# Topological Quantum Critical Points in Strong Coupling limits:
Global Symmetries and Strongly Interacting Majorana Fermions

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

(August 27, 2024)

###### Abstract

In this article, we discuss strong coupling limits of topological quantum
critical points (TQCPs) where quantum phase transitions between two
topological distinct superconducting states take place. We illustrate that
while superconducting phases on both sides of TQCPs spontaneously break same
symmetries, universality classes of critical states can be identified only
when global symmetries in topological states are further specified. In
dimensions $d=2,3$, we find that continuous $(d+1)$th order transitions at
weakly interacting TQCPs that were pointed out previously in the presence of
emergent Lorentz symmetry can be terminated by strongly interacting fixed
points of majorana fields. For $2d$ time reversal symmetry breaking TQCPs,
termination points are supersymmetric with ${\mathcal{N}}=4N_{f}={1}$ (where
$N_{f}$ is the number of four-component Dirac fermions and ${\mathcal{N}}$ is
the number of two-component real fermions) beyond which transitions are
discontinuous first order ones. For $2d$ time reversal symmetric TQCPs without
other global symmetries, termination points of $(d+1)$th order continuous
transition lines are generically conformal invariant without supersymmetry.
Beyond these strong coupling fixed points, there are first-order discontinuous
transitions as far as the protecting symmetry is not spontaneously broken but
no direct transitions if the protecting symmetry is spontaneously broken in
the presence of strong interactions. In this case, termination points can
become supersymmetric with ${\mathcal{N}}=4N_{f}=2$ only when interacting
majorana fermions further display an emergent global $U(1)$ symmetry. In $3d$,
strong coupling termination points can be further effectively represented by
new emergent gapless real bosons weakly coupled with free gapless majorana
fermions. However, in $1d$, time reversal symmetric $(d+1)$th continuous
transition lines of TQCPs are terminated by simple free majorana fermion fixed
points.

##  I Introduction

Quantum phase transitions between topological states with distinctly different
global topologiesSchnyder08 ; Kitaev09  have been fascinating. There are at
least two subclasses of such phenomena known to us so far. One are transitions
that are entirely driven and defined by a change of topologies or topological
invariants and otherwise do not exist in their non-topological counter-parts.
A familiar example of these unique transitions is the one due to increasing
interactions in a topological superconductor. Because there can be a change of
global topology when interactions allowed by symmetries increase, even though
states are ordered in the same way locally, there shall be a resultant bulk
transition beyond the standard Landau paradigm of order-disorder transitions.
These transitions usually turn out to be of relatively higher order. This is
in stark contrast to what happens in a non-topological s-wave superconductor
when one varies the same parameters. There, if one increases the interaction
strength from weak to strong, there are no transitions in conventional s-wave
superconductors, because an $s$-wave superconductor can only be topologically
trivial. So the very existence of such transitions in topological
superconductors entirely and crucially relies on the notion of global topology
of underlying superconductors and topological distinction of different states
Schnyder08 ; Kitaev09 ; Qi10 ; Qi08 ; Gurarie11 ; Volovik09 ; Wang12 ;
Volovik88 ; Volovik03 ; Read00 ; Qi11 ; Bernevig13 ; Hasan10 ; Fu08 . In other
words, global topologies and their characterization add a new dimension to the
parameter space along which phase transitions can occur. This new dimension is
beyond the standard Landau paradigm of order-disorder transitions.

The second subclass are transitions which can also occur in more conventional
non-topological states but for transitions in topological states, changes of
topologies are further involved. In all the cases we have examined, changes of
topologies result in distinct universality classes that are different from
their counter parts in non-topological systems. For instance, when an external
Zeeman field is applied, an s-wave superconductor can have a transition to an
FFLO state that further breaks translational symmetry in additional to U(1)
symmetry. In topological superconductors or superfluids, an external Zeeman
field can drive a time-reversal-invariant topological state at zero field into
a nodal phase with topological nodal points in its spectrum. The resultant
state is a topological state with perfect translational symmetry.

These TQCPs do not appear in the Landau paradigm of order-disorder phase
transitions because the states on both sides of TQCPs and TQCPs themselves
either all spontaneously break same symmetries or all have the same
symmetries. Therefore, TQCPs do not involve condensation of new bosonic
quantum fields or particles. It is exactly the global topology and topological
distinctions between different states that result in such a new class of
quantum critical points. And topological phase transitions associated with
these quantum critical points only exist at zero temperature.  For this
reason, we have referred them as topological quantum critical points, TQCPs.

As changes of topological invariants are discrete rather than continuous, one
might wonder how it is possible to describe TQCPs in terms of continuum
quantum fields. Such a possibility of a continuum quantum field theory
representation of topological quantum criticality can be most conveniently
understood via employing a standard adiabatic theorem.

Consider a fully gapped fermonic topological state protected by global
symmetries. One anticipates that it can maintain its topological distinction
under small Hamiltonian deformations within its symmetry class if the gap
remains open. A change in global topologies typically implies closing the gap.
If the gap indeed closes so to change global topologies or transitions are
continuous, then there shall be coalescing of gapped fermions into a ground
state near TQCPs. More importantly, in superconductors and fermonic
superfluids, elementary emergent particles that are relevant to generic TQCPs
are usually fermonic, not bosonic fields. In fact, they shall be real Majorana
fermions due to the emergent charge conjugation symmetry at U(1) symmetry
breaking TQCPs. So generically, if these topological quantum phase transitions
occur, we anticipate coalescing of real fermions into the ground state without
condensation of new bosonic quantum fields or spontaneously breaking
additional symmetries. This aspect is fundamentally different from the Landau
paradigm of order-disorder transitions. On the other hand, the above
observation does explicitly suggest effective quantum fields for coalesce
dynamics of gapped particles and topological quantum criticality in
superconductors.

One can further extend above arguments to situations where topological states
are gapless nodal phases. Gapless topological states are characterized by
topological invariants in an embedded subspace instead of a full physical
spaceSato17 ; Wen02 ; Kobayashi14 ; Zhao16 ; Meng12 ; Cho12 ; Sato06 ;
Schnyder11 ; Beri10 . Compared with fully gapped topological states, this
aspect of gapless phases makes a big difference in constructing quantum field
theory representations at TQCPs. At TQCPs, topological charges embedded in
different subspaces are created (or annihilated) and this is achieved again
via fermion coalescing. For instance, charges associated with nodal points can
be created in pairs at TQCPs when a Zeeman field is applied to a fully gapped
topological state. This happens when the gap closes at a point or a few
discrete points and real fermions coalesce along a particular direction of
Zeeman or spin-exchange fields. And different topologies of quantum nodal
phases naturally require different quantum field theory representations and
hence result in different universality classes. For instance, a nodal point
topology demands very different quantum Lifshitz majorana fields than a nodal
line topology would demand.

Perhaps the most remarkable consequence of gap closing in topological states
is proliferation of surface states into an interior of topological matter.
This is essential at TQCPs so that a state can topologically reconstruct in
the bulk and boundary simultaneously across a TQCP. Physically, these bulk
transitions are always accompanied or even heralded by surface quantum
criticality, a very distinct aspect of topological quantum criticality. A
continuum quantum field representation shall properly reflect such a bulk-
boundary correspondence at generic TQCPs.

These two particular aspects, one reflecting bulk topologies and the other
more on their consequences at boundaries, are both absent in other gap closing
phenomena in conventional non-topological transitions or in the standard
Landau paradigm. Actually these two aspects are what makes TQCPs outstanding.
Therefore, any appropriate quantum field theory representations for
topological TQCPs have to be in a class of field theories with boundary states
reflecting a corresponding change of topologies across TQCPs. This is one of
the guiding principles in explicit constructions, apart from more standard
symmetry considerations. The unique role of changes of topologies in these
zero temperature transitions therefore has to be encoded in these effective
fields.

In this article, we will focus on the roles of global symmetries on dynamics
of effective fields and hence TQCPs. In order-disorder phase transitions in
the standard Landau paradigm, symmetries of Hamiltonians and spontaneous
breaking of them play a paramount role in constructing effective quantum field
theories. At TQCPs that are not driven by further spontaneous breaking of
symmetries, one might wonder what is the role of symmetries then.

Although TQCPs considered below are entirely driven by changes of topologies,
not by condensation of bosonic particles, global symmetries do still play a
paramount role. That is perhaps not surprising from the point of view of
gapped topological states that are symmetry protected anywayChen10 ; Chen12 ;
Chen13 ; Kane05 ; Bernevig06b ; Fu07 ; Moore07 . Since topologies are
classified in terms of global symmetries and topological states are symmetry
protected, TQCPs that terminate those phases shall rely on protecting
symmetries as well. Indeed, dynamics of interacting fermion fields emerging at
TQCPs crucially depend on at least two classes of fundamental global
symmetries.

The first important symmetry is the continuous $U(1)$ symmetry associated with
charge conservation. Exactly at standard quantum critical points in the Landau
paradigm, continuous symmetries such as $U(1)$ symmetry is unbroken; they are
only broken in one of phases next to quantum critical points. TQCPs are unique
in a sense that such continuous global symmetries can be broken in the entire
vicinity of a TQCP and the entire quantum critical regime supports the same
local order.

If TQCPs are $U(1)$ symmetric, complex fermion fields that can exhibit global
$U(1)$ symmetries shall emerge and form a faithful representation. If $U(1)$
symmetry itself is spontaneously broken at TQCPs and in all quantum phases in
adjacent to them, one needs to adopt majorana fermion representations that are
suitable. Most of discussions in this article will be on $U(1)$ symmetry
broken superconductors or superfluids, although many discussions can be
extended to $U(1)$ symmetric TQCPs. Detailed dynamics of fermion fields can
further depend on other continuous symmetries. For instance, additional
continuous symmetry breaking such rotational ones at TQCPs and in their
proximity can result in distinct classes of scale symmetries that we can
associate with generalized fermionic Lifshitz fieldsYang21 ; Lifshitz41 .

The second class are discrete global symmetries such as time reversal symmetry
and/or emergent charge conjugation symmetry etc. These symmetries have played
a paramount role in defining topological winding numbers and classifying
topological states so naturally they also play an important role at TQCPs. In
Section IV, we will focus on such aspect at TQCPs. In general, a larger global
symmetry requires a bigger fermion representation. Physically, it naturally
leads to more emergent fermionic fields at TQCPs. The role of these global
symmetries is one of the main focuses of this article. We illustrate
dominating effects of global symmetries in both weakly and strongly
interacting limits, with an emphasis on the interplay between symmetries and
strong interactions.

In Sec II, we will introduce general phenomenologies of TQCPs. This part of
discussions are pedagogical. We summarize previous discussions on topological
state classifications via Green’s functions of interacting fermions and then
further extend and apply to near TQCPs. In Sec III, we identify a few main
issues in strong coupling limits that are focused on in this article. In Sec.
IV, we discuss the main role of global symmetries on dynamics at TQCPs and
illustrate an intimate relation between global symmetries and numbers of
emergent low energy majorana degrees of freedom. In Sec V, VI, we will
elaborate on dominating effects of global symmetries on TQCPS in various
strong coupling limits. In Section V, we focus on TQCPs in topological
superconductors with no global symmetries and illustrate that there shall
always be a transition between two topologically distinct states even in
strong coupling limits. However, continuous $(d+1)$ order transition lines
pointed out in a previous study in weakly coupling limitsYang19  shall be
terminated by a strong coupling supersymmetric conformal fixed point beyond
which transitions become first order ones. In Sec VI, we study TQCPs with a
single global symmetry (time reversal symmetry) but no other global
symmetries. We emphasize peculiar aspects of an emergent $U(1)$ symmetry when
interactions are local and various mass generation mechanisms that can
spontaneously break the protecting time reversal symmetry. We found that when
time reversal global symmetry is not spontaneously broken by strong
interactions, $(d+1)$th order transition lines are again terminated by a
conformal fixed point beyond which first order transitions occur. However,
when the protecting time reversal symmetry is broken spontaneously by strong
interactions, the continuous $(d+1)$th order transition line is simply
terminated by either supersymmetric or non-supersymmetric conformal field
fixed points. Beyond those points, two topological states can be deformed into
each other in the absence of protecting symmetries. In Sec. VII, we further
discuss some open questions on TQCPs in topological superconductors and
possible future directions. Sec.IX concludes our article.

##  II General Phenomenology of TQCPs

###  II.1 Quantum Field Representations for TQCPs

A TQCP, if exists, connects two topologically distinct states. To faithfully
reflect such an aspect of TQCPs in a quantum field representation, in
additional to standard symmetry considerations in constructing effective
theories, we can further require Green’s functions of interacting quantum
fields being a representation of a homotopy group that defines topologies or
topological invariants (see Fig.1). Similar ideas had been previously
introduced to understand topological quantum field representations for
topological insulators and Witten effects and axion dynamics near
boundariesQi08 ; Qi11 . One can also naturally apply it, as a general
principle, to TQCPs for the main purpose to understand bulk dynamics near a
TQCP and later on the bulk-boundary correspondence.

We first consider transitions between two fully gapped superconducting states,
one topological but the other is trivial. Let us define a tuning parameter
that drives such a transition as $m=\mu$ which depends on microscopic
parameters; TQCPs correspond to $\mu=\mu_{c}$. To apply to $(d+1)$ dimension
space-time of bulk states near TQCPs, we can further extend the space to
$(d+2)$ dimension by inclusion of an additional dimension of the general
tuning parameter $\mu$ along which a TQCP occurs. Green’s functions defined in
$(d+2)$ dimension, $G(i\Omega,{\bf k};\mu)$ are $M\times M$ complex matrices
where $M$ are numbers of bands under considerations. These functions shall
have well defined analytical structures in an imaginary frequency domain when
$\mu\neq\mu_{c}$ because all fermion fields are well gapped except at
$\mu=\mu_{c}$ or TQCPs. If they are also invertible when $\mu\neq\mu_{c}$,
they naturally form a representation of general linearized group
${\mathcal{G}L}(M,C)$ in $(d+2)$-dimension excluding a singular point of
$\mu=\mu_{c}$ and $\Omega={\bf k}=0$.

Therefore, at a $(d+1)$ dimension surface of such a $(d+2)$ dimension
hyperspace that excludes enclosed TQCPs (see Fig.1), $G(i\Omega,{\bf k};\mu)$
is smooth and can be thought as a mapping from a $(d+1)$ dimension surface to
$M\times M$ invertible complex matrices. This allows a classification of
quantum fields in terms of Green’s functions using $(d+1)$-dimensional sphere
embedded in $(d+2)$ dimension hyperspace. Quantum fields can be topologically
non-trivial if the Green’s functions $G(i\Omega,{\bf k};\mu)$ at a given
$\mu\neq\mu_{c}$ belong to nontrivial elements of $(d+1)$th homotopy group,
i.e. $\pi_{d+1}({\mathcal{G}L}(M,C))$. This classification of the Green’s
functions had been emphasized in quite a few previous studies of gapped
topological statesVolovik03 ; Volovik09 ; Qi11 ; Gurarie11 ; Wang12 . It can
also be applied to near TQCPs. And if $G(i\Omega,{\bf k};\mu>\mu_{c})$ are
associated with a different group element of $\pi_{d+1}({\mathcal{G}L}(M,C))$
compared with $G(i\Omega,{\bf k};\mu<\mu_{c})$, continuum quantum fields are
in different topological classes across $\mu_{c}$ and therefore can be applied
to study transitions involved changes of discrete topologies. This is a very
unique feature of topological quantum criticality. That is, effective quantum
fields for dynamics at TQCPs are representations of a topological group so to
reflect changes of topologies of underlying many-body states.

Simplest applications can be easily found in $2d$ time-reversal-symmetry
breaking superconductorsRead00 . In this case, one finds that quantum fields
for a TQCP form a nontrivial representation of
${\mathcal{\pi}}_{3}({\mathcal{G}L}(2,\mathbb{C}))=\mathbb{Z}$. At a surface
of a fixed chemical potential $\mu(\neq\mu_{c})$, the topological invariant
can be defined as ($k_{0}=\Omega$)

$\displaystyle N_{W}$ $\displaystyle=$ $\displaystyle\frac{1}{24\pi^{2}}\int
d{k_{x}}dk_{y}dk_{0}\epsilon_{\alpha\beta\gamma}$ $\displaystyle Tr$
$\displaystyle G\frac{\partial}{\partial
k_{\alpha}}G^{-1}G\frac{\partial}{\partial
k_{\beta}}G^{-1}G\frac{\partial}{\partial
k_{\gamma}}G^{-1},\alpha,\beta=0,x,y.$

This characterization can be extended to other even-spatial dimensions.

$N_{W}$ above can also be equivalently expressed in terms of the zero
frequency limit of the Green’s function, $G(0,{\bf k};\mu)$ without frequency
integration. Just as in the limit of mean-field or one-particle theory, one
can then completely isolate the frequency integration from the momentum
summation. The topological invariants above can then be more conveniently
expressed in terms of $G^{-1}_{0}({\bf k},\mu)=G^{-1}(0,{\bf k};\mu)$. It is
important to notice that although for a given $\Omega$, $G(i\Omega,{\bf
k};\mu\neq\mu_{c})$ are complex $M\times M$ matrices and are non-Hermitian,
$G_{0}({\bf k},\mu)$ are Hermitian following the following general relation

$\displaystyle G^{\dagger}(i\Omega,{\bf k};\mu)=G(-i\Omega,{\bf k};\mu).$ (2)

At a fixed chemical potential, $G_{0}({\bf k};\mu\neq\mu_{c})$ thus defines a
smooth mapping function from a $d$-momentum space to an effective Hamiltonian
manifold, $H_{M}$ which topologically differs from
${\mathcal{G}L}(M,\mathbb{C})$. This is most obvious if one takes the non-
interacting limit when $G^{-1}_{0}({\bf k})=H_{0}({\bf k})$. In the
interacting limit, $G^{-1}_{0}({\bf k};\mu\neq\mu_{c})$ further contains
effects of zero-frequency self-energies which are usually unknown but the
effective manifold $H_{M}$ shall be still be a hermitian one, the same as
$H_{0}({\bf k})$. So the topological charges above can also be conveniently
applied to represent a nontrivial mapping of the homotopy group
${\mathcal{\pi}}_{d=2}(H_{M})$. For $2d$ time reversal symmetry breaking
superconductors, $H_{M}$ is simply a two sphere $S^{2}$; and Eq.(LABEL:NW)
also represents topological charges of smooth mapping function
$G_{0}^{-1}({\bf k};\mu\neq\mu_{c})$ and reflects
${\mathcal{\pi}}_{2}(S^{2})=\mathbb{Z}$.

When applying to topological states with additional global symmetries such as
time reversal symmetries in odd spatial dimensions, one has to further tweak
the Green’s function winding number definition to properly reflect global
symmetries. In this case, it is more convenient to directly work with the zero
frequency Green’s function, $G^{-1}(0,{\bf k};\mu)^{-1}=G^{-1}_{0}({\bf
k};\mu)$ that takes into account interaction renormalization effects. Now to
apply to $d$-dimensional spatial space of bulk states instead of
$(d+1)$-dimensional space-time , we can again extend the space to $d+1$
dimension by inclusion of an additional dimension of the general tuning
parameter $\mu$ along which a TQCP occurs. $G_{0}({\bf k};\mu)$ defined in
$(d+1)$ dimension hyperspace, has well defined analytical structures when
$\mu\neq\mu_{c}$ because all fermion fields are well gapped except at
$\mu=\mu_{c}$ or TQCPs; they are also invertible when $\mu\neq\mu_{c}$ which
is the situation we intend to focus on.

At a $d$-dimension surface of such a $(d+1)$ dimension hyperspace that again
excludes enclosed TQCPs (see Fig.1), $G^{-1}_{0}({\bf k};\mu\neq\mu_{c})$ is
everywhere smooth and can be thought as a mapping from $d$-dimension surfaces
to a Hamiltonian manifold. This allows a classification of quantum fields
using a $d$-dimensional sphere embedded in $(d+1)$-dimension hyperspace. In
the presence of time reversal symmetry in superconductors and therefore a
chiral symmetry, $G^{-1}_{0}({\bf k};\mu)$ can always be cast into an off-
diagonal block matrix, with block matrices being $M\times M$ invertible
complex matrices, $Q({\bf k})$ and $Q^{\dagger}({\bf k})$. That is

$\displaystyle G^{-1}_{0}({\bf k};\mu)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}0&Q({\bf k})\\\ Q^{\dagger}({\bf k})&0\\\
\end{bmatrix}.$ (3)

This structure naturally appears in non-interacting or mean field theories and
can be shown to be true in the presence of interactions as far as the chiral
symmetry is presentVolovik03 ; Volovik09 ; Qi11 ; Gurarie11 ; Wang12 .

$G^{-1}_{0}({\bf k};\mu\neq\mu_{c})$ in Eq.(3) can be topologically non-
trivial if their off diagonal complex matrices $Q({\bf k};\mu)$ belong to
nontrivial elements of $d$-th homotopy group of mapping functions,
$\pi_{d}({\mathcal{G}L}(M,\mathbb{C}))$, instead of
$\pi_{d+1}({\mathcal{G}L}(M,\mathbb{C}))$ of space-time Green functions
discussed before. Note that $Q({\bf k};\mu)$ are $M\times M$ complex matrices
while the space-time Green’s function in this case, $G(i\Omega,{\bf k};\mu)$,
are $2M\times 2M$ invertible ones.

Simplest applications can be easily found in $3d$ time reversal symmetric
$p$-wave superfluids or superconductors isomorphic to ${}^{3}He$
B-phaseLeggett75 . In this case, $M=2$ and quantum fields for a TQCP shall
form a non-trivial representation of
${\mathcal{\pi}}_{d=3}({\mathcal{G}L}(2,\mathbb{C}))=\mathbb{Z}$;
${\mathcal{G}L}(2,\mathbb{C})$ now is the group for off-diagonal block complex
matrices $Q({\bf k})$ appearing in Green’s functions in the limit of zero
frequency. The topological invariant can be defined in terms of off-diagonal
block matrices $Q({\bf k};\mu)$ or formally in terms of zero frequency Green’s
function $G_{0}({\bf k};\mu\neq\mu_{c})$ itself which is more suitable for
discussions of interacting fields. In the later case, the winding number in
the presence of time reversal symmetry (and hence chiral symmetry) defined in
terms of $G_{0}({\bf k};\mu\neq\mu_{c})$ needs to be supplemented by a chiral
transformation $\Sigma$,

$\displaystyle N_{W}$ $\displaystyle=$ $\displaystyle\frac{1}{24\pi^{2}}\int
d{k_{x}}dk_{y}dk_{z}\epsilon_{\alpha\beta\gamma}$ $\displaystyle Tr$
$\displaystyle\Sigma G_{0}\frac{\partial}{\partial
k_{\alpha}}G_{0}^{-1}G_{0}\frac{\partial}{\partial
k_{\beta}}G^{-1}_{0}G_{0}\frac{\partial}{\partial
k_{\gamma}}G^{-1}_{0},\alpha,\beta=x,y,z.$

${\Sigma}={\mathcal{T}}{\mathcal{C}}$ is defined as a unitary chiral
transformation anti-commuting with $G^{-1}_{0}({\bf k};\mu)$ that now is
hermitian. The winding number defined in Eq.(LABEL:NW) would vanish if
$\Sigma$ were replaced with a unity matrix and hence it is crucial to have a
chiral transformation $\Sigma$ in its definition.

$\displaystyle{\Sigma}^{-1}G^{-1}_{0}({\bf k}){\Sigma}$ $\displaystyle=$
$\displaystyle-G^{-1}_{0}({\bf k}).$
$\displaystyle{\mathcal{T}}^{-1}G^{-1}_{0}({\bf k}){\mathcal{T}}$
$\displaystyle=$ $\displaystyle G^{-1}_{0}(-{\bf
k}),{\mathcal{C}}^{-1}G^{-1}_{0}({\bf k}){\mathcal{C}}=-G^{-1}_{0}(-{\bf k}).$

${\mathcal{T}}$, ${\mathcal{C}}$ are anti-unitary time reversal and charge
conjugation transformation respectively. In the case of $3d$ $p$-wave spin
triplet superconductors, the effective Hamiltonian manifold $H_{M}$ suggested
by hermitian matrix $G^{-1}_{0}({\bf k};\mu)$ can be defined on a three sphere
$S^{3}$. So $N_{W}$ in Eq.(LABEL:NW1) also equivalently represents
$\pi_{d=3}(S^{3})=\mathbb{Z}$.

Figure 1:  Standard cartoon pictures of changes of global topologies (of
integer groups); vertical direction represents a dimension of general tuning
parameter $m$ for TQCPs. TQCPs (as red spheres) are between fully gapped
topological superconductors (top) and trivial states (bottom) in a) and b).
Majorana quantum fields form a representation of either $\pi_{d+1}$homotopy
group of space-time Green’s functions in a $(d+1)$ dimension surface when $d$
is even (a)), or $\pi_{d}$ homotopy group of zero-frequency Green’s functions
in a $d$-dimension surface embedded in $(d+1)$ dimension when $d$ is odd (b)).
c) TQCPs between gapless topological superconductors and gapped states. For
time reversal symmetry breaking TQCPs, Quantum fields form a non-trivial class
of $\pi_{d-p}$, the $(d-p)$th homotopy group of space-time Green’s functions.
$d-p(>0)$ is the dimension of a subspace where topological invariants are
defined. $p=0,1$ are for a nodal point and nodal line phase respectively. In
three dimension, only $p=0$ nodal points are stable. d) is for time reversal
invariant case. Quantum fields belong to non-trivial classes of $\pi_{d-p-1}$,
the $(d-p-1)$th homotopy group of zero-frequency Green’s function or
invertible complex matrix $Q({\bf k})$. Nodal points are stable only in $d=2$
but not in three spatial dimensions.

For gapless topological phases, topological invariants are defined in an
embedded subspace or sphere with dimension lower than the physical space-time
or space dimensions. A TQPC connects two states with different topologies in
an embedded subsphere and again it crucially depends on the global time
reversal symmetry.

For time reversal symmetry breaking states, the embedded space shall be a
$[(d-p-1)+1]$ dimension space-time sphere transverse to a Fermi surface of
dimension $p$, $p=0,1,...,d-1$.. Quantum fields for a TQCP shall be in a non-
trivial representation of $[(d-p-1)+1]$th homotopy group of space-time Green’s
function for a fermi surface of dimension $p$. For instance, for a time
reversal symmetry breaking TQCP that results in emergence of a Fermi surface
or nodal structure of $p$ dimensionYang21 , topological charges can be defined
in a $(d-p)$-sphere or $S^{d-p}$. The space-time Green’s functions shall be in
a non-trivial class of homotopy group
${\mathcal{\pi}}_{d-p}({\mathcal{G}L}(M,\mathbb{C}))$ instead of
${\mathcal{\pi}}_{d+1}$ group for corresponding gapped topological states. And
for nodal point phases, $p=0$ and one is concerned with ${\mathcal{\pi}}_{d}$
group of the space-time Green’s function. This homotopy group of
superconducting Green’s functions (i.e. $M\leq 2$) is indeed a non-trivial
integer one, $\mathbb{Z}$ in $d=3$ indicating a stable time-reversal-symmetry
breaking nodal point superconducting phase in three spatial dimensions.

In the presence of time reversal symmetry, again a subspace of
$(d-p-1)$-spatial sphere transverse to a Fermi surface of dimension $p$ shall
be introduced. Zero frequency Green’s function $G_{0}({\bf k};\mu\neq\mu_{c})$
can then be studied in a sphere of $(d-p-1)$-dimension and mapping functions
shall be in a non-trivial class of $(d-p-1)$th homotopy group of the
corresponding invertible smooth mapping matrices $Q({\bf k})$ in Eq.(3). The
Green’s functions have to be in a non-trivial class of homotopy group of
${\mathcal{\pi}}_{d-p-1}({\mathcal{G}L}(M,\mathbb{C}))$. For $d=2$, this group
is a non-trivial integer group $\mathbb{Z}$ for nodal points ($p=0$) but in
$d=3$, only nodal lines ($p=1$) are topologically stable Beri10 ; Volovik03 ;
Wen02 ; Kobayashi14 .

Quantum fields we are dealing with are continuous and do not have same
ultraviolet structures as solid state Bloch bands, and don’t have well-defined
Chern numbers. However, by enforcing that they are in a topologically non-
trivial sector of corresponding homotopy groups of their Green’s functions,
one anticipates that continuum quantum fields are faithful representations of
TQCPs for changes of discrete topology. A TQCP defined in this way then
naturally connects two topologically distinct states in $d$-dimensions as
illustrated in Fig.1. This was the key element behind previous constructions
of effective field theories in quite a few different limitsYang19 ; Yang21 .
An important vindication for the validity of effective-field approaches to
changes of global topologies in quantum states is predictions of topological
boundary reconstructions across a TQCP which one usually can verify quite
straightforwardly by using its quantum field theories. This is what we now
turn to in the next subsection.

###  II.2 Bulk-Boundary Correspondence near TQCPs

If two topologically different quantum fields are fused together in space,
then spatial boundaries, or domain walls, separating them shall trap gapless
(majorana if $U(1)$ symmetry is broken) boundary modes as a result of change
of topologies in the physical space. Domain walls in a parameter space
therefore directly lead to spatial domain walls or boundaries and such a
phenomenon is robust and zero modes are usually stable as far as the
underlying protecting symmetries are respected and interactions are weak. This
feature, a bulk-boundary correspondence, is also an essential one in
discussions of gapped condensed matter topological states, either with $U(1)$
symmetry or with $U(1)$ symmetry broken.

As pointed out before, TQCPs are manifestation of changes of topologies. As we
had been employing quantum field theories to facilitate many discussions of
non-interacting gapped topological states, it is indeed not surprising and
also natural to employ continunmm fields to explore interacting physics at
gapless TQCPs if long wave length physics is the only aspect concerning us.
But there are at least three important caveats.

The first issue is related to definitions of topological invariants of
topological states that, often, are only meaningful when protecting symmetries
such as time-reversal symmetry are not broken. As topologies might not be well
defined when symmetries are broken by external fields, it has been confusing
how gapped states evolve when symmetries are broken and whether resultant
states shall still be (or not) related to a topological state. Continuum
quantum fields offer a continuous view about what happens with or without
protecting symmetries in the presence of interactions and provide very useful
insight on resultant states.

To elaborate on this, we take an example that had been examined before. If a
spin exchange or Zeeman field is introduced in a 3D time-reversal symmetric
topological superconductor, the concept of topological invariants defined
explicitly in terms of Kramer doublets is no longer applicable. However, the
effective field approach shows that i) there are no bulk transitions occur in
small fields; ii) however, topological surfaces are critical at zero field and
are gapped in various ways when time reversal symmetries are broken
explicitly; iii) bulk transitions occur at a finite Zeeman field exactly at
the moment when surface states undergo complete reconstruction. iv) the upper
critical dimension for such TQCPs is $1\frac{1}{2}$, below 2d and interactions
are irrelevant in 3d, implying a robust bulk-edge correspondence across
TQCPsYang21 . Aspect iii) is a principle feature of generic TQCPs. All bulk
phase transitions at TQCPs occur exactly when topological surface states
emerge or disappear, a bulk-boundary correspondence that appears at all
generic TQCPs we have examined so far. We have used it as another vindication
of continumm field approaches to TQCPs.

The second caveat is related to protecting global symmetries at TQCPs and
their roles in quantum dynamics of emergent quantum fields. We will discuss
them in details in the nest section. The conclusion is that global symmetries
set (minimal) numbers of fundamental fields that shall be present at gapless
TQCPs. Numbers of emergent gapless fields at TQCPs are independent of $N$, the
number of solid state Bloch bands that are actually present in microscopic
theories. For TQCPs breaking all global symmetries including time-reversal
ones and $U(1)$ symmetries, TQCPs can be effectively characterized by two-
component interacting majorana fields. For TQCPs with only time reversal
symmetry but breaking all other symmetries including global $U(1)$ symmetries
as in topological superconductors, TQCP dynamics can be represented with four-
component interacting majorana fields. Although quantum phase transitions are
not driven by local ordering and states on both sides of critical points can
both break the same symmetry spontaneously, dynamics at TQCPs do crucially
depend on global symmetries that remain unbroken at TQCPs.

Last issue is related to the second one but perhaps is more on strong coupling
limits which we will elaborate in the next subsection.

##  III Strongly Interacting Limits

Most discussions of solid state topological states, gapped insulators and
superconductors have been inspired by non-interacting physics. So are our
discussions on interacting TQCPs in the previous sections as well as
discussions on the bulk-boundary correspondence above. These discussions are
expected to be valid at least when interactions are not very strong.

Whether all or some of above discussions on topological states can be extended
from weakly interacting limits to certain strongly interacting limits and to
what extend are interesting questions and in general largely remain to be
answered. Most of this article is to deal this issue in a few broad classes of
topological superconductors. Particularly, we exclusively focus on continuous
evolutions of TQCPs when interactions of majorana fields allowed by global
symmetries become strong.

A very closely related question is what happens to bulk states when
interactions increase. One popular view is that as interactions increase in
gapped phases, gapless boundary states can open up a gap without breaking
protecting symmetries. So one approach that had been taken is then to
characterize interacting topological states by classifying topological ordered
boundaries and apply the bulk-boundary correspondence to learn about bulk
states. Those classifications provide general information on what are possible
in extreme limits; on this, we refer readers to those original
papersFidkowski11 ; Fidkowski13 ; Metlitski15 ; Wang15 ; Song17 . What kind of
concrete interactions and interacting Hamiltonians can lead to those strongly
interacting topological states are largely unknown to us.

Meanwhile, gapless bulk quantum fields at TQCPs can also be strongly
interacting although in high dimensions interactions are typically
perturbative and irrelevant in the long wavelength limit. The possibility to
extend discussions on TQCPs beyond weakly interacting limit is quite tempting.
How TQCPs in superconductors in strong coupling limits differ from weak
coupling limits is the key question we intend to answer in this article.

These discussions can further provide valuable detailed information on
dynamics of gapped topological states near TQCPs in strongly interacting
limits. So we anticipate gapless TQCPs are also excellent starting points to
further discussions on interaction dynamic effects of gapped bulk states and
limitations of various weakly-interacting pictures of TQCPs. We will return to
this when discussing some strong coupling fixed point descriptions of TQCPs in
superconductors and explicitly relate them to various new scenarios beyond
more standard TQCPs of weakly interacting majorana fermions.

##  IV The Role of Global Symmetries

Let us now turn to the main focus of the article, the role of global
symmetries. We consider a general quantum phase transition in a topological
state with $N(=2n)$ (including spins) continuum bands. More precisely, $N$ is
the number of complex fermion degrees relevant to our discussions. A TQCP
finds a representations in $2N=4n$-component majorana fields, $\chi$, with its
effective Hamiltonian constructed in the following way,

$\displaystyle H$ $\displaystyle=$ $\displaystyle\int d{\bf r}\chi^{T}({\bf
r}){\mathcal{H}}_{0}\chi({\bf r})+H_{I},\chi^{\dagger}(\bf r)=\chi^{T}(\bf
r);$ $\displaystyle{\mathcal{H}}_{0}$ $\displaystyle=$
$\displaystyle{\mathcal{P}}+{\mathcal{M}},{\mathcal{P}}^{\dagger}={\mathcal{P}}={\mathcal{P}}^{T},{\mathcal{M}}^{\dagger}={\mathcal{M}}=-{\mathcal{M}}^{T},$
$\displaystyle\chi^{T}$ $\displaystyle=$
$\displaystyle(\chi_{1},\chi_{2},...,\chi_{4n}),\\{\chi_{i}({\bf
r}),\chi_{j}({\bf r}^{\prime})\\}=\delta_{i,j}\delta({\bf r}-{\bf
r}^{\prime})$

Here the Hamiltonian matrix ${\mathcal{H}}_{0}$ is a $2N\times 2N$ hermitian
matrix and $\chi({\bf r})$ are $4n$ component real fermions. ${\mathcal{P}}$
is odd in momentum or gradient operators and is real and symmetric for a given
momentum. The mass matrix ${\mathcal{M}}$$(-\nabla^{2})$ is even in momentum
operators and is anti-symmetric, purely imaginary. For a $N=2n$-band state,
there are $N(2N-1)$ antisymmetric mass matrices and $N(2N+1)$ symmetric
hermitian matrices. There are total $4N^{2}$ Hermitian matrices for a general
construction. So ${\mathcal{M}}$ can be of a general form of

$\displaystyle{\mathcal{M}}=\sum_{k=1}^{N(2N-1)}A_{k}{\Gamma}_{k},{\Gamma}^{\dagger}_{k}={\Gamma}_{k}=-{\Gamma}^{T}_{k}$
(7)

where $\Gamma_{k}$, $k=1,...,N(2N-1)$ are anti-symmetric Hermitian matrices.
$H_{I}$ represents interactions among majorana particles or between majorana
fields and other bosonic dynamic degrees of freedom; we will further specify
its detailed structure in later sections.

For the purpose of our discussions, it is convenient to further arrange the
hermitian $2N\times 2N$ mass matrix ${\mathcal{M}}$ in a form of block
diagonal matrices after proper diagonalization. ${\mathcal{M}}$$({\bf p})$ in
the limit of zero momentum ${\bf p}=0$ can therefore be conveniently expressed
as a direct sum of diagonal mass matrices.

$\displaystyle{\mathcal{H}}_{0}$
$\displaystyle=\begin{pmatrix}\begin{bmatrix}\mathbb{P}_{11}&\mathbb{P}_{12}\\\
\mathbb{P}_{21}&\mathbb{P}_{22}\end{bmatrix}-\begin{bmatrix}\mathbb{M}_{11}&0\\\
0&\mathbb{M}_{22}\end{bmatrix}\end{pmatrix}$ (8)

while in general ${\mathcal{P}}$ does not have a diagonal form and can not be
expressed as a direct sum.

If TQCPs are time reversal invariant and if the time reversal symmetry is the
only global symmetry, Hermitian anti-symmetric mass matrices ${\mathcal{M}}$
near TQCPs shall have the following generic form of direct sum

$\displaystyle{\mathcal{M}}({\bf p})$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\mathbb{M}_{11}&0\\\
0&\mathbb{M}_{22}\end{bmatrix},\mathbb{M}_{22}=\mathbb{M}_{2}\oplus\mathbb{M}_{3}\dots\oplus\mathbb{M}_{N/2},$
$\displaystyle\mathbb{M}_{11}({\bf p}\rightarrow 0)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}m_{1}&0&0&0\\\ 0&m_{1}&0&0\\\ 0&0&-m_{1}&0\\\
0&0&0&-m_{1}\\\ \end{bmatrix},$ $\displaystyle\mathbb{M}_{i}({\bf
p}\rightarrow 0)$ $\displaystyle=$ $\displaystyle\begin{bmatrix}m_{i}&0&0&0\\\
0&m_{i}&0&0\\\ 0&0&-m_{i}&0\\\ 0&0&0&-m_{i}\\\ \end{bmatrix},i=2,...,{N/2}$ so
${\mathcal{M}}({\bf p})$ is a diagonal bloch matrix
$\displaystyle{\mathcal{M}}({\bf p})$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\mathbb{M}_{11}&0&\dots&0\\\
0&\mathbb{M}_{2}&\dots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\dots&\mathbb{M}_{\frac{N}{2}}\\\ \end{bmatrix}$

In Eq.(LABEL:mass), there are total $N=2n$ pairs of eigenvalues of $\pm
m_{i}$, $i=1,2,...,n=N/2$. Without losing generality, we have ordered the
eigen values as $0<m_{1}<m_{2}<...<m_{n}$ in the limit of ${\bf p}=0$. Two
fold-degeneracy in ${\mathcal{M}}$ reflects the time-reversal symmetry and the
reflection symmetry in $\pm m_{i}$ follows the charge-conjugation symmetry of
TQCPs considered here.

If the time reversal symmetry is absent and no other global symmetries are
present, the generic structure for the same $N$-band state shall be

$\displaystyle{\mathcal{M}}({\bf p})$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\mathbb{M}_{11}&0\\\
0&\mathbb{M}_{22}\end{bmatrix},\mathbb{M}_{22}=\mathbb{M}_{2}\oplus\mathbb{M}_{3}\dots\oplus\mathbb{M}_{N},$
$\displaystyle\mathbb{M}_{11}({\bf p}\rightarrow 0)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}m_{1}&0\\\ 0&-m_{1}\\\ \end{bmatrix},$
$\displaystyle\mathbb{M}_{i}({\bf p}\rightarrow 0)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}m_{i}&0\\\ 0&-m_{i}\\\ \end{bmatrix},i=2,...,{N}$
Again ${\mathcal{M}}({\bf p})$ is a diagonal block matrix
$\displaystyle{\mathcal{M}}({\bf p})$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\mathbb{M}_{11}&0&\dots&0\\\
0&\mathbb{M}_{2}&\dots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\dots&\mathbb{M}_{{N}}\\\ \end{bmatrix}$ (10)

where now they are $N$ pairs of non-degenerate eigenvalues $\pm m_{i}$,
$i=0,1,..N-1$ ordered from the smallest to the largest.

For a discussion on TQCPs, it is important to keep in mind that the proper
limit of ${\mathcal{M}}$ we need to take is

$\displaystyle m_{1}\rightarrow 0;\frac{m_{i}}{m_{1}}\rightarrow\infty\mbox{
for all $i\neq 1$.}$ (11)

Eq.(11) illustrates that only one mass gap, i.e. the smallest mass gap, closes
while all other mass gaps remain open as in a generic quantum critical point.

Therefore, $\mathbb{M}_{11}$ can be used to projected out a low energy
subspace for all TQCPs breaking time reversal symmetry. More important, it is
the dimension of $\mathbb{M}_{11}$ not that of the full mass matrix
${\mathcal{M}}$ that defines effective field theories for TQCPs. In other
words, $\mathbb{M}_{11}$ effectively defines a fundamental fermion
representation of global symmetries. The dimension of ${\mathcal{M}}_{11}$ is
$2\times 2$ for time-reversal symmetry breaking TQCPs.The simplest example in
this class can a 2D $p$-wave $Z_{2}$ topological superconductors with $N=2$ in
the presence of a Zeeman field. Breaking time reversal symmetry results in
$\mathbb{M}_{11}$ above and effectively projects out a time reversal symmetry
breaking $p+ip$ theory.

For time reversal invariant TQCPs, ${\mathbb{M}}_{11}$ is minimally $4\times
4$ and the minimum can be reached only when there are no any additional global
symmetries. The simplest example is a 3D time reversal odd-party p-wave
topological superconductor that is in a minimal representation to start with.
We anticipate other topological superconductors breaking all global symmetries
except the time reversal symmetry can be described by the same effective field
theory.

In both cases, the effective mass operators can be much smaller than $2N\times
2N$ mass matrices ${\mathcal{M}}$ of the original $N$-band complex fermions.
This is the one of main results of this article. We have found that the
dimension of effective fermion fields for TQCPs are largely independent of
$2N$, the number of microscopic real fermion bands involved, but only depends
on global symmetries present at TQCPs, a quite remarkable feature of
topological quantum criticality. Below we will show that strongly coupling
limit physics crucially depends on global symmetries because of the mass
operators discussed here.

All discussions on time reversal symmetric TQCPs can be easily generalized to
TQCPs with other global symmetries as far as mass operators can be reduced to
the same structure shown above. In this sense, time reversal symmetry is not
essential for our discussions although in this article we have chosen to work
with a situation where the only single global symmetry present at TQCPs
happens to be the time reversal one.

To illustrate this point further, we introduce two projection operators using
a direct sum form,

$\displaystyle P_{+}={\mathbb{I}_{l\times
l}}\oplus\mathbb{O}_{(2N-l)\times(2N-l)},$ $\displaystyle P_{+}\cdot
P_{-}=\mathbb{O}_{2N\times 2N},P_{+}+P_{-}=\mathbb{I}_{2N\times 2N}.$

where $l=4$ for time reversal symmetric TQCPs and $l=2$ for time reversal
symmetry breaking TQCPs. $\mathbb{I}$ and $\mathbb{O}$ are unity and null
matrices respectively. This is to be consistent with the dimension of
$\mathbb{M}_{11}$ that has been defined in Eq.(LABEL:mass),(10) with and
without global symmetries respectively.

Effective majorana fields, two- or four-component ones projected out by these
operators $P_{\pm}\chi({\bf r})=\chi_{\pm}$ shall have the following dynamics
in the limit of defined in Eq.(11),

$\displaystyle H_{eff}$ $\displaystyle=$ $\displaystyle=\int d{\bf
r}\chi^{T}_{+}({\bf r}){\mathcal{H}}_{e0}\chi_{+}({\bf
r})+H_{eI},\chi_{+}^{\dagger}(\bf r)=\chi_{+}^{T}(\bf r);$
$\displaystyle{\mathcal{H}}_{e0}$ $\displaystyle=$
$\displaystyle\mathbb{P}_{11}({\bf p})-\mathbb{P}_{12}({\bf
p})\mathbb{M}^{-1}_{22}({\bf p})\mathbb{P}_{21}({\bf p})+\mathbb{M}_{11}({\bf
p})+\dots$

where ${\bf p}=-i\nabla$ and the second term in ${\mathcal{H}}_{e0}$ that is
even in momenta shall be less relevant in the long wavelength limit than the
first one which can be linear in momenta.

The Lorentz symmetry can naturally emerge for real fermions in the low energy
limit near TQCPs which we will further explore below. If an emergent Lorentz
symmetry appears at a TQCP, ${\mathcal{P}}_{11}$ and ${\mathbb{M}}_{11}$ shall
have the following structures,

$\displaystyle{\mathbb{P}}_{11}=-i\nabla_{\alpha}\gamma_{\alpha},\mathbb{M}_{11}=\gamma_{0},\gamma_{\alpha}^{\dagger}=\gamma_{\alpha}=\gamma_{\alpha}^{T},\gamma_{0}^{\dagger}=\gamma_{0}=-\gamma^{T}_{0}$
$\displaystyle\\{\gamma_{\alpha},\gamma_{\beta}\\}=\delta_{\alpha\beta},\\{\gamma_{0},\gamma_{0}\\}=1,\\{\gamma_{\alpha},\gamma_{0}\\}=0;\alpha=1,..,d$
(14)

where $\gamma_{\alpha,0}$, $\alpha=1,...,d$ are anti-communting hermitian
matrices. Furthermore, $\gamma_{\alpha}$, $\alpha=1,..,d$ are real and
symmetric while $\gamma_{0}$ is purely imaginary and anti-symmetric.
Obviously, for time reversal symmetry breaking cases with 2-component majorana
fermions projected out by mass matrices ${\mathbb{M}}_{11}$, $\gamma_{i,0}$
shall be $2\times 2$ ones and can only be constructed in two spatial
dimensions or $d=2$ but not in $d=3$. With time reversal symmetry,
$\gamma_{i,0}$ shall be $4\times 4$ ones and can be constructed in both two
and three spatial dimensions or $d=2,3$.

In general, Eq.(14) can not be satisfied and there will be no emergent Lorentz
symmetry. Instead, generic mass terms can further break space-time Lorentz
symmetry or even spatial rotational symmetries leading to quantum Lifshitz
majorana fields (QLMFs). Eq.(LABEL:Heff) can also be applied to TQCPs
characterized by QLMFs; these TQCPs include transitions into various
topological nodal phases. A few concrete limits had been examined before in
Ref.Yang21  and Eq.(LABEL:Heff) is the most general description of those
dynamics.

Before concluding this section, we want to further remark on possible
extensions to other global symmetries that can be present in solid states. If
more global symmetries are present, we expect that the size of relevant mass
matrix ${\mathbb{M}_{11}}$, based on which $P_{+}$ is defined, becomes larger
while the size of ${\mathcal{M}}$ is fixed. More symmetries would lead to more
emergent fermion fields at TQCPs and more components of majorana fields. This
was also observed in a few concrete models studied in Ref.Yang21  and we do
not pursue further in this presentation.

Below we will discuss some manifestation of global symmetries in energetics
and dynamics.

##  V TQCPs near Strong Coupling Fixed Points I

Effective quantum field theories suggest that upper critical dimensions for
most TQCPs are usually lower than 2d so in high dimensions that our studies
are focused on, generic TQCPs are represented by weakly interacting majorana
fields. This leads to free majorana fermion universality classes and results
in higher order quantum phase transitions, typically higher than the second-
order Landau phase transitions. For TQCPs with Lorentz symmetry, transitions
are $(d+1)$th order ones for $d=2,3$Yang19 . Weak interactions also suggest
stability of topological surface states and robustness of surface-bulk
correspondence away from these TQCPs.

However, if majorana fields in a particular physical system happen to be
strongly interacting, then TQCPs in this limit can be strong coupling and
there can be higher emergent symmetries, higher than the simple generic scale
or scale-conformal symmetries that usually appear at critical points. We are
now turning to such limits. We will demonstrate below that these higher
emergent symmetries, or more specifically various supersymmetries, also
crucially depend on global symmetries discussed above.

When there are no global symmetries present at a TQCP and time reversal
symmetries are broken, at strong coupling fixed points majorana fermions alway
posscess an emergent supersymmetry with one supercharge, i.e. single (two-
component) majorana fermion with one emergent real scale field; they belong to
the simples ${\mathcal{N}}=4N_{f}={1}$ supersymmetry theory. $N_{f}$ refers to
the number of fermion degrees of freedom in term of $(3+1)D$ Dirac fermions
and ${\mathcal{N}}$ is the number of two-component real fermions. In other
words, there is only one single strong coupling fixed point in this limit.
Supersymmetry can also naturally emerge in strongly correlated nodal
phasesLee07 ; Balents98 , on topological surfacesGrover14 ; Ponte14 ; Li17  or
in other interesting correlated systemsYu10 ; Jian17 ; Zerf16 ; Fendley05 ;
Rahmani15 ; Affleck17 . Here we have found that the supersymmetry naturally
emerges at strong coupling fixed-points of TQCP dynamics if time reversal
symmetry is broken.

When a global symmetry is present and say a TQCP is time reversal symmetric,
there are multiple strong coupling fixed points. Usually, they only involve
one emergent real scalar field and are non-supersymmetric. However, when two
real scale fields are emergent in dynamics and both interact with majorana
fermions, the strong coupling fixed points of majorana fermions can then be
supersymmetric and belong to $N_{f}=\frac{1}{2}$ class. These discussions are
based on properties of interacting gapless majorana fermion fields and so are
also valid in the context of gapless topological states. Global symmetries can
be applied to classify strongly interacting gapless phases.

###  V.1 Time Reversal Symmetry Breaking TQCPs in $2d$ I: Local interactions

Let us start with the simplest limit of TQCPs when time reversal symmetry as
well as all other symmetries are all broken. The effective theory in $2d$ with
a local and rotational invariant interaction can be expressed in the following
representation;

$\displaystyle H$ $\displaystyle=$ $\displaystyle\int d{\bf r}[\chi^{T}({\bf
r})(\tau_{x}i\nabla_{z}-\tau_{z}i\nabla_{x})\chi({\bf r})$ $\displaystyle+$
$\displaystyle g\chi^{T}\tau_{y}\frac{\nabla}{i}\chi({\bf
r})\cdot\chi^{T}\tau_{y}\frac{\nabla}{i}\chi({\bf r})];$
$\displaystyle\chi^{T}$ $\displaystyle=$
$\displaystyle(\chi_{1},\chi_{2}),\\{\chi_{i}({\bf r}),\chi_{j}({\bf
r}^{\prime})\\}=\delta_{i,j}\delta({\bf r}-{\bf r}^{\prime}),i=1,2.$

where $\nabla=(\nabla_{x},\nabla_{y})$. Note that because only two-component
majorana fermions emerge in this limit, terms of a form of $\chi^{4}$ without
involving gradient operators vanish identically because of fermion statistics.
The four-fermi operator in Eq.(LABEL:H1) is the most relevant one for your
discussions.

The Hamiltonian for TQCPs in Eq.(LABEL:H1) displays an emergent $Z_{2}$
symmetry when $\chi$ transforms as

$\displaystyle\chi_{1}\rightarrow K\chi_{2},\chi_{2}\rightarrow-K\chi_{1},$
(16)

where $K$ is an action of complex conjugate. Under such a transformation,
$\tau$ matrices in Eq.(16) transform accordingly
$\tau_{x,z}\rightarrow-\tau_{x,z}$, $\tau_{y}\rightarrow-\tau_{y}$. The
interacting Hamiltonian for TQCPs is invariant under the transformation in
Eq.(16). This symmetry only emerges at TQCPs and is not a symmetry of adjacent
gapped topological states as a mass term breaks such a symmetry.

The emergent $Z_{2}$ symmetry differs from the time reversal symmetry which is
broken in our case. Time reversal transformation is defined in terms of

$\displaystyle\chi_{1}\rightarrow K\chi_{1},\chi_{2}\rightarrow-K\chi_{2}.$
(17)

Accordingly, $\tau_{x}\rightarrow-\tau_{x}$ and
$\tau_{z,y}\rightarrow\tau_{z,y}$ and the TQCP Hamiltonian in Eq.(LABEL:H1) is
not invariant under this transformation as stated before.

Under scale transformation, $g$ follows the following renormalization group
equation in terms of dimensionless coupling constant
$\tilde{g}=c_{d}g\Lambda^{d+1}$,where $c_{d}$ is a constant of order of unity
and in $d=2$, $c_{2}=\frac{1}{4\pi^{2}}$.

$\displaystyle\frac{d\tilde{g}}{dt}=(d+1)\tilde{g}+\tilde{g}^{2},t=\ln\Lambda;$
(18)

and we will mainly apply to $d=2$ in our discussions as time reversal symmetry
breaking topological superconductors only exist in $2d$. Here $\Lambda$ is a
running ultraviolet scale in the scale transformation ad $c_{d}$ is a constant
of order of unity and Eq.(18) is obtained in a one-loop approximation.

As suggested in Eq.(18), the free majorana fermion fixed point with
$\tilde{g}=0$ is infrared stable in $d=2$ as the scaling dimension of
interaction operator is $3$ (and scale dimension $0$ is for a marginal
interaction). It shows that as far as $\tilde{g}$ is small, weakly interacting
gapless majorana fermions can form a well defined phase. So $(d+1)$th order
transitions for $d=2$ time reversal symmetry breaking TQCPs Yang19  that
belong to free majorana fermion universality class shall be very robust as
weak interactions are highly irrelevant.

Although technically speaking Eq.(18) is perturbative, it is still suggestive
to extrapolate to strong coupling limits and to gain some insight about what
happens when $\tilde{g}$ is not too small. Eq.(18) dose imply a strong
coupling fixed point with $\tilde{g}^{*}=-(d+1)$. It appears that as far as
$\tilde{g}>\tilde{g}^{*}$, the free fermion fixed point at $\tilde{g}=0$
remains infrared stable, leading to a well-defined weakly interacting gapless
majorana phase boundary. The corresponding transitions of this universality
class is of $(d+1)$th order in $d=2$ as pointed out in a previous
articleYang19 .

However, in the strong coupling limit where $\tilde{g}<\tilde{g}^{*}=-(d+1)$,
$\tilde{g}$ can follows into $\tilde{g}^{*}=-\infty$. This indicates a
distinct strong coupling phase of majorana fermions where a mass gap of either
sign, $m$ or $-m$ is spontaneously created and fermions are fully gapped. The
ground state therefore shall be two-fold degenerate corresponding to two
masses $m$ and $-m$ and spontaneously break the emergent $Z_{2}$ symmetry at
TQCPs.

The strong coupling phase suggested above is an analogue of Gross-Neveau mass
generation well understood in quantum field theoriesGross74 ; Fei16 . Indeed,
if one further introduces $N$ flavours of majorana fermions that interact in
an identical way as shown in Eq.(LABEL:H1), then in the limit of large $N$,
one can obtain a renormalization equation similar to Eq.(18) above. One can
show that in the leading order of $(1/N)$, Eq.(18) can be applied to describe
renormalization of a coupling constant $\tilde{g}=c_{d}N^{2}g$ even when
$\tilde{g}$ is much larger than unity. So in the large $N$ limit, the strong
coupling physics conjectured above can be easily verified.

The strong coupling phase suggested above can be further connected to strongly
interacting lattice models of majorana fermions carefully studied
beforeRahmani15 ; Affleck17 . In $1d$, it had been shown explicitly that
strongly interacting fixed points implied in Eq.(18) with
$\tilde{g}^{*}=-(d+1)$ is directly related to a supersymmetric tri-critical
Ising point observed and confirmed in the lattice model.

Therefore, implications of the simple one-loop renormalization equation,
Eq.(18) are highly illuminating, shedding light on TQCPs in strong coupling
limits. It suggests that the weakly interacting $(d+1)$th order transition
line be terminated by a strong coupling fixed point beyond which there shall
be discontinuous first order transitions. To better understand strong coupling
physics of TQCPs without time reversal symmetry, and strong coupling fixed
points, below we introduce an extended model that is more suitable for current
studies. The model below can be viewed as an ultraviolet completion of the
infrared physics discussed so far. It not only leads to the same infrared
physics discussed above but also naturally extends the theory to further
higher inmtermediate energy scales which is fully consistent with the infrared
physics.

###  V.2 Time Reversal Symmetry Breaking TQCPs in $2d$ II: General discussions

Near a strong coupling fixed point and beyond, the long wave length physics is
expected to be naturally and more explicitly described by majorana fermions
interacting with emergent scalar fields. Therefore, to pursue discussions in
strong coupling limits, it is more convenient to introduce the following
extended model that is more suitable for such a purpose.

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0m}+H_{0s}+H_{I},$
$\displaystyle H_{0m}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}[\chi^{T}({\bf r})(\tau_{x}i\nabla_{z}-\tau_{z}i\nabla_{x})\chi({\bf r})$
$\displaystyle H_{0s}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}[\pi^{2}({\bf r})+\nabla\phi({\bf r})\cdot\nabla\phi({\bf
r})+M^{2}\phi^{2}({\bf r})];$ $\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\int d{\bf r}[g_{Y}\phi({\bf r})\chi^{T}({\bf
r})\tau_{y}\chi({\bf r})+g_{4}\phi^{4}({\bf r})];$ $\displaystyle[\phi({\bf
r})$ $\displaystyle,\pi({\bf r}^{\prime})]=i\delta({\bf r}-{\bf
r^{\prime}}),[\phi({\bf r}),\phi({\bf r^{\prime}}]=[\pi({\bf r}),\pi({\bf
r}^{\prime})]=0.$

where

$\displaystyle\chi^{T}=(\chi_{1},\chi_{2}),\\{\chi_{i}({\bf r}),\chi_{j}({\bf
r}^{\prime})\\}=\delta_{i,j}\delta({\bf r}-{\bf r}^{\prime}),i=1,2.$

The Hamiltonian, though breaking the time reversal symmetry, can have a
$Z_{2}$ symmetry; i.e. Eq.(LABEL:SUSY1) is invariant under the following
transformation

$\displaystyle\chi_{1}\rightarrow
K\chi_{2},\chi_{2}\rightarrow-K\chi_{1},\phi\rightarrow-\phi,\pi\rightarrow-\pi$
(21)

where $K$ is an action of complex conjugate. Again, under such a
transformation, $\tau_{x,z}\rightarrow-\tau_{x,z}$ and
$\tau_{y}\rightarrow-\tau_{y}$. The transformation is effectively a $\pi$
rotation around the $\tau_{y}$ axis followed by an operation of $K$.

The $Z_{2}$ symmetry defined above differs from the time reversal one as in
the later case, $\phi,\chi$ transform differently as

$\displaystyle\chi_{1}\rightarrow
K\chi_{1},\chi_{2}\rightarrow-K\chi_{2},\phi\rightarrow\phi,\pi\rightarrow\pi.$
(22)

As stated before, the Hamiltonian in Eq.(LABEL:SUSY1) breaks the time reversal
symmetry and is not invariant under the transformation in Eq.(22).

$Z_{2}$ symmetry in Eq.(21) can be more easily understood if one integrates
out fermions first and constructs an effective potential for $\phi$ field. The
effective potential is manifestly invariant under a standard reflection of a
real scalar field, $\phi\rightarrow-\phi$. The ground state is $Z_{2}$
invariant if $\langle\phi\rangle=0$ but otherwise is two-fold degenerate if
spontaneous symmetry breaking occurs and $\langle\phi\rangle\neq 0$. Note that
this is an emergent symmetry of interacting gapless majorana fermions exactly
at TQCPs but is not a symmetry of adjacent gapped topological states. One can
verify that a gapped topological state which is non-degenerate breaks this
$Z_{2}$ symmetry explicitly.

When $M^{2}$ is positive and finite, the ground state is non-degenerate and is
$Z_{2}$ symmetric as $\langle\phi\rangle=0$; majorana fermions remain
massless. One can further integrate over the gapped scalar fields $\phi$ and
generate four fermion interactions in $\chi$ fields similar to interactions in
Eq.(LABEL:H1). In the infrared limit (when momentum scale $\Lambda\ll M$) ,
fermion interactions mediated by the gapped scalar fields do have a simple
form of

$\displaystyle H_{I}=\frac{g^{2}_{Y}}{2M^{4}}\int d{\bf r}\nabla(\chi^{T}({\bf
r})\tau_{y}\chi({\bf r}))\cdot\nabla(\chi^{T}({\bf r})\tau_{y}\chi({\bf
r}))+...$ (23)

where we have muted higher order terms or other terms not relevant to the
following discussions. Again because of fermion statistics, terms like
$\chi^{4}$ without involving $\nabla\chi$ vanish identically and so there are
no contributions of order of $O(1/M^{2})$. Eq.(23) can be easily related to
$H_{I}$ in Eq.(LABEL:H1) if one sets

$\displaystyle g\sim-\frac{2g^{2}_{Y}}{M^{4}}.$ (24)

Therefore, majorana fermions interacting with massive scalar fields $\phi$ as
shown in Eq.(LABEL:SUSY1) are naturally related to interacting gapless
majorana fermions represented in Eq.(LABEL:H1). The limit with a large mass
gap $M^{2}(>0)$ corresponds to the weakly interacting limit with small $g$.
And at any finite mass gap $M^{2}$, the long wavelength limit of
Eq.(LABEL:SUSY1) is hence expected to be equivalent to Eq.(LABEL:H1). In the
infrared limit, we anticipate that the model with a finite mass $M^{2}$ again
follows into the infrared stable free majorana fixed point with gapped scale
fields that play little role in low energies. So Eq.(LABEL:SUSY1) is an
extended model of Eq.(LABEL:H1) and is more convenient for discussions of
strong coupling TQCPs.

Figure 2:  Phase diagrams of $Z_{2}\otimes Z_{2}$ (for ${\mathcal{T}}$ and
${\mathcal{R}}$) symmetric model in Eq.(LABEL:SUSY3) in a plane of
$X=M_{1}^{2}$, $Y=M_{2}^{2}$. Upper right corner in this digram is weakly
interacting. In the half plane of $Y>0$, states are time reversal invariant
(TRI) while in the half plane of $Y<0$, time reversal symmetry is broken
spontaneously (TRB). In the quadrant of $X<0,Y>0$, $Z_{2}$-reflection symmetry
is broken while in $X>0,Y>0$, states are $Z_{2}\otimes Z_{2}$ symmetric. In
$X>0,Y<0$, $Z_{2}$ symmetries of both ${\mathcal{T}}$ and ${\mathcal{R}}$ are
broken but the combined symmetry of ${\mathcal{T}R}$ remains; $Z_{2}\otimes
Z_{2}$ is spontaneously broken down to $Z_{2}$ for combined ${\mathcal{T}}$
and ${\mathcal{R}}$ transformation. $X=0,Y>0$ and $Y=0,X>0$ lines are for
$Z_{2}\otimes Z_{2}$ symmetric CFT states of Gross-Neveu fixed points or GNFP
classes in $2d$. In this plane, only $X=Y>0$ line is $U(1)$ symmetric if
$g_{Y1}=g_{Y2}$ (see Eq. (LABEL:SUSY3)). $U(1)$ symmetry is broken
spontaneously along $X=Y<0$ and broken explicitly in the rest of plane. Point
$X=Y=0$ in $2d$ is supersymmetric with $N_{f}=\frac{1}{2}$ and $U(1)$
symmetry; in $d=3$, it is infrared trivial. For $Y>0$, TQCPs are either
$(d+1)$th order ($X>0$) or first order ones ($X<0$). For $Y<0$, there shall be
no transitions because of spontaneously time reversal symmetry breaking in
this limit of strong interactions.

An interesting limit of Eq.(LABEL:SUSY1) is when $M^{2}$, the mass gap of
scalar fields also vanishes resulting in a scale invariant strong coupling
state. When $M^{2}=0$ in dimensions lower than $3d$ including $2d$, non-
interacting fixed point with $\tilde{g}^{*}_{Y}=0$ becomes infrared unstable.
The only infrared stable fixed point in $2d$ is strong coupling and is again
supersymmetricFei16 ; Grover14 ; Zerf16 .

The scale symmetric solution to Eq.(LABEL:SUSY1) when $M^{2}=0$ shall be
identified as an effective description of the anticipated strong coupling
fixed point $\tilde{g}^{*}$ discussed in the previous subsection. This fixed
point represents a conformal-field-theory (CFT) state with an additional
supersymmetry. Furthermore, the strong coupling fixed point is also $Z_{2}$
symmetric.

When $M^{2}$ further becomes negative, there is standard spontaneous symmetry
breaking of $Z_{2}$ symmetry in real scalar field $\phi$ and
$\langle\phi\rangle\neq 0$. This leads to a mass generation for majorana
fermions, $m\sim g_{Y}\langle\phi\rangle$ and two-fold degenerate ground
states in the strong coupling limit. We identify this as the strongly coupling
limit of gapless majorana fermions. By varying the mass gap $M^{2}$, we
anticipate that Eq.(LABEL:SUSY1) can be applied to capture both weak coupling
and strong coupling limits suggested by the renormalization group equations of
Eq.(LABEL:H1). And Eq.(LABEL:SUSY1) with $M^{2}=0$ shall be an effective
representation of the strong coupling point $\tilde{g}^{*}$ that turns out to
be conformal with an emergent supersymmetry.

We can apply these results of interacting gapless majorana fermions to examine
strong coupling TQCPs. In $2d$, the strong coupling fixed point above
indicates that the $(d+1)$th order transition line obtained in the weakly
interacting limit shall be terminated at $\tilde{g}^{*}$ with an emergent
supersymmetry. Beyond this point, a majorana mass, either positive or negative
is spontaneously generated by interactions. The mass $m$ is a function
$(\tilde{g}-\tilde{g}^{*})$ and ground state is two-fold degenerate. We
identified this as a first order transition line (See Fig. 3).

By passing, we want to point out that the field theory model itself can be
easily reduced to $d=1$ dimension which can be applied to study spinless time
reversal symmetric TQCPs in $1d$. The free fermion fixed points are again
stale. So the continuous $(d+1)$th or $2$nd order transition lines in this
case is again terminated by a supersymmetric fixed point beyond which first
order transitions occur. The conclusions here are qualitatively identical to
$2d$ time reversal symmetry breaking TQCPs.

##  VI  TQCPs near Strong Coupling Fixed Points II

###  VI.1 Time reversal invariant case I: Local interactions and emergent
$U(1)$ symmetry

With a single global symmetry, say time reversal symmetry, at a TQCP, the
minimal representation becomes a four-component majorana fermion field. If
there are no other global symmetries at TQCPs, only minimal four majorana
fields shall emerge at low energies relevant to our discussions. A generic
local interacting Hamiltonian in $3d$ with Lorentz symmetry can have the
following form,

$\displaystyle H$ $\displaystyle=$ $\displaystyle\int d{\bf r}[\chi^{T}({\bf
r})(\tau_{z}\otimes(\sigma_{x}{i\nabla_{z}}-\sigma_{z}{i\nabla_{x}})+\tau_{x}\otimes\mathbb{I}{i\nabla_{y}})\chi({\bf
r})$ $\displaystyle+$ $\displaystyle g_{1}\chi^{T}({\bf r})\tau_{y}\chi({\bf
r})\chi^{T}({\bf r})\tau_{y}\chi({\bf
r})];\mbox{}\chi^{T}=(\chi_{1\uparrow},\chi_{1\downarrow},\chi_{2\uparrow},\chi_{2\downarrow}).$

Indeed, Eq.(LABEL:TQCPTRI) can also be an effective theory for a topological
superconductor with p-wave pairing. One can reduce Eq.(LABEL:TQCPTRI) to $2d$
or $1d$ by muting one or two spatial gradient operators.

It is important to remark that if there are no other less relevant local
interactions involving gradient operators, Eq.(LABEL:TQCPTRI) as a model for
TQCPs has a hidden $U(1)$ symmetry although the $U(1)$ symmetry associated
with microscopic fermion conservation is spontaneously broken because of
superconductivity. And this emergent symmetry only appears exactly at TQCPs
but is not present in gapped phases adjacent to TQCPs.

To further visualize the symmetry, it is useful to introduce a unitary
rotation

$\displaystyle
U(\frac{2\pi}{3})=\frac{1}{2}[1+i(\tau_{x}\otimes\sigma_{y}+\tau_{y}+\tau_{z}\otimes\sigma_{y})]$
$\displaystyle
U^{\dagger}(\tau_{x}\otimes\sigma_{y},\tau_{y},\tau_{z}\otimes\sigma_{y})U\rightarrow(\tau_{y},\tau_{z}\otimes\sigma_{y},\tau_{x}\otimes\sigma_{y}).$
(26)

Correspondingly, $\chi\rightarrow U\chi$. Now in terms of $g_{0}=g_{1}/2$,
$H(\\{\chi\\})\rightarrow H^{\prime}(\\{\chi\\})$ and $H^{\prime}$ has the
following form

$\displaystyle H^{\prime}$ $\displaystyle=$ $\displaystyle\int d{\bf
r}[\chi^{T}({\bf
r})(\mathbb{I}\otimes(\sigma_{x}{i\nabla_{x}}+\sigma_{z}{i\nabla
z})+\tau_{y}\otimes\sigma_{y}{i\nabla_{y}})\chi({\bf r})$ $\displaystyle+$
$\displaystyle
g_{0}\chi^{T}\tau_{x}\otimes\sigma_{y}\chi\chi^{T}\tau_{x}\sigma_{y}\chi+g_{0}\chi^{T}\tau_{z}\otimes\sigma_{y}\chi\chi^{T}\tau_{z}\otimes\sigma_{y}\chi];$

Eq.(LABEL:SUSY2) is explicitly invariant under any rotation around $\tau_{y}$
and therefore actually has an emergent $U(1)$ symmetry. It suggests that this
shall be a unique property of interacting majorana fermions with local four-
fermion operators (without involving gradient operators).

Note that for the convenience of later presentations and for the transparency,
in deriving Eq.(LABEL:SUSY2), we have explicitly employed two different
representations for the local four-fermion interaction operator in
Eq.(LABEL:TQCPTRI). Since we deal with four component majorana fermions, one
can easily further show that the two four-fermion operators in
Eq.(LABEL:SUSY2) are actually identical and both are generated from a single
four fermion operator $g_{0}\chi^{T}\tau_{y}\chi\chi^{T}\tau_{y}\chi$.

We also have found that this tree level symmetry is respected when further
quantum corrections are taken into account. The one-loop renormalization group
equation for $g_{0}$ in Eq.(LABEL:TQCPTRI) can be easily obtained via
introducing $\tilde{g}_{0}(\Lambda)=c_{d}g_{0}\Lambda^{d-1}$ where for $d=2$,
$c_{2}=\frac{1}{4\pi^{2}}$;

$\displaystyle\frac{d\tilde{g}_{0}}{dt}=(d-1)\tilde{g}_{0}+\tilde{g}_{0}^{2},t=\ln\Lambda$
(28)

where $\Lambda$ is a running ultraviolet scale for the interacting
Hamiltonian. The free fermion fixed point $\tilde{g}=0$ above is stable in
$d=2,3$ where main applications had been foundYang19 ; $(d+1)$th order
transition lines of this universality class therefore shall persist in the
weakly interacting limit.

Eq.(28) itself also has strong coupling fixed points,
$\tilde{g}^{*}_{0}=-(d-1)$ in $2d$ and $3d$. Below we will mainly focus on
$2d$ which is most interesting. When $\tilde{g}\geq\tilde{g}^{*}$, we do not
expect spontaneously breaking of the emergent $U(1)$ symmetry as $\tilde{g}=0$
is an infrared stable fixed point. So we have arrived at a conclusion that for
TQCPs with time reversal global symmetry, strong coupling fixed points for
gapless majorana fermions at TQCPs appear to have an emergent $U(1)$ symmetry
as far as interaction operators are local four-fermion ones. In $2d$, this
turns out to have very important implications on the emergence of
supersymmetry at these TQCPs.

Indeed, one can further find, due to such an emergent $U(1)$ symmetry, that
$H^{\prime}$ in this representation is identical to the Hamiltonian of two-
component interacting Dirac fermions expressed in a majorana representation.
We therefore conclude that majorana fermions for TQCPs with time reversal
symmetry in Eq.(LABEL:TQCPTRI), are equivalent to, up to a unitary
transformation, conventional two component complex massless Dirac fermions
with local interactions. They shall be completely equivalent in dynamics
because the partition functions of two models are identical. More explicitly,
with time reversal symmetry at TQCPs, under the transformation defined in
Eq.(26), (LABEL:SUSY2),

$\displaystyle H\rightarrow
H^{\prime}(\\{U(\frac{2\pi}{3})\chi\\})=H_{2cD}(\\{\chi\\}),Z_{TQCP}=Z_{2cD};$
$\displaystyle H_{2cD}=\int d{\bf r}[\psi^{\dagger}({\bf r}){\bf\sigma}\cdot
i\nabla\psi({\bf
r})+g_{0}(\psi^{\dagger}\psi)^{2}],\psi^{T}=(\psi_{\uparrow},\psi_{\downarrow})$
(29)

where the subscribe $2cD$ refers two-component Dirac fermions,
$Z_{TQCP},Z_{2cD}$ are partition functions for TQCPs and 2-component Dirac
fermions respectively.

Eq.(29) appears to be surprising. TQCPs under considerations spontaneously
break the original underlying U(1) symmetry of complex fermions and charges
are not conserved because of superconductivity. Gapless Dirac fermions on the
other hand do not break the U(1) symmetry and fermion charges are conserved in
a Dirac model. The $U(1)$ symmetry discussed above is an emergent one only
appear in the long wavelength limit. Nevertheless, its infrared physics of
interacting gapless majorana fermions appears to be identical to complex Dirac
fermions because of this emergent symmetry. In $2d$, Dirac fermions with
attractive interactions are supersymmetric at their strongly interacting fixed
point. If the emergent $U(1)$ symmetry above indeed also appears at a
corresponding strong coupling fixed point, we intend to conclude $2d$ TQCPs
with time reversal symmetry where majorana fermions interact locally are also
supersymmetric at its strong coupling fixed points.

In the limit we are interested in, all other local terms of
$\chi^{6},\chi^{8}$ vanish identically. Any additional local interaction
operators have to have a quartic form of $\chi^{4}$ form further involving
gradient operators. But they are irrelevant in the infrared limit from the
renormalization point of view, So generally if we restrict ourself only to
local interactions, $U(1)$ symmetry appears to emerge in the infrared limit.

However, such an emergent $U(1)$ symmetry only appears when interactions can
be completely expressed in terms of local operators, an assumption that might
not be generic. It is more natural that microscopic interactions actually
break such a low energy symmetry at certain intermediate energy scales beyond
the effective theories of local interactions. In practical systems,
interactions can be further mediated by various dynamic fields. If we take
this point of view, then although effective theories with local interactions
suggest an emergent $U(1)$ symmetry, actual strong coupling fixed point
solutions generically shall not have such a $U(1)$ symmetry.

This is especially so because low energy symmetries at gapless strong coupling
fixed points shall also be consistent with high energy symmetries/asymmetries
of dynamic fields away from the strong coupling fixed points, in addition to
low energy emergent symmetries. And as the fixed points are approached, if
some of finite energy symmetries or asymmetries inherited by dynamic fields
become descending in the infrared limit of TQCPs, they can results in breaking
emergent $U(1)$ symmetries seen in the effective local theories above. Below
we are going to clarify this issue using a more general dynamic model. To
understand details near strong coupling fixed point $\tilde{g}^{*}$ in $2d$
and $3d$, it is again more convenient to apply a generalized model introduced
in the next subsection which captures both the infrared physics discussed so
far and additional higher intermediate energy sectors fully consistent with
the infrared physics. These intermediate energy sectors eventually descend to
low energy windows near strong coupling fixed points and dictate the
behaviours of TQCPs in strong coupling limits.

###  VI.2 Time reversal symmetry invariant case: General

To better understand the origin and consequence of emergent $U(1)$ symmetry
and limitation of the above analyses, we can further explore the TQCP dynamics
in an extended majorana fermion model by explicitly introducing two real
scalar fields $\phi_{1,2}=\phi_{1,2}^{\dagger}$,

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

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,

$\displaystyle[\phi_{i}({\bf r}),\pi_{j}({\bf
r}^{\prime})]=i\delta_{ij}\delta({\bf r}-{\bf r^{\prime}}),$
$\displaystyle[\phi_{i}({\bf r}),\phi_{j}({\bf r^{\prime}}]=[\pi_{i}({\bf
r}),\pi_{j}({\bf r}^{\prime})]=0,i=1,2$
$\displaystyle\chi^{T}=(\chi_{1\uparrow},\chi_{1\downarrow},\chi_{2\uparrow},\chi_{2\downarrow}).$
(31)

The Hamiltonian that describes TQCPs with time reversal symmetry is invariant
under time reversal transformation ${\mathcal{T}}$. In addition, just like an
emergent $U(1)$ symmetry in Eq.(LABEL:SUSY2) for TQCPs with local
interactions, it is also further invariant under an internal reflection
transformation ${\mathcal{R}}$. The reflection symmetry only emerges exactly
at TQCPs and it is not a symmetry of gapped phases adjacent to TQCPs which
only has ${\mathcal{T}}$-symmetry as we discussed in the previous section. So
at TQCPs, Eq.(LABEL:SUSY3) is invariant under a bigger symmetry group of
$Z_{2}\otimes Z_{2}$.

Under the time reversal transformation ${\mathcal{T}}$, these real fields
transform accordingly,

$\displaystyle\chi_{1}$ $\displaystyle\rightarrow$ $\displaystyle
i\sigma_{y}K\chi_{1},\chi_{2}\rightarrow-i\sigma_{y}K\chi_{2},$
$\displaystyle\phi_{1}$ $\displaystyle\rightarrow$
$\displaystyle\phi_{1},\phi_{2}\rightarrow-\phi_{2},\pi_{1}\rightarrow\pi_{1},\pi_{2}\rightarrow-\pi_{2}$
(32)

where $K$ is an action of complex conjugate. The Hamiltonian in
Eq.(LABEL:SUSY3) is manifestly time reversal symmetric.

Figure 3:  $2d$ TQCPs in strong coupling limits in the $m-g$ plane ( (a) and
(b) ), $m-Y$ plane (with $X=X_{0}>0$) (c) and $m-X$ plane (with $Y=Y_{0}>0$)
(d) . $m$ is the tuning fermion mass of TQCPs and $g$ is an effective local
interaction constant in Eq.(LABEL:SUSY1),(LABEL:SUSY2). $X=M_{1}^{2}$,
$Y=M_{2}^{2}$ are masses of scalar fields in dynamic models in
Eq.(LABEL:SUSY3). a) represents the results for $2d$ TQCPs wiithout time
reversal symmetries; b-d) describe $2d$ TQCPs with time reversal symmetry.
b-c) are scenarioes where time reversal symmetry is spontaneously broken in
strong coupling limit beyond the red termination points. d) is for TQCPs with
strong interactions that do not result in spontaneously time reversal symmetry
breaking. In a)- d), the thick (red) line represents a $(d+1)$th transition
lines obtained in Ref.Yang19  and dashed ones for first order transition
lines. The red dot separating them in a) is supersymmetric (SUSY) with
$N_{f}=\frac{1}{4}$ for TQCPs breaking time reversal symmetries but in d) is a
non-supersymmetric Gross-Neveu fixed point (GNFP) . In b), the red termination
point is supersymmetric with $N_{f}=\frac{1}{2}$ where majorana fermions
interact with an emergent $U(1)$ symmetry; beyond the termination point,
$U(1)$ symmetry is broken spontaneously. In b),c), we also show possible paths
around strong coupling fixed points that connect two topologically distinct
phases without closing the fermion mass gap because of spontaneously breaking
protecting time reversal symmetry beyond this point. (Time reversal symmetry
is broken spontaneously in shaded areas.)

In addition, Eq.(LABEL:SUSY3) is also invariant under the following reflection
transformation ${\mathcal{R}}$.

$\displaystyle\chi_{1}$ $\displaystyle\rightarrow$
$\displaystyle-\chi_{2},\chi_{2}\rightarrow\chi_{1}$ $\displaystyle\phi_{1}$
$\displaystyle\rightarrow$
$\displaystyle-\phi_{1},\pi_{1}\rightarrow-\pi_{1},\phi_{2}\rightarrow-\phi_{2},\pi_{2}\rightarrow-\pi_{2}$
(33)

In both $2d$ and $3d$, Eq.(LABEL:SUSY3) admits four phases depending on the
masses of scalar fields, $X=M_{1}^{2}$ and $Y=M_{2}^{2}$ (see Fig.2) and
spontaneous symmetry breaking of $Z_{2}\otimes Z_{2}$ symmetries in
Eq.(32),(33).

i) When $X>0,Y>0$, $\langle\phi_{1}\rangle=\langle\phi_{2}\rangle=0$ and the
phase is $Z_{2}\otimes Z_{2}$ symmetric. Majorana fermions remain gapless;
$X,Y\rightarrow\infty$ further corresponds to the weakly interacting limit
studied in Ref.Yang19 ;

ii) When $X<0,Y>0$, $\langle\phi_{1}\rangle\neq 0$ but
$\langle\phi_{2}\rangle=0$ and the phase breaks the reflection ${\mathcal{R}}$
symmetry but time reversal ${\mathcal{T}}$ symmetry remains unbroken. Symmetry
group $Z_{2}\otimes Z_{2}$ breaks down to $Z_{2}$ of ${\mathcal{T}}$ symmetry.
Majorana fermions acquire a time reversal invariant mass,
$g_{Y1}\langle\phi_{1}\rangle$ because of ${\mathcal{R}}$ symmetry breaking.
This is an analogue of the well-know Gross-Neveu mechanismGross74 ;

iii) When $X>0,Y<0$, $\langle\phi_{2}\rangle\neq 0$ but
$\langle\phi_{1}\rangle=0$ and the phase again breaks $Z_{2}\otimes Z_{2}$
symmetry to $Z_{2}$ of ${\mathcal{R}T}$. Although both $Z_{2}$ of time
reversal symmetry and $Z_{2}$ of reflection symmetry are spontaneously broken,
this phase of matter is still invariant under the combined transformation
${\mathcal{R}T}$ of time reversal ${\mathcal{T}}$ and reflection
${\mathcal{R}}$. Majorana fermions acquire a time reversal symmetry breaking
mass, $g_{Y2}\langle\phi_{2}\rangle$.

iv) When $X<0,Y<0$, $\langle\phi_{2}\rangle\neq 0$,
$\langle\phi_{1}\rangle\neq 0$. The phase spontaneously breaks $Z_{2}\otimes
Z_{2}$ symmetry and no symmetries remain. Majorana fermions acquire two masses
breaking both ${\mathcal{T}}$ and ${\mathcal{R}}$ symmetries. Apparently, only
iii) and iv) break the time reversal ${\mathcal{T}}$ -symmetry.

The upper critical dimension of Eq.(LABEL:SUSY3) is $3d$, below which free
majorana fermion fixed points are unstable when $M^{2}_{1}$ and/or $M^{2}_{2}$
becomes zero. So in $2d$, Eq.(LABEL:SUSY3) has the following strong coupling
fixed lines or points of majorana fermions when

$\displaystyle a)$ $X=0\neq Y>0$ or $M_{1}=0\neq M_{2}$ (34) $\displaystyle
b)$ $Y=0\neq X>0$ or $M_{2}=0\neq M_{1}$ $\displaystyle c)$ $X=Y=0$,
$M_{1}=M_{2}=0$.

All these lines or points are time reversal symmetric and reflection symmetric
and therefore are $Z_{2}\otimes Z_{2}$ symmetric gapless states. In $2d$,
$M_{1}=0$ while $M_{2}\neq 0$ or $M_{2}=0$ while $M_{1}\neq 0$ represent
conformal field theory states of Gross-Neveu fixed points (GNFP) in the model
and we call them GNFPs or lines. They correspond to $X=0,Y>0$ or $Y=0,X>0$
lines in Fig.2 and form boundaries of gapless weakly interacting majorana
fermions.

One can further verify that beyond the lines of fixed points a) or b), there
is mass generation as in the Gross-Neveau (GN) modelGross74 ; Fei16 . Beyond
the line of fixed points b) or fixed point c), time reversal symmetry breaking
masses are generated while beyond the line of points a), only time reversal
symmetric masses are generated. Along the line of $X=0,Y<0$ and $Y=0,X<0$,
majorana fermions remain massive; $X=0,Y<0$ is ${\mathcal{R}T}$ $Z_{2}$
symmetric and $Y=0,X<0$ is $Z_{2}$ time reversal symmetric.

In $3d$, phases displayed by Eq.(LABEL:SUSY3) are the same as in $2d$.
However, the gapless fixed lines or points identified in Eq.(34) appear to be
weakly interacting. In $3d$, $g_{Yi}$, $i=1,2$ actually flow into non-
interacting fixed points with $g_{Y1}=g_{Y2}=0$ instead of CFT fixed points.

Eq.(LABEL:SUSY3) generally does not have a $U(1)$ symmetry because of an
asymmetry between two masses $M_{1,2}$. It can have a similar emergent $U(1)$
symmetry under a rotation around $\tau_{y}$ only if $M_{1,2}$ two masses are
fully symmetric and $g_{Y1}=g_{Y2}$. For the purpose of our discussions below,
without loss of generality, let us focus on the limit of $g_{Y1}=g_{Y2}$ at
the moment although this condition can be relaxed later on when it comes to
supersymmetry.

To further this discussion, we introduce the following $U(1)$ transformation

$\displaystyle\chi^{\prime}=e^{i\tau_{y}\frac{\theta}{2}}\chi,\phi=e^{i\theta}\phi$
(35)

where $\phi=\phi_{1}+i\phi_{2}$ is a complex field representation for scalar
fields. In the gapped limit when both $M_{1}>M_{2}$ are nonzero, the long
wavelength physics is identical to, as anticipated, Eq.(LABEL:SUSY2) after
integration of massive scalar fields; $H$ in Eq.(LABEL:SUSY3) again has an
emergent $U(1)$ symmetry. However, in an intermediated scale between these two
mass scales (assuming $M_{1}>M_{2}$), $U(1)$ symmetry is always explicitly
broken. So for general masses $M_{1,2}$, $U(1)$ symmetry is absent in
Eq.(LABEL:SUSY3), unless $M_{1}=M_{2}$ (and $g_{Y1}=g_{Y2}$).

Only in the equal mass limit, $U(1)$ symmetry is present both in the infrared
limit of $\Lambda<M_{1}=M_{2}$ and ultraviolet limit $\Lambda>M_{1}=M_{2}$.
The $U(1)$ symmetry above the mass gap is unique to the case of $M_{1}=M_{2}$
and is crucial for the symmetry in the gapless limit. It is this ultraviolet
symmetry of gapped scalar fields that eventually emerge in the infrared limit
as the gaps become lower and close and the state becomes gapless.

Consequently, such a $U(1)$ symmetry in general is absent in the gapless limit
when only one of the masses, $M_{1}$ or $M_{2}$ becomes zero but the other one
is non-zero, i.e. at fixed points a) and b) above. In the gapless limit, one
can again verify that $H$ in Eq.(LABEL:SUSY3) is invariant under such a
unitary rotation only when $M^{2}_{1}=M^{2}_{2}\rightarrow 0$ as in case c).
In this limit, $g_{Y1}\rightarrow g_{Y2}$ as renormalization effects set in in
the infrared scale and don’t require fine tuning. Furthermore, the strong
coupling fixed point in this limit with a $U(1)$ symmetry shall also be
supersymmetric as pointed out in previous studiesLee07 ; Fei16  where strong
evidence has been presented for $2d$. In other words, in $2d$. among all fixed
points a), b), and c) listed in Eq.(34) below Eq.(LABEL:SUSY3), only the
supersymmetry fixed point c) has an emergent $U(1)$ symmetry. Later, we will
apply these results to studies on TQCPs.

So in $2d$, we intend to identify the fixed point in Eq.(28) that always has
an emergent $U(1)$ symmetry, with the supersymmetry fixed point in the more
general model of Eq.(LABEL:SUSY3) where full $U(1)$ symmetry only appears when
$M_{1}=M_{2}$. Especially, when $M^{2}_{1}=M^{2}_{2}=0$ or at $X=Y=0$ in
Fig.2, CFT states are a stable phase with an emergent supersymmetry of
$N_{f}=\frac{1}{2}$ for general $g_{Y1},g_{Y2}$; it effectively describes a
four-component majorana fermion field or a two-component Dirac field
interacting with a complex boson field $\Phi=\phi_{1}+i\phi_{2}$.

To summarize, we have observed that when a $U(1)$ symmetry emerges in a non-
local dynamic model, strong coupling fixed points of gapless majorana fermion
fields shall have, in addition to standard conformal symmetries, a
supersymmetry. At time reversal symmetric TQCPs defined in Eq.(LABEL:TQCPTRI),
we can say that supersymmetry naturally emerges in strongly interacting
majorana fermions as a result of an emergent U(1) symmetry.

In more generic cases when a) $M_{1}\neq M_{2}=0$ or b) $M_{1}=0\neq M_{2}$,
an asymmetry in masses forbids an emergent $U(1)$ symmetry in the gapless
limit. The strongly interacting gapless majorana fermions in a) and b) shall
just have standard scale-conformal symmetries as in GNFPs without additional
higher symmetries. $U(1)$ symmetries only emerge in the low wavelength limit
when both masses $M_{1,2}$ are finite but do not appear when one of masses
vanishes but the other remains finite. Therefore, GN fixed points in this
limit shall be identified as TQCPs with time reversal symmetry where
interactions between majorana fermions can not be simply characterized by
local four-fermion operators. In other words, when non-local interactions like
the ones in Eq.(LABEL:SUSY3) are present to break the emergent $U(1)$ symmetry
displayed in Eq.(LABEL:TQCPTRI), the corresponding dynamics in the gapless
limit shall be in a non-supersymmetric GNFP class.

Figure 4:  $1d$ (a),b)) and $3d$ (c), d)) time reversal symmetric TQCPs in
strong coupling limits. $m$ is the tuning fermion mass of TQCPs and $g$ is an
effective interaction constant in models with local interactions,
Eq.(LABEL:SUSY3),(LABEL:SUSY2), and $X=M_{1}^{2}$, $Y=M_{2}^{2}$ are masses of
scalar fields in dynamic models in Eq.(LABEL:SUSY3). a) is for $1d$ TQCPs when
interactions result in spontaneous mass generation but do not result in time
reversal symmetry breaking (i.e. symmetry breaking forced to be exactly along
$\phi_{1}$ direction in Eq.(LABEL:SUSY3)). The thick (red) line represents a
$(d+1)$th transition lines obtained in Ref.Yang19  and dashed ones for
discontinuous first order transition lines. b) represents $1d$ TQCPs with
local interactions (i.e. Eq.(3)) where time reversal symmetry is spontaneously
broken in strong coupling limits beyond the red termination point (because of
breaking of an emergent $U(1)$ symmetry). In $1d$, all the termination points
in a) and b) are free fermion fixed points (FFFP) unlike in $2d$. In c)(with
$X=X_{0}>0$) and d)(with $Y=Y_{0}>0$) for $3d$, termination points are free
fermion-boson fixed points (FFBFP) where new free gapless real bosons emerge
in the infrared limit but decoupled from the free gapless majorana fermions.
Paths in b) and c) connect two topological distinct states without closing the
fermion mass gap because of spontaneous breaking of protecting symmetry; i.e.,
two states can effectively be smoothly deformed into each other. (Time
reversal symmetry is broken spontaneously in shaded areas.)

##  VII Time reversal symmetric TQCPs in Strong Coupling Limits

In our previous studiesYang19 ; Yang21 , we have identified that a broad class
of TQCPs belong to weakly interaction majorana fermion classes, with or
without Lorentz symmetry. Now we are in a position to discuss those time
reversal symmetric TQCPs in strong coupling limits and what happens to the
free majorana fermion universality class. We will first focus on $2d$ which is
the most interesting limit and come back to $1d$ and $3d$ later. Discussions
in this section are entirely based on the results in Sec. VI on various strong
coupling fixed points of majorana fermions.

###  VII.1  $2d$ TQCPs in Strong Coupling Limits

In case a) in the previous section, we drive TQCPs from a weakly interacting
regime ($X>0,Y>0$ in Fig.2 ) into a strongly interacting regime by varying
$X=M^{2}_{1}$ across zero from a positive side $X=M^{2}_{1}>0$ while holding
$Y=M_{2}$ to be a constant. Strong interactions in $X<0$ side result in a
finite fermion mass, $g_{Y1}\langle\phi_{1}\rangle$, that respects the time
reversal symmetry. The ground state is gapped and two-fold degenerate because
of spontaneously $Z_{2}$ reflection symmetry breaking, and furthermore they
are time reversal symmetric. The $(d+1)$th transition line previously pointed
out in the weakly interacting limitYang19  is therefore terminated by a
standard GN fixed point of type a) beyond which transitions are first order
and discontinuous.

However, in case b) we drive TQCPs into strongly interacting limit by varying
$Y=M^{2}_{2}$ from a positive side across zero while holding $X=M^{2}_{1}$ to
be finite. when $-\infty<g^{2}_{Y2}/M^{2}_{2}<0$, time-reversal-symmetry
breaking mass of fermions, $g_{Y2}\langle\phi_{2}\rangle$, is spontaneously
generated which is qualitatively different from case a) where this symmetry is
respected. Strong interactions therefore can not only open a finite mass gap
of fermions at TQCPs but further lead to a connection between two otherwise
symmetry protected topological distinct phases via a symmetry breaking mass.
Topological charges without protecting symmetries are ill defined and one
anticipates that two topologically distinct superconducting states can be
effectively deformed into each other without closing the fermion mass gap in
this limitLP . The $(d+1)$th transition line is therefore simply terminated at
GNFPs of type b) in Eq.(34) and no direct transitions can occur beyond that
point.

In c), we drive TQCPs from the weakly interacting regime into strongly
interacting regimes by varying $M^{2}_{1}$ across zero while holding
$M^{2}_{1}/M^{2}_{2}$ to be exactly one. Time reversal symmetry breaking mass
is generated as a result of spontaneously breaking of $U(1)$ symmetry when
$\langle\phi_{1}\rangle^{2}+\langle\phi_{2}\rangle^{2}$ remains constant. This
mass aspect is similar to case b). So the $(d+1)$th transition line is again
simply terminated at fixed point type c). However because of an emergent
$U(1)$ symmetry, the termination point is now supersymmetric with
$N_{f}=\frac{1}{2}$. These results are summarized in Fig.3. Continuous weakly
interacting time reversal symmetric TQCPs in $d=2$ therefore can be terminated
by universal strong coupling fixed points a), b), c), beyond which there will
be either no direct transitions if the protecting symmetry is broken
spontaneously or discontinuous first order transitions. Details further depend
on dynamics and breaking of protecting time-reversal symmetry, and whether
there is an emergent supersymmetry.

###  VII.2 $1d$ TQCPs

So far, we have focused on time reversal symmetric TQCPs in $2d$. In $1d$,
there is only a free fermion fixed point in Eq.(LABEL:TQCPTRI),(28) with
$\tilde{g}=0$. Local interactions are marginal as suggested in
Eq.(LABEL:TQCPTRI), (28) and in this limit, there are no other strong coupling
fixed points as in a standard upper critical dimension. Especially,
interactions are marginally irrelevant when $\tilde{g}>0$ and marginally
relevant when $\tilde{g}<0$. This indicates that $\tilde{g}$ flows into
$\tilde{g}^{*}=-\infty$ when $\tilde{g}$ is negative and masses are
spontaneously generated even at weak coupling limit through the well-known
Gross-Neveu mechanismGross74 . On the other hand, free gapless majoarna
fermions are stable as far as $\tilde{g}>0$.

Therefore for time reversal symmetric TQCPs (spinful case only) in $1d$,
$(d+1)$th order or $2$nd order transition lines in $1d$ only exist when
$\tilde{g}>0$ and are terminated exactly at a free fermion fixed point
$\tilde{g}=0$. Beyond the termination point, there shall be first order
transitions if the protecting symmetry is unbroken spontaneously. However, if
interactions are strictly local, emergent $U(1)$ symmetry is broken beyond
point $\tilde{g}=0$ and then there will be no direct transitions between two
topologically distinct superconductors when $\tilde{g}<0$. Instead, two
topologically different states can again be smoothly deformed into each other
without closing the fermion mass gap. So the main difference between $1d$ and
$2d$ is that termination points in $1d$ can not be GNFP or SUSY fixed points
but are trivial free fermion fixed points (FFFP). Lines of $2$nd order TQCPs
are now terminated by free majorana fixed points beyond which there are either
$1$st order transitions or no direct transitions. At FFFP, dynamics are set by
weakly interacting gapless majorana fermions.

###  VII.3 $3d$ TQCPs in Strong Couling Limits

Just like in $2d$, there are strong coupling fixed points at
$\tilde{g}^{*}=-(d-1)$ in $3d$ in addition to the free fermion fixed point. So
again, strong coupling physics can be better understood in terms of emergent
scalar fields Eq.(LABEL:SUSY3). Eq.(LABEL:SUSY3) has the same phases as in
$2d$ as shown in Fig.2. However, it turns out in $3d$ that in the massless
limit of Eq.(LABEL:SUSY3) when $X=0$ or $Y=0$ or $X=Y=0$, the Yukawa coupling
constants $g_{Yi}$, $i=1,2$ flow into non-interacting fixed points instead of
GNFPs or SUSY fixed points and the theory in this limit becomes trivial.
Effectively, in $3d$ at strong coupling fixed points $\tilde{g}^{*}=-2$, there
are well defined emergent gapless real scalars fields but completely decoupled
from real fermions. Strong interactions between majorana fermions near
$\tilde{g}^{*}$ in $3d$ therefore mainly result in a new degree of freedom
represented by free gapless real bosons. The termination points of $(d+1)$th
order continuous transition line in $3d$ are now described by decoupled
gapless free majorana fermions and gapless free real bosons and we call it a
free fermion-boson fixed point (FFBFP). The gapless real bosons represent new
emergent degrees of freedom, unique to strongly coupled majorana fermions in
$3d$. So all the termination points (red dots in Fig. 3 ) discussed in $2d$
are still present but instead of being conformal GNFP, they become infrared
trivial FFBFP. This concludes our discussions on $3d$ TQCPs with time reversal
symmetries. And because of emergent real bosons, these termination points are
also different from $1d$ termination points which are represented by simple
free gapless real fermions, without emergent gapless real bosons. Results in
$1d$ and $3d$ are summarized in Fig.4.

##  VIII Open Questions on Topological Quantum Criticality with Larger Global
Symmetry Groups

So far we have concentrated on TQCPs that either break all symmetries in a
superconductor or with a (minimal) global symmetry of time reversal. We find
that TQCPs that break all global symmetries can be fully described by majorana
fermion fields of $N_{f}=\frac{1}{4}$ while TQCPs with time reversal symmetry
are characterized by majorana fields of $N_{f}=\frac{1}{2}$. Strong coupling
fixed points of these gapless majorana fermions terminate continuous $(d+1)$th
order phase transitions between topologically distinct states. Beyond these
strong coupling fixed points, for TQCPs without time reversal symmetry, mass
gap opens up signifying first-order phase transitions separated from the
weakly interacting $(d+1)$th order transitions by supersymmetric fixed points.

For time reversal symmetric TQCPs, if time reversal symmetry respecting
fermion mass gaps are opened up in strong coupling limits, ground states at
TQCPs are expected to be time reversal symmetric and are two fold degenerate
with $m=\pm|m|$. We again identify these states with first order transitions
separating topologically distinct phases. A CFT state of GNFP class separates
continuous $(d+1)$th order transitions line in the weakly interacting limit
from the discontinuous transition line.

We have also found that for time reversal symmetric TQCPs, time reversal
symmetry breaking majorana mass gaps can also be spontaneously generated in
the limit of strong coupling. In $2d$, the $(d+1)$th order transition line is
then completely terminated by strong coupling Gross-Neveu fixed points beyond
which states become simply connected without closing gaps. If majorana
fermions at TQCPs only interact locally, additional emergent $U(1)$ symmetry
leads to supersymmetric termination points. Spontaneous $U(1)$ symmetry
breaking simultaneously leads to breaking of protecting time reversal symmetry
of topological superconductors and topological states can again be deformed
smoothly.

What happens to superconducting TQCPs in $d=2,3$ with larger global symmetry
groups remains to be systematically studied in the future. When symmetry
groups get larger, degrees of freedom associated with majorana fields also
increase linearly leading to power-law large number of local interaction
operators. These operators can form different representations of subgroups of
global symmetries and dimensions of interaction parameter space become much
higher than one dimension or two dimensions that we have so far focused on in
this article. In that case, there are no fundamental principles that forbid
Green’s functions discussed in Sec. II from developing more peculiar
structures without spontaneously breaking protecting symmetries. For instance,
the bulk-(gapless) boundary correspondence in weakly interacting limits that
relies on an index theorem can be severely violated because of emergence of
zeros in Green’s functions in strongly interacting limitsGurarie11 . So an
interacting topological state with protecting symmetries might be deformed
into ones with different topological invariants without a transition.

As TQCPs appear to be infrared stable in $d=2,3$ where local fermion
interaction operators are infrared irrelevant, the bulk-gapless boundary
correspondence near weakly interacting TQCPs shall remain valid. If smooth
deformation between states in $d=2,3$ with different topological invariants
does happen while protecting symmetry is not spontaneously broken, one can
speculate that it likely involves some strong coupling fixed points where the
standard bulk-gapless boundary relation starts breaking down. Under which
specific conditions of interactions and when this can happen or can not happen
in a topological superconductor need to be investigated in the future.
Occurrence of smooth deformation in strong coupling limit also suggests that
the topological-invariant approach outlined in Sec. II in general is
inadequate and one has to introduce more general topological classification to
properly represent strongly interacting topologically matter with certain
global symmetriesFidkowski10 ; Fidkowski11 ; Fidkowski13 ; Metlitski15 ;
Wang15 ; Song17 ; Xu21 .

The possibility of such a scenairo can be closely related to the well-known
Fidkowski-Kitaev path of escape in one spatial dimensionFidkowski10 . There a
global time reversal symmetry is embedded in a much bigger ($so(8)$) symmetry
group unlike what we have in this article. Using eight chains of majorana
fermions, they have successfully constructed $so(7)$-subgroup invariant on-
site interaction operators embedded in $so(8)$ representations and utilized
them to determine an $so(7)$ symmetric line boundaries between first order
transitions and no-transition regions, in addition to an $so(8)$ line boundary
separating regions of second phase transitions and no-transitions. These
highly symmetric line boundaries have been applied to explicitly identify a
path that smoothly connects two gapped topologically different states in a
parameter space. In addition, emergent gauge symmetries and gauge fields
discussed in Ref.Bi19  can also be relevant to TQCPs in certain strong
coupling limits although it remains to be investigated when they become so in
topological superconductors and superfluids.

##  IX Conclusions

In conclusion, we have discussed the important role of global symmetries at
topological quantum critical points (TQCPs). The global symmetries determine
the numbers of emergent fermion fields at a generic TQCP. The number
corresponds to $N_{f}=\frac{1}{4}$ or a single two-component majorana field,
when all symmetries including $U(1)$ global gauge symmetry are broken, and is
$N_{f}=\frac{1}{2}$ or a four component majorana fermion field when a time
reversal symmetry is present at a TQCP without additional global symmetries.
This number will increase when more symmetries are present at TQCPs until it
reaches a maximum number of $N_{f}=N/2$ where $N(=2n)$ is the number of
microscopic complex fermions available at a TQCP. In general, the number of
emergent low energy majorana fermion fields at a TQCP is independent of the
number of complex fermions in a solid, $N$, and only depends on global
symmetries. We have presented our results in both cases with or without time
reversal symmetry. This is also expected to lead to distinctly different
thermodynamics and dynamics in quantum critical regimesYang21b .

We have also discussed strong coupling limits of TQCPs. We have found that
without time reversal symmetry, $2d$ TQCPs as strong coupling fixed points are
supersymmetric with $N_{f}=\frac{1}{4}$. Furthermore, the supersymmetric CFT
fixed point terminates the weakly interacting $(d+1)$th order continuous
transition line and separates it from a first order order transition line
beyond the supersymmetric state. Two topologically different superconductors
in this case are always separated by a transition line even in strong coupling
limits.

However, there are two different classes of strong coupling CFT states for
$2d$ TQCPs with one global symmetry of time reversal invariance. One of them
has an emergent $U(1)$ symmetry and is supersymmetric with
$N_{f}=\frac{1}{2}$. They describe strongly coupling physics of majorana
fermions only interacting locally near a time reversal symmetric TQCP. In
general, time reversal symmetric strong coupling TQCPs are CFT states without
higher symmetries such as supersymmetry or $U(1)$ symmetry; they correspond to
Gross-Neveu fixed points (GNFPs) with $Z_{2}\times Z_{2}$ symmetries of time
reversal ${\mathcal{T}}$ and reflection ${\mathcal{R}}$.

In the limit we have focused on if time reversal symmetry is not broken
spontaneously, two topological states are either separated by a $(d+1)$th
continuous transition line or a first order phase transition and can not be
deformed into each other smoothly. The strong coupling fixed points that
separate them are always GNFPs. However, if protecting time reversal symmetry
is spontaneously broken in the presence of strong interactions, then the
continuous transition lines emitted from weakly interacting limits will be
simply terminated beyond which states can be deformed into each other while
the fermion mass gap remains open. Termination points of $(d+1)$th continuous
transition lines in $2d$ can be either supersymmetric with $N_{f}=\frac{1}{2}$
or non-supersymmetric GNFPs depending on whether the time reversal symmetry
breaking occurs via spontaneous $U(1)$ symmetry breaking or not.

In $1d$, continuous transition lines of $(d+1)$th order time reversal
symmetric TQCPs are always terminated by a free majorana fermion fixed point
beyond which there is a first order phase transition if protecting symmetry is
not broken spontaneously or no direct transitions if symmetry is broken. In
$3d$ time reversal symmetric TQCPs, these termination points, beyond which
there are either first order transitions or no transitions again depending
symmetry breaking, are strongly interacting and can be effectively
characterized by emergent free gapless real bosons weakly coupled to free
gapless majorana fermions.

To summarize, at TQCPs without any global symmetries, we find that two
topologically distinct states are always separated by transitions even in
strong coupling limits. At TQCPs only with time reversal symmetry but no other
global symmetries, we find that two topologically distinct states are always
separated by transitions even in strong coupling limits we have focus on if
the protecting symmetry is not broken spontaneously. Smooth deformation
without gap closing becomes possible only when time reversal symmetry is
broken spontaneously.

The author would like to thank Ian Affleck, Zhengcheng Gu, Sung-Sik Lee and
Xiao-Gang Wen for inspiring discussions and Fan Yang for discussions on
regularizing interacting majorana fields. He also wants to thank the TDLi
workshop on ”New Frontiers in Extremely Strongly Interacting Matter” in July,
2021 where interactions and topology were discussed and the KITP workshop on
Topological Matter: Atomic, Molecular and Optical Systems (topology21) for its
hospitality. This project is in part support by a Professional Development
Grant from the University of British Columbia, and by an NSERC (Canada)
Discovery grant under contract RGPIN-2020-07070.

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