# An Interaction Bulk-Boundary Relation
and its Applications Towards Symmetry Breaking and Beyond

Saran Vijayan and 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 propose a simple but general scaling relation between
interactions in a gapped bulk topological matter and gapless interacting
surface states. We explicitly illustrate such a generic bulk-boundary relation
(BBR) for a few specific interactions in a topological quantum matter, where
we can perform dimensional reduction of a microscopic bulk theory to project
out interacting surfaces. We have examined renormalization effects of the
gapped bulk fermions on the interacting topological surface fermions. As
simple applications, we utilize effective interacting quantum fields implied
by BBR to explore feasibility of routes to various fascinating emergent
phenomena on surfaces including emergent Majorana fermions induced by
spontaneous symmetry breaking. We obtain sufficient conditions for these
interacting surface phenomena to take place. We have also found that for given
bulk electron-phonon interactions and when $\Omega_{D}\geq m$, the phonon-
mediated interactions on surface are strongest if the bulk Debye frequency
$\Omega_{D}$ matches $m$, the mass gap of the topological matter.

##  I Introduction

A hallmark manifestation of topology in quantum matter is the existence of
robust or protected boundary states in many topological quantum matter. One
very well-known one is conformal chiral Luttinger liquids along edges of
incompressible fractional quantum Hall states (FQHs)Wen1 ; Wen2 ; Chang . It
is generally believed that in topologically ordered states such as FQHs, there
is a general bulk-boundary correspondence between bulk states and conformal-
field theory boundaries and a boundary offers a unique holographic
representation of complicated interacting bulks.

In relatively more recent studies of topological statesKane05 ; Bernevig06a ;
Bernevig06b ; Fu07 ; Moore07 ; Qi10a ; Hasan10 ; Qi11 ; Bernevig  that are
usually defined in a non-interacting limit, relations between gapped bulks and
boundaries become trickier because of additional axes in the parameter space
that can be further assigned to interactions. A gapped bulk with gapless
surfaces imply that although interactions can be weak from a bulk perspective
and do not deform the bulk matter in topological non-trivial ways, it is
possible that they substantially reshape non-interacting surfaces. So gapless
surfaces that usually require various symmetry protections are in general much
less robust to interactions because of symmetry breaking phenomena.

For topological insulators that have been quite extensively studied over the
last decade or so, if the symmetry breaking is explicit, say either due to
magnetic coating or coating of a superconducting film that breaks the U(1)
symmetry, surfaces can be gapped and belong to a different class of boundary
statesFu08 ; Lutchyn10 ; Lutchyn11 . In fact, gapless surfaces can be thought
to be quantum critical near zero magnetic or zero tunneling of $U(1)$ symmetry
breaking fields; and there is a surface quantum criticality that one can
associate with tuning parameters that break various protecting symmetries.

If all protecting symmetries are fully respected by external fields,
interactions can in principle still lead to spontaneous symmetry breaking on
surfaces without deforming the gapped bulk in a non-trivial way. The one-to-
one bulk-boundary correspondence, i.e. the holographic principle suggested in
topological ordered FQHs therefore does not directly apply here. For instance,
at the simplest level, one can imagine that a weakly interacting bulk can have
either a standard weakly interacting gapless surface respecting all protecting
symmetries, or have a strongly interacting surface that spontaneously breaks
one of the protecting symmetries without coated external substrates. In the
latter case, topological surfaces are in a completely different phase so the
same bulk can have at least two very different boundaries further depending on
interactions, especially surface interactions.

It is fascinating that in theory, surfaces can be even topologically ordered
and further support fractionalized emergent new particles if all the
protecting symmetries are respected and if there is no spontaneous symmetry
breakingFidkowski ; Metlitski ; Wang1 ; Wang2 ; Song . The microscopic origin
of such exotic states hasn’t been fully understood. Numerical simulations of
the surface model suggest that it is unlikely to realize those states with
simple local interactionsNeupert . Nevertheless, if they occur in topological
surfaces, they must be driven by interactions, possibly long-range ones.

One very attractive feature of weakly or non-interacting gapless boundaries,
either $1D$ edges or $2D$ surfaces are that it does not have the standard
Fermion-doubling as in a corresponding bulk $1D$ or $2D$ respectivelyWu . If
ones’ goal is to utilize very unique boundaries of this type as building
blocks toward the more fascinating and exotic quantum matter with variable
holographic features, then perhaps one key practical step to take is to
further understand how topological surfaces are interacting. Specifically, as
all surfaces are boundaries of a bulk, one shall anticipate there is a general
principle that connects boundary interactions and bulk ones, hence a bulk-
boundary relation (BBR) of interactions. In this article from now on, we will
reserve the notion of BBR for this purpose only and focuses exclusively on the
relations between bulk and surface interactions.

The main object here is to illustrate a simple and practically very useful BBR
that one can apply to construct effective field theories of interacting
surface fermions given the inputs of bulk interactions. Specially, we will
establish concrete relations between interaction constants in the effective
field theories of surface fermions and the bulk interactions that are assumed
to be known.

The article is organized as follows. In Sec. II, we will put forward a
hypothesis BBR of which relates surface interactions to bulk ones. The
hypothesis is based on a simple phenomenology that all surface interactions
shall uniquely and explicitly depend on bulk interaction constants. When
implemented in topological states, one further notices that an energy gap of
bulk states, $m$, above which the bulk spectrum lies naturally defines an
infrared scale (IR) of bulk dynamics. Therefore at scale $m$, bulk
interactions that are usually introduced at a much higher ultraviolet scale
associated with the bulk bandwidth $W(\gg m)$ follows an infrared
renormalization flow induced by a set of Renormalization Group equations
(RGE).

On the other hand, the mass scale $m$ also defines the spatial extend of
surface states into the bulk interior and can be treated as a characteristic
scale of the momenta perpendicular to the surface. Therefore, $m$ appears as a
natural intermediate energy scale of gapless surface states around which
surface particles appear and interact. Interacting surface fermions at this
scale $m$ can differ very little from bulk fermions interacting at the same
scale.

We anticipate that the IR flow of RGE defined in the bulk can thus be matched
with surface interactions defined at scale $\Lambda_{s}=m$ for low energy
gapless surfaces(see Fig.1). This general matching condition can be employed
to quantify the principle of BBR and construct interaction constants in the
effective field theories of gapless surface fermions. For a few concrete
interactions, we derive specific relations between bulk and boundaries. In
Sec. III, we further apply a microscopic dimension reduction technique in a
few limiting cases to illustrate the general principle and offer more
quantitative features of the concrete relations. In Sec. IV, we discuss a
specific application of our results to surface spontaneous symmetry breaking
phenomena that lead to surface topological superconductivity and emergent
Majorana fermions. We further apply the same idea to investigate a possibility
of making supersymmetry holographic matter using interacting surfaces.

Figure 1:  Schematic of a portion of the energy spectrum of electrons in the
bulk(Brown lines) and the boundary(Green lines) in a gapped topological
insulator. The mass parameter $m$ defines an energy gap of bulk states. Bulk
electrons lie in the range $\Lambda_{b}\in[m,W]$, where W is the bulk UV cut-
off scale. The boundary states are gapless and low energy boundary electrons
of our special interests are approximately within the range of
$\Lambda_{s}\in[0,m]$. The velocity of fermions has been set to be unity.

##  II General Phenomenology and Matching Conditions

As surface states form boundaries of a bulk, it is tempting and possible to
relate effective surface fields to the microscopic bulk interactions in a
generic way. Let us consider a general Hamiltonian

$\displaystyle{\mathcal{H}}$ $\displaystyle=$
$\displaystyle{\mathcal{H}}_{bulk}+{\mathcal{H}}_{surf};$
$\displaystyle{\mathcal{H}}_{bulk}$ $\displaystyle=$
$\displaystyle{\mathcal{H}}_{bulk}(\\{G_{i}\\},i=1,..,N;W),$
$\displaystyle{\mathcal{H}}_{surf}$ $\displaystyle=$
$\displaystyle{\mathcal{H}}_{surf}(\\{g_{j}\\},j=1,..,M;W_{s})$

where ${\mathcal{H}}_{bulk(surf)}$ are the Hamiltonian acting on bulk
(surface) states. $\\{G_{i}\\},i=1,...,N$ and $\\{g_{j}\\},j=1,...,M$ are two
sets of interaction constants specifying bulk and surface interactions
respectively.

As stated before, $W$ is an UV scale for bulk physics which can be naturally
associated with the bandwidth. $W_{s}$ is an UV scale of surface states which
further depend on surface band microscopic structures as well as types of
interactions of interests. And typically, both $W$ and $W_{s}$ are much bigger
than the mass gap $m$.

The mass gap plays a paramount role in our discussions of BBR. For topological
states we will focus on, $m$ can be thought as an IR scale of bulk physics
above which all bulk states lie. On the other hand, the mass gap
$m\in[0,W_{s}]$ simultaneously is also a characteristic intermediate energy
scale of gapless surface dynamics because $\frac{1}{m}$ is set by the spatial
spread of surface states into the bulk interior. So indeed at this particular
scale of $m$, two dimensional surface states appear to be no different from
three dimensional bulk ones at the same scale. The matching between the IR
physics of the interacting bulk and the physics of interacting gapless surface
states at this special intermediate scale of $m$ is a key observation that
leads to the principle of interaction BBR.

There is an alternative and perhaps simpler view of BBR . For certain
discussions, it is even possible to treat $m$ simply as an effective UV scale
of interacting gapless surface fermions or $W_{s}\sim m$. And we can let the
scale transformation generated by RGEs of surface fermions run from a surface
IR scale of nearly zero energy, with some pre-assumed interactions, toward
some ultraviolet energies rather than the other way around. In that case, we
can conveniently match the IR oriented flow of the bulk interactions with the
UV oriented flow of the surface interactions at scale $m$. The match leads to
a surface theory formulated at the scale $m$ with bare interaction constants
determined by the matching conditions in BBR (see Fig.1, Fig.2).

In more generic situations where $W_{s}\geq m$, one can still apply BBR to
construct an effective field theory at an intermediate scale $m$. Once the
matching condition has been employed at scale $m$, following the UV flow
generated by RGEs (see more discussions in section IV), one can also easily
further extend the theory to the scale $W_{s}$, that is usually higher than
$m$ to construct the full surface field theory up to the UV scale $W_{s}$. Of
course if one wishes, one can also reverse the flow to scales smaller than $m$
back to infrared when needed and suitable. In this case, with the effective
surface field theory already fixed at the effective UV scale $W_{s}$ or an
intermediate scale $m$, one can find out the pre-assumed near zero energy
theories.

Figure 2:  A cartoon of a) bulk scale transformation from an $UV$ scale
$\Lambda_{b}=W$ to the bulk infrared scale $\Lambda_{b}=m$ (red online) and b)
surface scale transformation from an UV scale $\Lambda_{s}=W_{s}$ to
$\Lambda_{s}=m$ (red online) to an infrared scale $\Lambda_{s}\ll m$ (blue
online) of surface fermions in a 3D momentum space. In b), the scale
transformation is performed only in the two dimensional horizontal plane while
the vertical momentum remains fixed at a scale similar to $m$. a) and b)
suggest a bulk-boundary matching condition. The effective theories obtained in
a) and b) at scale of $m$ (red online) appear to be equivalent and so can be
matched to obtain a BBR relation between bulk and surface interactions. (See
the main text for details of BBR).

Formally speaking, following the general scheme of renormalization and
scaling, both bulk and surface fields can be equivalently redefined at an
arbitrary running UV energy scale $\Lambda$. In the rest of discussions, for
convenience and simplicity, for bulk interactions, $\Lambda$ is chosen to be
larger than $m$, while for surfaces, we restrict our discussions to energy
scales lower than $m$ (although all these conditions can be relaxed in
general). The effective theories defined at $\Lambda_{b,s}$, where subscripts
$b,s$ are for bulk and surface respectively, can be expressed as

$\displaystyle{\mathcal{H}}_{bulk}$ $\displaystyle=$
$\displaystyle\Lambda^{z}_{b}{\mathcal{F}}_{bulk}(\\{\tilde{G}_{i}(\Lambda_{b}),i=1,...,N\\},Z_{f}(\Lambda_{b});\frac{m}{\Lambda_{b}}<1),$
$\displaystyle{\mathcal{H}}_{surf}$ $\displaystyle=$
$\displaystyle\Lambda^{z}_{s}{\mathcal{F}}_{surf}(\\{\tilde{g}_{j}(\Lambda_{s}),j=1,..,M\\},Z_{f}(\Lambda_{s});\frac{m}{\Lambda_{s}}>1).$

Here we have defined $z$ as the dynamical critical exponent of our system
which will be set to be $z=1$. ${\mathcal{F}}_{bulk,surf}$ are dimensionless
and dimensionless coupling constants $\tilde{G}_{i}(\Lambda_{b}),i=1,..,N$,
and $\tilde{g}_{j}(\Lambda_{s}),j=1,...,M$ are further introduced to
facilitate discussions below. Their scale dependence on $\Lambda_{b,s}$ can be
obtained via the standard renormalization group analysis and details will be
presented later. $Z_{f}$ stands for the fermion field renormalization which
does not play significant role in our phenomenological discussions and we will
mute from now on.

An important phenomenology here is that bulk interactions defined at
$\Lambda_{b}=m$ in $\mathcal{H}_{bulk}$ shall describe, effectively and
precisely, interacting boundary dynamics of $\mathcal{H}_{surf}$ defined at
$\Lambda_{s}=m$. One clear evidence for this is that surface states can merge
into bulk spectrum at scale $m$ and there is no longer a clear separation
between bulk and surface states at that energy scale. Therefore, at least if
in the case when $N=M$ and if all bulk interactions adiabatically evolve into
surface counterparts as $\Lambda_{b}$ is lowered toward $m$, one can
anticipate surface interactions at that UV scale of $\lambda_{s}=m$ shall be
equivalent to bulk interactions. In this particular case, we have a rather
simple relation at the IR scale $\Lambda_{b}=m$,

$\displaystyle\tilde{g}_{i}(\Lambda_{s}=m)=\tilde{G}_{i}(\Lambda_{b}=m),i,j=1,...,N=M.$
(3)

In more general cases, Eq.3 shall be replaced by a more general relation,

$\displaystyle\tilde{g}_{i}(m)=\Pi_{i}(\\{\tilde{G}_{j}(m)\\},j=1,...,M),i=1,...,N$
(4)

where $\Pi_{i},i=1,..N$ are dimensionless functions that can be generated by
general matching conditions. Below, we will further illustrate explicitly
these functions in a few cases.

Figure 3:  The schematic of bulk-boundary relation of interactions. (a) and
(b) illustrate surface fermions interacting with higher energy bulk phonons or
photons without or with fermion polarization effects respectively. A fermion
loop in (b) indicates general bulk renormalization effects that lead to
suppression of coulomb interactions at lower energy scales. (c) illustrate
effective surface interactions in the infrared limit. The bulk-boundary
relation principle matches a)-b) and c) at a mass gap $m$ that effectively
defines a surface band width.  Figure 4:  Diagrams that contribute to the
renormalization group equations in Eq.20 . Solid lines are for fermions; wave
lines are for phonons and dashed ones for Coulomb interactions. a) and b) are
for phonon-electron interactions and c) and d) are for Coulomb interactions;
e) only contributes to the momentum dependence of fermion-phonon interactions
$G_{fp}$, and does not directly contribute to Eq.20. (The vertex correction
and fermion self-energy contributions are not shown here as they lead to less
relevant terms in the infrared limit and also do not contribute to our
analyses. )

###  II.1 Surface Coulomb Interactions

Now let us take 3D massive Dirac fermions with coulomb interactions as an
effective theory for 3D bulk topological states as the simplest example.
Surfaces are then naturally described by interacting two-component helical
fermions.

$\displaystyle{\mathcal{H}}_{bulk}$ $\displaystyle=$ $\displaystyle\int
d^{3}{\bf r}\psi_{b}^{\dagger}\left[v_{F}\sigma_{z}\otimes{\bf s}\cdot{\bf
p}-m\sigma_{x}\right]\psi_{b}+$ $\displaystyle+$ $\displaystyle e_{b}^{2}\int
d^{3}{\bf r}d^{3}{\bf r^{\prime}}\psi_{b}^{\dagger}({\bf r})\psi_{b}({\bf
r})\frac{1}{|{\bf r}-{\bf r}^{\prime}|}\psi_{b}^{\dagger}({\bf
r}^{\prime})\psi_{b}({\bf r}^{\prime}),$ $\displaystyle{\mathcal{H}}_{surf}$
$\displaystyle=$ $\displaystyle v_{F}\int d^{2}{\bf r}\psi_{s}^{\dagger}{\bf
s}\cdot{\bf p}\psi_{s}$ (5) $\displaystyle+$ $\displaystyle e_{s}^{2}\int
d^{2}{\bf r}d^{2}{\bf r^{\prime}}\psi_{s}^{\dagger}({\bf r})\psi_{s}({\bf
r})\frac{1}{|{\bf r}-{\bf r}^{\prime}|}\psi_{s}^{\dagger}({\bf
r}^{\prime})\psi_{s}({\bf r}^{\prime})$

where ${\bf p}$ is a momentum gradient operator defined either in a bulk or on
the top surface (and we only show one surface here). $\psi^{\dagger}_{s}$ is a
creation operator of two-component helical surface fermions that can be
projected out of four component bulk operators, i.e.
$\psi^{\dagger}_{s}={\mathcal{P}}_{+}\psi^{\dagger}_{b}$ and
${\mathcal{P}}_{+}=(1+\tau_{z})$. Finally, $e_{s,b}$ are surface and bulk
charges respectively; they are identical only at a classical level and can be
drastically different once quantum renormalization effects due to
polarizations are properly taken into account.

Dirac fermions with long range Coulomb interactions had been well understood
in previous studies of graphenegonzalez99 ; schmalian07 ; sdsharma ; lucas .
Very similar calculations lead to simple renormalization group equations for
${\mathcal{H}}_{b,s}$ that we list below. We express them in terms of
dimensionless coupling constants for the bulk (b) and surface (s)

$\displaystyle\alpha_{\eta}=\frac{e_{\eta}^{2}}{\hbar v_{F\eta}},\eta=b,s.$
(6)

In terms of these dimensionless constants, we have

$\displaystyle\frac{\partial\alpha_{\eta}}{\partial t_{\eta}}$
$\displaystyle=$ $\displaystyle c_{\eta}\alpha_{\eta}^{2},\eta=b,s$
$\displaystyle\alpha_{b}(t_{b}=0)$ $\displaystyle=$
$\displaystyle\alpha_{b0},\alpha_{s}(t_{s}=0)=\alpha_{s0}$ (7)

where $t_{\eta}=\ln\Lambda_{\eta}/\Lambda_{\eta 0}$ is a standard
dimensionless running scale defined in terms of $\Lambda_{\eta}$, $\eta=b,s$.
$c_{\eta}$, $\eta=b,s$ are constants that do not carry much significance
(Using Wilson-Fisher RG analysisPeshkin , we get $c_{s}=\frac{1}{4\pi^{2}}$
and $c_{b}=\frac{5}{12\pi^{2}}$). Note that for $\alpha_{b}$, $\Lambda_{b}$
runs from $m$ up to the ultraviolet scale $\Lambda_{b0}=W$ while for
$\alpha_{s}$, $\Lambda_{s}$ runs below $\Lambda_{s0}=m$ because surface states
are present below mass gaps. The second line in Eq.7 defines the flow initial
condition for bulk and surface interactions at their corresponding ultraviolet
scales of $\Lambda_{b0}=W$, $\Lambda_{s0}=m$.

In this case, there is single interaction constant. We expect bulk Coulomb
interactions adiabatically evolves into surface ones at matching scale
$\Lambda=m$. Therefore,

$\displaystyle\alpha_{s}(\Lambda_{s}=m)$ $\displaystyle=$
$\displaystyle\alpha_{b}(\Lambda_{b}=m)\mbox{ or }$
$\displaystyle\alpha_{s0}(\alpha_{b0},\frac{m}{W})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{b0}}{1+c_{b}\alpha_{b0}\ln\frac{W}{m}}$ (8)

where we have solved Eq.7 for bulk interactions to derive the second line.

The results in Eq.8 appear to be quite intuitive. It states that interacting
surface shall be naturally characterized by the low energy effective theory of
bulk interactions. In this particular case, surface interaction constant is
simply equal to, the running coupling constant of the bulk
$\alpha_{b}(\Lambda_{b})$ at matching scale $m$. As a consequence, surface
interactions (dimensionless one) have to be a simple function of two
dimensionless parameters: 1) bulk interaction constant $\alpha_{b0}$ and 2)
surface ”bandwidth” or band gap, $m$, measured in terms of bulk bandwidth $W$.
As we will see later, this feature is a robust one and appears in all the
cases we are examining. It is a simple but remarkable manifestation of the
principle of interaction BBR.

###  II.2 Charge and velocity renormalization in bulk and on surface

Although the renormalization equations for bulk and surface coupling constants
$\alpha_{b,s}$ are qualitatively similar, mechanisms leading to them are very
different. Namely, in the bulk, both charge and velocity renormalization
contribute to the renormalization of coupling constant $\alpha_{b}$ while on
surface, charge renormalization is absent and the renormalization of coupling
constant $\alpha_{s}$ is solely due to the velocity renormalization. The
absence of surface charge renormalization effects is generic of $2D$ Dirac
fermions with Coulomb interactions and they were studied and pointed out in
previous studies of graphene physicsgonzalez99 ; schmalian07 ; sdsharma ;
lucas .

In the bulk, we find the velocity and charge renormalization to the one-loop
approximation are given by

$\displaystyle\frac{\partial\tilde{e}_{b}^{2}}{\partial
t}=c_{be}\alpha_{b}\tilde{e}^{2}_{b},\frac{\partial\tilde{v}_{F}}{\partial
t}=-c_{bv}\alpha_{b}\tilde{v}_{F};c_{be},c_{bv}>0.$ (9)

Eq.9 leads to Eq.7 for $\alpha_{b}$ with $c_{b}=c_{be}+c_{bv}$. Here
$c_{be}=\frac{1}{6\pi^{2}}$ and $c_{bv}=\frac{1}{4\pi^{2}}$

One can verify the following solution for the charge and velocity,

$\displaystyle\frac{\tilde{e}^{2}_{b}}{\tilde{e}^{2}_{b0}}=(\frac{\alpha_{b}(\Lambda)}{\alpha_{b0}})^{A},\frac{\tilde{v}_{F}}{\tilde{v}_{F0}}=(\frac{\alpha_{b}(\Lambda)}{\alpha_{b0}})^{-B}.$
$\displaystyle\text{where}\,\,\,A=\frac{c_{be}}{c_{be}+c_{bv}},B=\frac{c_{bv}}{c_{be}+c_{bv}};A,B>0$
$\displaystyle\text{and}\,\,\,\,\alpha_{b}(\Lambda_{b})=\frac{\alpha_{b0}}{1+c_{b}\alpha_{b0}\ln\frac{W}{\Lambda_{b}}}.$
(10)

Note that $A+B=1$. Taking into account the values of $c_{be}$ and $c_{bv}$, we
find that $A=\frac{2}{5}$ and $B=\frac{3}{5}$

On surfaces, $c_{se}=A=0$ hence there is no charge renormalization. On the
other hand, $c_{sv}=c_{s}$ and $B=1$. The velocity increases logarithmically
as low energies are approached. Similar aspects were discussed in two
dimensional graphene.

###  II.3 Phonon-Mediated Attractive Surface Interactions

Next example we will focus on is possible phonon-mediated attractive surface
interactions. For the sake of simplicity, in this part of discussion, we are
going to assume this is the only way fermions interact and neglect other
interactions such as Coulomb ones. In next subsection, we will restore other
interactions. We will show that again surface interactions can be simply
related to two important aspects of topological matter (and these are the only
two factors relevant to surface interactions): 1) bulk phonon-interaction
constants $\tilde{G}_{fp0}$(dimensionless) defined at a scale of phonon
bandwidth or Debye frequency $\Omega_{D}$; 2) surface state bandwidth $m$
measured in terms of phonon bandwidth, $\frac{m}{\Omega_{D}}$ which we assume,
without losing generality, to be less than one.

$\displaystyle{\mathcal{H}}_{bulk}$ $\displaystyle=$
$\displaystyle{\mathcal{H}}_{f}+{\mathcal{H}}_{ph}+{\mathcal{H}}_{I}$
$\displaystyle{\mathcal{H}}_{f}$ $\displaystyle=$ $\displaystyle\int d^{3}{\bf
r}\psi_{b}^{\dagger}\left[v_{F}\sigma_{z}\otimes{\bf s}\cdot{\bf
p}-m\sigma_{x}\right]\psi_{b},$ $\displaystyle{\mathcal{H}}_{ph}$
$\displaystyle=$ $\displaystyle\int d^{3}{\bf r}[{\bf\pi}({\bf
r})^{2}+[\nabla{\bf\phi}({\bf r})]^{2}]$ $\displaystyle{\mathcal{H}}_{I}$
$\displaystyle=$ $\displaystyle G_{fp}\int d^{3}{\bf
r}\vec{\nabla}\cdot\vec{\phi}({\bf r})\psi_{b}^{\dagger}({\bf r})\psi_{b}({\bf
r}),$ $\displaystyle{\mathcal{H}}_{surf}$ $\displaystyle=$ $\displaystyle
v_{F}\int d^{2}{\bf r}\psi_{s}^{\dagger}{\bf s}\cdot{\bf p}\psi_{s}$
$\displaystyle+$ $\displaystyle g_{1}\int d^{2}{\bf r}\psi_{s}^{\dagger}({\bf
r})\psi_{s}({\bf r})\psi_{s}^{\dagger}({\bf r})\psi_{s}({\bf r}).$

$\psi_{b}$ and $\psi_{s}$ are the bulk and the surface Dirac fields
respectively. Here phonon fields are expressed in terms of ${\bf\phi}$ and
${\bf\pi}$ which form a pair of canonical dynamical fields, i.e.

$\displaystyle[{\bf\phi}_{p}({\bf r}),{\bf\pi}_{q}({\bf
r}^{\prime})]=i\hbar\delta_{p,q}\delta({\bf r}-{\bf r}^{\prime}).$ (13)

The bulk theory when casted at a running UV scale ${\Lambda_{b}}$ can be
characterized by a dimensionless renormalization charge
$\tilde{G}_{fp}(\Lambda_{b})=G_{fp}\Lambda_{b}^{(D_{b}-1)/2}$ with $D_{b}=3$
for bulk; surface interactions on the other hand can be characterized by
$\tilde{g}_{1}(\Lambda_{s})=g_{1}\Lambda_{s}^{d_{s}-1}$ with $d_{s}=2$ for
surfaces. Under scale transformation, they transform according to the
following equations;

$\displaystyle\frac{\partial\tilde{G}_{fp}}{\partial t_{b}}$ $\displaystyle=$
$\displaystyle\frac{D_{b}-1}{2}\tilde{G}_{fp}$
$\displaystyle\frac{\partial\tilde{g}_{1}}{\partial t_{s}}$ $\displaystyle=$
$\displaystyle\left(d_{s}-1\right)\tilde{g}_{1}+\tilde{g}_{1}^{2}$
$\displaystyle\tilde{G}_{fp}(t_{b}=0)$ $\displaystyle=$
$\displaystyle\tilde{G}_{fp0},\tilde{g}_{1}(t_{s}=0)=\tilde{g}_{10}$ (14)

where again $t_{b}=\ln(\Lambda_{b}/\Omega_{D})$ and
$t_{s}=\ln(\Lambda_{s}/m)$; $\Lambda_{b}\in[m,\Omega]$ and
$\Lambda_{s}\in[0,m]$. It is worth remarking that in this limit, the one-loop
contribution to the RGE of $\tilde{G}_{fp}$ in Eq.14, which is of order of
$\tilde{G}_{ph}^{3}$, vanishes identically because the electron-phonon
interaction vertex is further proportional to the momentum transfer.

As in conventional metals, electron-phonon interactions mediate an attractive
interaction between bulk Dirac fermions. We can characterize such interactions
by a conventional four-fermion operator in the bulk,

$\displaystyle{\mathcal{H}}^{eff}_{I}$ $\displaystyle=$ $\displaystyle
g_{b}\int d^{3}{\bf r}\psi_{b}^{\dagger}({\bf r})\psi_{b}({\bf
r})\psi_{b}^{\dagger}({\bf r})\psi_{b}({\bf r}),$
$\displaystyle\tilde{g}_{b}(\Lambda_{b})$ $\displaystyle=$
$\displaystyle-\tilde{G}^{2}_{pf}(\Lambda_{b}).$ (15)

where $\tilde{g}_{b}(\Lambda_{b})=g_{b}\Lambda_{b}^{D_{b}-1}$.

At $\Lambda_{b}=\Lambda_{s}=m$, we then apply the following matching condition
for $\tilde{g}_{1}(\Lambda_{s}=m)$, the phonon mediated interaction on
surface;

$\displaystyle\tilde{g}_{1}(\Lambda_{s}=m)=\tilde{g}_{b}(\Lambda_{b}=m)=-\tilde{G}^{2}_{fp}(\Lambda_{b}=m).$
(16)

This matching condition leads to the following relation between the surface
interactions and the bulk ones,

$\displaystyle\tilde{g}_{10}(\tilde{G}_{fp0},\frac{m}{\Omega_{D}})$
$\displaystyle=$
$\displaystyle-{\tilde{G}^{2}_{fp0}}(\frac{m}{\Omega_{D}})^{{D_{b}-1}}.$ (17)

One can obtain Eq.17 by solving Eq.14 with its ultraviolet boundary condition
defined at $t_{b}=0$ or $\Lambda_{b}=\Omega_{D}$. The structure is anticipated
as surface attractive interactions are mediated by electron-phonon
interactions and so $\tilde{g}_{10}$ is uniquely set by bulk characterization
of phonon fields, $\tilde{G}_{fp0}$ and relative surface bandwidth,
$m/\Omega_{D}$.

###  II.4 A more general case

Now we turn to a semi-realistic situation where both long range Coulomb
interactions and phonon interactions are present.

$\displaystyle{\mathcal{H}}_{bulk}$ $\displaystyle=$
$\displaystyle{\mathcal{H}}_{f}+{\mathcal{H}}_{ph}+{\mathcal{H}}_{I}$
$\displaystyle{\mathcal{H}}_{f}$ $\displaystyle=$ $\displaystyle\int d^{3}{\bf
r}\psi_{b}^{\dagger}\left[v_{F}\sigma_{z}\otimes{\bf s}\cdot{\bf
p}-m\sigma_{x}\right]\psi_{b},$ $\displaystyle{\mathcal{H}}_{ph}$
$\displaystyle=$ $\displaystyle\int d^{3}{\bf r}[{\bf\pi}({\bf
r})^{2}+[\nabla{\bf\phi}({\bf r})]^{2}]$ $\displaystyle{\mathcal{H}}_{I}$
$\displaystyle=$ $\displaystyle G_{fp}\int d^{3}{\bf
r}\vec{\nabla}\cdot\vec{\phi}({\bf r})\psi_{b}^{\dagger}({\bf r})\psi_{b}({\bf
r}),$ $\displaystyle+$ $\displaystyle e_{b}^{2}\int d^{3}{\bf r}d^{3}{\bf
r^{\prime}}\psi_{b}^{\dagger}({\bf r})\psi_{b}({\bf r})\frac{1}{|{\bf r}-{\bf
r}^{\prime}|}\psi_{b}^{\dagger}({\bf r}^{\prime})\psi_{b}({\bf r}^{\prime})$
$\displaystyle{\mathcal{H}}_{surf}$ $\displaystyle=$ $\displaystyle v_{F}\int
d^{2}{\bf r}\psi_{s}^{\dagger}{\bf s}\cdot{\bf p}\psi_{s}+g_{1}\int d^{2}{\bf
r}\psi_{s}^{\dagger}({\bf r})\psi_{s}({\bf r})\psi_{s}^{\dagger}({\bf
r})\psi_{s}({\bf r})$ (18) $\displaystyle+$ $\displaystyle e_{s}^{2}\int
d^{2}{\bf r}d^{2}{\bf r^{\prime}}\psi_{s}^{\dagger}({\bf r})\psi_{s}({\bf
r})\frac{1}{|{\bf r}-{\bf r}^{\prime}|}\psi_{s}^{\dagger}({\bf
r}^{\prime})\psi_{s}({\bf r}^{\prime}).$

To determine the effective surface field theories in this case, we again match
the renormalization flow of bulk interactions in the IR region near
$\Lambda_{b}=m$ with the renormalization flow of surface fermions in its UV
region $\Lambda_{s}=m$. As discussed before, the phonon mediated interactions
can be conveniently represented by an effective attractive interaction,
$\tilde{g}_{b}$, defined Eq.15. Therefore, BBR suggests a matching between
surface interactions and bulk ones at $m$, i.e.
$\tilde{g}_{1}(\Lambda_{s}=m)=\tilde{g}_{b}(\Lambda_{b}=m)$.

More specifically,

$\displaystyle\alpha_{s0}=\alpha_{s}(\Lambda_{s}=m)$ $\displaystyle=$
$\displaystyle\alpha_{b}(\Lambda_{b}=m),$
$\displaystyle\tilde{g}_{10}=\tilde{g}_{1}(\Lambda_{s}=m)$ $\displaystyle=$
$\displaystyle-\tilde{G}^{2}_{fp}(\Lambda_{b}=m).$ (19)

This can be easily implemented by working out similar renormalization group
equations for bulk interactions $\tilde{G}_{fp}$ and $\tilde{e}^{2}_{b}$ in
this case for $D_{b}=3$ or a $3D$ bulk.

$\displaystyle\frac{\partial\tilde{G}_{fp}}{\partial t_{b}}$ $\displaystyle=$
$\displaystyle\frac{D_{b}-1}{2}\tilde{G}_{fp}+c_{b}\alpha_{b}\tilde{G}_{fp},$
$\displaystyle\frac{\partial\alpha_{b}}{\partial t_{b}}$ $\displaystyle=$
$\displaystyle c_{b}\alpha^{2}_{b}$
$\displaystyle\tilde{G}_{fp}(t_{b}=\ln\frac{\Omega_{D}}{W})$ $\displaystyle=$
$\displaystyle\tilde{G}_{fp0},\alpha_{b}(t_{b}=0)=\alpha_{b0}.$ (20)

Note that $t_{b}$ is defined as $t_{b}=\ln\Lambda_{b}/W$ and so the boundary
conditions for $G_{fp}$ is set at $t_{b}=\ln\Omega_{D}/W\neq 0$. And we have
focused on the limit where $\Omega_{D}\ll W$ although one shall find it
straightforward to generalize to other limits.

The solutions to Eq.20 can be obtained and matching conditions in Eq.19 lead
to

$\displaystyle\alpha_{s0}(\alpha_{b0},\frac{m}{W})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{b0}}{1+c_{b}\alpha_{b0}\ln\frac{W}{m}}$
$\displaystyle\tilde{g}_{10}(\tilde{G}_{fp0},\frac{m}{\Omega_{D}})$
$\displaystyle=$
$\displaystyle\frac{-\tilde{G}^{2}_{fp0}}{\left[1+c_{b}\alpha_{b}(\Omega_{D})\ln\frac{\Omega_{D}}{m}\right]^{2}}(\frac{m}{\Omega_{D}})^{D_{b}-1}.$

When presenting $\tilde{g}_{10}$ in Eq.LABEL:match4, we have only kept the
lowest order contributions in terns of $\tilde{G}_{fp0}$.

It is worth emphasizing that two different coupling constants entering
Eq.LABEL:match4: 1) $\alpha_{b0}$, the bare coupling constant defined at
$\Lambda_{b}=W$, the bulk UV scale, which results in the effective surface
charges $\alpha_{s0}$ and 2) $\alpha_{b}(\Omega_{D})$ that is defined as

$\displaystyle\alpha_{b}(\Omega_{D})=\frac{\alpha_{b0}}{1+c_{b}\alpha_{b0}\ln\frac{W}{\Omega_{D}}},$
(22)

the renormalized coupling defined at Debye frequency $\Omega_{D}$. And
$\alpha_{b}(\Omega_{D})$ is much less than its bare value $\alpha_{b0}$ if
$\ln(W/\Omega_{D})\gg 1$ and $\alpha_{b0}$ is of order unity as in solid
states. The later coupling contributes to attractive interaction strength
$\tilde{g}_{10}$ that is further renormalized by Coulomb interactions. But its
contributions can be strongly suppressed when $\ln(W/\Omega_{D})$ is much
larger than unity, a limit we will focus on in later discussions of
applications. In this special limit, $\tilde{g}_{10}$ can be thought to be
entirely determined by phonon interactions and Coulomb interactions play
little role.

At last, surface interactions with $d_{s}=2$ follow the simple equations of

$\displaystyle\frac{\partial\tilde{g}_{1}}{\partial t_{s}}$ $\displaystyle=$
$\displaystyle({d_{s}-1})\tilde{g}_{1}+\frac{c_{sfp}}{\tilde{v}_{F}}\tilde{g}^{2}_{1}+c_{sg}\alpha_{s}\tilde{g}_{1}$
$\displaystyle\frac{\partial\alpha_{s}}{\partial t_{s}}$ $\displaystyle=$
$\displaystyle c_{s}\alpha^{2}_{s},\frac{\partial\tilde{v}_{F}}{\partial
t}=-c_{s}\alpha\tilde{v}_{F}$ $\displaystyle\tilde{g}_{1}(t_{s}=0)$
$\displaystyle=$
$\displaystyle\tilde{g}_{10},\alpha_{s}(t_{s}=0)=\alpha_{s0},\tilde{v}_{F}(t_{s}=0)=\tilde{v}_{F0}.$

where surface interactions at a UV scale $\Lambda_{s}=m$, $\tilde{g_{10}}$,
$\alpha_{s0}$, have already been specified in Eq.LABEL:match4. We find that
$c_{sg}=0$ as a consequence of absence of charge renormalization. Because
$\tilde{v}_{F}$ increases logarithmically in the IR limit, the quadratic term
for the $\beta$ function of $\tilde{g}_{1}$ is less relevant in the infrared
limit. However, in the later discussions in Section IV, when Coulomb
interactions are screened by an external metal, we can set
$c_{sfp}/\tilde{v}_{F}$ to be one because in that case, there is no
logarithmic growth in velocity $v_{F}$ and renormalization effects in
$\tilde{v}_{F}$ can be neglected.

Solutions to Eq.LABEL:RGEcphS can be adopted as a guiding principle for
discussions on possibilities of emergent surface topological superconductivity
and surface Majorana particles induced by attractive interactions or
spontaneous symmetry $U(1)$ breaking. They further provide valuable insight
into the feasibility of making supersymmetric holographic matter via surface
fermions.

But before we apply the principle of BBR to understand these issues, we will
further present a more quantitive implementation of BBR via a method of
dimension reduction. The principle of naturally emerges as results integration
over the degree of freedoms perpendicular to surfaces. For readers who are not
interested in this special technology and a microscopic verification of BBR,
they can skip the next section and go straight to Sec. IV for main predictions
of BBR .

##  III Dimensional Reduction

So far, we have introduced the general pedagogy of the interaction bulk-
boundary relation based on the renormalization group methodology. In this
section, we further establish the BBR using the technique of Dimensional
Reduction(DR). Using this technique, one can also explicitly take into account
the renormalization effect of bulk electrons on the surface field theory. In
other words, the dimensional reduction technique will provide microscopic
details of the bulk-boundary matching conditions stated at the beginning of
the previous section.

In this technique, we let bosonic fields (photons or phonons) that mediate
interactions between bulk or surface electrons be in the 3-dimensional bulk.
Surface fermions have finite spatial extension along the z-axis, which is
perpendicular to the topological insulator(TI) surface. Because the surface
fermions have finite spreads into the interior of the bulk, the bulk bosons
can mediate the interactions in the surface. This setup captures the essential
features of interactions between boundary fermions and bulk bosonic fields.
Later on, we integrate out the degrees of freedom perpendicular to the
surface, thus replacing the 3-dimensional bulk theory with an effective
2-dimensional surface field theory.

In the first part, we discuss the dimensional reduction of the domain wall
fermions mediated by the bulk boson fields at the tree level. Then, we take
into account the polarization effect of bulk electrons on the respective bulk
phonon and Coulomb fields. Thereafter, the dimensional reduction is then again
performed on the renormalized action of the domain wall fermions to the lowest
order in the bulk interactions strengths. Finally, we compare the result from
dimensional reduction with that from the RGE, derived in the previous section
using matching conditions at the bulk-boundary matching scale $m$. We find
that the results obtained in two different approaches match, proving that BBR
implemented through matching conditions at the mass scale provides an
efficient way to take into account the effect of bulk polarization on surface
interactions.

###  III.1 Domain wall model

The domain wall model for surface fermions is defined as follows: consider
that the TI surface is at $z=0$ plane. The spatial profile of bulk mass is
then,

$\displaystyle m(z)=\begin{cases}+m&0<z<+\infty\\\ -m&-\infty<z<0\end{cases}$
(24)

Throughout our calculations, we ignore the edge effects on the surface
physics, which means that the periodic boundary conditions are applied on the
2D surface. By plugging this mass function to the massive Dirac equation and
solving it, we obtain the wavefunction corresponding to domain wall fermions
given by,

$\displaystyle\Psi_{dw}(\textbf{r})_{k,\pm}$ $\displaystyle=$
$\displaystyle\frac{e^{i\textbf{k}_{\perp}.\textbf{r}}}{\sqrt{2}}\begin{bmatrix}1\\\
0\\\ 0\\\ \pm
e^{i\phi_{k_{\perp}}}\end{bmatrix}\sqrt{\frac{m}{v_{F}}}e^{-\frac{m}{v_{F}}|z|}$
(25) $\displaystyle=$
$\displaystyle\Psi_{s}(\textbf{r}_{\perp})_{k,\pm}\sqrt{\frac{m}{v_{F}}}e^{-\frac{m}{v_{F}}|z|}$

Here $k_{\perp}$ = ($k_{x}$, $k_{y}$), $\textbf{r}_{\perp}$ = ($x$, $y$) and
the angle $\phi_{k_{\perp}}=\text{tan}^{-1}\frac{k_{y}}{k_{x}}$.
$\Psi_{s}(\textbf{r}_{\perp})$ is the 4-component spinor wavefunction confined
to the 2D surface. Also, $v_{F}$ is the bare value of Fermi velocity. The
dispersion relation for the states are given by $E_{\pm}(k)=\pm
v_{F}|k_{\perp}|$. It is evident that the domain wall fermions are localized
near the surface at $z=0$, with the bulk mass term $m$ being the localization
parameter. In the limit of $m\rightarrow\infty$, the domain wall state evolves
itself into an ideal surface state with no tail end in the bulk. In the
opposite limit where $m\rightarrow 0$, the bulk gap closes, and the surface
states get hybridized with the bulk. In other words, there is no more bulk-
boundary distinction. Thus the domain wall model for boundary electrons has
the advantage that one could tune the localization of surface states as a
function of the bulk mass term. Closing the bulk gap automatically merges the
bulk and surface states.

Let us now write down the free-field theory of domain wall fermions and 2D
surface fermions. For this purpose, we shall define a pseudo-spin space with
the $4\times 4$ matrices $\Xi_{x}$, $\Xi_{y}$ and $\Xi_{z}$ as the generators.
These matrices have the definitions,

$\displaystyle\Xi_{x}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\tau_{x}\sigma_{x}-\tau_{y}\sigma_{y}\right),\,\,\Xi_{y}=\frac{1}{2}\left(\tau_{x}\sigma_{y}+\tau_{y}\sigma_{x}\right)$
$\displaystyle\Xi_{z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\tau_{z}\sigma_{0}+\tau_{0}\sigma_{z}\right)$
(26)

where $\tau_{\alpha}$ $(\alpha=x,y,z)$ and $\sigma_{\alpha}$ $(\alpha=x,y,z)$
are Pauli matrices defined in the particle-hole and the spin-1/2 spaces
respectively. It can be shown that the matrices $\Xi_{x}$, $\Xi_{y}$ and
$\Xi_{z}$ satisfy the $SU(2)$ algebra given by,
$\left[\Xi_{i},\Xi_{j}\right]=2i\epsilon_{ijk}\Xi_{k}$. In this Hilbert space,
the respective actions have the form,

$\displaystyle\mathcal{S}_{dwf}$ $\displaystyle=$ $\displaystyle\int
d^{3}\textbf{r}dt\,\psi^{\dagger}_{dw}(\textbf{r},t)\biggl{[}i\partial_{t}+iv_{F}\Xi_{x}\partial_{x}$
$\displaystyle+$ $\displaystyle
iv_{F}\Xi_{y}\partial_{y}+iv_{F}\partial_{z}+m(z)\biggr{]}\psi_{dw}(\textbf{r},t)$
$\displaystyle\mathcal{S}^{2D}_{surf}$ $\displaystyle=$ $\displaystyle\int
d^{2}\textbf{r}dt\,\psi_{s}^{\dagger}(\textbf{r},t)\biggl{[}i\partial_{t}+iv_{F}\Xi_{x}\partial_{x}+iv_{F}\Xi_{y}\partial_{y}\biggr{]}\psi_{s}(\textbf{r},t)$

where $m(z)$ has the form given in Eq.24. Hence $\psi^{\dagger}_{dw}$ describe
4-component gapless domain wall fermions defined in the 3D bulk while
$\psi^{\dagger}_{s}$ describes 4-component gapless surface fermions defined in
the 2D surface. Thus the only difference between them is that the domain wall
fermions have a finite spatial extension to the bulk which allow them to
interact with the bulk Coulomb and the phonon field.

###  III.2 Dimensional reduction I

In this sub-section, we write down the bare action that defines the
interaction between the domain wall fermions. The repulsive interaction
between domain wall fermions is mediated by the Coulomb potential and the
attractive interaction by phonons, the quanta of lattice vibrations. This
3-dimensional theory is then mapped into an effective 2D surface theory by
integrating out the degrees of freedom perpendicular to the surface.
Thereafter, we look at the limit of the resulting surface theory which can
lead to superconducting pairing and study the behavior of the two interactions
at the tree level. Following this, we find out a relation between the
dimensionless bulk and the surface coupling constants, thus verifying the
bulk-boundary matching conditions stated in Eq.LABEL:match4 at the tree level
limit.

The dynamics of domain wall fermions subject to the phonon-mediated
interaction and Coulomb repulsion are governed by the field theory:

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\mathcal{S}_{dwf}+\mathcal{S}_{e-ph,int}+\mathcal{S}_{cou,int}$
$\displaystyle\mathcal{S}_{e-ph,int}$ $\displaystyle=$
$\displaystyle-G^{2}_{fp}\int
d^{3}\textbf{r}\,d^{3}\textbf{r}^{\prime}\,dt\,dt^{\prime}\,\psi_{dw}^{\dagger}(\textbf{r},t)\psi_{dw}(\textbf{r},t)$
$\displaystyle
K_{e-ph}(\textbf{r}^{\prime}-\textbf{r},t^{\prime}-t)\psi_{dw}^{\dagger}(\textbf{r}^{\prime},t^{\prime})\psi_{dw}(\textbf{r}^{\prime},t^{\prime})$
$\displaystyle\mathcal{S}_{cou,int}$ $\displaystyle=$ $\displaystyle-\int
d^{3}\textbf{r}\,d^{3}\textbf{r}^{\prime}\,dt\,\psi_{dw}^{\dagger}(\textbf{r},t)\psi_{dw}(\textbf{r},t)$
(29)
$\displaystyle\frac{e^{2}_{b}}{|\textbf{r}-\textbf{r}^{\prime}|}\psi_{dw}^{\dagger}(\textbf{r}^{\prime},t)\psi_{dw}(\textbf{r}^{\prime},t)$

where $K_{e-ph}(\textbf{r},t)$ is the phonon propagator in the bulk of the
topological insulator. In momentum space, it has the expression,

$K_{e-ph}(\textbf{q},\omega)=\frac{v^{2}_{p}q^{2}}{\omega^{2}-v^{2}_{p}q^{2}}.$
(30)

Here $v_{p}$ is the phonon velocity. As discussed in the start of the section,
the effective surface action is derived by integrating out the degrees of
freedom perpendicular to the surface. That is,

$\displaystyle\mathcal{S}^{2D}$ $\displaystyle=$
$\displaystyle\int^{\infty}_{-\infty}dz\,\left[\mathcal{S}_{dwf}+\mathcal{S}_{e-ph,int}+\mathcal{S}_{cou,int}\right]$
(31) $\displaystyle=$
$\displaystyle\mathcal{S}^{2D}_{surf}+\mathcal{S}^{2D}_{e-ph,int}+\mathcal{S}^{2D}_{cou,int}$

The interaction term is now,

$\displaystyle\mathcal{S}^{2D}_{e-ph,int}$ $\displaystyle=$
$\displaystyle-\int
d^{2}\textbf{r}\,d^{2}\textbf{r}^{\prime}\,dt\,dt^{\prime}\,\psi_{s}^{\dagger}(\textbf{r},t)\psi_{s}(\textbf{r},t)$
$\displaystyle
K^{2D}_{e-ph}(\textbf{r}^{\prime}-\textbf{r},t^{\prime}-t)\psi_{s}^{\dagger}(\textbf{r}^{\prime},t^{\prime})\psi_{s}(\textbf{r}^{\prime},t^{\prime})$
$\displaystyle\mathcal{S}^{2D}_{cou,int}$ $\displaystyle=$ $\displaystyle-\int
d^{2}\textbf{r}\,d^{2}\textbf{r}^{\prime}\,dt\,\psi_{s}^{\dagger}(\textbf{r},t)\psi_{s}(\textbf{r},t)$
(33) $\displaystyle
K^{2D}_{cou}(\textbf{r}^{\prime}-\textbf{r})\psi_{s}^{\dagger}(\textbf{r}^{\prime},t)\psi_{s}(\textbf{r}^{\prime},t)$

where $K^{2D}_{e-ph}(\textbf{r},t)$ is now the effective phonon propagator on
the 2D surface. Similarly, $K^{2D}_{cou}(\textbf{r})$ is the effective surface
Coulomb propagator. In the momentum space, they have the form given by,

$\displaystyle K^{2D}_{e-ph}(v_{p}\textbf{q}_{\perp},\omega)$ $\displaystyle=$
$\displaystyle G^{2}_{fp}\int^{\infty}_{-\infty}$
$\displaystyle\frac{dq_{z}}{2\pi}\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}\frac{q_{\perp}^{2}+q^{2}_{z}}{\frac{\omega^{2}}{v^{2}_{p}}-q_{\perp}^{2}-q^{2}_{z}}$
(34) $\displaystyle K^{2D}_{cou}(\textbf{q}_{\perp})$ $\displaystyle=$
$\displaystyle
e^{2}_{b}\int^{\infty}_{-\infty}\frac{dq_{z}}{2\pi}\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}\frac{1}{q_{\perp}^{2}+q^{2}_{z}}$

where $M=\frac{2m}{v_{F}}$.

In the integral expression, the contribution from domain wall model
$\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}$ is in-fact a regularization
term that helps us get rid of ultraviolet divergences if any. If we take the
limit of $M\rightarrow\infty$, the domain wall wavefunction converges into a
strictly surface term. In the case of the Coulomb propagator, it is evident
from Eq.LABEL:qzintcou that this integral is not UV divergent and hence is
independent of $m$. Therefore, we could safely take the large $M$ limit and
ignore the regularization term.

But the integral involving bulk phonon propagator is UV divergent. Hence, we
must use the regularization term to integrate out the $q_{z}$ momentum
resulting in a linear dependence of $m$ . After taking all these into account,
the effective surface propagator for phonon-mediated interaction and Coulomb
interaction at the tree level turns out to be,

$K^{2D}_{e-ph}(v_{p}\textbf{q}_{\perp},\omega)=\\\
\frac{G^{2}_{fp}M}{4\left(\omega^{2}-v^{2}_{p}q^{2}_{\perp}\right)}\biggl{[}v^{2}_{p}q_{\perp}^{2}-\frac{v^{2}_{p}M^{2}\omega^{2}}{\left(Mv_{p}+\sqrt{v^{2}_{p}q_{\perp}^{2}-\omega^{2}}\right)^{2}}\biggr{]}$
(36) $\displaystyle
K^{2D}_{cou}(\textbf{q}_{\perp})=\frac{e^{2}_{b}}{2q_{\perp}}.$ (37)

$K^{2D}_{cou}$, the 2D surface Coulomb propagator, turns out to be
proportional to $\frac{1}{q_{\perp}}$ as expected, where $q_{\perp}$ is the
in-plane momentum.

As the main application of phonon-mediated interaction of $K^{2D}_{e-ph}$ is
superconductivity, we therefore restrict to the limit where the energy
exchanged between the electrons in the scattering process is less than
energies of most dominating phonons. These exchange processes of phonons also
lead to pairing near a Fermi surfaceabrikosov . On taking the limit $\omega\ll
v_{p}q_{\perp},m$, the second term in the bracket in Eq.36 can be ignored in
comparison to the first term, resulting in a rather simple form,

$\displaystyle K^{2D}_{e-ph}(v_{p}\textbf{q}_{\perp},\omega)$
$\displaystyle\approx$
$\displaystyle\frac{G^{2}_{fp}m}{2v_{F}}\frac{v^{2}_{p}q_{\perp}^{2}}{\omega^{2}-v^{2}_{p}q^{2}_{\perp}}.$
(38)

Thus the momentum and frequency dependence of the 2D surface propagator are
the same as its 3D bulk counterpart. It is well known that the phonon mediated
interactions are mostly dominated by phonons near
$v_{p}q_{\perp}\sim\Omega_{D}$, In the low energy window where
$\omega<m\ll\Omega_{D}$, we thus arrive at,

$\displaystyle K^{2D}_{e-ph}(\omega,v_{p}\textbf{q}_{\perp})$
$\displaystyle\sim$ $\displaystyle-\frac{G^{2}_{fp}m}{2v_{F}}$ (39)
$\displaystyle K^{2D}_{cou}(\textbf{q}_{\perp})$ $\displaystyle=$
$\displaystyle\frac{e^{2}_{b}}{2q_{\perp}}$ (40)

We find that, in this limit, the effective surface phonon propagator is short-
ranged in space. Upon comparing the surface action in Eqs.LABEL:dimredphon,33
with the interaction terms of the surface Hamiltonian $H_{surf}$ written down
in Eq.18, one finds that the surface coupling constants can be related to
their respective surface propagators as,

$\displaystyle
K^{2D}_{e-ph}(\omega,v_{p}\textbf{q}_{\perp})=g_{1}\,\,\,,\,\,\,\,q_{\perp}K^{2D}_{cou}(\textbf{q}_{\perp})=e^{2}_{s}.$
(41)

Using this relation, one can connect the bulk and the surface coupling
constants of the respective interactions. When represented in a dimensionless
form using the scaling analysis, the bare values of the surface and the bulk
coupling constants are found to be related in the following way at the tree-
level limit,

$\displaystyle\tilde{g}_{10}=-\frac{\tilde{G}^{2}_{fp0}}{2}\left(\frac{m}{\Omega_{D}}\right)^{2},\,\,\,\,\,\tilde{e}^{2}_{s0}=\frac{\tilde{e}^{2}_{b0}}{2}$
(42)

where $\tilde{g}_{10}=\tilde{g}(\lambda_{s}=m)$,
$\tilde{G}^{2}_{fp0}=\tilde{G}^{2}_{fp}(\Lambda_{b}=\Omega_{D})$
$\tilde{e}^{2}_{s0}=\tilde{e}^{2}_{s}(\Lambda_{s}=m)$ and
$\tilde{e}^{2}_{b0}=\tilde{e}^{2}_{b}(\Lambda_{b}=W)$ are defined as the bare
values of the dimensionless surface and the bulk coupling constants. Thus, up
to the tree-level limit, we are able to verify the matching conditions between
the respective surface and bulk interaction strengths stated in the previous
section(Eq.LABEL:match4).

Upon dimensional reduction, one also finds that the low frequency limit of
phonon-mediated interaction potential between surface fermions are attractive
and short-ranged in space. Thus, if the Coulomb interaction is sufficiently
suppressed in the long-wavelength scale, one can have a superconducting ground
state on the topological insulator surface. In the subsequent sub-sections, we
study the effect of bulk charge polarization on surface interactions using the
diagrammatic method followed by Dimensional Reduction(DR).

Figure 5: Feynman diagrams for (a) bulk charge polarization effects due to
Coulomb interactions and (b) contributions of Coulomb interactions to
renormalization of electron-phonon coupling constant. Dotted and wiggly lines
denote respectively Coulomb and phonon fields. Solid lines stand for bulk
electron propagators.

###  III.3 Bulk renormalization effect

The Coulomb and phonon fields could further nteract with massive Dirac
fermions in the bulk, resulting in a series of electron-hole excitations and
quantum renormalization effects. Here we study the one-loop renormalization
effect due to bulk electrons on the boson fields. Later, we will perform
dimensional reduction of the domain wall fermions mediated by these dressed
boson fields. The theory after dimensional reduction turns out to be an UV
theory of the surface, which includes the quantum corrections to the bulk-
boundary matching conditions written in the previous section at the tree-level
limit (Eq.42).

The field theory of bulk electrons of the topological state is governed by the
action

$\mathcal{S}^{b}_{dirac}=\int
d^{3}\textbf{r}\,dt\,\psi^{\dagger}_{b}(\textbf{r},t)\left(i\partial_{t}+iv_{F}\vec{\rho}.\vec{\nabla}-\eta
m\right)\psi_{b}(\textbf{r},t)$ (43)

where $\vec{\rho}$ and $\eta$ are $4\times 4$ matrices and are defined in
terms of the Dirac gamma matrices as $\vec{\rho}=\gamma^{0}\vec{\gamma}$ and
$\eta=\gamma^{0}$. $m$ is the mass gap of the bulk spectrum and $v_{F}$ is the
Fermi velocity of the electrons. Note that, the Dirac field $\psi_{b}(r,t)$
defined here is a 4-component spinor and satisfies the anti-commutation
relations.

The interaction between the bulk electrons is governed by the action,

$\mathcal{S}^{b}_{int}=\mathcal{S}^{b}_{cou,int}+\mathcal{S}^{b}_{e-ph,int}$

where

$\displaystyle\mathcal{S}^{b}_{cou,int}$ $\displaystyle=$ $\displaystyle-\int
d^{3}\textbf{r}\,d^{3}\textbf{r}^{\prime}\,dt\,\psi_{b}^{\dagger}(\textbf{r},t)\psi_{b}(\textbf{r},t)$
(44)
$\displaystyle\frac{e_{b}^{2}}{|\textbf{r}-\textbf{r}^{\prime}|}\psi_{b}^{\dagger}(\textbf{r}^{\prime},t)\psi_{b}(\textbf{r}^{\prime},t)$
$\displaystyle\mathcal{S}^{b}_{e-ph,int}$ $\displaystyle=$
$\displaystyle-G^{2}_{fp}\int
d^{3}\textbf{r}\,d^{3}\textbf{r}^{\prime}\,dt\,dt^{\prime}\,\psi_{b}^{\dagger}(\textbf{r},t)\psi_{b}(\textbf{r},t)$
$\displaystyle
K_{e-ph}(\textbf{r}^{\prime}-\textbf{r},t^{\prime}-t)\psi_{b}^{\dagger}(\textbf{r}^{\prime},t^{\prime})\psi_{b}(\textbf{r}^{\prime},t^{\prime})$

Both the electric charge and the Fermi velocity can get renormalized in a
3-dimensional bulk state. As said before, in this section, we will present the
standard one-loop renormalization. In this approximation, it is sufficient to
use just the bare value of Fermi velocity. This is because the effects of
Fermi self-energy renormalization will show up only in the higher order
contributions to the bulk charge renormalization.

The charge polarization(see fig.5) term to the one loop order in electron
bubble has the formPeshkin ,

$\displaystyle\Pi(\textbf{q})$ $\displaystyle=$
$\displaystyle\frac{e^{2}_{b0}}{2\pi^{2}v_{F}}\int^{1}_{0}d\beta\,\beta\left(1-\beta\right)\text{Log}\left[\frac{m^{2}+\beta(1-\beta)v^{2}_{F}q^{2}}{m^{2}+\beta(1-\beta)W^{2}}\right]$

Note that $v_{F}$ here is the bare value of Fermi velocity. The
renormalization condition imposed is that the bulk renormalized charge must
equal its bare counterpart at the bulk cut-off scale W, i.e
$\Pi(v^{2}_{F}q^{2}=W^{2})=0$. Since we are more interested in the case where
the energy scale W is much greater in magnitude than the bulk mass m, the bulk
polarization term takes the simple form,

$\displaystyle\Pi(\textbf{q})$ $\displaystyle\approx$
$\displaystyle\frac{e^{2}_{b0}}{2\pi^{2}v_{F}}\int^{1}_{0}d\beta\,\beta\left(1-\beta\right)\text{Log}\left[\frac{m^{2}+\beta(1-\beta)v^{2}_{F}q^{2}}{\beta(1-\beta)W^{2}}\right]$

As a result of the polarization, the bulk electric charge gets renormalized
and hence becomes a function of spatial momentum(see fig. 5(a)), given by,

$\displaystyle
e^{2}_{b}\left(\textbf{q}\right)=\frac{e^{2}_{b0}}{1-\Pi(\textbf{q})}$ (48)

where $e^{2}_{b0}$ is the bare charge of the bulk electrons at cut-off scale
$W$.

We have seen in the previous section that the Coulomb field can renormalize
the electron-phonon interaction vertex. The corresponding one-loop Feynman
diagram is shown in fig.5(b). The renormalized bulk electron-phonon coupling
constant has the form,

$\displaystyle
G_{fp}\left(\textbf{q}\right)=\frac{G_{fp0}}{1-\Pi^{eph}(\textbf{q})}$ (49)

$\Pi^{eph}(\textbf{q})$ has the same loop structure as the charge polarization
but with the UV cut off set at Debye frequency, $\Omega_{D}$, and $G_{fp0}$ is
the bare value of coupling constant at the cut-off scale. That is,

$\displaystyle\Pi^{eph}(\textbf{q})$ $\displaystyle=$
$\displaystyle\frac{e^{2}_{b}(\Omega_{D})}{2\pi^{2}v_{F}}\int^{1}_{0}d\beta\,\beta\left(1-\beta\right)$
(50)
$\displaystyle\text{Log}\left[\frac{m^{2}+\beta(1-\beta)v^{2}_{F}q^{2}}{\beta(1-\beta)\Omega_{D}^{2}}\right]$

In both $\Pi({\bf q})$ and $\Pi^{eph}({\bf q})$, there is a standard
logarithmic dependence on $m$ in the limit of zero momentum ${\bf q}=0$.

###  III.4 Dimensional Reduction II

Now we shall come back to talk about the domain wall fermions of the surface.
We have learned that the polarization effect of bulk fermions renormalize the
respective interaction strengths. These dressed boson fields could mediate
interactions between domain wall fermions, as shown in the schematic diagram
in fig.3. Following the same procedure as was done in the previous sub-
section, we perform dimensional reduction of the renormalized theory, to
arrive at the effective surface theory. The resulting surface field theory has
the form,

$\displaystyle\mathcal{S}^{2D}_{e-ph,int}$ $\displaystyle=$
$\displaystyle-\int
d^{2}\textbf{r}\,d^{2}\textbf{r}^{\prime}\,dt\,dt^{\prime}\,\psi_{s}^{\dagger}(\textbf{r}_{\perp},t)\psi_{s}(\textbf{r},t)$
$\displaystyle\mathcal{K}^{2D}_{e-ph}(\textbf{r}^{\prime}-\textbf{r},t^{\prime}-t)\psi_{s}^{\dagger}(\textbf{r}^{\prime},t^{\prime})\psi_{s}(\textbf{r}^{\prime},t^{\prime})$
$\displaystyle\mathcal{S}^{2D}_{cou,int}$ $\displaystyle=$ $\displaystyle-\int
d^{2}\textbf{r}\,d^{2}\textbf{r}^{\prime}\,dt\,\psi_{s}^{\dagger}(\textbf{r},t)\psi_{s}(\textbf{r},t)$
$\displaystyle\mathcal{K}^{2D}_{cou}\left(\textbf{r}^{\prime}-\textbf{r}\right)\psi_{s}^{\dagger}(\textbf{r}^{\prime},t)\psi_{s}(\textbf{r}^{\prime},t)$

where $\mathcal{K}^{2D}_{e-ph}(\textbf{r}_{\perp},t)$ and
$\mathcal{K}^{2D}_{cou}(\textbf{r}_{\perp})$ differ from the bare surface
propagators defined in the subsection B in such a way that these are obtained
by performing dimensional reduction on the dressed bulk propagators, that take
into account the bulk renormalization effect[defined in Eqs.
LABEL:selfenergy_cou, 50].

$\displaystyle\mathcal{K}^{2D}_{cou}(\textbf{q}_{\perp})$ $\displaystyle=$
$\displaystyle\int^{\infty}_{-\infty}\frac{dq_{z}}{2\pi}\frac{e^{2}_{b0}}{1-\Pi(\textbf{q})}\frac{1}{q_{\perp}^{2}+q^{2}_{z}}$
(52) $\displaystyle\mathcal{K}^{2D}_{e-ph}(\textbf{q}_{\perp},\omega)$
$\displaystyle=$
$\displaystyle\int^{\infty}_{-\infty}\frac{dq_{z}}{2\pi}\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}$
$\displaystyle\frac{G^{2}_{fp0}}{\left(1-\Pi^{eph}(\textbf{q})\right)^{2}}\frac{v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}{\omega^{2}-v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}$

Computing this integral is quite tricky. So, let us look for expansions that
help us get the lowest order corrections to bare result. In this case, we
perform a perturbative expansion of the denominator in powers of the bare bulk
coupling constant $\frac{e^{2}_{b0}}{v_{F}}$. This result in,

$\displaystyle\mathcal{K}^{2D}_{cou}(\textbf{q}_{\perp})$ $\displaystyle=$
$\displaystyle
e^{2}_{b0}\int^{\infty}_{-\infty}\frac{dq_{z}}{2\pi}\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}\frac{1}{q_{\perp}^{2}+q^{2}_{z}}\left[1+\Pi(\textbf{q})+....\right]=\mathcal{K}_{cou}^{2D(0)}(\textbf{q}_{\perp})+\mathcal{K}_{cou}^{2D(1)}(\textbf{q}_{\perp})+.....$
$\displaystyle\mathcal{K}^{2D}_{e-ph}(\textbf{q}_{\perp},\omega)$
$\displaystyle=$ $\displaystyle
G^{2}_{fp0}\int^{\infty}_{-\infty}\frac{dq_{z}}{2\pi}\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}\frac{v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}{\omega^{2}-v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}\left[1+2\Pi^{eph}(\textbf{q})+....\right]=\mathcal{K}_{e-ph}^{2D(0)}(\textbf{q}_{\perp})+\mathcal{K}_{e-ph}^{2D(1)}(\textbf{q}_{\perp})+....$

where the zeroth order terms $\mathcal{K}_{e-ph}^{2D(0)}(\textbf{q}_{\perp})$,
$\mathcal{K}_{cou}^{2D(0)}(\textbf{q}_{\perp})$ are the same as
$K_{e-ph}^{2D}(\textbf{q}_{\perp})$ and $K_{cou}^{2D(0)}(\textbf{q}_{\perp})$
derived in the previous sub-section and their exact expressions were given in
Eqs. 36, 37 respectively.

#### First order correction

Here we shall solve for $\mathcal{K}^{2D(1)}_{cou}(\textbf{q}_{\perp})$ and
$\mathcal{K}^{2D(1)}_{e-ph}(\textbf{q}_{\perp},\omega)$, the lowest order
quantum corrections to the bare propagators corresponding to the Coulomb and
phonon-mediated interaction respectively. Even though the complete solution of
the momentum integral is cumbersome, we limit ourselves to the case where the
magnitude of the in-plane momentum is close to the bulk mass gap and the
phonon frequency set to small, i.e. the static limit. The theory after
performing dimensional reduction will manifest itself as the UV theory of the
surface. Since the mass gap $m$ is the UV cut-off of the surface theory, it is
justified to take the in-plane momentum to be of the order of bulk mass. The
static limit suggesting pairing interactions that we will be interested in. In
other words, the first order quantum correction to the surface propagators at
the mass scale and the static limit manifest as the first-order correction to
the bare values of their respective surface coupling constants,

$\displaystyle
q_{\perp}\mathcal{K}^{2D(1)}_{cou}(\textbf{q}_{\perp})|_{v_{F}q_{\perp}\approx
m}$ $\displaystyle=$ $\displaystyle e^{2(1)}_{s0}$
$\displaystyle\mathcal{K}^{2D(1)}_{e-ph}(\textbf{q}_{\perp},\omega)|_{v_{F}q_{\perp}\approx
m,\omega\approx 0}$ $\displaystyle=$ $\displaystyle g^{(1)}_{10}$ (56)

After integrating out $q_{z}$, one finds the exact analytical form of
$e^{2(1)}_{s0}$ and $g^{(1)}_{10}$ as,

$\displaystyle e^{2(1)}_{s0}$ $\displaystyle=$
$\displaystyle\frac{e^{2}_{b0}}{2}\frac{e^{2}_{b0}}{6\pi^{2}v_{F}}\left[\log\frac{m}{W}+1.2611\right]$
(57) $\displaystyle g^{(1)}_{10}$ $\displaystyle=$
$\displaystyle-\frac{G^{2}_{fp0}m}{2v_{F}}\frac{e^{2}_{b}(\Omega_{D})}{3\pi^{2}v_{F}}\biggl{[}\log\frac{m}{\Omega_{D}}+1.06874\biggr{]}$
(58)

Details of the calculation is shown in Appendix A.

###  III.5 Comparison with RG results

In this section so far, we have used the technique of dimensional reduction to
project the renormalized bulk theory onto the surface. Thereafter, we have
derived the lowest order quantum correction due to the bulk electron
polarization to the surface field theory. In this sub-section, we will look at
the surface coupling constants derived in the previous section using RG
techniques and compare them with the results from dimensional reduction. At
the beginning of section II, we have stated that the effect of bulk
interaction can be taken into account for the surface theory by demanding that
the bulk and the boundary coupling constants must satisfy the matching
condition at the energy scale at which the boundary connects with the bulk,
which in this case is the bulk mass ${}^{\prime}m^{\prime}$. Our objective of
introducing the technique of dimensional reduction was to provide a
microscopic details of these bulk-boundary matching conditions in the case of
a 3D topological state.

The bulk-boundary matching condition is written down in Eq.LABEL:match4.
Expanding the relation to the first order in the bulk Coulomb interaction
strength $\frac{e^{2}_{b0}}{v_{F}}$, we arrive at

$\displaystyle\tilde{e}^{2}_{s0}(e_{b0},\frac{m}{W})$ $\displaystyle=$
$\displaystyle\tilde{e}^{2}_{b0}\left[1+\frac{\tilde{e}_{b0}^{2}}{6\pi^{2}v_{F}}\ln\frac{m}{W}+...\right]$
$\displaystyle\tilde{g}_{10}(\tilde{G}_{fp0},\frac{m}{\Omega_{D}})$
$\displaystyle=$
$\displaystyle-\tilde{G}^{2}_{fp0}\left(\frac{m}{\Omega_{D}}\right)^{2}\left[1+\frac{\tilde{e}_{b}^{2}\left(\Omega_{D}\right)}{3\pi^{2}v_{F}}\ln\frac{m}{\Omega_{D}}+..\right]$

On the other hand, from Eqs.42, 57 and 58, the dimensionless surface coupling
constants derived via the Dimensional Reduction technique to the lowest order
in quantum corrections are ,

$\displaystyle\tilde{e}^{2}_{s0}$ $\displaystyle=$
$\displaystyle\tilde{e}^{2(0)}_{s0}+\tilde{e}^{2(1)}_{s0}+.....$
$\displaystyle=$
$\displaystyle\frac{\tilde{e}^{2}_{b0}}{2}\biggl{[}1+\frac{\tilde{e}^{2}_{b0}}{6\pi^{2}v_{F}}\biggl{(}\ln\frac{m}{W}+1.2611\biggr{)}+...\biggr{]}$
$\displaystyle\tilde{g}_{10}$ $\displaystyle=$
$\displaystyle\tilde{g}^{(0)}_{10}+\tilde{g}^{(1)}_{10}+.......$ (60)
$\displaystyle=$
$\displaystyle-\frac{\tilde{G}^{2}_{fp0}}{2}\left(\frac{m}{\Omega_{D}}\right)^{2}\biggl{[}1+\frac{\tilde{e}^{2}_{b}(\Omega_{D})}{3\pi^{2}v_{F}}\biggl{(}\ln\frac{m}{\Omega_{D}}$
$\displaystyle+$ $\displaystyle 1.06874\biggr{)}+....\biggr{]}$

where the dimensionless form of the bulk coupling constants at their
respective UV cut-off scales has the definitions, $\tilde{e}_{b0}=e_{b0}$ and
$\tilde{G}_{fp0}=G_{fp0}\left(\frac{\Omega_{D}}{v_{F}}\right)^{(D_{b}-1)/2}$
respectively, with $D_{b}=3$ for bulk. Similarly, for the bare surface
coupling constants, $\tilde{e}_{s0}=e_{s0}$ and
$\tilde{g}_{10}=g_{10}\left(\frac{m}{v_{F}}\right)^{d_{s}-1}$ with $d_{s}=2$
for the surfaces.

Thus we find that, except for a numerical factor, the dimensional reduction
technique nicely recreates the RG results obtained by imposing the matching
conditions at the bulk-boundary matching scale $m$. Thus, we show that the
bulk-boundary correspondence indeed naturally emerges as a result of
integrating over the degrees of freedom perpendicular to the surfaces.

To conclude, in this section we utilized the technology of dimensional
reduction to show that the bulk-boundary matching conditions at the matching
scale is an efficient way to take into account the bulk renormalization effect
on the surface field theory. It should be noted that since the bulk mass $m$
is the UV cut-off scale for surface fermions, renormalization effects due to
surface fermions come into the picture once the energy scale is further
brought down towards an infrared regime of the surface theory. The RG flow of
the surface coupling constants as a result of surface fermion renormalization
has been written down in Eq.LABEL:RGEcphS in the previous section.

##  IV Applications

For a three dimensional topological bulk, a natural limit for us to focus on
is when $W$, the bulk bandwidth that can be associated with the UV scale of
Coulomb interaction is much higher than either the surface ”bandwidth” or
bandgap $m$ or $\Omega_{D}$, the Debye frequency that sets the bandwidth of
phonons. As seen in previous sections, this can result in substantial
suppression of long range Coulomb interactions at lower energy windows of $m$
or $\Omega_{D}$ due to strong renormalization effects.

However, even if phonon-mediated interactions dominate at the scale of
$\Omega_{D}$ due to strong suppression of long-range Coulomb interactions,
long range Coulomb interactions are represented by marginal surface operators
and can still become dominating again at asymptotic regimes where
$\Lambda_{s}\ll m,\Omega_{D}$. This hinders the emergence of surface
topological superconductivity and other related interaction phenomena. The
path suggested below involves two key ingredients.

Figure 6:  Schematic of renormalized Coulomb interactions (Green curve) and
phonon-mediated interactions (Blue one) vs the running scale $\Lambda$ in the
renormalization equations. In a)-c), we characterize the surface bandwidth
with a mass gap $m$, and bulk phonon bandwidth $\Omega_{D}$ and bulk fermion
band width $W$. a) the limit of $W\sim\Omega_{D}\gg m$; b) the limit of
$W\gg\Omega_{D},m$ and c) the limit of $W\gg\Omega_{D}\sim m$. In c), Coulomb
interactions further turn into dipole interactions when $\Lambda\ll m$ because
of a screening metal. Here $\alpha_{b0}$ is the bare Coulomb coupling constant
(see Eq.7) and $\tilde{G}_{fp0}$ the dimensionless electron-phonon one (see
Eq.14).

i) A route to avoid this issue of long-range Coulomb interaction is to further
add a metal on top of surfaces to effectively reduce long-range Coulomb
interactions into a dipole-dipole one (see fig.7). A Similar idea was proposed
in a previous studyPonte14 . Surface dipole-dipole interactions are
represented by an irrelevant operator that scales down much faster than
phonon-mediated interactions. So at least asymptotically, surfaces can then
interact mainly via phonon-mediated interactions. One shall anticipate that
there shall be spontaneous $U(1)$ symmetry breaking leading to surface
topological superconductivity that supports emergent quantum Majorana
particlesQiZhang ; kitaev .

ii) In the limit we are interested in, the Debye frequency $\Omega_{D}$ is
larger or much larger than the mass gap $m$, i.e. $\Omega_{D}\geq m$. Once
identifying surface interactions at the scale $m$ via matching conditions in
BBR, one can extract the bare interaction constant
${g}_{1}(\Lambda_{s}=\Omega_{D})$ (dimensionful) or
$\tilde{g}_{1}(\Omega_{D})$ (dimensionless one) via tracking the UV flow of
RGEs from the intermediate scale $m$ to $\Omega_{D}$.

If we focus on the weakly interacting limit, the procedure mentioned above is
particularly straightforward. The bare interaction constant is simply related
to $\tilde{g}_{1}(m)$ via $g_{1}(\Omega_{D})=\tilde{g}_{1}(m)/m^{d_{s}-1}$.
$\tilde{g}_{1}(\Omega_{D})$, the dimensionless phonon mediated interactions at
$\Lambda_{s}=\Omega_{D}$ are then linearly proportional to the interactions
defined at matching scale $\Lambda_{s}=m$ that had been obtained via the BBR.
This relation also follows the RGEs in Eq.65 obtained for surface fermions but
in the limit when $\tilde{g}_{1}^{2}$ term is muted to neglect weak surface
renormalization effects, i.e. following a tree level scaling.

Therefore, one finds that

$\displaystyle\tilde{g}_{1}(\Lambda_{s}=\Omega_{D})$ $\displaystyle=$
$\displaystyle\tilde{g}_{1}(\Lambda_{s}=m)(\frac{\Omega_{D}}{m})^{d_{s}-1}$
$\displaystyle=$
$\displaystyle-{\tilde{G}^{2}_{fp0}}(\frac{m}{\Omega_{D}})^{D_{b}-d_{s}}$

where $d_{s}=2$ is the dimension of surfaces and $D_{b}=3$ is the bulk
dimension.

For the purpose of realizing surface topological superconductivity and
possible supersymmetry surface holographic matter, one can further optimize
phonon mediated surface interactions by choosing a topological bulk so that

$\displaystyle m\approx\Omega_{D}\ll W.$ (62)

By doing so, one can achieve an optimal phonon mediated interaction for given
bulk fermion-phonon interactions, i.e

$\displaystyle\tilde{g}_{1}(\Lambda_{s}=\Omega_{D})=\tilde{g}_{10}(\tilde{G}_{fp0},\frac{m}{\Omega_{D}}=1)$
$\displaystyle=$ $\displaystyle-{\tilde{G}^{2}_{fp0}},$

Compared with the general result in Eq.LABEL:match4, $\tilde{g}_{10}$ is equal
to $\tilde{G}^{2}_{fp0}$ in this limit, where all other suppression factors
due to $\frac{m}{\Omega_{D}}\leq 1$ are simply reduced to unity, a maximum
value. So in the limit $\Omega_{D}\geq m$ and for a given dimensionless
electron-phonon interaction constant $\tilde{G}^{2}_{fp0}$, the surface
attractive interaction reaches a maximum when $m\sim\Omega_{D}$. Intuitively,
one can relate the matching of surface bandwidth $m$ with Debye frequency
$\Omega_{D}$ to one kind of dynamic resonance. Below we will focus on this
limit only.

Construction based on point i) and ii) leads to the following effective field
theories for interacting surfaces

Figure 7:  An illustrative picture representing the effect of a metal
substrate on top of a topological insulator(TI) surface.

$\displaystyle{\mathcal{H}}_{surf}$ $\displaystyle=$ $\displaystyle v_{F}\int
d^{2}{\bf r}\psi_{s}^{\dagger}{\bf s}\cdot{\bf p}\psi_{s}$ $\displaystyle+$
$\displaystyle g_{1}\int d^{2}{\bf r}\psi_{s}^{\dagger}({\bf r})\psi_{s}({\bf
r})\psi_{s}^{\dagger}({\bf r})\psi_{s}({\bf r})$ $\displaystyle+$
$\displaystyle p_{s}^{2}\int d^{2}{\bf r}d^{2}{\bf
r^{\prime}}\psi_{s}^{\dagger}({\bf r})\psi_{s}({\bf r})\frac{1}{|{\bf r}-{\bf
r}^{\prime}|^{3}}\psi_{s}^{\dagger}({\bf r}^{\prime})\psi_{s}({\bf
r}^{\prime}).$

Under scale transformation, the dimensionless coupling constants,
$\tilde{g}_{1}=g_{1}\Lambda^{d_{s}-1}$ and
$\tilde{p}^{2}_{s}=e^{2}_{s}(\Lambda_{s}/m)^{2}$, transform according to the
following one-loop renormalization group equations ($t_{s}=\ln\Lambda_{s}/m$),

$\displaystyle\frac{\partial\tilde{g}_{1}}{\partial t_{s}}$ $\displaystyle=$
$\displaystyle({d_{s}-1})\tilde{g}_{1}+\tilde{g}^{2}_{1}$
$\displaystyle\frac{\partial\tilde{p}^{2}_{s}}{\partial t_{s}}$
$\displaystyle=$ $\displaystyle 2\tilde{p}^{2}_{s}$
$\displaystyle\tilde{g}_{1}(t_{s}=0)$ $\displaystyle=$
$\displaystyle\tilde{g}_{10},\tilde{p}^{2}_{s}(t_{s}=0)=\tilde{e}^{2}_{s0}.$
(65)

Below we are going to discuss implications of the principle of BBR by
utilizing solutions to Eq.65

###  IV.1 Surface Topological Superconductivity and Majorana Fermions

If surfaces are dominated by phonon-mediated interactions, topological
superconductivity can naturally emerge in the presence of finite chemical
potential $\mu$ via conventional spontaneous $U(1)$ symmetry breaking. This
can happen even in weakly interacting limits (although practically finite
temperature measurements can put certain low bounds on interactions.) If
surfaces are weakly interacting, then Eq.LABEL:RGEcphS leads to the following
simple solution for $\Lambda_{s}\leq m$,

$\displaystyle\tilde{g}_{1}(\Lambda_{s})=\tilde{g}_{10}(\frac{\Lambda_{s}}{m}),\mbox{
or }\tilde{p}_{s}(\Lambda_{s})=\tilde{e}^{2}_{s0}(\frac{\Lambda_{s}}{m})^{2}.$
(66)

It is evident that if we choose to work with crystals with $m\sim\Omega_{D}\ll
W$, a sufficient condition to realize surface topological superconductivity
via interactions is

$\displaystyle\tilde{g}_{10}>\alpha_{s0},\mbox{ or
}{\tilde{G}^{2}_{fp0}}>\frac{\alpha_{b0}}{1+c_{b}\alpha_{b0}\ln\frac{W}{m}}$
(67)

where the second inequality in terms of bulk interaction parameters is
obtained by applying Eq.LABEL:match4.

The effective field theory that describe emergent topological
superconductivity phenomena and majorana field hosted in surfaces can be

${\mathcal{H}}_{surf}-\mu N=\\\ \int d^{2}{\bf
r}\chi_{s}^{T}\biggl{(}-i\mathbb{I}\otimes{\bf e}_{y}\times{\bf
s}\cdot{\bf\nabla}-\tau_{z}\otimes
s_{y}\Delta-\mu\tau_{y}\otimes\mathbb{I}\biggr{)}\chi_{s}\\\ +g_{M}\int
d^{2}{\bf r}\chi_{s}^{T}({\bf r})\tau_{y}\otimes\mathbb{I}\chi_{s}({\bf
r})\chi^{T}_{s}({\bf r})\tau_{y}\otimes\mathbb{I}\chi_{s}({\bf r})$ (68)

Here
$\chi_{s}^{T}=(\chi_{1\uparrow},\chi_{1\downarrow},\chi_{2\uparrow},\chi_{2\downarrow})$
are four-component spinfull majorana fermion fields that forms a natural
representation of two-component surface fermions. In Eq.68, we assume that the
surface is oriented normal to the y-direction. $\tau_{\alpha}$, $\alpha=x,y,z$
act on a subspace spanned by subscript indices $1,2$ and $s_{\alpha}$,
$\alpha=x,y,z$ act on spin indices $\uparrow,\downarrow$. And
$\chi_{s}$-fields satisfy the standard algebra of real-fermions

$\displaystyle\chi^{\dagger}_{s}({\bf r})=\chi^{T}_{s}({\bf
r}),\\{\chi_{s}({\bf r}),\chi_{s}({\bf
r}^{\prime})\\}=\mathbb{I}\otimes\mathbb{I}\delta({\bf r}-{\bf r}^{\prime})$
(69)

where two unity matrices are defined for two subspaces introduced above
respectively. In presenting Eq.68, we have restricted ourselves to density-
density interactions although it is possible to include other interactions
such as spin-density interactions. Detailed dynamics induced by other
interactions are not quite relevant to our current discussions of surface
topological superconductivity and will be presented in a separate article in
preparation.

Topological surface superconductivity belongs to a class of symmetry-protected
statesLudwig ; QiZhang ; kitaev  and it is stable with respect to weak
interactions represented by $g_{M}$. Finite pairing amplitude $\Delta$ in
Eq.68 induced by any weak interactions at a finite chemical potential, $\mu$,
is driven by the standard Cooper instability as in metals. On topological
surfaces, this not only breaks $U(1)$ symmetry spontaneously but further
results in emergent relativistic Majorana fields. On the other hand, if
$\Delta$ is taken to be zero, Eq.68 has an additional $U(1)$ invariance under
a rotation around $\tau_{y}$ axis. We identify it as a global $U(1)$ symmetry
of surface complex fermions and the theory restores U(1) symmetry as
anticipated.

###  IV.2 Supersymmetry Surface Holographic Matter

It is perhaps more interesting to consider the limit of strongly interacting
surfaces. This limit is associated with a phenomenon of emergent supersymmetry
suggested and studied before in a few different condensed matter systems
including topological surfacesLee07 ; Grover14 ; Ponte14 ; Jian15 ; Li17 ;
Li18 ; Jian17 . Supersymmetry states are conformal invariant and can be easily
identified with scale-invariant solutions of effective surface field theories
we are examining. Indeed, Eq.65 has a strong coupling fixed point atSachdev

$\displaystyle\tilde{g}_{1}^{*}=-({d_{s}-1}).$ (70)

We identify it as a quantum critical point (QCP) for transitions from a weakly
interacting gapless topological surface to a gapped topological surface
superconductor due to spontaneously $U(1)$ symmetry breaking at zero chemical
potential or $\mu=0$. The effective surface theories for such a QCP can be
obtained from Eq.68 by setting $\Delta=\mu=0$, i.e.,

$\displaystyle{\mathcal{H}}_{surf}$ $\displaystyle=$ $\displaystyle\int
d^{2}{\bf r}\chi_{s}^{T}\mathbb{I}\otimes{\bf e}_{y}\times{\bf
s}\cdot\frac{{\bf\nabla}}{i}\chi_{s}$ (71) $\displaystyle+$ $\displaystyle
g_{M}\int d^{2}{\bf r}[\chi_{s}^{T}({\bf
r})\tau_{x}\otimes{s_{y}}\chi_{s}({\bf r})\chi^{T}_{s}({\bf
r})\tau_{x}\otimes{s_{y}}\chi_{s}({\bf r})$ $\displaystyle+$
$\displaystyle\chi_{s}^{T}({\bf r})\tau_{z}\otimes{s_{y}}\chi_{s}({\bf
r})\chi^{T}_{s}({\bf r})\tau_{z}\otimes{s_{y}}\chi_{s}({\bf r})].$

Here we have used a representation for 4-fermion operators that is most
convenient for discussions on symmetries. In this representation, surfaces are
invariant under a $U(1)$ rotation generated around $\tau_{y}$, i.e.
$U(\theta)=e^{i\theta\tau_{y}/2}$.

As stated in the previous subsection, this is an emergent $U(1)$ symmetry that
can be identified with charge conservation. One can show explicitly this
describes surface complex fermions near a supersymmetry QCP. Eq.70 implies
that in the vicinity of strong coupling,

$\displaystyle{\tilde{G}^{2}_{fp0}}=(d_{s}-1),$ (72)

one can anticipate surfaces become a conformal field theory state with
supersymmetry.

##  V Conclusions

In conclusion, we have put forward in this article a general bulk-boundary
relation of interactions and have illustrated its application in a few
concrete limits. We have also investigated this principle in a diagrammatic-
based dimensional reduction approach. This general principle has been employed
to understand the conditions for spontaneous symmetry breaking to occur on
topological surfaces. We further discuss a potential route to supersymmetric
matter via strong electron-phonon interactions and additional metallic
screening of long-range Coulomb interactions. It is possible to apply the
principle to other approaches to topological surface symmetry breaking
phenomena. We plan to explore this direction in near future.

This project is funded by NSERC(Canada) discovery grant under
RGPIN-2020-07070. One of us (F.Z.) wants to thank discussions with Ian Affleck
and Fan Yang.

## Appendix A

Here we describe the step by step solution to the $q_{z}$ momentum integral in
Eq.LABEL:effsurfacepropCou, LABEL:effsurfacepropeph which contributes to the
quantum correction to the effective surface Coulomb and phonon propagators
respectively. Let us solve for the correction term to the Coulomb propagator
first.

### Evaluation of $\mathcal{K}_{cou}^{2D(1)}(\textbf{q}_{\perp})$

It was already mentioned in the main text that, in the case of Coulomb
propagator, the momentum integral is UV convergent. Therefore the
regularization term $\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}$ can be
safely ignored by taking the limit of $M\rightarrow\infty$ ( where
$M=\frac{2m}{v_{F}}$). This results, in the first order correction term to
have the form given by,

$\displaystyle\tilde{K}_{cou}^{2D(1)}(\textbf{q}_{\perp})$ $\displaystyle=$
$\displaystyle\frac{e^{4}_{b0}}{2\pi^{2}v_{F}}\int\frac{dq_{z}}{2\pi}\frac{1}{q_{\perp}^{2}+q^{2}_{z}}\Pi\left(q_{\perp},q_{z}\right)$
(73) $\displaystyle=$
$\displaystyle\frac{e^{4}_{b0}}{2\pi^{2}v_{F}}\int\frac{dq_{z}}{2\pi}\frac{1}{q_{\perp}^{2}+q^{2}_{z}}\int^{1}_{0}d\beta\,\beta\left(1-\beta\right)\text{Log}\left[\frac{m^{2}}{\beta(1-\beta)W^{2}}+\frac{v^{2}_{F}\left(q_{\perp}^{2}+q^{2}_{z}\right)}{W^{2}}\right]$
(74)

Figure 8: Key-Hole contour, with branch cut at $i\delta$

The integral has a branch cut at $q_{z}=\pm
i\sqrt{\frac{m^{2}}{v_{F}\beta\left(1-\beta\right)}+q^{2}_{\perp}}$. Likewise,
it has a poles at $q_{z}=\pm iq_{\perp}$.Therefore, the contour is chosen in
such a way that the integral runs from $-\infty$ to $\infty$ through real line
and the branch cut is from
$i\sqrt{\frac{m^{2}}{v^{2}_{F}\beta\left(1-\beta\right)}+q^{2}_{\perp}}$ to
$i\infty$. This type of Contour is called the Keyhole Contour, shown in fig.8.
The integral can be split into,

$\displaystyle\oint$ $\displaystyle=$
$\displaystyle\int^{+\infty}_{-\infty}+\int_{R}+I_{left}+I_{right}$ (75)

Here $I_{right}$ is the integral through the imaginary axis just along the
right side of it, while $I_{left}$ is the integral just along the left side.
The integral over the curve R and $\varepsilon$ will vanish in the limit
$R\rightarrow\infty$. Therefore, we are left with $I_{left}+I_{right}$ and the
contour integral in the LHS, where we could use residue theorem easily to
solve them. The LHS is given by,

$\oint\frac{dq_{z}}{2\pi}\frac{1}{q_{\perp}^{2}+q^{2}_{z}}\text{Log}\left[\frac{m^{2}}{\beta(1-\beta)W^{2}}+\frac{v^{2}_{F}\left(q_{\perp}^{2}+q^{2}_{z}\right)}{W^{2}}\right]\\\
=\frac{1}{2q_{\perp}}\log\frac{m^{2}}{\beta(1-\beta)W^{2}}$ (76)

Now, we focus on the integrals along the branch cut. First, let us call the
branch cut point as
$\delta=\sqrt{\frac{m^{2}}{v^{2}_{F}\beta\left(1-\beta\right)}+q^{2}_{\perp}}$.
Then $I_{left}$ and $I_{right}$ are given by,

$\displaystyle I_{right}$ $\displaystyle=$
$\displaystyle\int^{i\delta}_{i\infty}\frac{dq_{z}}{2\pi}\frac{1}{q_{\perp}^{2}+q^{2}_{z}}\left[\log\left(q_{z}-i\delta\right)+2\pi
i\right]$ (77) $\displaystyle I_{left}$ $\displaystyle=$
$\displaystyle\int^{i\infty}_{i\delta}\frac{dq_{z}}{2\pi}\frac{1}{q_{\perp}^{2}+q^{2}_{z}}\log\left(q_{z}-i\delta\right)$
(78)

Adding these two, we get,

$\displaystyle I_{right}+I_{left}$ $\displaystyle=$ $\displaystyle-2\pi
i\int^{i\infty}_{i\delta}\frac{dq_{z}}{2\pi}\frac{1}{q_{\perp}^{2}+q^{2}_{z}}$
(79) $\displaystyle=$ $\displaystyle-\frac{1}{2q_{\perp}}\log\frac{\
\frac{\delta}{q_{\perp}}+1}{\frac{\delta}{q_{\perp}}-1}$

where
$\frac{\delta}{q_{\perp}}=\sqrt{\left(\frac{m}{v_{F}q_{\perp}}\right)^{2}\frac{1}{\beta\left(1-\beta\right)}+1}$.

This result defines the bare theory of the surface Fermions interacting via
Coulomb field. Since the UV cut-off of surface theory is the bulk mass $m$, we
can set the in-plane momentum $v_{F}q_{\perp}\approx m$. In this limit, the
constant
$\frac{\delta}{q_{\perp}}\approx\sqrt{\frac{1}{\beta\left(1-\beta\right)}+1}$.
Thus the branch cut integral becomes,

$\displaystyle I_{right}+I_{left}$ $\displaystyle\approx$
$\displaystyle-\frac{1}{2q_{\perp}}\log\frac{\sqrt{\frac{1}{\beta\left(1-\beta\right)}+1}+1}{\sqrt{\frac{1}{\beta\left(1-\beta\right)}+1}-1}$
(80)

The first order correction to the surface Coulomb propagator as a result of
the bulk electron renormalization attains the asymptotic form,

$\displaystyle
q_{\perp}\mathcal{K}^{2D(1)}_{cou}(v_{F}\textbf{q}_{\perp})|_{v_{F}\textbf{q}_{\perp}\approx
m}$ $\displaystyle\approx$
$\displaystyle\frac{e^{2}_{b0}}{2}\frac{e^{2}_{b0}}{6\pi^{2}v_{F}}\left[\log\frac{m}{W}+1.2611\right]$

where the number 1.2611 is a result of the following integration,

$\displaystyle 1.2611$ $\displaystyle=$ $\displaystyle
3\int^{1}_{0}d\beta\beta(1-\beta)\biggl{[}\log\frac{\sqrt{\frac{1}{\beta\left(1-\beta\right)}+1}+1}{\sqrt{\frac{1}{\beta\left(1-\beta\right)}+1}-1}$
(82) $\displaystyle-$ $\displaystyle\log\beta(1-\beta)\biggr{]}$

### Evaluation of $\mathcal{K}^{2D(1)}_{e-ph}(\textbf{q}_{\perp})$

It is to be noted that unlike the Coulomb integral, we must include the
regularization term since the integral is UV divergent. This is evident from
the expression for effective phonon propagator in the 3D bulk given in Eq.30,
whose dimension is zero. Therefore, the first order quantum correction to the
surface phonon propagator has the form,

$\displaystyle\mathcal{K}^{2D(1)}_{e-ph}(\textbf{q}_{\perp})$ $\displaystyle=$
$\displaystyle\frac{e^{2}_{b}(\Omega_{D})}{\pi^{2}v_{F}}\int\frac{dq_{z}}{2\pi}\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}\frac{v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}{\omega^{2}-v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}\Pi(v_{F}q_{\perp},v_{F}q_{z})$
$\displaystyle=$
$\displaystyle\frac{e^{2}_{b0}(\Omega_{D})}{\pi^{2}v_{F}}\int\frac{dq_{z}}{2\pi}\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}\frac{v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}{\omega^{2}-v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}\int^{1}_{0}d\beta\,\beta\left(1-\beta\right)\text{Log}\left[\frac{m^{2}}{\beta(1-\beta)\Omega_{D}^{2}}+\frac{v^{2}_{F}\left(q_{\perp}^{2}+q^{2}_{z}\right)}{\Omega_{D}^{2}}\right]$
$\displaystyle=$
$\displaystyle\frac{e^{2}_{b0}(\Omega_{D})}{\pi^{2}v_{F}}\int\frac{dq_{z}}{2\pi}\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}\frac{v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}{\omega^{2}-v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}\int^{1}_{0}d\beta\,\beta\left(1-\beta\right)\text{Log}\frac{m^{2}}{\beta(1-\beta)\Omega_{D}^{2}}$
$\displaystyle+$
$\displaystyle\frac{e^{2}_{b0}(\Omega_{D})}{\pi^{2}v_{F}}\int\frac{dq_{z}}{2\pi}\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}\frac{v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}{\omega^{2}-v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}\int^{1}_{0}d\beta\,\beta\left(1-\beta\right)\text{Log}\left[1+\frac{v^{2}_{F}\beta(1-\beta)\left(q^{2}_{\perp}+q^{2}_{z}\right)}{m^{2}}\right]$
(84)

In the last step, we have written the $\log$ expression as a sum of two terms
for reasons that will be clear soon. Both the terms have poles in the Upper
Half Plane at $q_{z}=i\sqrt{v^{2}_{p}q_{\perp}^{2}-\omega^{2}},\pm iM$. While
the first term is smooth except for the poles in the Complex plane, the second
term has a branch cut at
$q_{z}=i\sqrt{\frac{m^{2}}{v^{2}_{F}\beta\left(1-\beta\right)}+q^{2}_{\perp}}$.
Solving the first term will give a logarithmic correction to the bare
propagator, with the log term being a ratio of the UV cut off scales for the
surface and the bulk, given by $m/\Omega_{D}$. The second term could be solved
only by using a branch cut integral just as in the previous case. Let us call
the integral expression to be $I$,

$I=\int\frac{dq_{z}}{2\pi}\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}\frac{v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}{\omega^{2}-v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}\\\
\log\left[1+\frac{v^{2}_{F}\beta(1-\beta)\left(q^{2}_{\perp}+q^{2}_{z}\right)}{m^{2}}\right]$
(85)

Consider the same Keyhole Contour given in fig.8. The integral over the curve
R will vanish in the limit $R\rightarrow\infty$ The LHS is just the sum of the
residues of the integrand at
$q_{z}=i\sqrt{q^{2}_{\perp}-\frac{\omega^{2}}{v^{2}_{p}}}$ and
$q_{z}=iM\sqrt{1-\epsilon}$ which is given by,

$\displaystyle\mathcal{C}$ $\displaystyle=$
$\displaystyle\oint\frac{dq_{z}}{2\pi}\left(\frac{M^{2}(1-\epsilon)}{q_{z}^{2}+M^{2}(1-\epsilon)}\right)^{2}\frac{v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}{\omega^{2}-v^{2}_{p}\left(q_{\perp}^{2}+q^{2}_{z}\right)}\log\left[1+\frac{v^{2}_{F}\beta(1-\beta)\left(q^{2}_{\perp}+q^{2}_{z}\right)}{m^{2}}\right]$
$\displaystyle=$
$\displaystyle\frac{M}{8}\left(\frac{2q^{2}_{\perp}}{\frac{\omega^{2}}{v^{2}_{p}}-q^{2}_{\perp}}-\frac{\frac{\omega^{2}}{v^{2}_{p}}}{\frac{\omega^{2}}{v^{2}_{p}}-q_{\perp}^{2}}\left(\frac{M^{2}}{\left(M-\sqrt{q_{\perp}^{2}-\frac{\omega^{2}}{v^{2}_{p}}}\right)^{2}}+\frac{M^{2}}{\left(M+\sqrt{q_{\perp}^{2}-\frac{\omega^{2}}{v^{2}_{p}}}\right)^{2}}\right)\right)\log\left[1+\frac{v^{2}_{F}\beta(1-\beta)}{m^{2}}\left(q_{\perp}^{2}-M^{2}\right)\right]$
$\displaystyle+$
$\displaystyle\frac{M^{3}}{8}\frac{\frac{\omega^{2}}{v^{2}_{p}}}{\frac{\omega^{2}}{v^{2}_{p}}-q_{\perp}^{2}}\left(\frac{1}{\left(M-\sqrt{q_{\perp}^{2}-\frac{\omega^{2}}{v^{2}_{p}}}\right)^{2}}-\frac{1}{\left(M+\sqrt{q_{\perp}^{2}-\frac{\omega^{2}}{v^{2}_{p}}}\right)^{2}}\right)\log\left[1+\frac{v^{2}_{F}\beta(1-\beta)}{m^{2}}\frac{\omega^{2}}{v^{2}_{p}}\right]$
$\displaystyle+$
$\displaystyle\frac{M}{4}\left(\frac{2q^{2}_{\perp}}{\frac{\omega^{2}}{v^{2}_{p}}-q^{2}_{\perp}}-\frac{\frac{\omega^{2}}{v^{2}_{p}}}{\frac{\omega^{2}}{v^{2}_{p}}-q_{\perp}^{2}}\left(\frac{M}{M+\sqrt{q_{\perp}^{2}-\frac{\omega^{2}}{v^{2}_{p}}}}+\frac{M}{M-\sqrt{q_{\perp}^{2}-\frac{\omega^{2}}{v^{2}_{p}}}}\right)\right)\frac{4\beta(1-\beta)}{1+\frac{v^{2}_{F}\beta(1-\beta)}{m^{2}}\left(q_{\perp}^{2}-M^{2}\right)}$
(87)

And the sum of the integrals along the branch cut, $I_{left}+I_{right}$ is
given by,

$\displaystyle I_{left}+I_{right}$ $\displaystyle=$ $\displaystyle-2\pi
i\int^{i\infty}_{i\delta}\frac{dq_{z}}{2\pi}\left(\frac{M^{2}}{q_{z}^{2}+M^{2}}\right)^{2}\frac{q_{\perp}^{2}+q^{2}_{z}}{\frac{\omega^{2}}{v^{2}_{p}}-\left(q_{\perp}^{2}+q^{2}_{z}\right)}$
$\displaystyle=$
$\displaystyle\frac{M^{4}}{\frac{\omega^{2}}{v^{2}_{p}}-q^{2}_{\perp}+M^{2}}\biggl{[}-\frac{\omega^{2}}{\frac{\omega^{2}}{v^{2}_{p}}-q^{2}_{\perp}+M^{2}}\left(\frac{1}{2M}\log\frac{\delta-M}{\delta+M}+\frac{1}{2\sqrt{q^{2}_{\perp}-\frac{\omega^{2}}{v^{2}_{p}}}}\log\frac{\delta-\sqrt{q^{2}_{\perp}-\frac{\omega^{2}}{v^{2}_{p}}}}{\delta+\sqrt{q^{2}_{\perp}-\frac{\omega^{2}}{v^{2}_{p}}}}\right)$
$\displaystyle+$
$\displaystyle\frac{M^{2}-q^{2}_{\perp}}{2M}\left(\frac{1}{2M^{2}}\log\frac{\delta+M}{\delta-M}-\frac{\delta}{M}\frac{1}{\delta^{2}-M^{2}}\right)\biggr{]}$
(89)

The resulting solution will define the first order quantum correction to the
bare theory of the surface Fermions interacting via lattice phonon field.
Since the UV cut-off of surface theory is the bulk mass $m$, we can set the
in-plane momentum $v_{F}q_{\perp}\approx m$. Also the phonon frequency is set
to $\omega\approx 0$, the limit which supports Superconducting pairing. In
this limit, the sum of the residues of the keyhole contour, whose full
expression is given in Eq.87 attains the limiting value,

$\displaystyle\mathcal{C}$ $\displaystyle\approx$
$\displaystyle-\frac{M}{4}\log\left[1-3\beta(1-\beta)\right]-\frac{M}{4}\frac{8\beta(1-\beta)}{1-3\beta(1-\beta)}$

Similarly, the sum of the integrals along the two sides of the branch cut
become,

$I_{right}+I_{left}=\\\
\frac{M}{4}\left[\log\frac{\frac{1}{2}\sqrt{\frac{1}{\beta(1-\beta)}+1}+1}{\frac{1}{2}\sqrt{\frac{1}{\beta(1-\beta)}+1}-1}-\frac{\sqrt{\frac{1}{\beta(1-\beta)}+1}}{\frac{1}{4\beta(1-\beta)}+\frac{1}{4}-1}\right]$
(91)

The solution of the integral denoted by $I$ given in eqn 85 is obtained by
subtracting the integral along the branch cuts from the sum of the residues.
Thus, we have,

$\displaystyle I$ $\displaystyle=$
$\displaystyle\mathcal{C}-I_{left}-I_{right}$ (92) $\displaystyle=$
$\displaystyle-\frac{M}{4}\biggl{[}2\log\left[2+\sqrt{1+\frac{1}{\beta(1-\beta)}}\right]+\log\beta(1-\beta)$
$\displaystyle-$
$\displaystyle\frac{2}{1+\frac{1}{2}\sqrt{\frac{1}{\beta(1-\beta)}+1}}\biggr{]}$

Finally, let us write down the first order correction to the surface phonon
propagator as a result of the bulk electron renormalization,

$\mathcal{K}_{e-ph}^{2D(1)}(\omega,v_{F}q_{\perp})|_{\omega\approx
0,v_{F}q_{\perp}=m}\approx\\\
-\frac{G^{2}_{fp0}m}{2v_{F}}\frac{e^{2}_{b}(\Omega_{D})}{3\pi^{2}v_{F}}\left[\log\frac{m}{\Omega_{D}}+1.06874..\right]$
(93)

where the numerical constant 1.06874 is the result of the following
integration,

$\displaystyle 1.06874$ $\displaystyle=$ $\displaystyle
6\int^{1}_{0}d\beta\beta(1-\beta)\biggl{[}\log\left[2+\sqrt{1+\frac{1}{\beta(1-\beta)}}\right]$
$\displaystyle-$
$\displaystyle\log\sqrt{\beta(1-\beta)}-\frac{2}{1+\frac{1}{2}\sqrt{\frac{1}{\beta(1-\beta)}+1}}\biggr{]}$

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