# New boundary criticality in topological phases

Xiaoyang Shen  Department of Physics, Tsinghua University, Beijing 100084,
China  Institute for Advanced Study, Tsinghua University, Beijing 100084,
China  Zhengzhi Wu  Institute for Advanced Study, Tsinghua University, Beijing
100084, China  Shao-Kai Jian<EMAIL_ADDRESS>Department of Physics and
Engineering Physics, Tulane University, New Orleans, Louisiana, 70118, USA

###### Abstract

We study the boundary criticality enriched by boundary fermions, which
ubiquitously emerge in topological phases of matter, with a focus on
topological insulators and topological superconductors. By employing
dimensional regularization and bosonization techniques, we uncover several
unprecedented boundary universality classes. These include the boundary Gross-
Neveu-Yukawa critical point and the special Berezinskii-Kosterlitz-Thouless
(BKT) transition, both resulting from the interplay between edge modes and
bulk bosons. We present a comprehensive sketch of the phase diagram that
accommodates these boundary criticalities and delineate their critical
exponents. Additionally, we explore a 1+1D conformal defect decorated with
fermions, where a defect BKT transition is highlighted. We conclude with a
discussion on potential experimental realizations of these phenomena.

Introductions.— Boundary physics holds a pivotal role across diverse fields,
such as the quantum Hall effect with boundary states in condensed matter
physics von Klitzing  _et al._ (1980); Tsui  _et al._ (1982); Laughlin (1981,
1983) and the AdS/CFT correspondence Maldacena (1998); Witten (1998); Gubser
_et al._ (1998) involving quantum field theory at the boundary of a
gravitational system in string theory Polchinski (1996); Takayanagi (2011);
Fujita  _et al._ (2011); Recknagel and Schomerus (2013); Izumi  _et al._
(2022). For critical phenomena, the renormalization group (RG) flow leads to
universality classes, where the system exhibits scale invariance. In this
context, the presence of boundaries enriches the universality class, giving
rise to unique boundary conformal field theories (BCFTs) Cardy (1984, 1989);
Affleck and Ludwig (1991). BCFT serves as a unified framework for
understanding boundary critical physics across diverse physical systems.
Hence, investigations into BCFT hold a broad appeal and bear wide-ranging
applications across numerous research disciplines Cardy (2008); Ge and Jian
(2024); Shen (2023); Andrei  _et al._ (2018); Ji and Chen (2024); Wu  _et al._
(2020); Brillaux and Fedorenko (2021); Ma  _et al._ (2022); Brillaux  _et al._
(2024).

An important research field in condensed matter physics that is closely
related to boundaries is the symmetry protected topological state (SPT) Kitaev
(2009); Ryu  _et al._ (2010); Gu and Wen (2009); Hasan and Kane (2010); Qi and
Zhang (2011); Chen  _et al._ (2012). The distinct feature of SPT from a
trivial state is the presence of gapless boundary modes. A well-known example
is the topological insulator Kane and Mele (2005); Bernevig  _et al._ (2006),
which is characterized by gapless fermions on the boundary. Because the
topological insulator has been realized experimentally König  _et al._ (2007);
Hsieh  _et al._ (2008), it is practically important to investigate the
interplay between BCFTs and the SPT. Notice that the notation of SPT has
recently been generalized to the realm of gapless phases of matter Scaffidi
_et al._ (2017); Verresen  _et al._ (2021); Yu  _et al._ (2024). Nevertheless,
the higher dimensional cases involving weakly interacting fermionic SPT states
have not been fully explored yet, especially in a theoretically controlled
way.

In this paper, we study the boundary criticalities enriched by boundary
gapless fermionic modes Liu  _et al._ (2021); Giombi  _et al._ (2022, 2023);
Barrat  _et al._ (2023); Shachar  _et al._ (2024); Herzog and Schaub (2023),
with an application to the topological insulators (TIs) and topological
superconductors (TSCs) protected by the time-reversal symmetry. It is well-
known that in the presence of a boundary, the Wilson-Fisher (WF) fixed point
can be enriched into ordinary, extraordinary, or special transition depending
on the coupling strength at the surface layer Diehl and Dietrich (1981, 1983);
Giombi and Khanchandani (2020); Padayasi  _et al._ (2022); Parisen Toldin and
Metlitski (2022); Metlitski (2022). A new ingredient in our model is that a
gapless mode, a Dirac fermion or Majorana fermion, resides on the boundary,
and enriches the transition via a Yukawa coupling at the boundary. By
implementing a $4-\epsilon$ RG calculation, we uncover a boundary Gross-Neveu-
Yukawa fixed point at the boundary of TSCs. On the other hand, in the case of
2+1D TIs, because the four-fermion interaction is marginal at its 1+1D edge,
we include it nonperturbatively by bosonization. In stark contrast to the
boundary transition in the $O(N)$ model, we find an unconventional ordinary
transition enriched by Luttinger liquid with a continuous boundary scaling
dimension. Interestingly, this unconventional ordinary transition terminates
at a Berezinskii–Kosterlitz–Thouless (BKT) like transition Kosterlitz and
Thouless (2018); Berezinsky (1971), which is termed as the special BKT
transition. We also extend our bosonization calculation to investigate the
1+1D conformal defect Chang  _et al._ (2023); Giombi and Liu (2023); Trépanier
(2023) decorated with fermions, and find a parallel scenario, where a distinct
conformal defect with a continuous scaling dimension terminates at a defect
BKT transition. Finally, we briefly discuss the experimental probes of various
boundary criticalities, and propose possible experimental realizations.

Figure 1:  (a) RG flow parametrized by $\tilde{\lambda}$ and $g$.
$\tilde{\lambda}$ stands for $3\lambda/(4\pi)^{2}$. (b) Phase diagram of
boundary criticality in TSCs as functions of the bulk coupling and the
boundary enhancement. BD(BO)/SD(SO)+gapless/gapped stands for bulk disordered
(bulk ordered)/surface disordered (surface ordered) with gapless/gapped
boundary fermion. The multicritical point corresponds to the boundary Gross-
Neveu-Yukawa fixed point illustrated in (a).

Boundary Ising criticality.— We briefly review the boundary Ising critical
theory. The field theory of an Ising model in a $d$-dimensional Euclidean
spacetime $\mathcal{M}$ with a boundary $\partial\mathcal{M}$ is given by

$S_{\text{b}}=\int_{\mathcal{M}}d^{d}x\left[\frac{1}{2}(\partial\phi)^{2}+\frac{\lambda}{4!}\phi^{4}\right]+\int_{\partial\mathcal{M}}d^{d-1}xh\phi^{2}\,,$
(1)

where $\phi$ denotes a real scalar field,
$(\partial\phi)^{2}=\sum_{\mu=0}^{d-1}(\partial_{\mu}\phi)^{2}$ and $\lambda$
is the coupling strength. The second term denotes a surface term, with $h$
being a surface mass 111Some boundary symmetry-breaking coupling that drives
the system to the normal universality class is neglected. See Diehl Diehl and
Dietrich (1981, 1983) for a detailed discussion.. The bulk is at the WF fixed
point, and in the RG calculation, it does not receive corrections from the
surface. Hence, depending on the surface mass $h$, there are distinct boundary
universality classes: the special transition corresponds to vanishing surface
mass $h=0$ 222Strictly speaking, the surface mass is located at some
$h=h_{\text{sp}}$ for the special transition., and the ordinary and
extraordinary transition correspond, respectively, to a disordered and ordered
surface, together with $h\to+\infty$ and $h\to-\infty$.

Boundary GNY criticality.— Our model starts by introducing the fermion,
$\psi$, on the boundary, coupling to the surface order parameter,

$\begin{split}S_{\text{bGNY}}=&\int
d^{d}x\left[\frac{1}{2}(\partial\phi)^{2}+\frac{\lambda}{4!}\phi^{4}\right]\\\
&+\int
d^{d-1}x\left(\bar{\psi}\not{\partial}\psi+g\phi\bar{\psi}\psi+h\phi^{2}\right)\,,\end{split}$
(2)

where $g$ denotes the boundary Yukawa coupling, and $\psi$ denotes a two-
component surface Majorana fermion of a TSC or a two-component surface Dirac
fermion of a TI. We use the convention: $\bar{\psi}=\psi^{\dagger}\gamma^{0}$
,$\not{\partial}=\sum_{\mu=0}^{2}\partial_{\mu}\gamma^{\mu}$, and
$\gamma^{0}=\sigma_{z},\gamma^{1}=-\sigma_{y},\gamma^{2}=\sigma_{x}$. Since we
are interested in the BCFT, the bulk is tuned to the WF critical point with
$\phi$ being the Ising order parameter field. When the Ising field $\phi$
orders, it will break the time reversal symmetry and consequently gap out the
edge mode $\psi$. In general, the boson velocity $v_{b}$ and fermion velocity
$v_{f}$ can be different. However, it can be shown that there exists a stable
fixed point $(v_{f}/v_{b})_{*}=1$, implying an emergent Lorentz invariance at
the boundary (see Supplemental Material Sec. I). The emergent Lorentz symmetry
allows us to scale $v_{f}=v_{b}=1$.

At the special transition $h=0$, dimension regularization with $\epsilon=4-d$
leads to the following the RG equations (see Supplemental Material Sec. II),

$\displaystyle\frac{dg}{dl}$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{2}g-\frac{2}{3\pi^{2}}g^{3}+\frac{1}{32\pi^{2}}\lambda
g\,,$ (3) $\displaystyle\frac{d\lambda}{dl}$ $\displaystyle=$
$\displaystyle\epsilon\lambda-\frac{3}{16\pi^{2}}\lambda^{2}\,.$ (4)

Note that the second equation is exactly the 1-loop RG equation for the bulk
WF fixed point because the bulk does not receive renormalization from the
boundary. The bulk coupling, however, renormalizes the boundary Yukawa
coupling. The RG equations have four types of fixed points: a UV stable
Gaussian fixed point, a WF fixed point, which is unstable along $g$ direction,
a mixed Yukawa fixed point with $\lambda_{*}=0$ and
$g_{*}^{2}=\frac{3\pi^{2}\epsilon}{4}$ Herzog and Huang (2017), which is
unstable along $\lambda$ direction, and an IR stable boundary Gross-Neveu-
Yukawa (GNY) fixed point with $\lambda_{*}=\frac{(4\pi)^{2}\epsilon}{3}$ and
$g_{*}^{2}=\pi^{2}\epsilon$. The RG flow is shown in Fig. 1 (a). At the
boundary GNY fixed point, the critical exponent for the boundary order
parameter and the boundary correlation length are unchanged, i.e.,
$\tilde{\Delta}_{\hat{\phi}}=\frac{d-2}{2}-\frac{\epsilon}{6}$,
$\frac{1}{\nu_{\text{bdy}}}=2-\frac{\epsilon}{3}$, but most interestingly, we
obtain a new anomalous dimension of the fermion,
$\Delta_{\psi}=\frac{d-1}{2}+\frac{\epsilon}{12}$.

The above perturbative $\epsilon$ expansion is trustworthy for small
$\epsilon$. We believe the boundary GNY fixed point can be extrapolated to
$\epsilon=1$ for a 2+1D bulk system with a 1+1D edge in the context of
topological superconductivity. More precisely, it describes the special
transition at a spontaneous time reversal symmetry breaking transition of a
2+1 dimensional topological superconductor. In this case, the time reversal
symmetry protects gapless edge Majorana fermions, which are coupled to the
boundary order parameter via a boundary Yukawa coupling. When the time
reversal symmetry is spontaneously broken in the bulk, the gapless edge
fermion is no longer protected. Based on the phase diagram of the boundary
critical Ising model, one can sketch the phase diagram enriched by the edge
mode, as shown in Fig. 1 (b). The phase diagram is controlled under two
parameters, the bulk coupling $J$ and the ratio between the surface coupling
$J_{1}$ and bulk coupling $\kappa\equiv J_{1}/J$, sometimes dubbed as the
surface enhancement. Consider a weak bulk coupling $J<J_{c}$ and a weak
surface enhancement $\kappa<\kappa_{c}$, both bulk and surface are disordered.
As the bulk coupling increases to $J>J_{c}$, the bulk and surface order
simultaneously, and the ordinary transition emerges at $J=J_{c}$. At the
ordinary transition, thanks to the Dirichlet boundary condition, the boson
gradient $\partial_{z}\phi$ couples to the gapless boundary fermions, and the
coupling turns out to be irrelevant. As a result, the fermion is a spectator
at the ordinary transition. On the other hand, for a large surface enhancement
$\kappa>\kappa_{c}$, the critical value of the coupling at the surface is less
than that in the bulk. In this case, starting from a weak bulk coupling
$J<J_{c}$, as the bulk coupling increases, the surface will order with the
bulk remaining disordered. At this surface transition point, one can solely
utilize the surface order parameter $\hat{\phi}$ and edge Majorana mode $\psi$
to construct an effective field theory. The theory then flows to the
celebrated fixed point with an emergent $\mathcal{N}=1$ supersymmetry or
tricritical Ising critical point Friedan  _et al._ (1985); Grover  _et al._
(2014); Jian  _et al._ (2015); Fei  _et al._ (2016); Li  _et al._ (2017). As
the bulk coupling keeps increasing and reaches the critical strength of the
bulk, an extraordinary phase transition takes place. Since the fermion is
gapped, it is not involved in the extraordinary transition. Now, most
interestingly, the ordinary transition and extraordinary transition merge at a
multicritical point, $J=J_{c}$ and $\kappa=\kappa_{c}$, the special fixed
point. As we discussed above, the special fixed point decorated by the edge
fermions is described by the boundary GNY fixed point.

Figure 2: Phase diagram for boundary criticality in TIs as a function of the bulk coupling $J$ and the boundary enhancement $\kappa$. Note that the ordinary transition is also enriched by gapless boundary fermions. Boundary universality class | $\Delta_{\psi}$ | $\tilde{\Delta}_{\hat{\phi}}$ | $1/\nu_{\text{bdy}}$
---|---|---|---
Boundary GNY ($\epsilon=4-d$) | $\frac{d-1}{2}+\frac{\epsilon}{12}$ | $\frac{d-2}{2}-\frac{\epsilon}{6}$ | $2-\frac{\epsilon}{3}$
Special BKT ($K_{c}=2-\Delta_{\text{ord}}$) | $\frac{1}{2}+\frac{1}{4}\left(K_{c}+\frac{1}{K_{c}}-2\right)$ | $K_{c}$ | $\xi\propto e^{\frac{b}{\sqrt{K-K_{c}}}}$
Defect BKT ($K_{c}^{\prime}=2-\Delta_{\text{defect}}$) | $\frac{1}{2}+\frac{1}{4}\left(K_{c}^{\prime}+\frac{1}{K_{c}^{\prime}}-2\right)$ | $K_{c}^{\prime}$ | $\xi\propto e^{\frac{b^{\prime}}{\sqrt{K-K_{c}^{\prime}}}}$
Table 1: Critical exponents at the boundary criticality in topological
systems. The $\Delta_{\text{ord}}$ ($\Delta_{\text{defect}}$) denotes the
scaling dimension of the boundary (defect) order parameter.
$\tilde{\Delta}_{\hat{\phi}}$ denotes the scaling dimension in the fermion
enriched boundary transition. $\Delta_{\text{ord}}=1.2626$ from Monte Carlo
simulation Deng  _et al._ (2005) in $d=2+1$, and
$\Delta_{\text{defect}}=1-\frac{\epsilon}{6}$ from a $d=4-\epsilon$
calculation Giombi and Liu (2023). $b,b^{\prime}$ are nonuniversal constants
and $d$ is the bulk dimension.

Special BKT transition.— While the boundary GNY fixed point can be realized in
the transition in TSCs, for 2+1D TIs protected by time-reversal symmetry, a
crucial issue arises at its boundary, i.e., four-fermion interactions are
marginal and consequently should be considered 333Four fermion interaction,
such as $\int d^{2}x(\psi^{\dagger}\partial\psi)^{2}$, is irrelevant for the
Majorana fermion. To include the effect of four-fermion interactions, instead
of a $4-\epsilon$ RG calculation, we analyze the boundary theory directly in
1+1D by bosonization Fradkin (2013); Wu  _et al._ (2006). More precisely,
there are two marginal four-fermion interactions respecting time-reversal
symmetry, the forward-scattering and the two-particle backward-scattering
interactions permitted at half-filling Wu  _et al._ (2006). The former can be
lumped non-perturbatively into the Luttinger parameter, while the latter is
presented by a vortex operator. The boundary theory after bosonization is (see
Supplemental Material Sec. III)

$\begin{split}S_{\text{f}}=&\int
d^{2}x\Big{[}\frac{1}{2K}(\partial\varphi)^{2}+g_{1}\phi\cos(\sqrt{4\pi}\varphi)\\\
&+g_{2}\cos\left(\sqrt{16\pi}\varphi\right)\Big{]}\,,\end{split}$ (5)

where $\varphi$ ($\phi$) denotes the bosonization field (the boundary order
parameter), $K$ and $g_{2}$ present the Luttinger parameter and the strength
of two-particle backward scattering, respectively, and $g_{1}$ is originated
from the Yukawa coupling between the fermion and boson.

Again, we consider the bulk boson is at the WF fixed point. To proceed, we can
hypothetically decouple the bulk boson from the boundary fermion at the
outermost layer, i.e., $g_{1}=0$. We expect this hypothetical bulk system
without the outermost boundary layer to be described by the ordinary boundary
universality class. We then apply the conformal perturbation theory at this
ordinary transition fixed point for $g_{1}\neq 0$. Because the boundary
fermion will not affect the bulk RG flow, the relevant RG equations,
consisting of only boundary couplings, yield

$\displaystyle\frac{dK}{dl}=-\pi^{2}g_{1}^{2}K^{2}-4\pi^{2}g_{2}^{2}K^{2}\,,$
(6) $\displaystyle\frac{dg_{1}}{dl}=(2-\Delta_{\hat{\phi}}-K)g_{1}\,,$
$\displaystyle\frac{dg_{2}}{dl}=(2-4K)g_{2}\,,$

where $\Delta_{\hat{\phi}}=\Delta_{\text{ord}}$ is the scaling dimension of
the boundary order parameter for ordinary transition without fermions. Without
coupling to bulk bosons, i.e., $g_{1}=0$, the first and the third RG equations
describe a BKT transition at $K=1/2$ for the boundary fermion Wu  _et al._
(2006); Li and Yao (2017). This can be understood as a conventional BKT
transition: the repulsive four fermion interaction increases, which reduces
$K$ from its noninteracting value $K=1$, then as the interaction strength
exceeds a critical value, $K<1/2$, it triggers a run-away flow for $g_{2}$
that condenses a boundary order.

Interestingly, with a nontrivial coupling between the gapless bulk boson and
the boundary fermion, i.e., $g_{1}\neq 0$, the second RG equation is the new
input for the boundary criticality. It indicates that when $K\leq K_{c}\equiv
2-\Delta_{\text{ord}}$, the coupling $g_{1}$ becomes relevant. Hence, it is
important to compare these two critical couplings $K=1/2$ and
$K_{c}=2-\Delta_{\text{ord}}$. A relatively accurate estimate of surface
scaling dimension at the ordinary transition from Monte Carlo simulation is
$\Delta_{\text{ord}}\approx 1.2626$ Deng  _et al._ (2005). Hence, using this
value, the critical strength is $K_{c}\approx 0.7374>1/2$, indicating a
different BKT transition that happens at a weaker interaction strength.
Actually, in the region $K<K_{c}$ where $g_{1}$ is relevant, it is natural to
ask whether the theory flows to a gapped phase or a strongly coupled fixed
point. To capture the behavior at the strong coupling region precisely, we
perform the RG calculation up to the third order Amit  _et al._ (1980). The
higher order RG flow shows that it flows to the massive phase instead of the
strongly coupled fixed point (see Supplemental Material Sec. IV).

We can clearly understand how the edge state enriches the boundary criticality
when the bulk is at the WF fixed point: At a weak boundary interaction,
$K>K_{c}$, the presence of a Luttinger liquid at the edge renders a continuous
scaling dimension for the surface order parameter
$\tilde{\Delta}_{\hat{\phi}}=K$. Note that we use
$\tilde{\Delta}_{\hat{\phi}}$ to denote the scaling dimension of the boundary
order parameter, i.e., $\cos(\sqrt{4\pi}\varphi)$, at the fermion enriched
boundary transition. This is an unconventional ordinary transition enriched by
the gapless edge state. As the interaction increases to $K=K_{c}$, a BKT
transition takes place, and subsequently, as it keeps increasing, i.e.,
$K<K_{c}$, the boundary boson orders spontaneously and the edge fermion is
gapped out. Notice that for $K<K_{c}$, the theory is described by the Ising
extraordinary transition because the edge state is gapped. Hence, $K=K_{c}$
marks a special transition separating the unconventional ordinary transition
and the Ising extraordinary transition, so we term it special BKT transition.
The critical exponents at this special BKT transition are given by
$\tilde{\Delta}_{\hat{\phi}}=K_{c}$,
$\Delta_{\psi}=\frac{1}{2}+\frac{1}{4}\left(K_{c}+\frac{1}{K_{c}}-2\right)$,
and the correlation length scales as
$\xi\propto\exp\left[\frac{b}{\sqrt{K-K_{c}}}\right]$, where $b$ is a
nonuniversal number, as $K$ approaches the critical point $K_{c}$. With this
understanding, the phase diagram is sketched in Fig. 2.

Defect criticality in topological systems.— We extend our calculation to
topological phases with a 1+1D defect Metlitski (2022); Chang  _et al._
(2023); Giombi and Liu (2023); Trépanier (2023). We briefly review the
conformal defect in an Ising theory. The action with an 1+1D mass defect in a
$d$ dimensional critical Ising theory reads

$S_{\text{d}}=\int_{\mathcal{M}}d^{d}x\left[\frac{1}{2}(\partial\phi)^{2}+\frac{\lambda}{4!}\phi^{4}\right]+\int_{\mathcal{D}}d^{2}xh\phi^{2}\,,$
(7)

where $h$ in the second term denotes the defect mass. Instead of a
codimensional one boundary described in Eq. (1), it describes of a 1+1D
defect, $\mathcal{D}$, embedded in a $d$ dimensional Euclidean spacetime
$\mathcal{M}$. In $4-\epsilon$ regularization, the RG equation for $\lambda$
is the same as Eq. (4), while the RG equation for $h$ reads Giombi and Liu
(2023)

$\frac{dh}{dl}=\epsilon h-\frac{h^{2}}{\pi}-\frac{\lambda h}{16\pi^{2}}\,,$
(8)

which reveals a nontrivial conformal defect at a stable fixed point
$h_{*}=\frac{2\pi\epsilon}{3}$. At this fixed point, the scaling dimension of
the defect order parameter
$\Delta_{\text{defect}}=\frac{d-2}{2}+\frac{h_{*}}{\pi}=1-\frac{\epsilon}{6}$
.

We assume that a gapless fermion is located at the defect, and couples to the
defect order via a Yukawa term, i.e.,
$S_{\text{f}}=\int_{\mathcal{D}}d^{2}x\left(\bar{\psi}\not{\partial}\psi+g\phi\bar{\psi}\psi\right)$.
The convention of Dirac matrix is the same as in Eq. (2). We then employ the
bosonization technique to study the effect of a decorated fermion on the
conformal defect. The bosonized action is the same as Eq. (5), in which a
marginal four-fermion interaction is included via a Luttinger parameter $K$.
The theory leads to the same RG equations as listed in Eq. (6), except now
$\Delta_{\hat{\phi}}=\Delta_{\text{defect}}$. It implies a defect BKT
transition at $K_{c}^{\prime}=2-\Delta_{\text{defect}}=1-\frac{\epsilon}{6}$,
where we use the result of the $4-\epsilon$ calculation for
$\Delta_{\text{defect}}$ in the last equality. Crucially, the critical
Luttinger parameter $K_{c}^{\prime}$ is again greater than $1/2$, i.e.,
$K_{c}^{\prime}>\frac{1}{2}$, for $\epsilon\leq 1$, which in turn indicates
that the critical interaction strength for the defect BKT transition is weaker
than the critical strength for local interactions.

Hence, there is a parallel scenario for the fermion enriched conformal defect,
in which the bulk is always located at the WF fixed point: At weak
interactions, $K>K_{c}^{\prime}$, the conformal defect is enriched to have a
continuous scaling dimension $\tilde{\Delta}_{\hat{\phi}}=K$ before a defect
BKT transition taking place at $K=K_{c}^{\prime}$. The critical exponents at
this defect BKT transition are summarized in Table 1. After the BKT
transition, the Ising field orders on the defect and the fermion opens up a
gap. The defect is then described by a normal universality class Giombi and
Liu (2023).

It is also direct to analyze the defect criticality in the TSCs in the same
fashion as the boundary case. Notice that the defect becomes an interface at
$2+1$D bulk. The special interface fixed point can be identified by directly
performing the standard dimensional regularization in the co-dimension one
interface.

Concluding remarks.— Possible experimental probes of the novel boundary
criticality include scanning tunneling microscopy (STM) Binnig and Rohrer
(1983, 1987); Eggert (2000) and transport techniques Kane and Fisher (1992);
Li  _et al._ (2015). Since the experimental realization of topological
superconductors remains less well understood, we focus on topological
insulators. The ordinary transition enriched by Luttinger liquid possesses an
unconventional Yukawa term. This can be observed in a transport experiment via
a subleading correction $\delta G$ to the quantized conductance $G_{0}$ Maslov
and Stone (1995); Maslov (1995); Lezmy  _et al._ (2012), $G=G_{0}-\delta G$.
The subleading contribution respects the scaling law, $\delta G\sim
E^{2\Delta_{\text{Yukawa}}-2}+E^{2\Delta-2}$, where the energy scale $E$ is
given by the larger of either temperature or voltage, and
$\Delta_{\text{Yukawa}}=\Delta_{\text{ord}}+K$ and $\Delta=4K$. As
$\Delta_{\text{Yukawa}}<\Delta$ in the ordinary transition, the Yukawa scaling
can be extracted from the leading term in $\delta G$. Moreover, regarding the
special BKT transition, a key feature distinct from the conventional one is
the critical Luttinger parameter, $K_{c}$. This can be observed in
differential tunneling conductance using STM, $\frac{dI}{dV}\sim
E^{2\Delta_{\psi}-1}$. Finally, given that boundary criticality is an
interacting effect and involves the breaking of time-reversal symmetry,
possible material candidates include topological Kondo insulators, such as
SmB6 Dzero  _et al._ (2010); Alexandrov  _et al._ (2013); Dzero  _et al._
(2016), and magnetic topological insulators, such as MnBi2Te4 Otrokov  _et
al._ (2019); Bernevig  _et al._ (2022).

To conclude, we provide a comprehensive study of boundary and defect
criticality in topological phases. We identify several novel boundary
criticalities summarized in Table 1, and sketch the phase diagrams. The
fermion mode is treated as a perturbation to the original fixed point. Hence,
it remains an interesting future question to see how the boundary criticality
is modified in the strong coupling region. Moreover, there still exist many
intriguing problems regarding the fermion-enriched boundary criticality. For
instance, it would be valuable to investigate the higher-order boundary
criticality, which can be realized in the higher-order topological insulators
Benalcazar  _et al._ (2017); Schindler  _et al._ (2018). Besides, the study of
the boundary criticality with $N>1$ flavor boson would be important, as some
novel phase structures may take place for a general value of $N$ Harribey  _et
al._ (2023); Sun  _et al._ (2023); Metlitski (2022).

Acknowledgments. We thank Chi-Ming Chang, Yang Ge for helpful discussions. The
work is supported in part by a startup grant from Tulane University. Z. Wu
acknowledges the support from the Shuimu Fellow Foundation at Tsinghua
University.

## References

  * von Klitzing  _et al._ (1980) K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
  * Tsui  _et al._ (1982) D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).
  * Laughlin (1981) R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).
  * Laughlin (1983) R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
  * Maldacena (1998) J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998), arXiv:hep-th/9711200 .
  * Witten (1998) E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998), arXiv:hep-th/9802150 .
  * Gubser  _et al._ (1998) S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B 428, 105 (1998), arXiv:hep-th/9802109 .
  * Polchinski (1996) J. Polchinski, in  _Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 96): Fields, Strings, and Duality_ (1996) pp. 293–356, arXiv:hep-th/9611050 .
  * Takayanagi (2011) T. Takayanagi, Phys. Rev. Lett. 107, 101602 (2011), arXiv:1105.5165 [hep-th] .
  * Fujita  _et al._ (2011) M. Fujita, T. Takayanagi, and E. Tonni, Journal of High Energy Physics 2011 (2011), 10.1007/jhep11(2011)043.
  * Recknagel and Schomerus (2013) A. Recknagel and V. Schomerus,  _Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2013).
  * Izumi  _et al._ (2022) K. Izumi, T. Shiromizu, K. Suzuki, T. Takayanagi, and N. Tanahashi, Journal of High Energy Physics 2022 (2022), 10.1007/jhep10(2022)050.
  * Cardy (1984) J. L. Cardy, Nucl. Phys. B 240, 514 (1984).
  * Cardy (1989) J. L. Cardy, Nucl. Phys. B 324, 581 (1989).
  * Affleck and Ludwig (1991) I. Affleck and A. W. W. Ludwig, Phys. Rev. Lett. 67, 161 (1991).
  * Cardy (2008) J. Cardy, “Boundary conformal field theory,”  (2008), arXiv:hep-th/0411189 [hep-th] .
  * Ge and Jian (2024) Y. Ge and S.-K. Jian, (2024), arXiv:2403.18691 [cond-mat.stat-mech] .
  * Shen (2023) X.-Y. Shen, (2023), arXiv:2308.12598 [hep-th] .
  * Andrei  _et al._ (2018) N. Andrei, A. Bissi, M. Buican, J. Cardy, P. Dorey, N. Drukker, J. Erdmenger, D. Friedan, D. Fursaev, A. Konechny, C. Kristjansen, I. Makabe, Y. Nakayama, A. O’Bannon, R. Parini, B. Robinson, S. Ryu, C. Schmidt-Colinet, V. Schomerus, C. Schweigert, and G. Watts, “Boundary and defect cft: Open problems and applications,”  (2018), arXiv:1810.05697 [hep-th] .
  * Ji and Chen (2024) W. Ji and X. Chen, “Topological defects of 2+1d systems from line excitations in 3+1d bulk,”  (2024), arXiv:2407.02488 [cond-mat.str-el] .
  * Wu  _et al._ (2020) X.-C. Wu, Y. Xu, H. Geng, C.-M. Jian, and C. Xu, Phys. Rev. B 101, 174406 (2020).
  * Brillaux and Fedorenko (2021) E. Brillaux and A. A. Fedorenko, Phys. Rev. B 103, L081405 (2021).
  * Ma  _et al._ (2022) R. Ma, L. Zou, and C. Wang, SciPost Phys. 12, 196 (2022).
  * Brillaux  _et al._ (2024) E. Brillaux, A. A. Fedorenko, and I. A. Gruzberg, Phys. Rev. B 109, 174204 (2024).
  * Kitaev (2009) A. Kitaev, AIP Conference Proceedings 1134, 22 (2009), https://pubs.aip.org/aip/acp/article-pdf/1134/1/22/11584243/22_1_online.pdf .
  * Ryu  _et al._ (2010) S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Ludwig, New Journal of Physics 12, 065010 (2010).
  * Gu and Wen (2009) Z.-C. Gu and X.-G. Wen, Physical Review B 80 (2009), 10.1103/physrevb.80.155131.
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Reviews of Modern Physics 83, 1057–1110 (2011).
  * Chen  _et al._ (2012) X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Science 338, 1604 (2012), https://www.science.org/doi/pdf/10.1126/science.1227224 .
  * Kane and Mele (2005) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).
  * Bernevig  _et al._ (2006) B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006), https://www.science.org/doi/pdf/10.1126/science.1133734 .
  * König  _et al._ (2007) M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007), https://www.science.org/doi/pdf/10.1126/science.1148047 .
  * Hsieh  _et al._ (2008) D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature 452, 970 (2008).
  * Scaffidi  _et al._ (2017) T. Scaffidi, D. E. Parker, and R. Vasseur, Phys. Rev. X 7, 041048 (2017).
  * Verresen  _et al._ (2021) R. Verresen, R. Thorngren, N. G. Jones, and F. Pollmann, Phys. Rev. X 11, 041059 (2021).
  * Yu  _et al._ (2024) X.-J. Yu, S. Yang, H.-Q. Lin, and S.-K. Jian, Phys. Rev. Lett. 133, 026601 (2024).
  * Liu  _et al._ (2021) S. Liu, H. Shapourian, A. Vishwanath, and M. A. Metlitski, Phys. Rev. B 104, 104201 (2021), arXiv:2104.15026 [cond-mat.str-el] .
  * Giombi  _et al._ (2022) S. Giombi, E. Helfenberger, and H. Khanchandani, Journal of High Energy Physics 2022, 18 (2022), arXiv:2110.04268 [hep-th] .
  * Giombi  _et al._ (2023) S. Giombi, E. Helfenberger, and H. Khanchandani, Journal of High Energy Physics 2023, 224 (2023), arXiv:2211.11073 [hep-th] .
  * Barrat  _et al._ (2023) J. Barrat, P. Liendo, and P. van Vliet, arXiv e-prints , arXiv:2304.13588 (2023), arXiv:2304.13588 [hep-th] .
  * Shachar  _et al._ (2024) T. Shachar, R. Sinha, and M. Smolkin, (2024), arXiv:2404.18403 [hep-th] .
  * Herzog and Schaub (2023) C. P. Herzog and V. Schaub, Journal of High Energy Physics 2023 (2023), 10.1007/jhep02(2023)129.
  * Diehl and Dietrich (1981) H. W. Diehl and S. Dietrich, Zeitschrift fur Physik B Condensed Matter 42, 65 (1981).
  * Diehl and Dietrich (1983) H. W. Diehl and S. Dietrich, Zeitschrift fur Physik B Condensed Matter 50, 117 (1983).
  * Giombi and Khanchandani (2020) S. Giombi and H. Khanchandani, Journal of High Energy Physics 2020, 10 (2020), arXiv:1912.08169 [hep-th] .
  * Padayasi  _et al._ (2022) J. Padayasi, A. Krishnan, M. Metlitski, I. Gruzberg, and M. Meineri, SciPost Physics 12, 190 (2022), arXiv:2111.03071 [cond-mat.stat-mech] .
  * Parisen Toldin and Metlitski (2022) F. Parisen Toldin and M. A. Metlitski, Phys. Rev. Lett. 128, 215701 (2022), arXiv:2111.03613 [cond-mat.stat-mech] .
  * Metlitski (2022) M. Metlitski, SciPost Physics 12, 131 (2022).
  * Kosterlitz and Thouless (2018) J. M. Kosterlitz and D. J. Thouless, in  _Basic Notions Of Condensed Matter Physics_ (CRC Press, 2018) pp. 493–515.
  * Berezinsky (1971) V. L. Berezinsky, Sov. Phys. JETP 32, 493 (1971).
  * Chang  _et al._ (2023) C.-M. Chang, J. Chen, and F. Xu, SciPost Phys. 15, 216 (2023), arXiv:2208.02757 [hep-th] .
  * Giombi and Liu (2023) S. Giombi and B. Liu, Journal of High Energy Physics 2023, 4 (2023), arXiv:2305.11402 [hep-th] .
  * Trépanier (2023) M. Trépanier, JHEP 09, 074 (2023), arXiv:2305.10486 [hep-th] .
  * Note (1) Some boundary symmetry-breaking coupling that drives the system to the normal universality class is neglected. See Diehl Diehl and Dietrich (1981, 1983) for a detailed discussion.
  * Note (2) Strictly speaking, the surface mass is located at some $h=h_{\text{sp}}$ for the special transition.
  * Herzog and Huang (2017) C. P. Herzog and K.-W. Huang, Journal of High Energy Physics 2017, 189 (2017), arXiv:1707.06224 [hep-th] .
  * Friedan  _et al._ (1985) D. Friedan, Z.-a. Qiu, and S. H. Shenker, Phys. Lett. B 151, 37 (1985).
  * Grover  _et al._ (2014) T. Grover, D. N. Sheng, and A. Vishwanath, Science 344, 280 (2014), arXiv:1301.7449 [cond-mat.str-el] .
  * Jian  _et al._ (2015) S.-K. Jian, Y.-F. Jiang, and H. Yao, Phys. Rev. Lett. 114, 237001 (2015).
  * Fei  _et al._ (2016) L. Fei, S. Giombi, I. R. Klebanov, and G. Tarnopolsky, PTEP 2016, 12C105 (2016), arXiv:1607.05316 [hep-th] .
  * Li  _et al._ (2017) Z.-X. Li, Y.-F. Jiang, and H. Yao, Phys. Rev. Lett. 119, 107202 (2017).
  * Deng  _et al._ (2005) Y. Deng, H. W. J. Blöte, and M. P. Nightingale, Phys. Rev. E 72, 016128 (2005).
  * Note (3) Four fermion interaction, such as $\intop\ilimits@d^{2}x(\psi^{\dagger}\partial\psi)^{2}$, is irrelevant for the Majorana fermion.
  * Fradkin (2013) E. Fradkin,  _Field Theories of Condensed Matter Physics_ (2013).
  * Wu  _et al._ (2006) C. Wu, B. A. Bernevig, and S.-C. Zhang, Phys. Rev. Lett. 96, 106401 (2006).
  * Li and Yao (2017) Z.-X. Li and H. Yao, Phys. Rev. B 96, 241101 (2017).
  * Amit  _et al._ (1980) D. J. Amit, Y. Y. Goldschmidt, and G. Grinstein, J. Phys. A 13, 585 (1980).
  * Binnig and Rohrer (1983) G. Binnig and H. Rohrer, Surface Science 126, 236 (1983).
  * Binnig and Rohrer (1987) G. Binnig and H. Rohrer, Rev. Mod. Phys. 59, 615 (1987).
  * Eggert (2000) S. Eggert, Phys. Rev. Lett. 84, 4413 (2000).
  * Kane and Fisher (1992) C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 68, 1220 (1992).
  * Li  _et al._ (2015) T. Li, P. Wang, H. Fu, L. Du, K. A. Schreiber, X. Mu, X. Liu, G. Sullivan, G. A. Csáthy, X. Lin, and R.-R. Du, Phys. Rev. Lett. 115, 136804 (2015).
  * Maslov and Stone (1995) D. L. Maslov and M. Stone, Phys. Rev. B 52, R5539 (1995).
  * Maslov (1995) D. L. Maslov, Phys. Rev. B 52, R14368 (1995).
  * Lezmy  _et al._ (2012) N. Lezmy, Y. Oreg, and M. Berkooz, Phys. Rev. B 85, 235304 (2012).
  * Dzero  _et al._ (2010) M. Dzero, K. Sun, V. Galitski, and P. Coleman, Phys. Rev. Lett. 104, 106408 (2010).
  * Alexandrov  _et al._ (2013) V. Alexandrov, M. Dzero, and P. Coleman, Phys. Rev. Lett. 111, 226403 (2013).
  * Dzero  _et al._ (2016) M. Dzero, J. Xia, V. Galitski, and P. Coleman, Annual Review of Condensed Matter Physics 7, 249 (2016).
  * Otrokov  _et al._ (2019) M. M. Otrokov, I. I. Klimovskikh, H. Bentmann, D. Estyunin, A. Zeugner, Z. S. Aliev, S. Gaß, A. Wolter, A. Koroleva, A. M. Shikin, _et al._ , Nature 576, 416 (2019).
  * Bernevig  _et al._ (2022) B. A. Bernevig, C. Felser, and H. Beidenkopf, Nature 603, 41 (2022).
  * Benalcazar  _et al._ (2017) W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Science 357, 61 (2017), https://www.science.org/doi/pdf/10.1126/science.aah6442 .
  * Schindler  _et al._ (2018) F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Science Advances 4 (2018), 10.1126/sciadv.aat0346.
  * Harribey  _et al._ (2023) S. Harribey, I. R. Klebanov, and Z. Sun, JHEP 10, 017 (2023), arXiv:2307.00072 [hep-th] .
  * Sun  _et al._ (2023) Y. Sun, M. Hu, Y. Deng, and J.-P. Lv, Phys. Rev. Lett. 131, 207101 (2023), arXiv:2301.11720 [cond-mat.stat-mech] .
  * Giombi  _et al._ (2016) S. Giombi, I. R. Klebanov, and G. Tarnopolsky, J. Phys. A 49, 135403 (2016), arXiv:1508.06354 [hep-th] .
  * Fei  _et al._ (2015) L. Fei, S. Giombi, I. R. Klebanov, and G. Tarnopolsky, JHEP 12, 155 (2015), arXiv:1507.01960 [hep-th] .

Supplemental Material

##  I Mixed Yukawa fixed point and emergent Lorentz symmetry

As a warm-up, we first consider a free scalar theory in the bulk coupled to a
Dirac fermion in a codimension-1 boundary. Such a theory can describe, for
instance, the tricritical point in a 3+1D topological insulator. The scalar
field lives in a $d$-dimensional half-infinite Euclidean spacetime,
$\mathcal{M}$, spanned by
$x^{0}\equiv\tau,x^{1},...,x^{d-2}\in(-\infty,\infty)$ and $x^{d-1}\equiv
y\in(0,\infty)$. While the Dirac fermion lives on the boundary of this half-
infinite spacetime, $\mathcal{\partial M}$, with
$\tau,x^{1},...,x^{d-2}\in(-\infty,\infty)$, and $y=0$. Note that $\tau$
stands for the imaginary time. The Euclidean action is

$S=\int_{\mathcal{M}}d^{d}x\frac{1}{2}\left((\partial_{\tau}\phi)^{2}+v_{B}^{2}(\nabla\phi)^{2}\right)+\int_{\partial\mathcal{M}}d^{d-1}x(\bar{\psi}(\gamma^{0}\partial_{0}+v_{F}\gamma\cdot\partial)\psi-g\phi\bar{\psi}\psi)\,,$
(S1)

where $\phi$, $\psi$ denotes the scalar field and the surface Dirac fermion
with velocity $v_{B}$ and $v_{F}$, respectively. $g$ denotes the Yukawa
interaction strength,
$\gamma\cdot\partial=\sum_{i=1}^{d-2}\gamma^{i}\partial_{i}$ with
$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\delta^{\mu\nu}$, $\mu,\nu=0,...,d-2$, and
$\bar{\psi}=\psi^{\dagger}\gamma^{0}$.

Since the theory preserves translation symmetry in $x^{0,...,d-2}$ directions,
we can work in the momentum space for those coordinates (frequency space for
$\tau$), while in the real space for the $y$ coordinate. Notice that the
fermion is localized on the surface, its free propagator is

$G_{\psi}(p)=i\frac{\gamma^{0}\omega+v_{F}\gamma\cdot\bf{p}}{\omega^{2}+v_{F}^{2}{\bf
p}^{2}}\,,$ (S2)

where $\omega$ denotes the frequency. $\gamma\cdot{\bf
p}=\sum_{i=1}^{d-2}\gamma^{i}p_{i}$ with $p_{i}$ being the momentum in the
$x^{i}$ coordinate, and ${\bf p}^{2}\equiv\sum_{i=1}^{d-2}p_{i}^{2}$.

For the bulk boson,
$(\partial_{\tau}^{2}+\sum_{i}v_{B}^{2}\partial_{i}^{2})G_{\phi}=0$, in the
presence of a boundary at $y=0$, the propagator can be derived via the image
method:

$G_{\phi}(x-x^{\prime},y,y^{\prime})=b_{0}\left(\frac{1}{\left(\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}\right)^{\frac{d-2}{2}}}+\frac{\lambda}{\left(\left(x-x^{\prime}\right)^{2}+\left(y+y^{\prime}\right)^{2}\right)^{\frac{d-2}{2}}}\right)\,.$
(S3)

$b_{0}=\frac{\Gamma(\frac{d-2}{2})}{4\pi^{d/2}}$ is a coefficient that is set
to make the coefficient of the propagator in the momentum space unity. Here
$x=(v_{B}\tau,{\bf x})$ is a shorthand notation for coordinates parallel to
the boundary. $\lambda$ is given by the boundary condition: $\lambda=-1$ for
Dirichlet boundary condition $\phi(x,y=0)=0$ and $\lambda=1$ for Neumann
boundary condition $\partial_{y}\phi(x,y=0)=0$. Here we will focus on the
Neumann boundary condition $\lambda=1$. After Fourier transformation for
coordinates parallel to the boundary, the free propagator becomes

$G_{\phi}(p,y,y^{\prime})=\frac{e^{-\sqrt{\omega^{2}+v_{B}^{2}{\bf
p}^{2}}|y-y^{\prime}|}+e^{-\sqrt{\omega^{2}+v_{B}^{2}{\bf
p}^{2}}(y+y^{\prime})}}{2\sqrt{\omega^{2}+v_{B}^{2}{\bf p}^{2}}}\,.$ (S4)

Since we are considering the surface criticality, we can restrict to $y=0$. In
this case, we define the free propagator for boson at the surface as

$G_{\phi}(p)=G_{\phi}(p,0,0)=\frac{1}{\sqrt{\omega^{2}+v_{B}^{2}{\bf
p}^{2}}}\,.$ (S5)

The scaling dimension of Yukawa coupling is given by $[g]=\frac{1}{2}(d-4)$.
Hence, we will work at $d=4-\epsilon$, and in this case, the boson self-
interaction at the boundary $\int_{\partial\mathcal{M}}d^{d-1}x\phi^{4}$ is
irrelevant. To conduct the one-loop renormalization group, we will consider
the one-loop diagram for the self-energy of the boson and fermions
respectively, and the Yukawa vertex. We will first assume the Lorentz symmetry
by $v_{B}=v_{F}$ and set $v_{B}=1$ without loss of generality to obtain a
stable fixed point. And then we will show that at this stable fixed point the
assumption is valid by calculating the RG flow of $v_{B}$ and $v_{F}$.

To simplify the notation, we will use the following shorthand notation

$G_{\psi}(p)=i\frac{\not{p}}{p^{2}}\,,\quad G_{\phi}(p)=\frac{1}{|p|}\,,$ (S6)

where $\not{p}=\gamma^{0}\omega+\gamma\cdot\bf{p}$, $p^{2}=\omega^{2}+{\bf
p}^{2}$ and $|p|=\sqrt{p^{2}}$.

Supplementary Figure S1: (a) Scalar one loop propagator correction; (b)
Fermion one loop propagator correction; (c) One loop vertex correction. The
Black dashed line represents the projected boson. The Red dashed line
represents the bulk boson. Balck solid arrow represents the fermion.

The self-energy for the boson is

$\Pi_{\phi}(q)=(-1)g^{2}\int\frac{d^{d-1}p}{(2\pi)^{d-1}}\frac{\operatorname{tr}[i\not{p}i(\not{p}+\not{q})]}{p^{2}(p+q)^{2}}=g^{2}\frac{2^{5-2d}\pi^{2-\frac{d}{2}}}{\cos\left(\frac{\pi
d}{2}\right)\Gamma\left(\frac{d}{2}-1\right)}q^{d-3}\,.$ (S7)

Here we choose $\operatorname{tr}(\gamma^{\mu}\gamma^{\nu})=2\delta^{\mu\nu}$.
There is no singularity at $d=4-\epsilon$ for $\phi$ and thus $Z_{\phi}$
should be finite. The self-energy for the fermion is

$\displaystyle\Pi_{\psi}(q)$
$\displaystyle=g^{2}\int\frac{d^{d-1}p}{(2\pi)^{d-1}}\frac{(i\not{p})}{p^{2}|p-q|}$
(S8)
$\displaystyle=ig^{2}\frac{4^{2-d}\pi^{\frac{1-d}{2}}\Gamma\left(2-\frac{d}{2}\right)\Gamma(d-2)}{\Gamma\left(d-\frac{3}{2}\right)}\frac{\not{q}}{q^{4-d}}\,,$

and in $d=4-\epsilon$, the singular terms can be extracted,

$\Pi_{\psi}(q)=i\not{q}g^{2}\left[\frac{1}{6\pi^{2}\epsilon}+\frac{1}{36\pi^{2}}\left(10-3\gamma-3\log\left(q^{2}/\pi\right)\right)\right]\,.$
(S9)

The one-loop vertex is given by:

$\Gamma\left(k,q\right)=g^{3}\int\frac{d^{d-1}p}{(2\pi)^{d-1}}\frac{i(\not{p}+\not{k})i(\not{p}+\not{q})}{(p+k)^{2}(p+q)^{2}|p|}\approx-g^{3}\frac{1}{2\pi^{2}\epsilon}+\text{
finite }.$ (S10)

$Z$ factors can then be extracted via this one-loop calculation:

$\displaystyle Z_{\psi}=1-g^{2}\frac{1}{6\pi^{2}\epsilon}\,,$ (S11)
$\displaystyle Z_{\phi}=1+g^{2}(\mathcal{O}(1))\,,$ $\displaystyle
Z_{g}=1+g^{2}\frac{1}{2\pi^{2}\epsilon}\,.$

The bare coupling is associated with renormalized coupling via

$g_{\text{bare}}Z_{\phi}^{1/2}Z_{\psi}=g\mu^{\epsilon/2}Z_{g}\,.$ (S12)

This gives the beta function

$\beta=-\frac{\epsilon}{2}g+\frac{2}{3\pi^{2}}g^{3}+\mathcal{O}\left(g^{4}\right)\,.$
(S13)

For $\epsilon>0$, the Gaussian fixed point is unstable, and there is a stable
nontrivial IR fixed point $g_{*}^{2}=\frac{3\pi^{2}}{4}\epsilon$.

Let us now include the renormalization of velocity, i.e. we consider
$v_{F}\neq v_{B}$. Again, we calculate the contribution of the three one-loop
diagrams. The one-loop self-energy for bosons is

$\displaystyle\Pi_{\phi}(\nu,{\bf
p})=2g^{2}\int\frac{d^{d-2}q}{(2\pi)^{d}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{\omega(\omega+\nu)+v_{F}^{2}\textbf{q}_{j}(\textbf{p}_{j}+\textbf{q}_{j})}{\left(\omega^{2}+v_{F}^{2}\textbf{q}^{2}\right)\left[(\omega+\nu)^{2}+v_{F}^{2}(\textbf{p}+\textbf{q})^{2}\right]}$
(S14)
$\displaystyle=\frac{g^{2}}{2}\int_{0}^{1}dx\int\frac{d^{d-2}q}{(2\pi)^{d}}\left[\frac{v_{F}^{2}\textbf{q}^{2}+x(1-x)\left(\nu^{2}+v_{F}^{2}\textbf{p}^{2}\right)-x(1-x)\nu^{2}+v_{F}^{2}\textbf{q}^{2}-x(1-x)v_{F}^{2}\textbf{p}^{2}}{\left[v_{F}^{2}\textbf{q}^{2}+x(1-x)\left(\nu^{2}+v_{F}^{2}\textbf{p}^{2}\right)\right]^{3/2}}\right]$
$\displaystyle=\frac{2\Gamma(\frac{5}{2}-d)(d-2)g^{2}}{(4\pi)^{\frac{d-1}{2}}v_{F}^{d-2}}\int_{0}^{1}dx\frac{1}{\left[x(1-x)(\nu^{2}+v_{F}^{2}\textbf{p}^{2})\right]^{\frac{3}{2}-d}}$
$\displaystyle=-\frac{g^{2}}{8v_{F}^{2}}\sqrt{\nu^{2}+v_{F}^{2}\textbf{p}^{2}}+\mathcal{O}(\epsilon)\,.$

The one-loop self-energy for fermions is

$\displaystyle\Pi_{\psi}(\nu,\textbf{q})$
$\displaystyle=ig^{2}\int\frac{dp^{d-2}}{(2\pi)^{d-2}}\int\frac{d\omega}{2\pi}\frac{\omega\gamma_{0}+v_{F}\textbf{p}_{j}\gamma^{j}}{(\omega^{2}+v_{F}^{2}\textbf{p}^{2})\sqrt{(\omega+\nu)^{2}+v_{B}^{2}(\textbf{p}+\textbf{q})^{2}}}$
(S15)
$\displaystyle=\frac{ig^{2}}{2}\int\frac{dp^{d-2}}{(2\pi)^{d-2}}\int\frac{d\omega}{2\pi}\int_{0}^{1}dx\frac{\omega\gamma_{0}+v_{F}\textbf{p}_{j}\gamma^{j}}{\left[(1-x)(\omega^{2}+v_{F}^{2}\textbf{p}^{2})+x((\omega+\nu)^{2}+v_{B}^{2}(\textbf{p}+\textbf{q})^{2})\right]^{\frac{3}{2}}}\,.$

To proceed, let
$\textbf{p}+\frac{xv_{B}^{2}}{(1-x)v_{F}^{2}+xv_{B}^{2}}\textbf{q}\rightarrow\textbf{p}$,
$\omega+x\nu\rightarrow\omega$ and rearrange the integral, we have

$\displaystyle\frac{ig^{2}}{2}\int\frac{dp^{d-2}}{(2\pi)^{d-2}}\int\frac{d\omega}{2\pi}\int_{0}^{1}\frac{dx}{\sqrt{x}}\frac{\omega\gamma_{0}+v_{F}p_{j}\gamma_{j}}{\left[(1-x)(\omega^{2}+v_{F}^{2}p^{2})+x((\omega+\nu)^{2}+v_{B}^{2}(p+q)^{2})\right]^{\frac{3}{2}}}$
(S16) $\displaystyle=$
$\displaystyle\frac{ig^{2}}{2}\int\frac{dp^{d-2}}{(2\pi)^{d-2}}\int\frac{d\omega}{2\pi}\int_{0}^{1}\frac{dx}{\sqrt{x}}\frac{(\omega-x\nu)\gamma_{0}+v_{F}\left(p_{j}-\frac{xv_{B}^{2}}{(1-x)v_{F}^{2}+xv_{B}^{2}}q_{j}\right)\gamma_{j}}{\left[\omega^{2}+x(1-x)\nu^{2}+((1-x)v_{F}^{2}+xv_{B}^{2})p^{2}+\frac{x(1-x)v_{B}^{2}v_{F}^{2}}{((1-x)v_{F}^{2}+xv_{B}^{2})}q^{2})\right]^{\frac{3}{2}}}$
$\displaystyle=$
$\displaystyle\frac{ig^{2}}{2\pi}\int\frac{dp^{d-2}}{(2\pi)^{d-2}}\int_{0}^{1}\frac{dx}{\sqrt{x}}\frac{-x\nu\gamma_{0}+v_{F}\left(p_{j}-\frac{xv_{B}^{2}}{(1-x)v_{F}^{2}+xv_{B}^{2}}q_{j}\right)\gamma_{j}}{\left[x(1-x)\nu^{2}+((1-x)v_{F}^{2}+xv_{B}^{2})p^{2}+\frac{x(1-x)v_{B}^{2}v_{F}^{2}}{((1-x)v_{F}^{2}+xv_{B}^{2})}q^{2})\right]}$
$\displaystyle=$ $\displaystyle
ig^{2}\frac{2\Gamma\left(2-\frac{d}{2}\right)}{(4\pi)^{\frac{d}{2}}}\int_{0}^{1}\frac{dx}{\sqrt{x}}\frac{-x\nu\gamma_{0}+v_{F}\left(-\frac{xv_{B}^{2}}{(1-x)v_{F}^{2}+xv_{B}^{2}}q_{j}\right)\gamma_{j}}{\left[\frac{x(1-x)\nu^{2}}{(1-x)v_{F}^{2}+xv_{B}^{2}}+\frac{x(1-x)v_{B}^{2}v_{F}^{2}}{((1-x)v_{F}^{2}+xv_{B}^{2})^{2}}q^{2}\right]^{2-\frac{d}{2}}((1-x)v_{F}^{2}+xv_{B}^{2})}$
$\displaystyle=$
$\displaystyle-\frac{ig^{2}}{\epsilon}\left[\frac{\nu\gamma_{0}\left(\sqrt{v_{B}^{2}-v_{F}^{2}}-v_{F}\arctan(\frac{\sqrt{v_{B}^{2}-v_{F}^{2}}}{v_{F}}\right)}{2\pi^{2}(v_{B}^{2}-v_{F}^{2})^{\frac{3}{2}}}-\frac{q_{j}\gamma_{j}\left(v_{F}-\frac{v_{B}^{2}\arctan\frac{\sqrt{v_{B}^{2}-v_{F}^{2}}}{v_{F}}}{\sqrt{v_{B}^{2}-v_{F}^{2}}}\right)}{4\pi^{2}v_{F}(v_{B}^{2}-v_{F}^{2})}\right]+\mathcal{O}(1)$
$\displaystyle=$
$\displaystyle-\frac{ig^{2}}{4\pi^{2}\epsilon}\left[\nu\gamma_{0}\left(\frac{2}{(v_{B}^{2}-v_{F}^{2})}-\frac{2v_{F}\arctan\frac{\sqrt{v_{B}^{2}-v_{F}^{2}}}{v_{F}}}{(v_{B}^{2}-v_{F}^{2})^{\frac{3}{2}}}\right)+q_{j}\gamma_{j}\left(-\frac{1}{(v_{B}^{2}-v_{F}^{2})}+\frac{v_{B}^{2}\arctan\frac{\sqrt{v_{B}^{2}-v_{F}^{2}}}{v_{F}}}{v_{F}(v_{B}^{2}-v_{F}^{2})^{\frac{3}{2}}}\right)\right]+\mathcal{O}(1)\,.$

It is direct to verify that when $v_{F}=v_{B}=1$, the result nicely coincides
with Eq. (S10). Now, we can directly extract the $Z$ factors:

$\displaystyle Z_{\phi}$ $\displaystyle=1+g^{2}\mathcal{O}(1)\,,$ (S17)
$\displaystyle Z_{v_{B}}$
$\displaystyle=Z_{\phi}^{-\frac{1}{2}}\left(1+g^{2}\mathcal{O}(1)\right)\,,$
$\displaystyle Z_{\psi}$
$\displaystyle=1-\frac{g^{2}}{4\pi^{2}\epsilon}\left(\frac{2}{(v_{B}^{2}-v_{F}^{2})}-\frac{2v_{F}\arctan\frac{\sqrt{v_{B}^{2}-v_{F}^{2}}}{v_{F}}}{(v_{B}^{2}-v_{F}^{2})^{\frac{3}{2}}}\right)+\mathcal{O}(1)\,,$
$\displaystyle Z_{v_{F}}$
$\displaystyle=Z_{\psi}^{-1}\left(1-\frac{g^{2}}{4\pi^{2}\epsilon}\left(\frac{1}{(v_{B}^{2}-v_{F}^{2})}-\frac{v_{B}^{2}\arctan\frac{\sqrt{v_{B}^{2}-v_{F}^{2}}}{v_{F}}}{v_{F}(v_{B}^{2}-v_{F}^{2})^{\frac{3}{2}}}\right)+\mathcal{O}(1)\right)\,.$

$Z_{\phi}$ and $Z_{v_{B}}$ are both finite, which means that the boson
velocity does not receive the renormalization from the boundary. With

$v_{F,\text{bare}}=Z_{v_{F}}Z_{\psi}^{-1}v_{F}\,,$ (S18)

the beta function for $v_{F}$ can be evaluated as

$\beta_{v_{F}}=-\frac{3g^{2}}{8\pi^{2}(v_{B}^{2}-v_{F}^{2})}+\frac{g^{2}(2v_{F}+\frac{v_{B}^{2}}{v_{F}})\arctan\frac{\sqrt{v_{B}^{2}-v_{F}^{2}}}{v_{F}}}{8\pi^{2}(v_{B}^{2}-v_{F}^{2})^{\frac{3}{2}}}\,.$
(S19)

At the stable fixed point $g_{*}^{2}$, the fixed point value of $v_{F}$ can be
directly solved to be $v_{B}$, which indicates an emergent Lorentz symmetry.
Therefore, it justifies our assumption.

Now setting $v_{F}=v_{B}=1$, the boundary field $\psi$ acquires an anomalous
dimension,

$\hat{\eta}_{\psi}=\mu\frac{\partial}{\partial\mu}\log
Z_{\psi}^{\frac{1}{2}}=\frac{g^{2}}{12\pi^{2}}=\frac{\epsilon}{16}\,,$ (S20)

in the last step, we have substituted the fixed point value $g^{2}_{\ast}$.

##  II Boundary Gross-Neveu-Yukawa Universality class

In this section, we generalize our previous calculation to include a self-
interaction for the bulk scalar field. The corresponding Euclidean action
reads

$S=\int
d^{d}x\left(\frac{1}{2}(\partial\phi)^{2}+\frac{1}{4!}\lambda\phi^{4}\right)+\int
d^{d-1}x\left(\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-g\phi\bar{\psi}\psi\right),$
(S21)

where $\gamma^{\mu}\partial_{\mu}=\sum_{i=0}^{d-2}\gamma^{i}\partial_{i}$ and
$\int d^{d}x\phi^{4}$ is the self-interaction for the bulk boson with strength
given by $\lambda$. Notice that we again assume the Lorentz symmetry, and will
justify this assumption later.

The scaling dimensions are $[\lambda]=4-d,[g]=2-\frac{d}{2}$. Hence, for
$d=4$, the boson self-interaction $\phi^{4}$ and the Yukawa coupling are both
tree-level marginal, which allows us to perform a RG calculation at
$d=4-\epsilon$. For the same reason, we neglect the surface
$\int_{\partial\mathcal{M}}d^{d-1}x\phi^{4}$. According to the previous study,
while the boundary theory will not renormalize from the bulk theory, the bulk
theory can lead to nontrivial correction to the boundary theory. At one-loop
level, no Feynman graphs involved both Yukawa vertex and $\phi^{4}$ vertex
exist. However, the $\phi^{4}$ vertex will contribute to the self-energy of
the boundary boson by the tadpole diagram which we denote
$\Pi_{\phi}^{\phi^{4}}$ and result in a surface anomalous dimension,

$\displaystyle\Pi^{\phi^{4}}_{\phi}(p)$
$\displaystyle=\frac{\lambda}{2}\int\frac{d^{d-1}k}{(2\pi)^{d-1}}\int_{0}^{\infty}dy\frac{e^{-|p|y}}{|p|}\frac{1+e^{-2|k|y}}{2|k|}\frac{e^{-|p|y}}{|p|}$
(S22)
$\displaystyle=\frac{\lambda}{8}\int\frac{d^{d-1}k}{(2\pi)^{d-1}}\left(\frac{1}{|p|^{3}|k|}+\frac{1}{(|k|+|p|)|k||p|^{2}}\right)$
$\displaystyle=\frac{\lambda}{8}\frac{2\pi^{\frac{d-1}{2}}}{\Gamma(\frac{d-1}{2})}\int\frac{dk}{(2\pi)^{d-1}}\left(\frac{|k|^{d-3}}{|p|^{3}}+\frac{|k|^{d-3}}{(|p|+|k|)|p|^{2}}\right)\,.$

The superscript indicates the contribution from the $\phi^{4}$ vertex.
According to Veltman’s formula, the first term vanishes and the second term is

$\frac{\lambda}{4}\frac{\pi^{\frac{d-1}{2}}}{\Gamma(\frac{d-1}{2})(2\pi)^{d-1}}\frac{1}{p^{5-d}}\int_{0}^{1}duu^{2-d}(1-u)^{d-3}=\frac{\lambda}{4}\frac{\Gamma(3-d)\Gamma(d-2)\pi^{\frac{d-1}{2}}}{\Gamma(1)\Gamma(\frac{d-1}{2})(2\pi)^{d-1}}\frac{1}{p^{5-d}}\,,$
(S23)

where we set $u=\frac{|p|}{|k|+|p|}$. Expanding as a series of $\epsilon$, we
arrive at

$G_{\phi}^{\phi^{4}}(p)=-\frac{\lambda}{16\pi^{2}|p|}\frac{1}{\epsilon}\,.$
(S24)

This modifies $Z_{\phi}$

$Z_{\phi}=1+\frac{\lambda}{16\pi^{2}}\frac{1}{\epsilon}\,.$ (S25)

Hence, the modified $\beta$ function for Yukawa coupling $g$ and the $\beta$
function for $\lambda$, which gives the usual Wilson-Fisher fixed point, are
respectively given by:

$\begin{split}&\beta_{g}=-\frac{\epsilon}{2}g+\frac{2}{3\pi^{2}}g^{3}-\frac{1}{32\pi^{2}}\lambda
g\,,\\\
&\beta_{\lambda}=-\epsilon\lambda+\frac{3}{(4\pi)^{2}}\lambda^{2}\,.\end{split}$
(S26)

From the second equation, it is clear that the IR fixed point of $\lambda$ is
given by the usual Wilson-Fisher point
$\lambda_{*}=\frac{(4\pi)^{2}\epsilon}{3}$ by the bulk theory. Back to the
first equation, we find a new fixed point for $g$ at
$g_{*}^{2}=\pi^{2}\epsilon$. We call this new fixed point the boundary Gross-
Neveu-Yukawa fixed point. It is a stable fixed point from the Jacobian

$M_{ij}=\frac{\partial\beta_{i}}{\partial g_{j}},\quad
M=\left[\begin{array}[]{cc}-\epsilon+\frac{3}{8\pi^{2}}\lambda_{*}&0\\\
\frac{1}{32\pi^{2}}g_{*}&-\frac{\epsilon}{2}-\frac{1}{32\pi^{2}}\lambda_{*}+\frac{2}{\pi^{2}}g_{*}^{2}\end{array}\right]\,,$
(S27)

whose eigenvalues, i.e., $\frac{4}{3}\epsilon,\epsilon$, are all positive,
meaning that this fixed point is stable. At this new fixed point, the fermion
anomalous dimensions are

$\eta_{\psi}=\frac{g_{*}^{2}}{12\pi^{2}}=\frac{\epsilon}{12}\,.$ (S28)

##  III Special BKT phase transition

In this section, we consider the 3-dimensional Ising boundary criticality in
the presence of boundary fermion. As explained in the main text, our strategy
is to perturb the ordinary fixed point by coupling with a boundary fermion.
After bosonization of the Yukawa coupling between the Ising field and the
boundary fermion,

$S_{\text{f}}=\int
dxd\tau\left[\frac{1}{2K}\left(\frac{1}{v_{f}}(\partial_{\tau}\varphi)^{2}+v_{f}(\partial_{x}\varphi)^{2}\right)+g\phi\cos(\sqrt{4\pi}\varphi)\right]$
(S29)

where $\phi$ denotes the Ising field with scaling dimension
$\Delta_{\hat{\phi}}$, and $\varphi$ denotes the bosonization field. $K$ and
$g$ are the Luttinger parameter and the Yukawa coupling, respectively, and the
velocity of the fermion is $v_{f}$. Without loss of generality, we can rescale
the bulk boson velocity to be $v_{b}=1$, the velocity ratio
$v_{f}=v_{f}/v_{b}$ then enters into the RG flow of velocity. The OPE of the
vertex operator is now

$\displaystyle:\phi V_{1}(x)::\phi V_{-1}(0):=$ (S30)
$\displaystyle\frac{C_{\phi}}{\left(x^{2}+v_{f}^{2}\tau^{2}\right)^{K}\left(x^{2}+\tau^{2}\right)^{\Delta_{\phi}}}\left[1+i\sqrt{4\pi}\left(x\partial_{x}\varphi(0)+\tau\partial_{\tau}\varphi(0)\right)\right.\left.-\frac{4\pi}{2}\left(x^{2}\left(\partial_{x}\varphi\right)^{2}+\tau^{2}\left(\partial_{\tau}\varphi\right)^{2}+2x\tau\partial_{x}\varphi\partial_{\tau}\varphi\right)\right]$

where $\Delta_{\hat{\phi}}$ is the scaling dimension of the boundary boson
$\phi$, and $b$ is the coefficient in the correlation function of the boundary
boson: $G_{\phi}(x)=\frac{C_{\phi}}{x^{2\Delta_{\hat{\phi}}}}$. For
simplicity, we choose to renormalize the two-point function of $\phi$, namely
set $C_{\phi}=1$. The consistent OPE coefficients can be derived from the
standard conformal perturbation theory Giombi  _et al._ (2016); Fei  _et al._
(2015). We need not worry about the issue of normalization, since the results
can be related by properly rescaling $g$. The renormalized action takes the
form of

$\begin{split}\delta S&=\frac{g^{2}}{8}\int dxd\tau\int_{a^{2}<x^{\prime
2}+\tau^{\prime
2}<a^{2}(1+2d\ell)}dx^{\prime}d\tau^{\prime}\left[\left(x^{\prime
2}(\partial\varphi(x))^{2}+\tau^{\prime
2}\left(\partial_{\tau}\varphi(x)\right)^{2}\right)\frac{4\pi}{\left(x^{\prime
2}+v_{f}^{2}\tau^{\prime 2}\right)^{K}\left(x^{\prime 2}+\tau^{\prime
2}\right)^{\Delta_{\phi}}}\right]\\\ &=dlg^{2}\int dxd\tau\left[2\pi
v_{f}A(v_{f})(\partial\varphi(x))^{2}+\frac{2\pi}{v_{f}}B(v_{f})\left(\partial_{\tau}\varphi(x)\right)^{2}\right]\,,\end{split}$
(S31)

where the two functions $A(v_{f}),B(v_{f})$ are defined as

$A\left(v_{f}\right)=\frac{1}{4v_{f}}\int_{0}^{2\pi}d\theta\frac{\cos^{2}\theta}{\left(\cos^{2}\theta+v_{f}^{2}\sin^{2}\theta\right)^{K}}\,,\qquad
B\left(v_{f}\right)=\frac{v_{f}}{4}\int_{0}^{2\pi}d\theta\frac{\sin^{2}\theta}{\left(\cos^{2}\theta+v_{f}^{2}\sin^{2}\theta\right)^{K}}\,.$
(S32)

So we have the following RG equations

$\displaystyle\frac{dK}{dl}=-2\pi\left(A(v_{f})+B(v_{f})\right)g^{2}K^{2}\,,$
(S33)
$\displaystyle\frac{dv_{f}}{dl}=2\pi\left(A(v_{f})-B(v_{f})\right)g^{2}v_{f}K\,,$
$\displaystyle\frac{dg}{dl}=(2-\Delta_{\hat{\phi}}-K)g\,.$

We can check the RG equations at $v_{f}=1$, and now
$A(v_{f}=1)=B(v_{f}=1)=\frac{\pi}{4}$. If we consider the boundary of $2+1$d
TIs, then $\Delta_{\hat{\phi}}=\Delta_{\text{ord}}$, and we have:

$\displaystyle\frac{dK}{dl}=-\pi^{2}g^{2}K^{2}\,,$ (S34)
$\displaystyle\frac{dg}{dl}=(2-\Delta_{\hat{\phi}}-K)g\,.$

Considering the two-particle backscattering term
$g_{2}\cos(\sqrt{16\pi}\varphi)$, the RG equation yields

$\displaystyle\frac{dK}{dl}=-\pi^{2}g^{2}K^{2}-4\pi^{2}g_{2}^{2}K^{2}\,,$
(S35) $\displaystyle\frac{dg}{dl}=(2-\Delta_{\hat{\phi}}-K)g\,,$
$\displaystyle\frac{dg_{2}}{dl}=(2-4K)g_{2}\,.$

For the defect case, the coefficients of the projected boson propagator should
be twice larger than the boundary case. And as we mentioned before, it can be
simply incorporated by redefining $g$ without changing the physics.

##  IV Higher order special BKT phase transition

In this section, we evaluate the RG flow of the special BKT phase transition
up to the third order Amit  _et al._ (1980). We start with the action

$S=\int
d^{2}x\left[\frac{1}{2}(\partial\varphi)^{2}+\frac{g}{K}\phi\cos(\sqrt{K}\varphi)\right]+S_{\text{bulk}}\,,$
(S36)

where $S_{\text{bulk}}$ is the action of $\phi$ in the bulk. Notice here, we
have redefined the field and coupling constant for simplicity as compared to
the action in the main text. We choose such a convention for the reason that
the log divergence can be absorbed into two independent renormalization
factors $Z_{g},Z_{\varphi}$.

$\displaystyle g_{\text{bare}}=Z_{g}g\,,\quad
K_{\text{bare}}=Z_{\varphi}^{-1}K\,.$ (S37)

Supplementary Figure S2: (a) Feynman diagram for the zero-order vertex
correction $\Gamma^{(0)}$ ($g\phi\varphi^{2}$ as example and up to three
loops). The red external line is the $\phi$ entry and black is the $\varphi$
entry. The black solid vertex represents the dressed vertex. The black solid
internal line is the propagator of $\varphi$. (b) Feynman diagram for the
second order self-energy $\Sigma^{(2)}_{\varphi}(p)$. The dashed line
represents $\cosh G_{\varphi}-1$ and the wavy line represents $\sin
G_{\varphi}-G_{\varphi}$. (c) Typical Feynman diagrams that contribute to the
log divergence in the third order of vertex correction $\Gamma^{(3)}_{g}$.
Diagrams related by symmetry operations are neglected here. The complete
expressions of the diagrams are listed in Eq. (S47).

The propagator of $\sqrt{K}\varphi$ is

$G_{\varphi}(x,a)=K\left.\int\frac{d^{2}p}{(2\pi)^{2}}\frac{\mathrm{e}^{\mathrm{i}py}}{p^{2}+m_{0}^{2}}\right|_{y^{2}=x^{2}+a^{2}}=\frac{K}{2\pi}K_{0}\left[m_{0}\left(x^{2}+a^{2}\right)^{1/2}\right]\sim\frac{K}{4\pi}\ln\left(x^{2}+a^{2}\right)\,,$
(S38)

where $m_{0}$ is the IR cutoff and $a$ is the UV cutoff. In the case of a
small IR cutoff, the propagator behaves logarithmically. $K_{0}(r)$ is the
second-type modified Bessel function. Notice that the $\phi$ is not normalized
by $\varphi$, we first trigger the flow of the bulk to infrared and then
consider its effect on the surface. The $\phi$ propagator is chosen to be
$\frac{1}{(x^{2}+a^{2})^{2\Delta_{\phi}}}$. Our strategy is to expand the
vertex
$\frac{g}{K}\phi\cos(\sqrt{K}\varphi)=\sum_{n=0}^{\infty}\frac{g}{K}\phi\frac{(-1)^{n}(\sqrt{K}\varphi)^{2n}}{(2n)!}$
and resum all the log-divergence up the higher order of $g$. We first focus on
the tree-level of $g$, which originates from the vertex correction illustrated
in Fig. S2 (a). After resummation, the correction is

$g_{\text{bare}}=\left.a^{\Delta_{\hat{\phi}}-2}\exp\left(-\frac{1}{2}G_{\varphi}(x)\right)g\right|_{x=0}=a^{K/4\pi+\Delta_{\hat{\phi}}-2}g\,.$
(S39)

We proceed to the higher-order calculation of the RG. In the following
calculation, we perturb near $K_{c}=2-\Delta_{\hat{\phi}}$ and define
$\delta=K/4\pi-2+\Delta_{\hat{\phi}}$. The log divergence in the 1PI diagrams
for the $\varphi$ propagator in Fig. S2 (b). The self-energy of
$\sqrt{K}\varphi$ takes the form

$\Sigma_{\varphi}^{(1)}(p)=0\,,$ (S40)
$\displaystyle\Sigma^{(2)}_{\varphi}(p)$
$\displaystyle=-\frac{g_{\text{bare}}^{2}}{K}\int
d^{2}x\left(e^{ipx}G_{\phi}(x)\sinh G_{\varphi}(x)-G_{\phi}(x)\cosh
G_{\varphi}(x)\right)$ (S41) $\displaystyle\sim-\frac{1}{2}(1+2\delta\log
a)\int d^{2}x\left(e^{ipx}-1\right)e^{G_{\varphi}(x)}G_{\phi}(x)$
$\displaystyle=-\frac{1}{4}\frac{1}{4\pi(2-\Delta_{\hat{\phi}})}(1-\frac{1}{2-\Delta_{\hat{\phi}}}\delta+2\delta\log
a)g^{2}\int dx\frac{2\pi p^{2}x^{3}}{(x^{2}+a^{2})^{2+\delta}}$
$\displaystyle\sim\frac{g^{2}p^{2}}{8(2-\Delta_{\hat{\phi}})}\left[1-\left(1+\frac{1}{2-\Delta_{\hat{\phi}}}\right)\delta\right]\log
a\,.$

By requiring

$G_{\varphi}(p)^{-1}=Z_{\varphi}p^{2}+\Sigma_{\varphi}(p)$ (S42)

is finite, we extract $Z_{\varphi}$ factor

$Z_{\varphi}=1-\frac{g^{2}}{8(2-\Delta_{\hat{\phi}})}\log
a+\frac{(3-\Delta_{\hat{\phi}})g^{2}\delta}{8(2-\Delta_{\hat{\phi}})^{2}}\log
a,$ (S43)

Using Eq. (S37), we have the beta function of $\delta$

$\beta_{\delta}=(1+\delta)\frac{\partial Z_{\varphi}}{\partial\log
a}=-\frac{g^{2}}{8(2-\Delta_{\hat{\phi}})}+\frac{g^{2}\delta}{8(2-\Delta_{\hat{\phi}})}.$
(S44)

We now implement the calculation of beta function of $g$ up to
$\mathcal{O}(g^{3})$. The higher-order diagrams that have the proper log
divergence are illustrated in Fig. S2 (c). They take the form of

$\Gamma_{g}^{(1)}(0)=g_{\text{bare}}\,,$ (S45) $\Gamma_{g}^{(2)}(0)=0\,,$
(S46)
$\displaystyle\Gamma_{g}^{(3)}(0)\sim\frac{g_{\text{bare}}^{3}}{K^{2}}\int
d^{2}xd^{2}y[\sinh G_{\varphi}(x-y)(\cosh G_{\varphi}(y)-1)\cosh
G_{\varphi}(x)G_{\phi}(x)+\sinh G_{\varphi}(x-y)\cosh
G_{\varphi}(y)G_{\phi}(y)(\cosh G_{\varphi}(x)-1)$ (S47)
$\displaystyle\left.+\sinh G_{\varphi}(x-y)G_{\phi}(x-y)(\cosh
G_{\varphi}(x)-1)(\cosh G_{\varphi}(y)-1)\right]$
$\displaystyle-\frac{g_{\text{bare}}^{3}}{K^{2}}\int d^{2}xd^{2}y\left[(\cosh
G_{\varphi}(x-y)-1)\sinh G_{\varphi}(x)G_{\phi}(x)\sinh G_{\varphi}(y)+(\cosh
G_{\varphi}(x-y)-1)\sinh G_{\varphi}(x)\sinh G_{\varphi}(y)G_{\phi}(y)\right.$
$\displaystyle\left.+\cosh G_{\varphi}(x-y)G_{\phi}(x-y)\sinh
G_{\varphi}(x)\sinh G_{\varphi}(y)\right]$
$\displaystyle-\frac{1}{2}\frac{g_{\text{bare}}^{3}}{K^{2}}\int
d^{2}xd^{2}y[(\cosh G_{\varphi}(x-y)-1)(\cosh G_{\varphi}(y)-1)\cosh
G_{\varphi}(x)G_{\phi}(x)$ $\displaystyle+(\cosh G_{\varphi}(x-y)-1)(\cosh
G_{\varphi}(x)-1)\cosh G_{\varphi}(y)G_{\phi}(y)\left.+(\cosh
G_{\varphi}(x)-1)(\cosh G_{\varphi}(y)-1)\cosh
G_{\varphi}(x-y)G_{\phi}(x-y)\right]$
$\displaystyle\frac{1}{2}\frac{g_{\text{bare}}^{3}}{K^{2}}\int
d^{2}xd^{2}y\left[\sinh I(x-y)\sinh I(x)G_{\phi}(x)\sinh I(y)+\sinh
I(x-y)\sinh I(x)\sinh I(y)G_{\phi}(y)\right.$ $\displaystyle\left.+\sinh
I(x-y)G_{\phi}(x-y)\sinh I(x)\sinh I(y)\right]\,$

The summation of the terms that have the proper log divergence is

$\frac{1}{4}\frac{(g_{\text{bare}})^{3}}{K^{2}}\int
d^{2}xd^{2}yG_{\phi}(x)e^{G_{\varphi}(x)+G_{\varphi}(y)-G_{\varphi}(x+y)}$
(S48)

where we have used the exchange symmetry $x\leftrightarrow y$ and
$y\leftrightarrow-y$ in the integral. The integral is divided into three
parts:

$\text{Region 1}:|y|,|x|<\Delta\,,\quad\text{Region
2}:|x|<\Delta,|y|>\Delta\,,\quad\text{Region 3}:|x|>\Delta,|y|<\Delta\,.$
(S49)

We first perform the evaluation of the integral in the first region. In region
I, we can approximate both $G_{\varphi}(x)$, $G_{\varphi}(y)$ and
$G_{\varphi}(x+y)$ into the log function, and directly work out the integral

$\displaystyle J_{1}$
$\displaystyle=\frac{1}{4}\frac{(g_{\text{bare}})^{3}}{K^{2}}\int_{\mathcal{R}_{1}}d^{2}xd^{2}yG_{\phi}(x)e^{G_{\varphi}(x)+G_{\varphi}(y)-G_{\varphi}(x+y)}$
(S50)
$\displaystyle=\frac{1}{4}\frac{(g_{\text{bare}})^{3}}{K^{2}}\int_{\mathcal{R}_{1}}d^{2}xd^{2}y\frac{\left((x-y)^{2}+a^{2}\right)^{2-\Delta_{\phi}}}{\left(x^{2}+a^{2}\right)^{2}\left(y^{2}+a^{2}\right)^{2-\Delta_{\phi}}}$
$\displaystyle\sim\frac{1}{64}g^{3}\left(\log\left(\frac{a}{\Delta}\right)^{2}+\log^{2}\left(\frac{a}{\Delta}\right)^{2}\right)\,.$

The integral at the second line cannot explicitly be worked out for general
$\Delta_{\hat{\phi}}$, hence we would approximate $\Delta_{\hat{\phi}}\approx
1$ because the estimate of $\Delta_{\text{ord}}$ from Monte Carlo simulation
is $\Delta_{\text{ord}}\approx 1.2626$. The contribution from region 2 is
identical to region 3, and we perform the Taylor expansion for small $x$, say
$G_{\varphi}(x+y)=G_{\varphi}(y)+hG_{\varphi}^{\prime}(y)+\frac{1}{2}h^{2}G_{\varphi}^{\prime\prime}(y)$,
where $h=2xy\cos\theta+x^{2}$. The integrals are evaluated to be

$\displaystyle J_{2}=J_{3}$
$\displaystyle=\frac{1}{4}\frac{\left(g_{\mathrm{bare}}\right)^{3}}{K^{2}}\int_{\mathcal{R}_{2}}d^{2}xd^{2}yG_{\phi}(x)e^{G_{\varphi}(x)+G_{\varphi}(y)-G_{\varphi}(x+y)}$
(S51)
$\displaystyle=\frac{\pi}{2}\frac{g_{\text{bare}}^{3}}{K^{2}}\int_{0}^{\Delta}d^{2}xx^{2}G_{\phi}(x)e^{G_{\varphi}(x)}\int_{\Delta}^{\infty}ydy\int_{0}^{2\pi}d\theta\left(-G_{\varphi}^{\prime}(y)+2y^{2}\cos\theta^{2}G_{\varphi}^{\prime}(y)^{2}-2y^{2}\cos\theta^{2}G_{\varphi}^{\prime\prime}(y)\right)$
$\displaystyle\sim-\frac{1}{64}g^{3}\log\left(\frac{a^{2}}{\Delta^{2}}\right)\int_{\Delta}^{\infty}dy\left(2yK_{0}^{\prime}(y)^{2}-yK_{0}^{\prime\prime}(y)\right)$
$\displaystyle=-\left.\frac{1}{64}g^{3}\log\left(\frac{a^{2}}{\Delta^{2}}\right)\left(y^{2}\left(K_{1}^{2}-K_{0}K_{2}(y)\right)+yK_{1}(y)\right)\right|_{\Delta}^{\infty}$
$\displaystyle\sim\frac{1}{64}g^{3}\log\left(\frac{a^{2}}{\Delta^{2}}\right)\left(2+2\log\left(\Delta\right)\right)\,.$

Summing over contributions from all regions, the divergence is

$J=J_{1}+J_{2}+J_{3}\sim\frac{5}{32}g^{3}\log a+\frac{1}{16}g^{3}\log^{2}a\,.$
(S52)

Supplementary Figure S3: Flow pattern of the special BKT phase transition in
from the higher order RG equation.

Hence the $Z_{g}$ is

$Z_{g}=1+\frac{5}{32}g^{3}\log a\,.$ (S53)

Finally, the higher order beta function for $\delta$ and $g$ is

$\displaystyle\beta_{\delta}$
$\displaystyle=-\frac{g^{2}}{8}+\frac{g^{2}\delta}{8}\,,$ (S54)
$\displaystyle\beta_{g}$ $\displaystyle=-\delta
g-\frac{5}{32(2-\Delta_{\hat{\phi}})^{2}}g^{3}\,.$

Recall that $\delta=K/4\pi-2+\Delta_{\hat{\phi}}$, by rescaling $K/4\pi\to
K,g^{2}/8\to g^{2}$, the flow pattern at the $K-g$ plane is shown in Fig. S3,
at the strongly coupled region, the RG flows to $K=0$, which corresponds to
the gapped phase.


# Active Learning for NLP with Large Language Models

Xuesong Wang This is the project report for ECE 7500 at Wayne State University
for Fall 2023 semester. Department of Electrical and Computer Engineering
Wayne State University
Detroit, MI, USA
<EMAIL_ADDRESS>
###### Abstract

Human annotation of training samples is expensive, laborious, and sometimes
challenging, especially for Natural Language Processing (NLP) tasks. To reduce
the labeling cost and enhance the sample efficiency, Active Learning (AL)
technique can be used to label as few samples as possible to reach a
reasonable or similar results. To reduce even more costs and with the
significant advances of Large Language Models (LLMs), LLMs can be a good
candidate to annotate samples. This work investigates the accuracy and cost of
using LLMs (GPT-3.5 and GPT-4) to label samples on 3 different datasets. A
consistency-based strategy is proposed to select samples that are potentially
incorrectly labeled so that human annotations can be used for those samples in
AL settings, and we call it mixed annotation strategy. Then we test
performance of AL under two different settings: (1) using human annotations
only; (2) using the proposed mixed annotation strategy. The accuracy of AL
models under 3 AL query strategies are reported on 3 text classification
datasets, i.e., AG’s News, TREC-6, and Rotten Tomatoes. On AG’s News and
Rotten Tomatoes, the models trained with the mixed annotation strategy
achieves similar or better results compared to that with human annotations.
The method reveals great potentials of LLMs as annotators in terms of accuracy
and cost efficiency in active learning settings.

###### Index Terms:

natural language processing, large language model, active learning, annotation

##  I Introduction

Supervised deep learning requires a large amount of ground truth labels.
Usually the samples are labeled by humans. However, annotation by human
annotators is expensive, laborious, and sometimes challenging, which is
especially true in Natural Language Processing (NLP) systems [1, 2, 3]. On
Google Cloud Platform111https://cloud.google.com/ai-platform/data-
labeling/pricing, the text classification labeling costs for 1000 units (50
words per unit) $129 in Tier 1 and $90 for Tier 2. In [3], the authors assumes
$0.11 per 50 tokens, and I will follow the same estimates in this work.

To reduce the labeling cost and enhance the sample efficiency, Active Learning
(AL) technique can be utilized, assuming not all samples are equal [1]. AL
algorithms start the model training using a few initially labeled samples,
iteratively select most representative and(or) informative samples to be
labeled by annotator, add the newly-labeled samples into the training set to
continue training or train from scratch, and stop until stopping criteria is
met. It can achieve similar or even better results compared to passive
training which trains the model using the entire training set [2], but the
labeling cost is much lower.

To reduce the annotation cost, some researchers have proposed to use Large
Language Models (LLMs), e.g., GPT-3, to label the instances thanks to the few-
shot (in-context) learning capability of LLMs [3]. It costs 50% to 96% less to
use labels from GPT-3 than using labels from humans to reach the same
performance on multiple NLU and NLG tasks [3]. It is found that data labeled
by GPT-3 is usually more noisy than human-labeled data, and the authors in [3]
use the confidence score of GPT-3 API to measure the uncertainty of the model
output. If the output confidence score is low, the sample will be labeled by
human instead. As a result, the samples that labeled by GPT-3 and humans are
combined to train the model to achieve a better performance with the limited
labeling budget. When using LLMs to label the instances, the demonstration in
the prompt plays an important role in the accuracy of the annotation. Authors
of [4] investigated effects of the number of examples in the demonstration and
the strategies to select examples.

In order to explore the performance of using latest LLMs to label instances
for AL in order to reduce even more costs of labeling, this paper utilizes
GPT-3.5 and GPT-4 to label training samples. Two sets of experiments are
designed. First, GPT-3.5 and GPT-4 are utilized to label samples in 3 datasets
using different number of examples in the demonstration and different
demonstration example selection strategies. Because the GPT-3.5 and GPT-4 APIs
do not have the confidence score output parameter anymore, a consistency-based
strategy is proposed to find the samples that the GPT-3.5 or GPT-4 is not
confident in. The annotation accuracy, actual cost and the inconsistency rate
are reported. The second set of experiments are using GPT-3.5 in AL
experiments. For those samples that GPT-3.5 are uncertain, we use human
annotations; the GPT-3.5 annotations are used for the rest of the samples. We
report the accuracy of AL experiments and compare the model performance
between the models using human annotations only and mixed annotations.

The paper is organized as follows: the related literature is briefly reviewed
in section II, followed by the detailed introduction of the proposed method in
section III. In section IV, the experiment setup is given, and the experiment
results are illustrated and analyzed in section V. Finally, the limitation,
future work are discussed and the conclusion is drawn in section VI.

##  II Related Work

A general procedure of AL is illustrated in Table I [1]. Most of the
literature focus on the query strategy design. One dimension is
informativeness of the samples. For instance, we can select samples based
output uncertainty, like Least Confidence [5], Prediction Entropy [6], and
Breaking Ties [7]. We can also use multiple models to select the most
disagreed samples [8]. Instances that would most strongly impact the model
should be considered informative, therefore, we can use the norm of gradient
to select the instances [9]. Considering informativeness only might introduce
sampling bias and the selection of outliers, thus the representativeness
should be taken into consideration as well [1]. For instance, we can select
instances that are more representative of the unlabeled set [10, 11], or
instances that are different from labeled instances [12, 13, 14]. Moreover, we
can take a hybrid approach to combine the two metrics together [15, 16].
Authors of [2] compared 5 query strategies on 5 text classification datasets
using 2 transformer models. In this work, 3 of those 5 query strategies are
compared, i.e., random (as baseline), Least Confidence [5], and Breaking Ties
[7], on 3 of the 5 datasets, i.e., AG’s News [17], Rotten Tomatoes Movie
Reviews [18], and TREC-6 [18], using 1 transformer model, i.e., DistilRoBERTa
[19].

Using GPT-3 to label instances was proposed in [3] with a price of $0.04 per
1k tokens [3]. To reduce the labeling noise in the GPT-3 annotation, they
proposed to measure the model uncertainty using API output parameter
$logprobs$. If $logprobs$ is low, the sample will be labeled by humans instead
of using GPT-3 label. The labels created by GPT-3 and humans are mixed to
train the model. They found that it costs 50% to 96% less to use the labels
from GPT-3 than using labels from humans on a variety of NLU and NLG tasks to
reach the same performance. The goal of this work is to test the latest
GPT-3.5 and GPT-4 models, whereas the $logprobs$ output parameter is not
present anymore. A consistency-based approach is proposed and introduced in
III.

Benefit from few-shot (in-context) learning capability of GPT-3.5 and GPT-4
models, LLMs can be used to annotate samples directly without finetuning, with
just some demonstration examples in the prompt [4]. Authors of [4] explored
the effects of demonstration example selection strategies, and they found that
selecting the examples that are most similar to the test sample can reach the
highest accuracy. In this work, three strategies are compared on GPT-3.5 and
GPT-4 models as well, i.e., $random$ (as baseline), examples that cost fewest
tokens ($min\\_token$), and examples that are most similar to the test
examples ($max\\_similarity$).

TABLE I: General Procedure of Active Learning _Input:_ An unlabeled data pool
$U$
---
_Output:_ Final labeled dataset $L$ and trained model $M$
_Procedure:_
1: Initialize $L$ and $U$
2: train $M$ using $L$
3: while not stop_criterion() do:
4: samples $I$ $\leftarrow$ query($M$,$U$,$L$)
5: annotate samples $I$
6: $U=U-I$, $L=L+I$
7: train $M$ using $L$
_return:_ $L$, $M$

##  III Method

In this section, the annotation method using GPT-3.5 and GPT-4 will be
introduced. Then consistency-based uncertain sample selection strategy is
proposed.

###  III-A Annotation by GPT-3.5 and GPT-4

The OpenAI Chat Completion API222https://platform.openai.com/docs/api-
reference/chat is used for the annotation because the legacy completion API is
retiring. The pricing of the GPT-3.5 and GPT-4 models is listed in Table II.
It requires a list of messages in the request, and returns a message, as shown
in Table III. The system prompt should be tuned for each different dataset in
terms of the type of the texts, number of categories, and the class names. The
API has a parameter called $response\\_format$ which can be used to ensure the
JSON format of the response. As we can see, $N*C$ demonstration examples are
included in the messages given by the role user, where $N$ is the number of
examples of each class included in the demonstration, and $C$ is the number of
classes. Given a pool of labeled examples, we have different strategies to
select a subset of them to be included in the demonstration. We can randomly
select $N$ samples for each class ($random$), select the $N$ samples that
consume fewest tokens ($min\\_token$), or select $N$ samples that are most
similar to the sentence to label ($max\\_similarity$).

TABLE II: Pricing of GPT-3.5 and GPT-4 Model | Input | Output
---|---|---
gpt-3.5-turbo-1106 | $0.0010 / 1K tokens | $0.0020 / 1K tokens
gpt-4-1106-preview | $0.01 / 1K tokens | $0.03 / 1K tokens
TABLE III: Example Annotation Process using OpenAI Chat Completion API Role | Content
---|---
Input Messages
system |  You have been trained to classify movie reviews.The user will give you some movie reviews, and you will classify the reviews into 2 categories: Negative, Positive. You should use the given class names only. The examples are given by the user and they should be used as your references. You should respond in a json format with the class names only.
user | 1\. first example sentence; 2. second example sentence;
user | 1\. Negative; 2. Positive;
user | 1\. sentence to label; 2. sentence to label;
Output Messages
assistant | 1\. Negative; 2. Positive;

###  III-B Consistency-base Uncertain Sample Selection Strategy

Unlike GPT-3 completion API which is retiring, the Chat Completion API does
not have the output parameter $logprobs$ to indicate the confidence of the
model output. I propose to use a consistency-base method to measure the model
uncertainty on the output samples. Two parameters, i.e., $n$ and $temperature$
are tuned to let the API return $n$ results with variations controlled by
$temperature$. Larger $temperature$ creates more variations, and larger $n$
consumes more tokens. If there is any inconsistency among the $n$ results on
any samples, those samples will be marked as _inconsistent_ and when we
calculate the accuracy of the annotation, these samples are not counted. Over
the active learning process, the inconsistent samples will be labeled by
humans.

##  IV Experiment Setup

The first experiment set is designed to test and analyze the accuracy and the
cost of GPT-3.5 and GPT-4 to label instances. Different number of examples are
included in the demonstration and different demonstration example selection
strategies are compared. The second experiment set compares the effects of
GPT-3.5 annotator on the model performance in active learning settings.

###  IV-A Experiment Set 1: Annotation by GPT-3.5 and GPT-4

The set of experiments are conducted on 3 text classification datasets: AG’s
News [17], TREC-6 and Rotten Tomatoes Movie Reviews [18]. For each dataset,
125 samples are randomly selected in a stratified manner, i.e., keep the ratio
of the samples across categories. And a subset of stratified 25 samples over
the 125 samples are taken as the initial set, from which the demonstration
examples will be selected. We report the accuracy and cost on the other 100
samples. Because all the datasets have ground truth labels for training,
validation and test sets, we can use the ground truth to measure the accuracy.
To test the effect of the number of examples in demonstration, we tested
$N={1,2,3,4,5}$. In order to determine how I should select the demonstration
examples from the initial set, three demonstration example selection
strategies are compared: $random$, $min\\_token$, and $max\\_similarity$. When
calculating the cost in USD($), we use the pricing in Table II and the
consumed input and output token numbers in the API’s output. Due to the budget
limitation, all the experiment results are reported over 1 run, instead of
multiple runs. To measure the inconsistency, we let API return $n=3$ results,
and $temperature$ is set to 0.2. We also report the inconsistency rate which
indicates in the active learning settings, how many samples need to be labeled
by humans.

###  IV-B Experiment Set 2: Active Learning using Mixed Labels

The set of experiments are designed to compare the performance of active
learning using human annotations only and using mixed annotations from GPT-3
and humans. The same 3 datasets are used in this experiment set. The model
trained in active learning process is DistilRoBERTa [19], and 3 query
strategies are tested, i.e., $random$, $Least\ Confidence$ [5] and $Breaking\
Ties$ [7]. All the layers are finetuned from pretrained model and the model is
re-trained after each query iteration. The mean accuracy on test set and the
mean Area Under Curve (AUC) of the test accuracy curve are reported over 3
runs. The model training has an early stop policy which will stop the model
training if the model does not improve performance over a few epochs. For each
experiment, we query 10 times, and 50 samples each query, totally 500 samples.
10% of the labeled samples are used as validation set during model training.
The initial set contains 25 samples, which are randomly stratified sampled
instances. For GPT-3.5 labeler, $N$=3 for each class, and the demonstration
selection strategy is $min\\_token$.

###  IV-C Datasets

The 3 datasets used in this paper are AG’s News [17], TREC-6 and Rotten
Tomatoes Movie Reviews [18]. The AG’s News dataset is to classify the news
topics into 4 categories, i.e., World, Sports, Business, and Sci/Tech. The
dataset contains 120k samples in the training set, and 7600 in test set. The
TREC-6 dataset is to classify questions into 6 coarse labels, i.e.,
Abbreviation, Entity, Description, Human, Location, and Numeric. The training
set contains 5452 samples and the test set contains 500 samples. The Rotten
Tomatoes dataset is to classify movie reviews into negative and positive
classes. The training set contains 9596 samples and the test set contains 1066
samples. All the pre-processing procedure is aligned with [2].

##  V Results

In this section, the results of two sets of experiments are discussed and
analyzed.

###  V-A Results of Experiment Set 1

####  V-A1 Cost Efficiency

The costs of labeling 100 samples using GPT-3.5 and GPT-4 are shown in Fig 1
and Fig 2, respectively. And the costs of labeling such 100 samples by human
annotators are listed in Fig IV, assuming the same pricing listed in [3]. We
can see that the labeling cost by GPT-3 is about 1/1000 of human annotations.
And the cost of GPT-4 is about 10 times of GPT-3.5. Providing more examples in
the demonstration consumes more tokens, thus causing more costs. In addition,
we can see that $min\\_token$ demonstration selection strategy costs less than
or equal to other strategies.

Figure 1: Cost of labeling 100 samples by GPT-3.5 Figure 2: Cost of labeling 100 samples by GPT-4 TABLE IV: Cost of Labeling 100 Samples by Humans Dataset | Rotten Tomatoes | AG’s News | TREC-6
---|---|---|---
Cost ($) | 11.02 | 12.16 | 11.00

####  V-A2 Accuracy

The annotation accuracy of GPT-3.5 and GPT-4 on 100 examples are shown in Fig
3 and Fig 4, respectively. We can see that on AG’s News and TREC-6, GPT-4 has
higher accuracy, sometimes even close to 100% accurate; however, on Rotten
Tomatoes, GPT-4 has slightly lower accuracy compared to GPT-3.5, which
deserves further investigation. The improvement on TREC-6 is significant,
which indicates that on some difficult dataset, GPT-4 might have greater
potentials. In terms of the number of demonstrations for each class $N$, I do
not see a definite trend in the results. Among 3 demonstration selection
strategies, $min\\_token$ achieves highest or close to highest accuracy.

Figure 3: Accuracy of labeling 100 samples by GPT-3.5 Figure 4: Accuracy of
labeling 100 samples by GPT-4

####  V-A3 Inconsistency Rate

The inconsistency rates using GPT-3.5 and GPT-4 are shown in Fig 5 and Fig 6,
respectively. We can see that on TREC-6, GPT-3.5 has much higher inconsistency
rate than GPT-4, indicating that it requires more human annotation if GPT-3.5
is used in the mixed labeling strategy. On the other datasets, GPT-4 has
slightly lower inconsistency rate than GPT-3.5. We should note that the model
showing inconsistency is better than making $consistent$ mistakes where
multiple responses make the same mistakes. For $consistent$ mistakes, we
cannot detect them since we do not have the ground truth for them during
active learning settings. I do not see a correlation between the number of
demonstration examples $N$ and the inconsistency rate; however, the
$min\\_token$ strategy has highest or close to highest inconsistency rate,
which coincides with the findings in [4].

Figure 5: Inconsistency rate of labeling 100 samples by GPT-3.5 Figure 6:
Inconsistency rate of labeling 100 samples by GPT-4

###  V-B Results of Experiment Set 2

The test accuracy and AUC of the test accuracy curve in active learning
experiment sets are shown in Table V. As we can see that for AG’s News and
Rotten Tomatoes, the mixed labels from GPT-3.5 and humans can reach similar or
even better results compared to using human annotations only. However, on
TREC-6, the accuracy is significant lower than human annotation trained
models, which might be explained by the low accuracy of the GPT-3.5
annotations. Among 3 query strategies, Breaking Ties achieved the best
accuracy, but very close to Least Confidence, which is aligned with [2].

TABLE V: Test Accuracy and AUC in Active Learning

Dataset | Query | Test Accuracy | Test Acc AUC
---|---|---|---
Strategy | Human | Mixed | Human | Mixed
AG | BT | 0.90 | 0.88 | 0.85 | 0.84
LC | 0.89 | 0.87 | 0.82 | 0.82
R | 0.88 | 0.87 | 0.84 | 0.83
RT | BT | 0.84 | 0.81 | 0.76 | 0.77
LC | 0.83 | 0.84 | 0.76 | 0.78
R | 0.82 | 0.83 | 0.78 | 0.78
TREC-6 | BT | 0.94 | 0.74 | 0.85 | 0.69
LC | 0.94 | 0.70 | 0.81 | 0.65
R | 0.82 | 0.75 | 0.82 | 0.67

  * 1

AG: AG’s News, RT: Rotten Tomatoes Movie Review

  * 2

BT: Breaking Ties, LC: Least Confidence, R: Random

##  VI Discussion and Conclusion

In this paper, we have explored the accuracy and cost efficiency of GPT-3.5
and GPT-4 as sample annotators for active learning. Through the first set of
experiments, we have found that GPT-3.5 has lower cost and reasonable accuracy
on easy tasks; on the contrary, GPT-4 has higher cost and performs better on
harder tasks, e.g., TREC-6 question classification. With the proposed
consistency-based label fixing strategy, we have demonstrated the potential of
the mixed annotations over the human annotations. The experiments have shown
that by using GPT-3.5 annotation and fixing incorrectly labeled samples with
human annotation, the accuracy of the active learning models can be close to
or even higher than human label trained models on easy tasks, e.g., movie
review and news topic classifications. This work has offered insights for
future dataset annotation process that GPT-3.5 or GPT-4 can be the initial
annotator to reduce the cost, without losing too much accuracy.

Despite the interesting results the experiments have given, caution should be
taken when using the proposed method. We should note that some datasets might
be included in the training set of GPT-3.5 and GPT-4, therefore, they might
not perform as well as shown in this paper when inference on new datasets. The
reason why GPT-3.5 and GPT-4 behaves worse on TREC-6 deserves further
investigation. The consistently mislabeled samples are not detectable in this
work. A possible solution might be to randomly select a few samples from those
samples that we think GPT-3.5 or GPT-4 label correctly, and label them by
human efforts. However, this might cause higher costs. But anyways, this is
all about trade-off between costs and accuracy.

## References

  * [1] Z. Zhang, E. Strubell, and E. Hovy, “A survey of active learning for natural language processing,” in _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , Y. Goldberg, Z. Kozareva, and Y. Zhang, Eds. Abu Dhabi, United Arab Emirates: Association for Computational Linguistics, Dec. 2022, pp. 6166–6190. [Online]. Available: https://aclanthology.org/2022.emnlp-main.414
  * [2] C. Schröder, A. Niekler, and M. Potthast, “Revisiting uncertainty-based query strategies for active learning with transformers,” in _Findings of the Association for Computational Linguistics: ACL 2022_. Dublin, Ireland: Association for Computational Linguistics, May 2022, pp. 2194–2203. [Online]. Available: https://aclanthology.org/2022.findings-acl.172
  * [3] S. Wang, Y. Liu, Y. Xu, C. Zhu, and M. Zeng, “Want to reduce labeling cost? GPT-3 can help,” in _Findings of the Association for Computational Linguistics: EMNLP 2021_ , M.-F. Moens, X. Huang, L. Specia, and S. W.-t. Yih, Eds. Punta Cana, Dominican Republic: Association for Computational Linguistics, Nov. 2021, pp. 4195–4205. [Online]. Available: https://aclanthology.org/2021.findings-emnlp.354
  * [4] K. Margatina, T. Schick, N. Aletras, and J. Dwivedi-Yu, “Active learning principles for in-context learning with large language models,” _arXiv preprint arXiv:2305.14264_ , 2023.
  * [5] D. D. Lewis and W. A. Gale, “A sequential algorithm for training text classifiers,” in _Proceedings of the 17th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval_ , ser. SIGIR ’94. Berlin, Heidelberg: Springer-Verlag, 1994, p. 3–12.
  * [6] A. Holub, P. Perona, and M. C. Burl, “Entropy-based active learning for object recognition,” in _2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops_ , 2008, pp. 1–8.
  * [7] T. Luo, K. Kramer, S. Samson, A. Remsen, D. Goldgof, L. Hall, and T. Hopkins, “Active learning to recognize multiple types of plankton,” in _Proceedings of the 17th International Conference on Pattern Recognition, 2004. ICPR 2004._ , vol. 3, 2004, pp. 478–481 Vol.3.
  * [8] H. S. Seung, M. Opper, and H. Sompolinsky, “Query by committee,” in _Proceedings of the Fifth Annual Workshop on Computational Learning Theory_ , ser. COLT ’92. New York, NY, USA: Association for Computing Machinery, 1992, p. 287–294. [Online]. Available: https://doi.org/10.1145/130385.130417
  * [9] B. Settles, M. Craven, and S. Ray, “Multiple-instance active learning,” _Advances in neural information processing systems_ , vol. 20, 2007.
  * [10] V. Ambati, S. Vogel, and J. Carbonell, “Active learning and crowd-sourcing for machine translation,” in _Proceedings of the Seventh International Conference on Language Resources and Evaluation (LREC’10)_ , N. Calzolari, K. Choukri, B. Maegaard, J. Mariani, J. Odijk, S. Piperidis, M. Rosner, and D. Tapias, Eds. Valletta, Malta: European Language Resources Association (ELRA), May 2010. [Online]. Available: http://www.lrec-conf.org/proceedings/lrec2010/pdf/244_Paper.pdf
  * [11] Y. Zhao, H. Zhang, S. Zhou, and Z. Zhang, “Active learning approaches to enhancing neural machine translation,” in _Findings of the Association for Computational Linguistics: EMNLP 2020_ , T. Cohn, Y. He, and Y. Liu, Eds. Online: Association for Computational Linguistics, Nov. 2020, pp. 1796–1806. [Online]. Available: https://aclanthology.org/2020.findings-emnlp.162
  * [12] M. Eck, S. Vogel, and A. Waibel, “Low cost portability for statistical machine translation based on n-gram frequency and TF-IDF,” in _Proceedings of the Second International Workshop on Spoken Language Translation_ , Pittsburgh, Pennsylvania, USA, oct 24-25 2005. [Online]. Available: https://aclanthology.org/2005.iwslt-1.7
  * [13] M. Bloodgood and C. Callison-Burch, “Bucking the trend: Large-scale cost-focused active learning for statistical machine translation,” in _Proceedings of the 48th Annual Meeting of the Association for Computational Linguistics_. Uppsala, Sweden: Association for Computational Linguistics, Jul. 2010, pp. 854–864. [Online]. Available: https://aclanthology.org/P10-1088
  * [14] A. Erdmann, D. J. Wrisley, B. Allen, C. Brown, S. Cohen-Bodénès, M. Elsner, Y. Feng, B. Joseph, B. Joyeux-Prunel, and M.-C. de Marneffe, “Practical, efficient, and customizable active learning for named entity recognition in the digital humanities,” in _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , J. Burstein, C. Doran, and T. Solorio, Eds. Minneapolis, Minnesota: Association for Computational Linguistics, Jun. 2019, pp. 2223–2234. [Online]. Available: https://aclanthology.org/N19-1231
  * [15] J. Zhu, H. Wang, T. Yao, and B. K. Tsou, “Active learning with sampling by uncertainty and density for word sense disambiguation and text classification,” in _Proceedings of the 22nd International Conference on Computational Linguistics (Coling 2008)_ , D. Scott and H. Uszkoreit, Eds. Manchester, UK: Coling 2008 Organizing Committee, Aug. 2008, pp. 1137–1144. [Online]. Available: https://aclanthology.org/C08-1143
  * [16] F. Zhdanov, “Diverse mini-batch active learning,” _arXiv preprint arXiv:1901.05954_ , 2019.
  * [17] X. Zhang, J. Zhao, and Y. LeCun, “Character-level convolutional networks for text classification,” in _Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 1_ , ser. NIPS’15. Cambridge, MA, USA: MIT Press, 2015, p. 649–657.
  * [18] B. Pang and L. Lee, “Seeing stars: Exploiting class relationships for sentiment categorization with respect to rating scales,” in _Proceedings of the 43rd Annual Meeting on Association for Computational Linguistics_ , ser. ACL ’05. USA: Association for Computational Linguistics, 2005, p. 115–124. [Online]. Available: https://doi.org/10.3115/1219840.1219855
  * [19] V. Sanh, L. Debut, J. Chaumond, and T. Wolf, “Distilbert, a distilled version of bert: smaller, faster, cheaper and lighter,” _arXiv preprint arXiv:1910.01108_ , 2019.