
r}[\chi^{T}((\tau_{x}{i\nabla_{x}}-\tau_{z}{i\nabla_{z}})\otimes\sigma_{z}+1\otimes\sigma_{x}{i\nabla_{y}})\chi+g_{1}\chi^{T}\tau_{y}\otimes\sigma_{z}\chi\chi^{T}\tau_{y}\otimes\sigma_{z}\chi+g_{2}\chi^{T}1\otimes\sigma_{y}\chi\chi^{T}1\otimes\sigma_{y}\chi];$
(31)

C1: Dual of Model A1

$\displaystyle H_{DTS1}=\frac{1}{2}\int d{\bf
r}[\chi^{T}((\tau_{x}{i\nabla_{x}}-\tau_{z}{i\nabla_{z}})\otimes\sigma_{x}-\sigma_{z}\otimes
1{i\nabla_{y}})\chi+g_{1}\chi^{T}\tau_{y}\otimes\sigma_{x}\chi\chi^{T}\tau_{y}\otimes\sigma_{x}\chi+g_{2}\chi^{T}1\otimes\sigma_{y}\chi\chi^{T}1\otimes\sigma_{y}\chi];$
(32)

D: Dual Model of B as Gapless Nodal Phase with P-symmetry

The $\tau-\sigma$ dual of Weyl fermions is exactly an EFT for gapless
superconducting nodal phases with parity-symmetry. It has the following
explicit form,

$\displaystyle H_{NP}=\frac{1}{2}\int d{\bf
r}[\chi^{T}(\tau_{x}{i\nabla_{x}}+\tau_{z}{i\nabla_{z}})\otimes\mathbb{I}+\tau_{y}\otimes\sigma_{y}{i\nabla_{y}})\chi+g_{1}\chi^{T}\tau_{y}\otimes\sigma_{x}\chi\chi^{T}\tau_{y}\otimes\sigma_{x}\chi+g_{2}\chi^{T}\tau_{y}\otimes\sigma_{z}\chi\chi^{T}\tau_{y}\otimes\sigma_{z}\chi].$
(33)

Indeed, $H_{NP}$ in Eq.(33) are related to $H_{DTS}$ and $H_{DTS1}$ via
$F$-rotations, $U_{F}$ defined below

$\displaystyle U_{F}($ $\displaystyle{\bf n}$
$\displaystyle,\theta)=\cos\frac{\phi}{2}-i\sin\frac{\phi}{2}(n_{x}\tau_{y}\otimes\sigma_{x}+n_{y}\sigma_{y}+n_{z}\tau_{y}\otimes\sigma_{z});$
(34) $\displaystyle{\bf n}=\frac{1}{\sqrt{3}}(1,1,1),\phi=\pm\frac{2\pi}{3}.$

##  VI Emergent $U(1)$ symmetries and conserved charges in EFTs

Now we turn to hidden $U(1)$-symmetries that can emerge in gapless superfluids
or superconductors. At first sight, superfluids break the standard $U(1)$
global symmetries spontaneously. So the hidden $U(1)$ symmetries generally
differ from the usually $U(1)$ gauge symmetry that leads to particle number or
charge conservation. However, it turns out that the standard $U(1)$ charge-
symmetry is one of six class more general possible emergent $U(1)$ symmetries
in gapless superconductors or superfluids. In fact, the charge $U(1)$ symmetry
group as well other more general hidden $U(1)$ symmetries are all embedded in
an $Spin(4)$ group or an $SU(2)\otimes SU(2)$ group. They are represented by
invariance of the Hamiltonians under of one of the special $K$-type or
$F$-type $SU(2)$ rotations and therefore we can name these hidden symmetries
as $\tau$ or $\sigma$ symmetries in general.

Although the physical significance of emergent $U(1)$ symmetry depends on how
emergent particles are connected to physical fermions we can probe and
measure, in this section we will carry out discussions using EFTs introduced
before without specifying the relations between emergent fermions and physical
complex fermionic particles we start with. The connection between these two
will be explored in the next section when we explore physical consequence.

###  VI.1 $U_{0}(1)$ symmetry and chiral charge conservation

To start with, we focus on the simplest EFT, $H_{Weyl}$ in Eq.(29). Eq.(29)
has a free or non-interacting real fermions fixed point that is infrared
stable in spatial dimensions higher than one or $d>1$. The dynamics of gapless
fermions in the infrared limit turn out to be equivalent to simple single copy
of non-interacting Weyl fermions as the partition functions of these two
systems are identical.

This is a rather peculiar limit as physically a single copy of Weyl fermions
with a given chirality are usually forbidden in a bulk $3D$ because of the
well-known fermion double problemsNielson81 . Nevertheless, dynamics of single
copy of Weyl fermions can appear in an effective theory of the bulk of gapless
superfluids (See below for more elaborated discussions). The emergent symmetry
discuss here is an asymptotic one even when interactions are included. We
first exam the fixed point Hamiltonian with $g_{1,2}$ set to be zero.

The $U(1)$ global symmetry is generated by the operator

$\displaystyle Q_{0}=Q^{\tau}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int d{\bf r}q^{\tau}_{2}({\bf
r}),U_{2}=\exp(i\frac{\phi}{2}Q_{0});$ $\displaystyle q^{\tau}_{2}({\bf r})$
$\displaystyle=$ $\displaystyle\chi({\bf r})\tau_{y}\otimes 1{\chi({\bf r}}).$
(35)

Hamiltonian $H_{Weyl}$ remains invariant under this $U(1)$ transformation and
$Q_{0}$ is a conserved charge at the non-interacting fixed point when
$g_{1}=g_{2}=0$. This is an obvious result as $H_{Weyl}$ also coincides with a
free complex fermion fixed point. For this reason, the model doesn’t break the
$U(1)$ global charge symmetry and the $U(1)$ charge, $Q_{0}$, shall be
conserved.

This transformation is a special $F$-rotation around the axis ${\bf
n}=(0,1,0)$, i.e. $F_{2}=\tau_{y}\otimes 1$ and therefore results in a
rotation in $F_{1}-F_{3}$ plane. The other two charges of $F_{1},F_{3}$ are
rotated accordingly,

$\displaystyle Q^{\tau}_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d{\bf r}q^{\tau}_{1}({\bf r}),q^{\tau}_{1}({\bf r})=\chi({\bf
r})\tau_{x}\otimes\sigma_{y}{\chi({\bf r}});$ $\displaystyle Q^{\tau}_{3}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int d{\bf r}q^{\tau}_{3}({\bf
r}),q^{\tau}_{3}({\bf r})=\chi({\bf r})\tau_{z}\otimes\sigma_{y}{\chi({\bf
r}});$ $\displaystyle Q^{\tau}_{1}$ $\displaystyle\rightarrow
Q^{{}^{\prime}\tau}_{1}=\cos{\phi}Q^{\tau}_{1}+\sin{\phi}Q^{\tau}_{3},$
$\displaystyle Q^{\tau}_{3}$ $\displaystyle\rightarrow
Q^{{}^{\prime}\tau}_{3}=-\sin{\phi}Q^{\tau}_{1}+\cos{\phi}Q^{\tau}_{3}.$ (36)

In addition, under the time reversal transformation,

$Q^{\tau}_{1,2}\rightarrow
Q^{\tau}_{1,2},Q^{\tau}_{3}\rightarrow-Q^{\tau}_{3}.$ (37)

Finally, as expected

$\displaystyle[Q^{\tau}_{i},Q^{\tau}_{j}]=2i\epsilon_{ijk}Q_{k}^{\tau},i,j,k=1,2,3.$
(38)

Weyl fermions with single chirality as suggested in $H_{Weyl}$ surprisingly
appear as emergent fermions in three dimensional topological superfluids. The
corresponding EFTs can have an infrared unstable strong coupling fixed point
in spatial dimensions lower than three. At those fixed points, there is an
additional emergent supersymmetry and states have higher symmetries than the
non-interacting fixed point. Beyond that point, $U(1)$ symmetry of $Q_{0}$ is
broken spontaneously and states are gapped. In this case, they can form a
topological superconducting state with time reversal symmetry if the symmetry
is broken along the direction of $Q^{\tau}_{1}$ or $F_{1}$ as $Q^{\tau}_{1}$
is even under the time reversal transformation. However, if the symmetry is
not broken strictly along the direction of $Q^{\tau}_{1}$, then the T-symmetry
can also be broken and the gapped states will be superconducting without
symmetry protected topological features.

$Q_{0}$ conservation here is obviously closely related to chiral symmetries
discussed in quantum field theories and implies the conservation of Weyl
fermions, either right handed or left handed. However, there is a very
important fundamental difference between the physics in superfluids and in the
standard quantum field theory. In quantum field theories, fermions with both
chiralities do appear coupled at certain ultraviolet scales and chiral
anomalies induced by topological instantons eventually reduce the
$U_{L}(1)\otimes U_{R}(1)$ to the simple $U_{L+R}(1)$ charge gauge
symmetryt'Hooft76 . The right handed or left handed charges are not separately
conserved as a result of chiral anomalies.

However, in Eq.(29), when taken as an effective field theory of superfluids,
only right handed (or left handed) fermions with single chirality can emerge
because the underlying topological superfluid state has odd parity under the
parity transformation. That is the order parameter is odd under the parity
transformation

$\displaystyle\Delta_{\alpha\beta}(-{\bf p})=-\Delta_{\alpha\beta}({\bf
p}),\alpha,\beta=\uparrow,\downarrow.$ (39)

The ground state transforms non-trivially under this parity transformation.
Therefore, in a given superfluid state where symmetries are broken
spontaneously, only one chirality can emerge in effective field theories.
Fermions with opposite chirality only live in a different superfluid state or
universe that is related to the one under consideration via a parity
transformation.

Unless these two ground states can be connected by quantum tunnelling
processes, dynamics in a given superfluid can only be related to either left-
handed or right handed ones but not both. In $(3+1)$D, $U(1)$ gauge monopoles
are absent enforcing single chirality of real fermions. On the other hand,
dynamics in two different superfluids related by parity transformation can
both be represented by the single Weyl cone dynamics, left or right. Apart
from that, they are identical.

For this reason, emergent chiral charges, as emergent fermions, can be
conserved without usual anomalies extensively discussed in quantum field
theories. This emergent symmetry and conservation law appear to be more robust
than the chiral charge conservation that only appears at a classical level in
quantum field theories of Yang-Mills fieldst'Hooft76 .

Finally, a generator of emergent symmetry always commutes with the Hamiltonian
of EFTs, i.e. commutes with the non-interacting fixed point Hamiltonian that
exhibits the Lorentz symmetry. So the generator also commutes with both spin
rotation subgroup generators, $S^{ij}$ and boost operators $S^{0i}$, which are
basically constructed out of the free Hamiltonian (see discussions on
$\alpha,\beta$ matrices Section II.). As it commutes rather than anti-
commuting with $S^{0i}$, such symmetry generators are not Lorentz invariant
and do always break the Lorentz symmetry.

Before leaving this section, we want to point out that Weyl physics is also
useful for discussions on helium-3 A phaseLeggett75 ; Volovik03  where two
nodal points emerge on two opposite sides of a Fermi surface. In that case,
two copies of two-component-complex fermions, or, eight real fermions are
needed to describe dynamics. Each nodal point forms a representation of left
and right Weyl fermions respectively as a result of fermion doubling. From
fermion doubling point of view, helium-3 A phase is a conventional state with
expected numbers of fermi degrees of freedom while EFTs here do not have the
issue of fermion doubling. That is EFTs discussed here and below only carry
half of degrees of freedom in helium-3 A phase in low energy sectors so
avoiding the fermion doubling problem.

###  VI.2 $U_{1,3}(1)$ symmetry and charge conservation of $Q_{1,3}$

Now we are on the course to explore much less obvious emergent $U(1)$
symmetries in other more subtle cases presented in the previous section. The
emergent symmetries in Eq.(20) and Eq.(27) can be carried out in a similar way
as we have done in the previous subsection.

For the EFT in Eq.(20),

$\displaystyle Q^{\tau}_{1}=\frac{1}{2}\int d{\bf r}q^{\tau}_{1}({\bf
r}),U_{1}=\exp(i\frac{\phi}{2}Q^{\tau}_{1}),$ $\displaystyle q^{\tau}_{1}({\bf
r})=\chi({\bf r})\tau_{x}\otimes\sigma_{y}{\chi({\bf r}})$ (40)

defines a $U(1)$ transformation that leaves the EFT invariant. Charge $Q_{1}$
is therefore conserved.

For the EFT in Eq.(27),

$\displaystyle Q^{\tau}_{3}=\frac{1}{2}\int d{\bf r}q^{\tau}_{3}({\bf
r}),U_{3}=\exp(i\frac{\phi}{2}Q^{\tau}_{3}),$ $\displaystyle q^{\tau}_{3}({\bf
r})=\chi({\bf r})\tau_{z}\otimes\sigma_{y}{\chi({\bf r}})$ (41)

defines a $U(1)$ gauge transformation that leaves the EFT invariant. Charge
$Q^{\tau}_{3}$ is conserved.

$Q^{\tau}_{1,2,3}$ are three possible conserved charges of real fermions;
their corresponding symmetry groups are embedded in an $SU(2)$ subgroup of
$\tau$-type, dual to spin rotation $SU(2)$ subgroup of $S^{ij}$. And for a
given gapless state of superfluid, only one of them can be conserved. When
this emergent $U(1)$ symmetry is further broken, superfluids become gapped and
none of the charges above are conserved.

###  VI.3 $U_{1,2,3}$ symmetry and $\tau-\sigma$ duality

Because of the $\tau-\sigma$ duality, there are three $\sigma$ charges defined
below

$\displaystyle Q^{\sigma}_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d{\bf r}\chi({\bf r})\tau_{y}\otimes\sigma_{x}{\chi({\bf
r})},U_{1}=\exp(i\frac{\phi}{2}Q^{\sigma}_{1}),$ $\displaystyle
Q^{\sigma}_{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int d{\bf
r}\chi({\bf r}){I}\otimes\sigma_{y}\chi({\bf
r}),U_{2}=\exp(i\frac{\phi}{2}Q^{\sigma}_{2}),$ $\displaystyle Q^{\sigma}_{3}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int d{\bf r}\chi({\bf
r})\tau_{y}\otimes\sigma_{z}\chi({\bf
r}),U_{3}=\exp(i\frac{\phi}{2}Q^{\sigma}_{3}).$

Three $U(1)$ transformations $U_{1,2,3}$ above leave EFTs in
Eq.(31),(32),(33), respectively, invariant. And so $Q^{\sigma}_{1,2,3}$ are
conserved respectively. Again if such a symmetry is broken, gapless states
need to be fully gapped and there will be no more emergent $U(1)$ symmetries
or conserved charges.

Just like $Q^{\tau}_{i}$, $Q^{\sigma}_{i}$, $i=1,2,3$ satisfy the usual
$SU(2)$ algebras. Furthermore, they all commute with each of $Q^{\tau}_{i}$,
$i=1,2,3$;

$\displaystyle[Q^{\sigma}_{i},Q^{\sigma}_{j}]=2i\epsilon_{ijk}Q^{\sigma}_{k},[Q^{\sigma}_{i},Q^{\tau}_{j}]=0,i,j,k=1,2,3.$
(43)

###  VI.4 Emergent $U(1)$ symmetries: Effects of interactions

So far, we have discussed emergent $U(1)$ symmetries exactly at a free fermion
fixed point, the asymptotic theory. Now we turn to the most subtle effects of
interactions and intend to answer the question of whether these surprising
symmetries are generic.

In the presence of interactions, various $U(1)$ symmetries discussed above
become much less obvious and in general there are no fundamental reasons why
they shall be present. As we will see below, they can still emerge but only in
the infrared limit where energy scales are low enough so that other less
relevant irrelevant operators that we haven’t included in
Eq.(20),(27),(29),(31),(32),(33) play little role. However, once in an
intermediate energy window where those muted interactions become important,
$U(1)$ symmetries discussed above are no longer valid and dynamics start
substantially deviating from what we have described above.

At first sight, EFTs with interactions $g_{1,2}$ appear to break $U(1)$
symmetries that are present in the non-interacting limit of $g_{1,2}=0$.
However, the key observation is that away from the non-interacting fixed
point, these most relevant irrelevant interaction operators are actually
invariant under those $U(1)$ rotations. This is largely because there is only
one single most relevant irrelevant 4-fermion operator (i.e.without gradient
operators) we can construct for four-component real fermions. And so all
interaction operators appearing in our EFTs are identical; i.e.

$\displaystyle H_{Ix}$ $\displaystyle=$ $\displaystyle H_{Iy}=H_{Iz}$
$\displaystyle H_{Ix}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}\chi^{T}\tau_{x}\otimes\sigma_{y}\chi\chi^{T}\tau_{x}\otimes\sigma_{y}\chi,$
$\displaystyle H_{Iy}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}\chi^{T}\tau_{y}\otimes I\chi\chi^{T}\tau_{y}\otimes I\chi,$ $\displaystyle
H_{Iz}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}=\chi^{T}\tau_{z}\otimes\sigma_{y}\chi\chi^{T}\tau_{z}\otimes\sigma_{y}\chi$
(44)

As far as only such interactions that are entirely local in $\chi$ fields are
present in EFTs, $U(1)$ symmetries are still emergent just like in the non-
interacting fixed point.

However, we do not expect these symmetries are present at higher energy scales
when further moving away from the infrared fixed point and other less relevant
irrelevant interaction operators play more important roles. One such example
are interaction operators of the form,

$\displaystyle H_{LR}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}\chi^{T}\tau_{z}\otimes\sigma_{x}\nabla\chi\chi^{T}\tau_{z}\otimes\sigma_{x}\nabla\chi$

where the subscript (LR) implies less relevant. Presence of such an
interaction operator involving gradient fields $(\chi\nabla\chi)^{2}$
invalidates the emergent $U(1)$ symmetry. However, the scaling dimensions of
such operators are higher than $H_{I}$ in Eq.(44), i.e.

$\displaystyle Dim[H_{I}]=d-1,Dim[H_{LR}]=d+1,$ (46)

near the infrared stable fixed point. Hence, $U(1)$ symmetry always emerges as
an asymptotic symmetry in the infrared limit.

Below,we summarize our main conclusions. $U(1)$ symmetry become emergent if
one of the following conditions is met:

a) microscopic UV theories further indicate only a contact interaction with
zero range appears at ultraviolet scales of EFTs to forbid the presence of
operators of the form of $H_{LR}$ or alike in our EFTs;

b) the $U(1)$ symmetry is also a fundamental one suggested by the intrinsic
symmetry of the underlying physics;

c) in the infrared limit near the non-interacting fixed point when $H_{LR}$
are strongly suppressed; remaining interactions without gradient fields do
exhibit the same $U(1)$ symmetry as the fixed point.

Both a) and b) are additional inputs to EFTs, rather than intrinsic of EFTs
themselves. On the other hand generically, c) can always be satisfied as far
as we lower the energy scales so to close to the infrared fixed point.

For instance, if $H_{Weyl}$ is intended as a real Fermion representation of
free Weyl fermions, then $U(1)$ symmetry itself is a fundamental global
symmetry that EFTs shall also have and the emergent symmetry is the same as
the fundamental symmetry. The global gauge invariance enforces a $U(1)$
symmetry, or the condition b) is satisfied. And $U(1)$ symmetry is preserved
by all allowed interaction operators, not surprisingly. In this case, the
$U(1)$ symmetry not only appears near the infrared fixed point but also in
intermediate or higher energy scales and the corresponding $U(1)$ charge
conservation becomes exact.

However, if $H_{Weyl}$ is intended as a representation of a gapless superfluid
which already breaks the $U(1)$ global symmetry associated with charges, then
the $U(1)$ symmetry shown above is an emergent symmetry, either specifically
for certain interactions so that the condition a) is met, or alway appearing
in the infrared limit as stated in c). In the following section, we will
discuss the generic case c) so that $U(1)$ symmetry does emerge in low energy
sectors whenever EFTs are applicable.

##  VII Emergent $U(1)$ symmetries and applications in gapless superfluids or
superconductors

Now we are going to apply these emergent symmetries and conserved charges and
find out physical consequences in more specific systems. We will focus on a
few mostly commonly seen states although it can be easily generalized to many
other possible states. Again in this section, we have assumed an infrared
limit where interactions are fully captured by the discussions in the previous
section.

###  VII.1 Applications to Strong coupling limit of p-wave superfluids with
$T$ reversal symmetries

As stated before, EFT in Eq.(20) can be directly applied to understand time-
reversal-symmetric spinful superfluids or superconductors. $\chi$ fields are
related to physical complex fermion fields $\psi$ in a standard way,

$\displaystyle\psi_{\alpha}({\bf r})=\frac{1}{\sqrt{2}}[\chi_{1\alpha}({\bf
r})+i\chi_{2\alpha}({\bf r})],\alpha=\uparrow,\downarrow.$ (47)

Using the above convention, we can identify Eq.(20) as an EFT for a $p$-wave
superfluid or superconductor. Furthermore, each EFT representation of real
fermions that is generated by $F$ or $K$ rotations discussed before shall be
uniquely related a distinct physics system.

In Eq.(20), the spin rotation generators $S^{ij}$ of the emergent Lorentz
symmetry in this superfluid are defined by $K$-rotations in Eq.(III) or by
$Q^{\sigma}_{1,2,3}$. On the other hand, $Q^{\tau}_{1,2,3}$ are related to the
generators of the dual group, $S^{ij}_{D}$ or $F$ rotations.

Three spin rotationally invariant $\tau$-generators or charges for $F$
rotations in terms of complex fermions are

$\displaystyle Q^{\tau}_{1}$ $\displaystyle=$ $\displaystyle\frac{i}{2}\int
d{\bf r}[\Psi^{T}({\bf r})\sigma_{y}{\Psi({\bf r})}-\Psi^{\dagger}({\bf
r})\sigma_{y}{{\Psi^{\dagger}}^{T}({\bf r})}]$ $\displaystyle Q^{\tau}_{2}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int d{\bf r}[\Psi^{\dagger}({\bf
r}){\Psi({\bf r})}-\Psi({\bf r}){\Psi^{\dagger}({\bf r})}]$ $\displaystyle
Q^{\tau}_{3}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int d{\bf
r}[\Psi^{T}({\bf r})\sigma_{y}{\Psi({\bf r})}+\Psi^{\dagger}({\bf
r})\sigma_{y}{{\Psi^{\dagger}}^{T}({\bf r})}].$ (48)

where $\Psi^{T}=(\psi_{\uparrow},\psi_{\downarrow})$.

In gapless $p$-wave T-invariant superfluids that naturally appear in strong
coupling limits near topological phase transitions, only $Q^{\tau}_{1}$ is
conserved due to the emergent $U(1)$ symmetry. This charge conservation
becomes exact in the limit of contact interactions but for more generic
interactions it is an emergent symmetry in the infrared limit. When vacuum
expectation values of $Q^{\tau}_{2}$ or $Q^{\tau}_{3}$ are non-zero, the state
becomes fully gapped and $Q^{\tau}_{1}$ charge is no longer conserved.

###  VII.2 Applications to Nodal phases in Superfluids

Next, we look into the EFT in Eq.(33). Using the identification scheme for
complex fermions above, we verify that the EFT can be for a $3D$ nodal phase
physically induced by a strong magnetic field along the $y$-direction with a
parity symmetry. When the magnetic field is decreased, the nodal phase will
undergo a quantum Lifshitz transition of free majorana class into a gapped
superconductorYang21 . At zero field, the gapped superconductor can be a
topological one with time reversal symmetry.

In this particular case of Eq.(33), $\sigma_{y}=-1$ means left-handed (L)
nodal point while $\sigma_{y}=+1$ for right handed (R) nodal point and
$\sigma_{x,y,z}$ acting on the pseudo spin space of LR. The emergent $U(1)$
symmetry in EFTs is due to the underlying azimuthal symmetry around the
direction of $y$. Unlike in the previous subsection, the spin rotation
subgroup $S^{ij}$ of the emergent Lorentz symmetry in this superfluid turns
out to be defined by $F$-rotations in Eq.(III) or $Q^{\tau}_{1,2,3}$.

On the other hand, $Q^{\sigma}_{1,2,3}$ are all rotationally invariant under
the action of spin group $S^{ij}$ or in this case, $F$-rotations generated by
$Q^{\tau}_{1,2,3}$. In fact, they are directly related to the generators of
the dual group, $S^{ij}_{D}$ or $K$ rotations and hence all commute with
$Q^{\tau}_{1,2,3}$. These three $U(1)$ charges of $\sigma$-type defined in
Eq.(LABEL:sigmac) are now directly related to charges at left- and right-
handed nodal points. Introduce a pseudo spin representation for left and right
fields $\psi_{L,R}$ and four-component real fermion $\chi$-fields,

$\displaystyle\chi^{T}_{1}({\bf r})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}[\Psi^{T}+\Psi^{\dagger}],\chi^{T}_{2}({\bf
r})=-\frac{i}{\sqrt{2}}[\Psi^{T}-\Psi^{\dagger}]$ $\displaystyle\Psi^{T}({\bf
r})$ $\displaystyle=$
$\displaystyle\frac{1}{{2}}[(1,i)\psi_{R}+(1,-i)\psi_{L}],$ (49)

where we have defined pseudo spin real fermion fields as
$\chi^{T}_{1}=(\chi_{1\uparrow},\chi_{1\downarrow})$,
$\chi^{T}_{2}=(\chi_{2\uparrow},\chi_{2\downarrow})$. Following
Eq.(LABEL:sigmac),(49), we find that in terms of complex fermions,

$\displaystyle Q^{\sigma}_{1}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}\Psi^{\dagger}({\bf r})\sigma_{x}{\Psi({\bf r})},$ $\displaystyle
Q^{\sigma}_{2}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}\Psi^{\dagger}({\bf r})\sigma_{y}{\Psi({\bf r})}=Q_{R}-Q_{L},$
$\displaystyle Q^{\sigma}_{3}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}\Psi^{\dagger}({\bf r})\sigma_{z}{\Psi({\bf r})}.$

More explicitly,

$\displaystyle Q^{\sigma}_{1}$ $\displaystyle=$ $\displaystyle i\int d{\bf
r}[\psi^{\dagger}_{L}({\bf r}){\psi_{R}({\bf r})}-\psi^{\dagger}_{R}({\bf
r}){\psi_{L}({\bf r})}],$ $\displaystyle Q^{\sigma}_{2}$ $\displaystyle=$
$\displaystyle\int d{\bf r}[\psi^{\dagger}_{R}({\bf r}){\psi_{R}({\bf
r})}-\psi^{\dagger}_{L}({\bf r}){\psi_{L}({\bf r})}]=Q_{R}-Q_{L},$
$\displaystyle Q^{\sigma}_{3}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}[\psi^{\dagger}_{L}({\bf r}){\psi_{R}({\bf r})}+\psi^{\dagger}_{R}({\bf
r}){\psi_{L}({\bf r})}].$ (51)

where $Q_{L}=\int d{\bf r}\psi^{\dagger}_{L}\psi_{L}$ is the number of complex
fermions at L-nodal point and $Q_{R}=\int d{\bf r}\psi^{\dagger}_{R}\psi_{R}$
the fermion charge at $R$-nodal point.

For a nodal phase, the emergent $U(1)$ symmetry leads to the conservation of
$Q^{\sigma}_{2}$ or $Q_{R}-Q_{L}$ as the Hamiltonian in Eq.(33) remains
invariant under the rotations of $Q^{\sigma}_{2}$. This is a surprising
result: superfluids break the conventional global $U(1)$ symmetry and so
$Q^{\tau}_{2}=Q_{L}+Q_{R}$ is not conserved. Nevertheless, the difference
between $Q_{L}$ and $Q_{R}$ is conserved.

Microscopically, one can also verify that this conservation law becomes exact
when there are no back scattering terms from two left particles to right ones
or vice versa; this condition leads to the $U(1)$ symmetry of
$Q^{\sigma}_{2}$. Practically, this can be achieved by forbidding Umklapp
processes.

Figure 2:  An $U(1)$ symmetry emerges in the infrared limit (IR) in an EFT
that has a generic $Z_{2}$ symmetry up to an ultraviolet (UV) scale. Red dot
indicates an infrared stable fixed point that dictates possible emergent
symmetries in the limit of IR. The stability of the fixed point implies the
robustness of emergent $U(1)$ symmetries.

###  VII.3 $U(1)$ Gauge Symmetry

So far we have treated the emergent symmetry as a global one and discovered
various $U(1)$ charges carried by real fermions. One can further extend the
discussions and impose a $U(1)$ gauge symmetry of EFTs discussed before. A
gauge symmetry leads to $U(1)$ charge-current conserved dynamics and below we
briefly identify these currents.

We again focus on the limit where the spin group of the Lorentz transformation
of $S^{ij}$ is an $SU(2)$ group isomorphically generated by $K$-type
generators,

$\displaystyle
K_{1}=\tau_{y}\otimes\sigma_{x},K_{2}=\tau_{y}\otimes\sigma_{z},K_{3}=1\otimes\sigma_{y}.$
(52)

Concrete examples of this are the models in Eq.(20), (27),(29).

In the case of Eq.(20) that describes dynamics in gapless odd-parity
superconductors, the emergent $U(1)$ gauge symmetry is generated by the
following local transformation

$\displaystyle U_{1}=\exp[-i\phi({\bf
r},t)Q_{1}^{\tau}],Q_{1}^{\tau}=\frac{1}{2}\int d{\bf r}\chi^{T}({\bf
r})\tau_{x}\otimes\sigma_{y}\chi({\bf r}).$

A gauge invariant EFT suggests a standard minimum coupling (mc) between the
real fermion fields $\chi({\bf r})$ and emergent gauge fields $(A_{0},{\bf
A})$, i.e.

$\displaystyle H_{mc}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int d{\bf
r}[A_{0}\rho({\bf r})+{\bf A}\cdot{\bf J}({\bf r})];\rho({\bf
r})=\frac{1}{2}\chi^{T}({\bf r})\tau_{x}\otimes\sigma_{y}\chi({\bf r}),$
$\displaystyle{\bf J}_{x}({\bf r})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\chi^{T}({\bf r})\tau_{y}\otimes\sigma_{x}\chi({\bf
r}),{\bf J}_{y}({\bf r})=\frac{1}{2}\chi^{T}({\bf
r})I\otimes\sigma_{y}\chi({\bf r}),{\bf J}_{z}({\bf
r})=\frac{1}{2}\chi^{T}({\bf r})\tau_{y}\otimes\sigma_{z}\chi({\bf r}).$ (54)

In terms complex fermions, these charge and current of emergent $U(1)$ gauge
symmetry correspond to

$\displaystyle\rho({\bf r})=\frac{i}{2}[\Psi^{T}({\bf r})\sigma_{y}{\psi({\bf
r})}-\Psi^{\dagger}({\bf r})\sigma_{y}{{\Psi^{\dagger}}^{T}({\bf r})}];{\bf
J}_{x}({\bf r})=\Psi^{\dagger}({\bf r})\sigma_{x}{\Psi({\bf r})},{\bf
J}_{y}({\bf r})=\Psi^{\dagger}({\bf r})\sigma_{y}{\Psi({\bf r})},{\bf
J}_{z}({\bf r})=\Psi^{\dagger}({\bf r})\sigma_{z}{\Psi({\bf r})}$ (55)

where the $U(1)$ current density in this case is exactly a spin density field
(up to a $1/2$ factor) while charge density is related to singlet
condensation.

Real fermions are electrically charge neutral so do not respond to electric or
magnetic fields. However, the identifications in Eq.(55) imply possible
interactions, analogous to electric or magnetic ones, between real fermions in
gapless superfluids or superconductors and spin singlet condensates, as well
as spin density structures. Potential applications and other implications will
be further discussed in a separated paper that is under preparation.

##  VIII Emergent $Z_{2}$ symmetry in gapless superconductor or superfluids

As stated in the previous section, a $U(1)$ symmetry can emerge in gapless
superfluid or superconductors only if one of the following three conditions is
met:

i) exactly at an infrared stable non-interacting fixed point of EFTs in
dimensions higher than $(1+1)D$;

ii) near the infrared fixed point where the less relevant irrelevant gradients
fields are negligible and the remaining most relevant irrelevant interactions
are invariant under $U(1)$ transformation.

iii) if interactions are strictly zero-range contact ones and the emergent
symmetry is a result of this specific type interactions.

iv) if interactions in EFTs have to be subject to a $U(1)$ symmetry constraint
as a consequence of intrinsic symmetries of microscopic UV model.

If neither of these requirements is satisfied, strictly speaking there can be
no emergent $U(1)$ symmetries in quantum dynamics of EFTs. But are there
remaining emergent symmetries for more general interactions in gapless
superfluids? If there are, what are they?

Below we will discuss one generalization of the above EFTs to include not only
interactions that is completely local in $\chi({\bf r})$-fields without
gradient fields but also interactions that are non-local in space or time.

Once interactions are generalized beyond the local fields of $\chi^{4}$-form,
instead, there can be a lower emergent symmetry that the $U(1)$ symmetry can
be reduced to. In other words, in more generic interacting systems, $U(1)$
symmetry only emerges at infrared energy scales when $H_{LR}$ are entirely
renormalized to almost zero but at intermediate or higher energy scales, it
breaks down to a smaller group. In the example below, we will show explicitly
that $U(1)$ symmetry evolves into $Z_{2}$ in higher energy scales.

###  VIII.1 Generalized Interaction Model

To better understand the origin and consequence of emergent $U(1)$ symmetry
and limitation of the above analyses, we further explore an extended
interacting real fermion model by explicitly introducing two real scalar
fields $\phi_{1,2}=\phi_{1,2}^{\dagger}$, We will see explicitly that $U(1)$
symmetry in EFTs discussed in the previous section appears to be an infrared-
limit symmetry or an asymptotic symmetry of a more general quantum dynamics
with lower symmetries.

To further carry out discussions, we illustrate our main point by working with
an EFT of the following form although one can easily arrive at the same
conclusion by working with other rotated EFTs.

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H_{I};$ $\displaystyle
H_{0}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int d{\bf r}[\chi^{T}({\bf
r})(\mathbb{I}\otimes(\sigma_{x}{i\nabla_{x}}+\sigma_{z}{i\nabla_{z}})+\tau_{y}\otimes\sigma_{y}{i\nabla
y})\chi({\bf r})$ $\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{i=1}^{2}\pi^{2}_{i}({\bf
r})+\nabla\phi_{i}({\bf r})\cdot\nabla\phi_{i}({\bf
r})+M^{2}_{i}\phi_{i}^{2}({\bf r})],$ $\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\int d{\bf r}[\sum_{i=1}^{2}g_{Yi}\phi_{i}({\bf r})\chi^{T}({\bf
r})\tau_{i}\otimes\sigma_{y}\chi({\bf r})+g_{4}(\sum_{i=1,2}\phi_{i}^{2}({\bf
r}))^{2}],$

where $\tau_{1,2}=\tau_{x,z}$, $M_{1,2}$ are masses of real scalar fields
$\phi_{1,2}$ respectively. And we have further set the speeds of real scalar
fields, $v_{1,2}=1$ so to have a desired emergent Lorentz symmetry. Two real
scalar fields $\phi_{1,2}$ and four-component majorana fields in
Eq.(LABEL:SUSY3) are defined in a standard way,

$\displaystyle[\phi_{i}({\bf r}),\pi_{j}({\bf
r}^{\prime})]=i\delta_{ij}\delta({\bf r}-{\bf r^{\prime}}),$
$\displaystyle[\phi_{i}({\bf r}),\phi_{j}({\bf r^{\prime}}]=[\pi_{i}({\bf
r}),\pi_{j}({\bf r}^{\prime})]=0,i=1,2;$
$\displaystyle\chi^{T}=(\chi_{1\uparrow},\chi_{1\downarrow},\chi_{2\uparrow},\chi_{2\downarrow}).$
(57)

In the special limit when $M_{1}=M_{2}$ and $g_{Y1}=g_{Y2}$, the model has a
$U(1)$ symmetry and remains invariant if we make the following $U(1)$
transformation

$\displaystyle\chi$
$\displaystyle\rightarrow\chi^{\prime}=U(\phi)\chi=\exp(i\frac{\phi}{2}Q^{\tau}_{2})\chi,$
(58) $\displaystyle\Phi$
$\displaystyle\rightarrow\Phi^{\prime}=U_{B}(\phi)\Phi=\exp(-i\phi)\Phi,\Phi=\phi_{1}+i\phi_{2}.$

However, generically two masses $M_{1,2}$ are not equal so the EFT is not
$U(1)$ invariant. Instead, it is invariant under the following discrete
transformation or a $\pi$-rotation,

$\displaystyle U(\pi)=\exp(i\frac{\pi}{2}Q^{\tau}_{2}),U_{B}=-1.$ (59)

The transformation $U(\pi)$ is purely real. Under this transformation,

$\displaystyle U^{\dagger}(\pi)q^{\tau}_{1}({\bf r})U(\pi)$ $\displaystyle=$
$\displaystyle-q^{\tau}_{1}({\bf
r}),U^{\dagger}(\pi)Q^{\tau}_{1}U(\pi)=-Q^{\tau}_{1};$ $\displaystyle
U^{\dagger}(\pi)q^{\tau}_{3}({\bf r})U(\pi)$ $\displaystyle=$
$\displaystyle-q^{\tau}_{3}({\bf
r}),U^{\dagger}(\pi)Q^{\tau}_{3}U(\pi)=-Q^{\tau}_{3}.$ (60)

One can verify that for arbitrary mass ratio $M_{1}/M_{2}$, the Hamiltonian is
indeed invariant under this transformation of $\chi$ fields combined with a
reflection of $\phi$ fields, i.e.

$\displaystyle\chi_{1\alpha}$ $\displaystyle\rightarrow$
$\displaystyle\chi_{2\alpha},\chi_{2\alpha}\rightarrow-\chi_{1\alpha},\alpha=\uparrow,\downarrow;$
$\displaystyle\phi_{1}$ $\displaystyle\rightarrow$
$\displaystyle-\phi_{1},\pi_{1}\rightarrow-\pi_{1},\phi_{2}\rightarrow-\phi_{2},\pi_{2}\rightarrow-\pi_{2}.$
(61)

In the limit when the running scale $\Lambda\ll M_{1,2}$, the massive bosons
fields can be integrated out resulting in an EFT involving 4-fermion
operators. In the infrared limit $\Lambda/M_{1,2}\rightarrow 0$, the most
relevant interactions in EFT are exactly of the local $\chi^{4}$-fields and a
higher $U(1)$ symmetry emerges in this limit as a result. However, when
$\Lambda$ is comparable or larger than $M_{1,2}$, only $Z_{2}$ symmetry can be
present in these intermediate scales.

Figure 3:  Breaking of emergent symmetries along different axis defined by
generalized mass operators $\\{Q^{\tau}_{i},i=1,2,3\\}$ (top three in red) and
its dual of $\\{Q^{\sigma}_{i},i=1,2,3\\}$ (bottom three in blue). At each dot
on the circle or along each direction, an expectation value of one of these
six generators of group $Spin(4)=SU(2)\otimes SU(2)$ (that are isomorphic to
$SO(4)$ ones) becomes non-zero. The gapless state labelled as $0$ in the
centre has the highest emergent symmetries, with both a) Lorentz symmetry and
b) $U(1)$ (or $Z_{2}$ symmetry). If the spin rotation subgroup of EFTs of
state $0$ is defined by generators $\\{Q^{\sigma}_{i},i=1,2,3\\}$ or $K$-type
rotations, then only two of top three states $1,2,3$ are invariant under the
Lorentz group $SO(3,1)$ but both break the emergent $U(1)$ or $Z_{2}$
symmetry. The third one is $SO(3)$ rotationally invariant but fully preserves
the infrared $U(1)$ symmetry. On the other hand, all the bottom three states
break the $SO(3,1)$ Lorentz symmetry to an $SO(2)$ rotation one but again all
display an emergent $U(1)$ symmetry.

###  VIII.2 $Z_{2}$ symmetry breaking

Unlike the emergent $U(1)$ symmetry near the infrared free fixed point, this
discrete $Z_{2}$ reflection symmetry appears to be a more generic emergent
symmetry in gapless topological superfluids or superconductors with emergent
Lorentz symmetries as it does not require fine tuning of $M_{1,2}$. It can
emerge at higher energies away from the infrared limit of $\Lambda\ll
M_{1,2}$. The emergent symmetries are again absent whenever the states are
gapped and EFTs become massive.

In addition, Eq.(LABEL:SUSY3) can also exhibit a ${\cal Z}_{2}$-time reversal
symmetry. Further combined with this discrete time reversal symmetry, the
model then is $Z_{2}\otimes{\cal Z}_{2}$ symmetric.

Indeed, in the strong coupling limit of EFTs, the discrete
$Z_{2}\otimes{\mathcal{Z}}_{2}$ symmetry can be spontaneously broken resulting
in massive real fermions. This mechanism of mass generation is related to
Gross-Neveu physics that had been well understood for Dirac fermionsGross74 ;
Fei16 . For this to occur, one of the fields of $Q^{\tau}_{1}$ or
$Q^{\tau}_{3}$ has to condense. That is

$\displaystyle\langle Q^{\tau}_{1}\rangle\neq 0,\mbox{ and/or}\langle
Q^{\tau}_{3}\rangle\neq 0.$ (62)

And if $Q^{\tau}_{3}$ is involved in the ${Z}_{2}$-reflection symmetry
breaking, the time reversal symmetry is also broken as $Q^{\tau}_{3}$ is odd
under the time reversal transformation. However, if only condensation of
$Q^{\tau}_{1}$ is involved in $Z_{2}$-reflection symmetry breaking, the time
reversal symmetry is still intact as $Q^{\tau}_{1}$ is singlet under time
reversal transformation.

If $M_{1}=M_{2}=0$ and so $U(1)$-symmetry rather than $Z_{2}\otimes{\cal
Z}_{2}$ happens to be emergent at the strong coupling fixed point, a
supersymmetry along with conformal symmetry shall also appear, similar to a
strong coupling fixed point in a complex fermion nodal phaseLee07 ; Balents98
; Fei16 . This feature had been recently applied to understand a strong
coupling limit of topological quantum criticality with time reversal
symmetryZhou22 .

##  IX Emergent $U(1)$ symmetry without Lorentz symmetry

As mentioned in the previous section, there are six classes of $U(1)$
symmetries embedded in a $SU(2)\otimes SU(2)$ group generated by
$Q^{\tau}_{i},Q^{\sigma}_{i}$, $i=1,2,3$ that double covers an $SO(4)$ group.
For extended models with non-local interactions between $\chi$ fields, six
$\pi$-rotations generated by these operators then define six different $Z_{2}$
symmetries for different systems in even high energy scales.

Below we list the dynamic consequence of breaking a $U(1)$ or $Z_{2}$ symmetry
starting from gapless Lorentz symmetric superfluids which have such a $U(1)$
or $Z_{2}$ symmetry. The effects of breaking emergent symmetry only depend on
how these operators transform under the Lorentz group generated by
$(S^{0i},S^{ij})$, rotation group of $S^{ij}$, and a $Z_{2}$ group.

We now fix the spin rotation subgroup of the Lorentz group to be
$\sigma$-type. More explicitly, $s_{i}=\frac{1}{2}\epsilon_{ijk}S^{jk}$,
$i,j,k=x,y,z$ are related to $Q^{\sigma}_{i}$, $i=1,2,3$. Then, the dual of
$S^{ij}$, $S^{ij}_{D}$ are associated to $Q^{\tau}_{i}$, $i,j=1,2,3$.

$\displaystyle\epsilon_{ijk}S^{jk}=Q^{\sigma}_{i},\epsilon_{ijk}S^{jk}_{D}=Q^{\tau}_{i};i=x,y,z.$
(63)

a) Only two out of three $Q^{\tau}_{i}$, $i=1,2,3$ are Lorentz invariant. For
EFTs in Eq.(20), they are $Q^{\tau}_{1,3}$; when $\langle
Q^{\tau}_{i}\rangle\neq 0$, $i=1,3$, a mass gap opens up but with full Lorentz
symmetry of $SO(3,1)$. However, $Z_{2}$ symmetry is alway broken. (See Fig.3)

b) The third $Q^{\tau}_{i}$ is rotation invariant but not Lorentz invariant.
This is also the generator for $U(1)$ emergent symmetry group. For EFTs in
Eq.(20), this operator is $Q^{\tau}_{1}$. When $\langle
Q^{\tau}_{1}\rangle\neq 0$, a spherical Fermi surface emerges while the state
remains gapless. The Lorentz symmetry of $SO(3,1)$ is broken but the subgroup
$SO(3)$ rotation symmetry remains. In addition, $Z_{2}$ or $U(1)$ symmetry in
the infrared remains unbroken.

c) Three $Q^{\sigma}_{i}$, $i=1,2,3$, that are dual to $Q^{\tau}_{i},i=1,2,3$,
are also the generators of spin rotation subgroup of $SO(3,1)$. So they not
only fully break the Lorentz symmetry $SO(3,1)$, and also break the subgroup
rotation symmetry $SO(3)$. When $\langle Q^{\sigma}_{i}\rangle\neq 0$,
$i=1,2,3$, gapless real fermions appear at two nodal points along one of the
$x,y,z$ directions, i.e. a nodal phase emerges. Furthermore, $U(1)$ symmetry
remains as $Q^{\sigma}_{i}$, $i=x,y,z$ are all singlets of $\tau$-symmetry
group which defines the emergent $U(1)$ symmetry group in the current
discussion. (See Fig.3)

In this particular case, although the Lorentz symmetry (of $\sigma$ type) is
broken, there can be an emergent Lorentz symmetry of $\tau$-type reappearing
in the reconstructed EFT. That is, in contrast to Eq.(63)

$\displaystyle\epsilon_{ijk}S^{jk}=Q^{\tau}_{i},\epsilon_{ijk}S^{jk}_{D}=Q^{\sigma}_{i};i=x,y,z.$
(64)

Overall, non-zero expectation values of these operators can be either due to
applied external fields that explicitly break these symmetries or due to
strong interactions that break these symmetries spontaneously. Either way,
gapless superconducting states will become either fully gapped states with
Lorentz symmetry but breaking the emergent $Z_{2}$ or infrared $U(1)$ symmetry
or other gapless states that break the full Lorentz symmetry but preserve the
emergent infrared $U(1)$ symmetry. Note that all the gapless states under our
considerations, with or without Lorentz symmetry, have emergent $Z_{2}$ or
even $U(1)$ symmetry in the infrared limit.

Figure 4:  Schematic of dispersion relations in different gapless superfluids
or superconductors with emergent $U(1)$ symmetries. a) Gapless states with
highest symmetries, both $SO(3,1)$ Lorentz symmetry and an emergent $U(1)$
symmetry (corresponding to state 0 in the centre of Fig.3); b) Gapless states
with either $SO(3)$ or $SO(2)$ rotation symmetries but no full Lorentz
symmetry due to condensation of one of Lorentz non-invariant charges in
$\\{Q^{\tau}_{i},Q^{\sigma}_{i}\\}$, $i=1,2,3$. All these states have the same
emergent $U(1)$ symmetry as the parent state in a); c) Gapped states that
break emergent $U(1)$ symmetries in a) but with full $SO(3,1)$ Lorentz
symmetry. The dashed horizontal line indicates zero energy and the spectrum
below the line is related to the above via a standard charge conjugation
transformation in superfluids.

##  X Conclusion

In conclusion, we have investigated various emergent $U(1)$ or $Z_{2}$
symmetries that appear at different energy scales in gapless superconductors
or superfluids and studied relations between them. We show that emergent
$U(1)$ symmetries in these gapless states can be conveniently characterized by
$S^{ij}_{D}$, a group dual to $S^{ij}$, the spinor rotation subgroup of the
emergent Lorentz symmetry group. Generators of symmetry groups are always
singlets under the action of $S^{ij}$ of the rotation subgroup but are not
invariant under the Lorentz group. They can appear either in form of $U(1)$
symmetry in infrared limits or as a $Z_{2}$ symmetry in even intermediate
energy scales.

All the $U(1)$ symmetries are embedded in an $Spin(4)=SU(2)\otimes SU(2)$
group that double covers an $SO(4)$ group. $Spin(4)$ group has been identified
as a product of an $SU(2)$ Lorentz spin rotation subgroup generated by
$S^{ij}$, $i\neq j=x,y,z$, which is associated with $F$-(or $K$-)rotations and
its dual $SU(2)$ rotation group induced by $S^{ij}_{D}$, $i\neq j=x,y,z$ that
is associated with $K$\- (or $F$-)rotations. For real fermions, $K$ and
$F$-rotationss or $SU_{K,F}(2)$ transformations are generated by purely
imaginary hermitian operators that are algebraically defined in
Eqs.(III),(11). And all $U(1)$ emergent symmetry groups are shown to be spin
rotation singlets under the spin rotation of $S^{ij}$.

When the symmetry is broken spontaneously due to strong interactions, the
states become fully gapped while preserving the Lorentz symmetry. However,
these emergent $U(1)$ global symmetries always remain if strong interactions
break emergent Lorentz $SO(3,1)$ symmetries spontaneously and only result in
other gapless states with much lower symmetries such as $SO(3)$ or $SO(2)$
rotational symmetries. As an open question, it is interesting to see what
happens to emergent symmetries and whether emergent symmetry groups become
bigger when the degrees of freedom of real fermions, or the central charges
further increase to $4N$ with $N>1$.

There had also been quite a few intensive efforts to generalize the concept of
gapped topological states or phases put forwarded before to gapless
limitsKestner11 ; Fidkowski11 ; Sau11 ; Cheng11 ; Kraus13 ; Ortiz14 ; Ruhman15
; Iemini15 ; Keselman15 ; Kainaris15 ; Montorsi17 ; Wang17 ; Kane17 ;
Scaffidi17 ; Guther17 ; Verresen18 ; Jiang18 ; Verresen19 ; Thorngren21 . And
the vast majority of these efforts have been mainly on one-dimensional states.
One interesting question that had been asked recently in Ref.Thorngren21  is
whether there are intrinsic topological states that can only be defined in the
gapless limit and there are no gapped counter-parts.

The main efforts in this article are to illustrate unique emergent symmetries
in high dimensional gapless states. These may be important observations to
future studies of gapless topological states and orders. Broadly speaking, it
remains to be fully understood whether these or other unique emergent
symmetries in gapless states that always appear to be broken in the gapped
limit are essential for general understanding of topological physicsChen10 ;
Chen12 ; Chen13 ; Wen17 . The other interesting issue is possible local gauge
symmetries associated with the emergent global symmetries discussed so far
Witten18 . If we enforce such a local gauge symmetry, real fermions considered
above can be further interacting with emergent dynamic $U(1)$ gauge fields or
$Z_{2}$ gauge fields. These gauge fields can be in either weakly interacting
phases or strong coupling confining phases. In general, between these two
phases there can even be highly symmetric quantum critical points with
emergent gauge fields, similar to effective field theories discussed recently
in Ref. Bi19 ; Ji20 . Perhaps an important question is under which conditions,
such gauge fields and their interactions become practically relevant in
studies of practical gapless superconductors or superfluids and what are
potential implications.

The author wants to thank Z.-C. Gu, G. W. Semenoff, X. G. Wen and F. Yang for
discussions on symmetries in gapless states. This project is in part supported
by NSERC, Canada under a Discovery grant under RGPIN-2020-07070.

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