# Magnetic impurities at quantum critical points: large-$N$ expansion and SPT
connections

Shang Liu  Department of Physics, Harvard University, Cambridge, MA 02138, USA
Hassan Shapourian  Microsoft Station Q, Santa Barbara, CA 93106, USA
Department of Physics, Harvard University, Cambridge, MA 02138, USA
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA
02139, USA  Ashvin Vishwanath  Department of Physics, Harvard University,
Cambridge, MA 02138, USA  Max A. Metlitski  Department of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA

###### Abstract

In symmetry protected topological (SPT) phases, the combination of symmetries
and a bulk gap stabilizes protected modes at surfaces or at topological
defects. Understanding the fate of these modes at a quantum critical point,
when the protecting symmetries are on the verge of being broken, is an
outstanding problem. This interplay of topology and criticality must
incorporate both the bulk dynamics of critical points, often described by
nontrivial conformal field theories, and SPT physics. Here, we study the
simplest nontrivial setting - that of a 0+1 dimensional topological mode - a
quantum spin - coupled to a 2+1D critical bulk. Using the large-$N$ technique
we solve a series of models which, as a consequence of topology, demonstrate
intermediate coupling fixed points. We compare our results to previous
numerical simulations and find good agreement. We also point out intriguing
connections to generalized Kondo problems and Sachdev-Ye-Kitaev (SYK) models.
In particular, we show that a Luttinger theorem derived for the complex SYK
models, that relates the charge density to particle-hole asymmetry, also holds
in our setting. These results should help stimulate further analytical study
of the interplay between SPT physics and quantum criticality.

###### Contents

  1. I Introduction
  2. II Impurity Models and Summary of Results
  3. III Spinor Impurity with Vector Bulk Fields
    1. III.1 Fermion correlation functions
    2. III.2 Impurity susceptibility
  4. IV Spinor Impurity with Tensor Bulk Fields
    1. IV.1 Impurity spin correlation function
    2. IV.2 Impurity susceptibility
  5. V Antisymmetric Tensor Impurity with Tensor Bulk Fields
    1. V.1 Scaling ansatz
    2. V.2 Correlator at nonzero temperature and susceptibility
  6. VI A $\mathbb{Z}_{2}$ symmetric impurity in the $O(3)$ model
    1. VI.1 Large $N$ generalization
      1. VI.1.1 Beyond perturbation theory
  7. VII Comparison with Related Results
    1. VII.1 Impurity problems at $N=3$
    2. VII.2 Sachdev-Ye-Kitaev models
  8. VIII Conclusion
  9. A Analytic Expressions for the Static Susceptibility
  10. B A Luttinger Theorem for the Impurity Problem
  11. C Perturbative calculations for the $\mathbb{Z}_{2}$ symmetric impurity

##  I Introduction

Following the classification of topological phases of non-interacting fermions
in various symmetry settings and dimensions Kitaev  _et al._ (2009); Ryu  _et
al._ (2010); Hasan and Kane (2010); Qi and Zhang (2011), which include
topological insulators and superconductors, attention has turned to symmetry
protected topological (SPT) states in interacting systems. Despite the
presence of potentially strong interactions, a key simplification of SPTs is
the existence of an energy gap and a unique ground state in the absence of
boundaries. There too, significant progress has been achieved Turner and
Vishwanath (2013); Wen (2017); Senthil (2015) in classifying and
characterizing fundamental properties of these phases. SPT phases exhibit many
interesting topological phenomena such as robust gapless modes on the surface
or on topological defects.

In recent years, the question of stability of topological phenomena to the
closing of the energy gap has begun to be explored Kestner  _et al._ (2011);
Scaffidi  _et al._ (2017); Kainaris  _et al._ (2017); Chen  _et al._ (2018);
Verresen  _et al._ (2018); Jiang  _et al._ (2018); Parker  _et al._ (2018);
Keselman  _et al._ (2018); Jones and Verresen (2019); Verresen  _et al._
(2019); Verresen (2020). Surprisingly, various situations where SPT physics
survives the closing of an energy gap have been identified, which have been
dubbed gapless SPTs. In particular, examples have been found where the surface
modes of an SPT (partially) survive right at the bulk ordering transition,
where one of the protecting symmetries is on the verge of breaking
spontaneously. This has been shown to lead to unconventional boundary
criticality Scaffidi  _et al._ (2017); Parker  _et al._ (2018); Verresen  _et
al._ (2017, 2018, 2019); Verresen (2020). Most analytical approaches so far
focus on 1+1 dimensions where we have greater analytical control of 2D
conformal field theories (CFTs) and where numerical simulations are readily
accessible. Much less is known in higher dimensions. However, one numerical
study of a 2+1D strongly interacting gapless SPT is found in Zhang and Wang
(2017) (see also Ding  _et al._ (2018); Weber  _et al._ (2018); Weber and
Wessel (2019); Zhu  _et al._ (2021)), where the transition between a 2+1D
Affleck-Lieb-Kennedy-Tasaki (AKLT) phase, a topological paramagnet, and a
symmetry-breaking phase was studied. Although the bulk criticality was
identical to the ordering transition out of a trivial paramagnet, i.e. the
$O(3)$ Wilson-Fisher critical point, boundary critical behavior distinct from
the “ordinary" universality classCardy (1996); Diehl (1997) of the classical
$O(3)$ model was observed. It, however, remains to be understood to what
extend this “non-ordinary" boundary criticality is a consequence of the bulk
AKLT physics.Weber and Wessel (2019); Metlitski (2020)

Here, we seek to analyze examples where a gapless boundary or a zero-mode
associated with a defect leaves behind a nontrivial fingerprint even after
coupling to a critical bulk. In particular, we focus on 2+1D interacting
bosonic systems. One may attempt to analyze the setup in the aforementioned
numerical work Zhang and Wang (2017); Ding  _et al._ (2018); Weber  _et al._
(2018); Weber and Wessel (2019): coupling a 1+1D gapless boundary theory
originating from the SPT edge modes to the $O(3)$ critical bulk. However, this
is a hard problem on two accounts. First, even the problem of boundary
critical phenomena in 3D classical statistical mechanics is only now being
understood in models with continuous symmetries Metlitski (2020); Parisen
Toldin (2021); Hu  _et al._ (2021), even in the absence of SPT physics.
Second, one must add quantum effects i.e. SPT physics to the problem. Here we
will consider a simpler version of the problem which is both tractable and
displays remarkably rich behavior. Instead of terminating the bulk on a 1+1D
boundary, we consider a 0+1D topological mode bound to a topological defect in
the bulk. Our analysis sheds light on the closely related question of the
screening of impurity spins at magnetic quantum critical points, and allows us
to compare our results with numerical studies on spin models.

It is helpful to have a concrete lattice model in mind. Imagine a 2+1D
spin-$1$ Heisenberg model $H=\sum_{ij}J_{ij}{\bm{S}}^{i}\cdot{\bm{S}}^{j}$
with antiferromagnetic couplings as shown in Fig. 1a. The couplings on
horizontal bonds are $J$, while the couplings on vertical bonds are
$J^{\prime}$. There are no couplings beyond the nearest-neighbor ones. Now
suppose $J^{\prime}$ is much smaller than $J$, then the system is in a 2+1D
AKLT or Haldane phase, a topological paramagnet protected by the $SO(3)$ spin
rotation and vertical lattice translation symmetries. Indeed, if we create a
vertical edge, then this edge will host a dangling spin-$1/2$ chain which can
not be gapped out by adjusting the boundary couplings as long as the spin
rotation and vertical translation symmetries are preserved. We may also choose
the protecting symmetries to be spin rotation together with the translation
generated by $T_{x}T_{y}$ which moves a lattice site to its next-nearest
neighbor in the upper right. For the same reason, an edge parallel to the
vector $(1,1)$ (compatible with $T_{x}T_{y}$) will be gapless given these
symmetries. For our purpose here, consider a dislocation defect with a $(1,1)$
Burgers vector as shown in Fig. 1b, which may be regarded as a flux of the
translation symmetry generated by $T_{x}T_{y}$. It is not hard to see that
this defect supports a single dangling spin-$1/2$, a 0+1D topological mode. We
can gradually increase the value of $J^{\prime}$ up to $J$, and at some point
there will be a transition from the topological paramagnet phase to the Néel
ordered phase. This transition is in the $O(3)$ universality class as
confirmed numerically in Ref. Matsumoto  _et al._ , 2001. Right at the
critical point, the spin-$1/2$ on the defect and the gapless bulk degrees of
freedom are strongly coupled to one another, and we inquire about their fate
in the low energy limit. This is equivalent to an impurity problem where a
spin-$1/2$ impurity is immersed into the 2+1D critical bulk. Two remarks are
in order. (i) In a physical lattice, the dislocation will carry a logarithmic
elastic energy and will locally distort the couplings far away from the defect
core. Here, we will consider an idealized model where locally the couplings
away from the defect core are not distorted and one can detect the dislocation
only by encircling it. (ii) We have chosen a special dislocation Burgers
vector such that the Néel order parameter $\vec{\phi}$ acquires no phase by
winding around the defect point. In other words, the $O(3)$ critical bulk sees
no branch cut due to the dislocation. If the Burgers vector is $(0,1)$, then
the dislocation leads to a $\mathbb{Z}_{2}$ defect line where the order
parameter $\vec{\phi}$ twists $\vec{\phi}\mapsto-\vec{\phi}$ across the branch
cut.

We expect the impurity not to be fully screened since the $SO(3)$ bulk
symmetry fractionalizes into $Spin(3)=SU(2)$ on the impurity site and the bulk
does not have the right degrees of freedom to cancel the spinor impurity. It
is then interesting to understand the dynamics of this impurity, for example
its nontrivial spin susceptibility due to interaction with the bulk degrees of
freedom. Moreover, one may ask what happens if we replace the
spin-$\frac{1}{2}$ impurity by an integer spin eg. spin-$1$; is there an even-
odd effect? In this paper, we address these questions using the large-$N$
technique and at the same time present several models with an unscreened
impurity which are analogous to the gapless SPT states studied earlier. Our
work provides a novel perspective on the interplay between topology and 2+1D
quantum criticality.

Figure 1: (a) A spin-$1$ antiferromagnetic Heisenberg model. The Heisenberg
couplings on horizontal bonds are $J$, while the couplings on vertical bonds
are $J^{\prime}$. When $J^{\prime}/J$ is sufficiently small, the system is in
a topological paramagnetic phase protected by the $SO(3)$ spin rotation
symmetry and translation symmetry. As we increase $J^{\prime}/J$ up to $1$,
there is a transition to the Néel ordered phase. (b) A dislocation defect with
a $(1,1)$ Burgers vector, which supports a spin-$1/2$ topological mode in the
paramagnetic phase.

The impurity problems of interest in this paper resemble the Kondo effect,
especially the multichannel generalizations Emery and Kivelson (1992); Affleck
(1995); Parcollet  _et al._ (1998). However, an important difference is that
the noninteracting bulk electrons are here replaced by strongly coupled
critical modes. This further implies that the gapless modes are intrinsically
2+1D and do not admit a dimensional reduction.

The problem of a spin-$S$ impurity coupled to the $O(3)$ Wilson-Fisher bulk
CFT was analyzed in groundbreaking work in Sachdev  _et al._ (1999); Vojta
_et al._ (2000), in the context of an impurity spin coupled to critical
fluctuations. Utilizing a $D=4-\epsilon$ spacetime dimension expansion, an
interacting fixed point near the decoupled one was found for small $\epsilon$,
where the impurity is not fully screened and exhibits a Curie form static
susceptibility at finite temperature. See also Sachdev (2001); Sachdev and
Vojta (2003) for alternative approaches.

In this work, we analyze three $SO(N)$ symmetric generalizations of the above
$SO(3)$ problem in the large-$N$ limit. Nontrivial fixed points in which the
impurity is not fully screened are found in all these models for the impurity
transforming both in the spinor (projective) and non-projective
representations of $SO(N)$. To make a contrast, we also consider a fourth
slightly different model where a fully-screened fixed point can be realized
for impurities transforming in non-projective representations of $SO(N)$. This
is in accord with the expectation that while an impurity transforming
projectively under $SO(N)$ can never be screened by the $O(N)$ Wilson-Fisher
critical bulk, whether a non-projective impurity is screened depends on the
details of the Hamiltonian near the impurity.

The large-$N$ approach adopted here has the general advantage of generating
one-parameter families of new models, being exactly solvable in a strongly-
coupled limit and being able to access non-perturbative fixed points (if any).
In fact, an $SU(N)$ large-$N$ analysis was also applied in Vojta  _et al._
(2000) albeit with a noninteracting bulk. We point out however that in our
calculation which includes bulk interactions, a logarithmic UV divergence
which is present in the free case is cancelled, leading to finite results for
the spin susceptibility. A few other impurity problems with interacting
critical bulk were investigated in Florens  _et al._ (2006, 2007), where the
impurity spin is not protected and a fully-screened phase appears in the phase
diagram. One impurity model to be studied below is similar to that in Florens
_et al._ (2006), except we study a model with $SO(N)$ rather than $SU(N)$
symmetry.

Finally, let us highlight an unexpected connection. Two of our impurity
problems have interesting links to the real and complex Sachdev-Ye-Kitaev
(SYK) models Kitaev (2015a, b); Maldacena and Stanford (2016); Sachdev (2015);
Gu  _et al._ (2020), respectively. In particular, in the latter case, we prove
a Luttinger theorem relating the spin representation, expressed in terms of a
parton ‘charge density’ in the UV to the particle-hole asymmetry of the IR
Green’s function, which is exactly the same as the so-called charge formula in
the complex SYK model Georges  _et al._ (2001); Sachdev (2015); Gu  _et al._
(2020). This is a nontrivial matching because the charge formula contains an
anomalous term that is not present, for example, in the multichannel $SU(N)$
Kondo model Parcollet  _et al._ (1998).

In the next section, we give more detailed motivations and explanations for
our impurity models and summarize our main results. The following four
sections examine the dynamics of these models in detail. In Section VII, we
compare with related results including different approaches to the $SO(3)$
problem mentioned above, as well as the results of large scale quantum Monte
Carlo numerics, and the SYK models. We conclude and discuss possible future
directions in Section VIII.

##  II Impurity Models and Summary of Results

Sec. | Impurity | Bulk |  Coupling at $\bm{r=0}$ |  Low-Energy Properties
---|---|---|---|---
III | Spinor ($\gamma_{0}$, $\gamma_{\alpha\geq 1}$) | $\phi_{\alpha}$ | $(\mathrm{i}\gamma_{0}\gamma_{\alpha})\phi_{\alpha}$ |  $[\gamma_{0}]=1/2$, $[\gamma_{\alpha\geq 1}]=0$, $\chi_{\rm imp}={\mathcal{C}}_{\rm cp}/T$ (cf. Eq. 34). Impurity not fully screened.
IV | Spinor ($\gamma_{0}$, $\gamma_{\alpha\geq 1}$) | $\phi_{\alpha\beta}$ | $(\mathrm{i}\gamma_{\alpha}\gamma_{\beta})\phi_{\alpha\beta}$ |  $[\gamma_{0}]=0$, $[\gamma_{\alpha\geq 1}]=1/4$, $\chi_{\rm imp}={\mathcal{C}}_{\rm cp}/T$ (cf. Eq. 52). Impurity not fully screened.
V |  Antisym. $\nu N$-Tensor ($c_{\alpha}$, $c^{\dagger}_{\alpha}$) | $\phi_{\alpha\beta}$ | $c^{\dagger}_{\alpha}c_{\beta}\phi_{\alpha\beta}$ |  $[c_{\alpha}]=1/4$, $\chi_{\rm imp}={\mathcal{C}}_{\rm cp}/T$ (cf. Fig. 7). Impurity not fully screened.
VI |  Traceless Sym. $N_{b}$-Tensor ($b_{\alpha}$, $b^{\dagger}_{\alpha}$) | $\phi_{\alpha}$ | $b^{\dagger}_{\alpha}b_{\beta}\phi_{\alpha}\phi_{\beta}+\cdots$ |  $N=3$: for $S=1$ ($N_{b}=1$), a fully screened fixed point is conjectured to be present in the phase diagram, but for $S\geq 2$ this fixed point is not found. $N\geq 4$: a fully screened fixed point is present in the phase diagram.
Table 1: Summary of the impurity models. The first column indicates the
section numbers. The second column contains the $SO(N)$ symmetry
representation of the impurity Hilbert space as well as the parton degrees of
freedom. $\gamma$, $c$ and $b$ represent Majorana fermion, canonical complex
fermion and canonical complex boson, respectively. Bulk fields are listed in
the third column, either an $O(N)$ vector $\phi_{\alpha}$ or an antisymmetric
tensor $\phi_{\alpha\beta}$. For the latter case, the bulk critical point
being considered is fine tuned to have $O(N(N-1)/2)$ symmetry as explained in
the main text. The fourth column shows the bulk-impurity coupling at
$\bm{r}=0$, where repeated indices are summed over from $1$ to $N$. The low-
energy property of these impurity problems are given in the last column, where
$[\cdot]$ denotes the scaling dimension and $\chi_{\rm imp}$ is the impurity
static susceptibility defined in (25). The scaling dimension results for the
first three models are valid in the $N\rightarrow\infty$ limit. In the last
model and when $N=3$, we actually allow the impurity to have an arbitrary spin
quantum number $S\geq 1$, as explained in the main text.

We consider a spin-$1/2$ impurity coupled to a 2+1D gapless bulk described by
the critical $O(3)$ non-linear $\sigma$-model (NL$\sigma$M). More precisely,
we write down the following Euclidean action:

$\displaystyle S$ $\displaystyle=S_{\rm b}+S_{\rm imp}+S_{\rm cp},$ (1)
$\displaystyle S_{\rm b}$
$\displaystyle=\frac{1}{2}\int\mathrm{d}^{3}x\left[\sum_{\alpha=1}^{3}(\partial_{\mu}\phi_{\alpha})^{2}+\lambda(x)\left(\sum_{\alpha=1}^{3}\phi_{\alpha}^{2}-\frac{3}{g}\right)\right],$
(2) $\displaystyle S_{\rm imp}$
$\displaystyle=\frac{1}{4}\int\mathrm{d}\tau\sum_{i=0}^{3}\gamma_{i}(\tau)\partial_{\tau}\gamma_{i}(\tau),$
(3) $\displaystyle S_{\rm cp}$
$\displaystyle=J\int\mathrm{d}\tau\sum_{\alpha=1}^{3}\mathrm{i}\gamma_{0}(\tau)\gamma_{\alpha}(\tau)\phi_{\alpha}(\tau,{\bm{r}}=0).$
(4)

Here $\lambda(x)$ is a Lagrange multiplier field enforcing the NL$\sigma$M
constraint $\phi_{\alpha}\phi_{\alpha}=\frac{3}{g}$, and $g$ is the bulk
coupling constant. We use a Majorana representation of the spin impurity;
there are four Majorana operators $\gamma_{i}\leavevmode\nobreak\ (i=0,1,2,3)$
acting on a four-dimensional extended Hilbert space and we project onto one of
the two fermion-number parity sectors, say
$P_{f}=-\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}=1$ for concreteness. The spin
operators which generate the $SO(3)$ symmetry on the impurity spin Hilbert
space are represented as
$S_{\alpha\beta}=-\frac{1}{2}\mathrm{i}\gamma_{\alpha}\gamma_{\beta}$ for
$\alpha,\beta=1,2,3$ and, specially for the $SO(3)$ case, we define
$S_{\alpha}=\frac{1}{2}\epsilon^{\alpha\beta\gamma}S_{\beta\gamma}\propto\gamma_{0}\gamma_{\alpha}$,
e.g. $S_{1}=S_{23}\propto\gamma_{0}\gamma_{1}$. It is clear that
$\gamma_{\alpha}$ and therefore $\gamma_{0}\gamma_{\alpha}$ for $\alpha=1,2,3$
transform under the vector representation of $SO(3)$. However,
$\gamma_{0}\gamma_{\alpha}$ have to transform trivially under the
$\mathbb{Z}_{2}$ inversion subgroup of the $O(3)$ symmetry since
$\prod_{\alpha=1}^{3}(\gamma_{0}\gamma_{\alpha})\propto P_{f}$ and we restrict
to a certain fermion-number parity. As a result, the coupling term $S_{\rm
cp}$ breaks the $O(3)$ symmetry down to $SO(3)$. It is important to note that
the $SO(3)$ symmetry is represented projectively on the impurity. The two
different signs of $J$ are equivalent by the field redefinition
$\phi\mapsto-\phi$ which is a symmetry of the bulk action $S_{\rm b}$.

This $SO(3)$ problem was first analyzed in $D=4-\epsilon$ spacetime dimensions
in Sachdev  _et al._ (1999); Vojta  _et al._ (2000) for a general spin-$S$
impurity. They identified an interacting fixed point near the decoupled one
for small $\epsilon$, where the impurity is not fully screened and exhibits a
Curie form static susceptibility at low temperature. In this work, we study
several $SO(N)$ generalizations of this problem in the large-$N$ limit. For
comparison, we also study a slightly different model that enjoys the
$\phi\mapsto-\phi$ symmetry of the bulk.

Let $N\in 2\mathbb{Z}+1$. The most straightforward $SO(N)$ symmetric
generalization of this impurity problem is to consider the critical $O(N)$
NL$\sigma$M in the bulk and an $SO(N)$ spinor impurity. More precisely, we
have vector fields $\phi_{\alpha}\leavevmode\nobreak\ (\alpha=1,2,\cdots,N)$
in the bulk. On the impurity site, we introduce Majorana operators
$\gamma_{0}$, $\gamma_{\alpha}\leavevmode\nobreak\ (\alpha=1,2,\cdots,N)$ and
project onto a certain fermion-number parity sector, say
$P_{f}=(-\mathrm{i})^{(N+1)/2}\gamma_{0}\gamma_{1}\cdots\gamma_{N}=1$. The
most relevant coupling between the impurity and the bulk is proportional to
$\mathrm{i}\gamma_{0}\gamma_{\alpha}\phi_{\alpha}(\bm{r}=0)$. It is obvious
that this new model reduces to the original one at $N=3$.

The above generalization, though simple, has one disadvantage: it is hard to
include other types of impurity. For example, when $N=3$, we may instead
consider a spin-$1$ impurity, but it does not have a direct generalization
with bulk being the $O(N)$ NL$\sigma$M. Indeed, in order to couple to the
vector fields $\phi_{\alpha}$, one needs impurity _operators_ which transform
as an $SO(N)$ vector, but there is no such operator when the impurity _Hilbert
space_ is in the vector representation. There is an alternative large-$N$
model where we introduce antisymmetric matrix fields in the bulk which
transform in the $SO(N)$ adjoint representation, namely
$\phi_{\alpha\beta}\leavevmode\nobreak\ (\alpha,\beta=1,2,\cdots,N)$ and
$\phi^{T}=-\phi$. There are now $N(N-1)/2$ independent bulk $\phi$ fields and
we require fine-tuning to the critical point with an emergent $O(N(N-1)/2)$
symmetry. On the impurity site, we may have the same $SO(N)$ spinor
representation and the Majorana operators $\gamma_{0},\gamma_{\alpha}$
subjected to a fermion-number parity projection. The most relevant bulk-
impurity coupling is
$\mathrm{i}\gamma_{\alpha}\gamma_{\beta}\phi_{\alpha\beta}(\bm{r}=0)$. It is
straightforward to check that this new large-$N$ problem also reduces to the
original one at $N=3$ since vector and adjoint representations are equivalent
for $SO(3)$. This new bulk theory has the disadvantage of extra fine-tuning;
one at least needs to turn off the single-trace term
$\operatorname{Tr}(\phi^{4})$ in the bulk. However, it enables us to consider
impurities in different $SO(N)$ representations as motivated earlier. Indeed,
$\phi_{\alpha\beta}$ can directly couple to the $so(N)$ generators which exist
in any representation.

With these motivations, in this paper, we consider the following three
large-$N$ generalizations of the original $SO(3)$ problem:

  * 1.

An $SO(N)$ spinor impurity coupled to vector bulk fields $\phi_{\alpha}$.

  * 2A.

An $SO(N)$ spinor impurity coupled to adjoint bulk fields
$\phi_{\alpha\beta}$.

  * 2B.

A totally antisymmetric $\nu N$-tensor impurity coupled to adjoint bulk
fields.

For all these three models, we find nontrivial fixed points where the impurity
is not fully screened by the bulk and these can be explained by symmetry
fractionalization arguments. To make a contrast with these intermediate-
coupling fixed points, we also introduce the fourth model:

  * 3.

An impurity in the $O(N)$ model respecting the $\phi\mapsto-\phi$ symmetry.

Interestingly, this model exhibits very different behaviors for $N=3$ and
$N\geq 4$. For $N=3$, we allow the impurity to be in an arbitrary spin-$S$
representation with $S\geq 1$ and find that the low-energy physics has a
sensitive dependence on $S$. More precisely, when $S=1$, there is an IR stable
fully screened fixed point, but when $S\geq 2$ (including integer $S$), the
fully screened fixed point becomes inaccessible in our analytic approach,
instead, we find a runaway flow to some yet unknown strong coupling fixed
point. For $N\geq 4$, the impurity is taken to transform in the traceless
symmetric tensor representation of $SO(N)$ with $N_{b}$ indices. Here we find
a fully screened fixed point for all values of $N_{b}$.

The implications of our results for $N=3$ are the following. When a spin-$1/2$
impurity is coupled to the $O(3)$ critical bulk, our findings for Models 1 and
2A imply the impurity to be not fully screened. If we instead consider a
spin-$1$ impurity, we conclude from Models 2B and 3 that there exist both a
stable not fully screened fixed point and a fully screened fixed point.

In Section VII.1, we compare our findings with $\epsilon$-expansion and QMC
results on impurity problems at $N=3$. In Section VII.2, we show that Model 2A
and 2B are closely connected with the real and complex SYK models,
respectively. We give a brief summary of our main results in Table 1.

##  III Spinor Impurity with Vector Bulk Fields

In this section, we consider an $SO(N)$ spinor impurity coupled to the
critical $O(N)$ NL$\sigma$M, namely the following Euclidean action.

$\displaystyle S$ $\displaystyle=S_{\rm b}+S_{\rm imp}+S_{\rm cp},$ (5)
$\displaystyle S_{\rm b}$
$\displaystyle=\frac{1}{2}\int\mathrm{d}^{3}x\left[\sum_{\alpha=1}^{N}(\partial_{\mu}\phi_{\alpha})^{2}+\lambda(x)\left(\sum_{\alpha=1}^{N}\phi_{\alpha}^{2}-\frac{N}{g}\right)\right],$
(6) $\displaystyle S_{\rm imp}$
$\displaystyle=\frac{1}{4}\int\mathrm{d}\tau\sum_{i=0}^{N}\gamma_{i}(\tau)\partial_{\tau}\gamma_{i}(\tau),$
(7) $\displaystyle S_{\rm cp}$
$\displaystyle=J\int\mathrm{d}\tau\sum_{\alpha=1}^{N}\mathrm{i}\gamma_{0}(\tau)\gamma_{\alpha}(\tau)\phi_{\alpha}(\tau,{\bm{r}}=0),$
(8)

where $N$ is assumed to be odd. We use a Majorana representation of the
impurity where a fermion-number parity projection is implied. This projection
may seem troubling, but fortunately, it is in fact unnecessary for our
purpose. There is a $\mathbb{Z}_{2}$ unitary transformation, $\gamma_{0}$
together with the $\phi\mapsto-\phi$ bulk symmetry operation, which flips the
fermion-number parity but commutes with the Hamiltonian. As a consequence, the
correlation function of any set of operators which have even total charge
under this $\mathbb{Z}_{2}$ transformation takes exactly the same value in the
two fermion-number parity sectors and there is no need to apply the
projection; examples include the two-point functions of the spin operator
$S_{\alpha\beta}\propto\gamma_{\alpha}\gamma_{\beta}$ or the $SO(N)$ vector
operator $\mathrm{i}\gamma_{0}\gamma_{\alpha}$. We will therefore consider the
extended problem where the fermion operators are physical since for the
quantities of interest, the projection to the fixed fermion-number parity
sector makes no difference. Notice that the sign of $J$ in $S_{\rm cp}$ is
irrelevant since it can be flipped by the $\phi\mapsto-\phi$ bulk symmetry.
The large-$N$ limit of the bulk action $S_{\rm b}$ is well-known Moshe and
Zinn-Justin (2003). In particular, at zero temperature and when $g$ is
slightly above the critical value $g_{c}$, the two-point correlation function
of $\phi$ is given by

$\displaystyle g_{\phi}(x)$ $\displaystyle\equiv\left\langle
T\phi_{\alpha}(x)\phi_{\alpha}(0)\right\rangle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (\text{$\alpha$ arbitrary})$
$\displaystyle=\int\frac{\mathrm{d}\omega\mathrm{d}^{2}p}{(2\pi)^{3}}\frac{1}{\omega^{2}+p^{2}+m^{2}}\mathrm{e}^{-\mathrm{i}\omega\tau+\mathrm{i}\bm{p}\cdot\bm{r}},$
(9)

where $m=\xi^{-1}\sim g-g_{c}$ is the inverse correlation length. For later
convenience, let us introduce some related notation. We define
$g^{\lambda}_{\phi}(x,x^{\prime})$ as a generalized two-point function whose
matrix inverse is

$\displaystyle(g^{\lambda}_{\phi})^{-1}(x,x^{\prime})=-\partial^{2}\delta(x-x^{\prime})+\lambda(x)\delta(x-x^{\prime}).$
(10)

In the special case $x=(\tau,0)$ and $x^{\prime}=(\tau^{\prime},0)$, we denote
$g_{\phi}(x-x^{\prime})$ and $g^{\lambda}_{\phi}(x,x^{\prime})$ by
$g_{\phi,0}(\tau-\tau^{\prime})$ and
$g^{\lambda}_{\phi,0}(\tau,\tau^{\prime})$, respectively.

Now we apply standard path integral manipulations to the action $S$, similar
to the ones described in Parcollet  _et al._ (1998). First we integrate out
all $\phi$ fields in the bulk, leading to a nonlocal four-fermion interaction
on the impurity. Next, we decouple this interaction into
$\gamma_{0}\gamma_{0}$ and $\gamma_{\alpha}\gamma_{\alpha}$ channels using a
Hubbard-Stratonovich transformation. This process introduces bosonic bilocal
fields $Q(\tau,\tau^{\prime})$ and $\bar{Q}(\tau,\tau^{\prime})$ that are
antisymmetric by definition. Finally, we integrate out all fermion fields. The
impurity part of the final action reads

$\displaystyle S^{\prime}$
$\displaystyle=\frac{1}{2J^{2}}\int\mathrm{d}\tau\mathrm{d}\tau^{\prime}\bar{Q}(\tau,\tau^{\prime})\left[g_{\phi,0}^{\lambda}(\tau,\tau^{\prime})\right]^{-1}Q(\tau,\tau^{\prime})$
$\displaystyle-\ln\operatorname{Pf}\left[\frac{1}{2}\partial_{\tau}\delta(\tau-\tau^{\prime})-Q(\tau,\tau^{\prime})\right]$
$\displaystyle-N\ln\operatorname{Pf}\left[\frac{1}{2}\partial_{\tau}\delta(\tau-\tau^{\prime})-\bar{Q}(\tau,\tau^{\prime})\right].$
(11)

Here, $\left[g^{\lambda}_{\phi,0}(\tau,\tau^{\prime})\right]^{-1}$ is the
ordinary number inverse instead of the matrix inverse. There is also a bulk
part of the action that we did not show and it has an overall $N$ factor. If
we choose $J=J_{0}/\sqrt{N}$, then the first term contains a factor of $N$
which suppresses the fluctuations of $Q$ and $\bar{Q}$ in the large-$N$ limit
and with a proper choice of the path integral contours. We thus expect the
existence of a controlled large-$N$ limit where $Q$, $\bar{Q}$ and $\lambda$
all become classical and their expectation values can be solved from the
saddle-point equations. We assume the time translation symmetry. Let
$G_{0}(\tau)=\left\langle T\gamma_{0}(\tau)\gamma_{0}(0)\right\rangle$ and
$G(\tau)=\left\langle T\gamma_{\alpha}(\tau)\gamma_{\alpha}(0)\right\rangle$
for an arbitrary $\alpha=1,\cdots,N$ be the Majorana Green’s functions. As
$N\rightarrow\infty$, we have

$\displaystyle G^{-1}_{0}(\tau,\tau^{\prime})$
$\displaystyle=\frac{1}{2}\partial_{\tau}\delta(\tau-\tau^{\prime})-Q(\tau,\tau^{\prime}),$
(12) $\displaystyle G^{-1}(\tau,\tau^{\prime})$
$\displaystyle=\frac{1}{2}\partial_{\tau}\delta(\tau-\tau^{\prime})-\bar{Q}(\tau,\tau^{\prime}),$
(13)

where $G^{-1}_{0}(\tau,\tau^{\prime})$ is the matrix inverse of
$G_{0}(\tau,\tau^{\prime})\equiv G_{0}(\tau-\tau^{\prime})$, and similarly for
$G^{-1}$. Anticipating that $G_{0}$ and $G$ are of order $1$, we have the
following saddle-point equations for $Q$ and $\bar{Q}$ at infinite $N$:

$\displaystyle\bar{Q}(\tau,\tau^{\prime})$ $\displaystyle=0,$ (14)
$\displaystyle Q(\tau,\tau^{\prime})$
$\displaystyle=J_{0}^{2}g^{\lambda}_{\phi,0}(\tau,\tau^{\prime})G(\tau-\tau^{\prime}),$
(15)

where we used

$\displaystyle\frac{\delta\ln\operatorname{Pf}G^{-1}}{\delta\bar{Q}(\tau,\tau^{\prime})}=\frac{1}{2}\operatorname{Tr}\left(G\frac{\delta
G^{-1}}{\delta\bar{Q}(\tau,\tau^{\prime})}\right)=G(\tau-\tau^{\prime}),$ (16)

keeping in mind that
$\bar{Q}(\tau,\tau^{\prime})=-\bar{Q}(\tau^{\prime},\tau)$ are not independent
from each other. These equations imply that $Q$ is of order $1$ while
$\bar{Q}$ is of order $1/N$. As a consequence, we see from the action
$S^{\prime}$ that the impurity has no order-$1$ correction to the bulk
$\lambda(x)$; we can simply set $\lambda$ to its uniform bulk value $m^{2}$
and replace $g^{\lambda}_{\phi,0}$ by $g_{\phi,0}$.

###  III.1 Fermion correlation functions

The fermion Green’s functions can be easily solved from the self-consistent
equations. We focus on the bulk critical point $g=g_{c}$. First consider the
simplest situation $T=0$ where we set $m=0$. $\bar{Q}=0$ implies that
$G(\tau)=\operatorname{sgn}(\tau)$ is free. Using
$g_{\phi,0}(\tau)=1/(4\pi|\tau|)$, we obtain the solution for
$Q(\tau-\tau^{\prime})\equiv Q(\tau,\tau^{\prime})$:

$\displaystyle Q(\tau)=\frac{J_{0}^{2}}{4\pi\tau}\quad{\rm or}\quad
Q(\mathrm{i}\omega)=\frac{1}{4}\mathrm{i}J_{0}^{2}\operatorname{sgn}(\omega).$
(17)

The solution for $G_{0}$ is given by

$\displaystyle
G_{0}(\mathrm{i}\omega)^{-1}=-\frac{1}{2}\mathrm{i}\omega-Q(\mathrm{i}\omega).$
(18)

We are mostly interested in the long-time or small-frequency regime, namely
$\tau^{-1}\ll\Lambda_{\rm UV}$ where $\Lambda_{\rm UV}$ is the UV energy scale
including both $J_{0}^{2}$ and the bulk high-energy cutoff. Thus we may omit
the $\mathrm{i}\omega/2$ term on the right-hand side of the above equation and
get

$\displaystyle G_{0}(\tau)=\frac{4}{\pi
J_{0}^{2}\tau}\quad(T=0,\leavevmode\nobreak\ \tau^{-1}\ll\Lambda_{\rm UV}).$
(19)

We have found that at the new fixed point,
$\gamma_{\alpha}\leavevmode\nobreak\ (\alpha=1,\cdots,N)$ remains free (zero
scaling dimension), while $\gamma_{0}$ acquires a scaling dimension $1/2$. One
can easily compute the leading-order correlation functions of more fermions
using Wick’s theorem. In particular, the $SO(N)$ vector operators
$\mathrm{i}\gamma_{0}\gamma_{\alpha}$ which directly couple to the
$\phi$-fields in the bulk have a scaling dimension $1/2$, same as the leading-
order dimension of $\phi_{\alpha}$.

Next, consider nonzero temperature with $T,\tau^{-1}\ll\Lambda_{\rm UV}$. When
$T>0$, a nonzero mass $m=\mu T$ is generated for the bulk $\phi$ bosons with
$\mu=2\ln\left[(\sqrt{5}+1)/2\right]\approx 0.96$. Using

$\displaystyle g_{\phi,0}(\tau)$
$\displaystyle=\frac{1}{\beta}\sum_{\omega_{n}}\int\frac{\mathrm{d}^{2}p}{(2\pi)^{2}}\frac{1}{\omega^{2}_{n}+p^{2}+m^{2}}\mathrm{e}^{-\mathrm{i}\omega_{n}\tau}$
(20)
$\displaystyle=\int\frac{\mathrm{d}^{2}p}{(2\pi)^{2}}\frac{\cosh\left[\left(\frac{1}{2}\beta-|\tau|\right)\sqrt{p^{2}+m^{2}}\right]}{2\sqrt{p^{2}+m^{2}}\sinh\left(\frac{1}{2}\beta\sqrt{p^{2}+m^{2}}\right)},$
(21)

we found in the small frequency regime,

$\displaystyle
G_{0}(\mathrm{i}\omega_{n})=\frac{1}{J_{0}^{2}}\Phi_{0}(\mathrm{i}\tilde{\omega}_{n}),$
(22)

where

$\displaystyle\tilde{\omega}_{n}\equiv\omega_{n}/T,\quad\Phi_{0}(\mathrm{i}\tilde{\omega}_{n})=-\frac{1}{\Psi(\mathrm{i}\tilde{\omega}_{n})},$
(23)

and

$\displaystyle\Psi(\mathrm{i}\tilde{\omega}_{n})=\int_{0}^{\infty}\frac{\mathrm{d}x}{4\pi}\leavevmode\nobreak\
\frac{\mathrm{i}\tilde{\omega}_{n}\coth\left(\frac{1}{2}\sqrt{x+\mu^{2}}\right)}{\sqrt{x+\mu^{2}}\left(\tilde{\omega}_{n}^{2}+x+\mu^{2}\right)}.$
(24)

Note that the dimensionless function $\Phi_{0}$ does not dependent on $J_{0}$
or the temperature $T$; one can also write in the time domain
$G_{0}(\tau)=(T/J_{0}^{2})\Phi_{0}(\tau T)$ with
$\Phi_{0}(\mathrm{i}\tilde{\omega}_{n})\equiv\int_{0}^{1}\mathrm{d}(\tau
T)\mathrm{e}^{\mathrm{i}\omega_{n}\tau}\Phi_{0}(\tau T)$. As
$n\rightarrow\infty$, $\Phi_{0}(\mathrm{i}\tilde{\omega}_{n})$ approaches its
asymptotic form $4\mathrm{i}\operatorname{sgn}(\tilde{\omega}_{n})$ expected
from the zero-temperature Green’s function. $\gamma_{\alpha}$ with $\alpha\geq
1$ are still free with $G(\tau)=\operatorname{sgn}(\tau)$ or
$G(\mathrm{i}\omega_{n})=2\mathrm{i}/\omega_{n}$.

The equality between the scaling dimension of the $SO(N)$ vectors
$\mathrm{i}\gamma_{0}\gamma_{\alpha}$ and that of the $\phi$ fields in the
bulk seems to imply that the impurity completely merges into the bulk.
However, this is not the case. One simple evidence is that the $so(N)$
generators $\mathrm{i}\gamma_{\alpha}\gamma_{\beta}\leavevmode\nobreak\
(\alpha,\beta\geq 1)$ on the impurity site have zero scaling dimension. In the
following, we will compute a physical response of the impurity showing that it
is not fully screened by the bulk.

###  III.2 Impurity susceptibility

The static susceptibility $\chi$ of the system to a uniform magnetic field in
general takes the form:

$\displaystyle\chi=\mathcal{A}\chi_{\rm b}+\chi_{\rm imp},$ (25)

where $\mathcal{A}$ is the spatial area. As the temperature $T$ goes to zero,
the bulk susceptibility $\chi_{\rm b}$ goes to zero linearly in $T$ Chubukov
and Sachdev (1993). However, as we will show below, the impurity
susceptibility $\chi_{\rm imp}$ actually diverges as $T^{-1}$, which means the
impurity spin does not merge into the bulk.

Let us first determine how to couple a slowly-varying magnetic field to the
system. In the $SO(3)$ case, the magnetic field has three components
$H_{\alpha}(x)$ with $\alpha=1,2,3$ and microscopically couples to each spin
degree of freedom $\mathcal{S}_{\alpha}$, either in the bulk or at the
impurity site, as $-\sum_{\alpha}H_{\alpha}(x)\mathcal{S}_{\alpha}$. For the
impurity site, we just replace $\mathcal{S}_{\alpha}$ by $S_{\alpha}$, and for
the bulk spins, the correct modification of the action turns out to be the
replacement Sachdev (2011)

$\displaystyle\partial_{\tau}\phi_{\alpha}\mapsto\partial_{\tau}\phi_{\alpha}-\mathrm{i}(\bm{H}\times\bm{\phi})_{\alpha}.$
(26)

In the $SO(N)$ case, we generalize the magnetic field to an antisymmetric
matrix $H_{\alpha\beta}$ with $\alpha,\beta=1,\cdots,N$ which has $N(N-1)/2$
independent components and reduces to a vector field in the $SO(3)$ case by
$H_{\alpha}=\frac{1}{2}\epsilon^{\alpha\beta\gamma}H_{\beta\gamma}$. The
coupling of $H_{\alpha\beta}$ to the impurity site is

$\displaystyle\Delta\mathcal{L}_{\rm
imp}=\frac{1}{2}\sum_{\alpha<\beta}H_{\alpha\beta}(\mathrm{i}\gamma_{\alpha}\gamma_{\beta}),$
(27)

and the coupling to the bulk is

$\displaystyle\partial_{\tau}\phi_{\alpha}\mapsto\partial_{\tau}\phi_{\alpha}+\mathrm{i}H_{\alpha\beta}\phi_{\beta},$
(28)

both of which reduce to the correct form at $N=3$.

To compute the susceptibility $\chi$, we take $H_{12}(x)=-H_{21}(x)=h$ to be a
constant and turn off other components of $H_{\alpha\beta}$. Then $\chi$ is
given by

$\displaystyle\chi=T\left.\frac{\partial^{2}\ln Z}{\partial
h^{2}}\right|_{h=0}=T\frac{1}{Z}\left.\frac{\partial^{2}Z}{\partial
h^{2}}\right|_{h=0},$ (29)

where we used $(\partial Z/\partial h)|_{h=0}=0$ since the $SO(N)$ symmetry
can reverse the sign of $h$. The additional terms in the Lagrangian now take
the form

$\displaystyle\Delta\mathcal{L}_{\rm
b}=2\mathrm{i}h(\partial_{\tau}\phi_{1})\phi_{2}-\frac{1}{2}h^{2}\left(\phi^{2}_{1}+\phi^{2}_{2}\right),$
$\displaystyle\Delta\mathcal{L}_{\rm
imp}=\frac{1}{2}h(\mathrm{i}\gamma_{1}\gamma_{2}).$ (30)

First consider the decoupled system where $J_{0}=0$. It is straightforward to
show that $\chi_{\rm imp}=1/(4T)\equiv\mathcal{C}_{\rm free}/T$ for all $N$.
This coincides with the well-known result $S(S+1)/(3T)$ for a free $SO(3)$
spin with $S=1/2$. Next consider the much more complicated coupled system.
Depending on whether $\partial/\partial h$ acts on $\Delta\mathcal{L}_{\rm b}$
or $\Delta\mathcal{L}_{\rm imp}$, the impurity susceptibility naturally
separate into three terms:

$\displaystyle\chi_{\rm imp}=\chi_{\rm b,b}+2\chi_{\rm b,imp}+\chi_{\rm
imp,imp}.$ (31)

These three terms can be computed order-by-order in $1/N$. At order-$1$, we
found that only $\chi_{\rm imp,imp}$ contributes:

$\displaystyle\chi_{\rm
imp,imp}^{(0)}=\frac{1}{4\beta}\sum_{\omega}G(\mathrm{i}\omega_{n})G(-\mathrm{i}\omega_{n})=\frac{1}{4T}.$
(32)

This is the same as a free spin, a not surprising result since we have found
that $\gamma_{\alpha}$ with $\alpha\geq 1$ are free. We have thus shown that
the impurity is not screened by the bulk; it is in fact free to the leading
order. We can go to the next order to reveal a nontrivial interaction effect.

Figure 2: (a) Feynman diagrams at order $1$ and $1/N$ for the impurity
susceptibility $\chi_{\rm imp}$. Single circles are bulk source terms of the
form $(\partial_{\tau}\phi)\phi$, double circles are bulk source terms of the
form $\phi^{2}$ which couples to $h^{2}$ and black solid dots are impurity
source terms of the form $\gamma\gamma$ (cf. Eq. (30)). The solid lines are
propagators for $\gamma_{\alpha}$ with $\alpha\geq 1$ and the dashed lines are
$\gamma_{0}$ propagators. $\phi$ boson propagators are represented by double
lines (solid and dashed). (b) The shaded bubble which appears in the first
panel. The wavy line represents the propagator of the Lagrange multiplier
field $\lambda$ and is of order $1/N$.

Fig. 2a presents all the diagrams of order $1$ and $1/N$ for $\chi_{\rm imp}$,
where the shaded bubble is explained in panel b. Computing all the diagrams in
Fig. 2, we found that

$\displaystyle\chi_{\rm imp}=\frac{\mathcal{C}_{\rm cp}}{T}.$ (33)

There is no anomalous dimension in $T$, but the Curie coefficient is
renormalized, coinciding with the qualitative behavior obtained from
$\epsilon$-expansion in Vojta  _et al._ (2000). Let us avoid going into the
full details of this calculation, but only note that there are remarkable
divergence cancellations. One cancellation is between $\chi_{\rm imp,imp}$ and
$2\chi_{\rm b,imp}$, both of which are logarithmically divergent. The other
happens in the computation of the shaded bubble as depicted in Fig. 2b and
appearing in $\chi_{\rm b,b}$. There are two terms contributing to this shaded
bubble, one is the fermion bubble and the other is the $1/N$ correction to the
$\lambda$ expectation value. Both of them contain logarithmic UV divergences,
but their sum is finite. The interaction effect of the bulk theory, more
precisely the $\lambda\phi^{2}$ vertex, plays an important role for this
latter cancellation. In Vojta  _et al._ (2000), a similar fermion bubble
divergence showed up in the $SU(N)$ large-$N$ approach but since the bulk
there was assumed to be free, this divergence remained un-cancelled and would
result in a log-divergence of the susceptibility. In fact, the model in Ref.
Vojta  _et al._ (2000) is more closely related to the case of tensor bulk
fields to be discussed later. The analytic expressions for $\mathcal{C}_{\rm
cp}$ can be found in Appendix A. Numerically we obtained

$\displaystyle\mathcal{C}_{\rm cp}=\left[1+\frac{1}{N}\left(-0.4187\pm 9\times
10^{-5}\right)\right]\mathcal{C}_{\rm free},$ (34)

where $\mathcal{C}_{\rm free}=1/4$ has been defined above. If one plugs in
$N=3$, this gives $\mathcal{C}_{\rm cp}=0.8604\leavevmode\nobreak\
\mathcal{C}_{\rm free}$. Numerically, infinite frequency sums are truncated by
some maximal frequency $\tilde{\omega}_{\rm max}$. The error above comes from
an extrapolation to $\tilde{\omega}_{\rm max}=\infty$.

The impurity being not fully screened is a consequence of symmetry
fractionalization. The $SO(N)$ symmetry in the bulk fractionalizes into a
$Spin(N)$ symmetry on the impurity site, therefore the bulk does not have the
right degrees of freedom to fully screen the impurity. More explicitly, the
counterclockwise spin rotation by $\theta$ on the $\alpha\beta$-plane is
represented on the impurity Hilbert space as
$R_{\alpha\beta}(\theta)=\exp(-\mathrm{i}\theta
S_{\alpha\beta})=\cos(\theta/2)-\sin(\theta/2)\gamma_{\alpha}\gamma_{\beta}$,
where we used
$S_{\alpha\beta}=-\frac{1}{2}\mathrm{i}\gamma_{\alpha}\gamma_{\beta}$. We see
that the $2\pi$ rotation (on an arbitrary plane) acts as $-1$ on the impurity.
On the contrary, a $2\pi$ rotation is equivalent to the identity in the
$SO(N)$ group, for example it acts trivially on the bulk fields
$\phi_{\alpha}$. Hence the impurity spins furnish a projective representation
of $SO(N)$.

##  IV Spinor Impurity with Tensor Bulk Fields

In this section, we consider an alternative generalization of the $N=3$
problem described earlier in Section II. We demand the bulk scalar fields to
transform under the _adjoint_ representation of $SO(N)$, instead of the vector
representation. Note that these two are equivalent in the special case $N=3$.
We therefore introduce an antisymmetric matrix field
$\phi_{\alpha\beta}\leavevmode\nobreak\ (\alpha,\beta=1,\cdots,N)$ which
satisfies $\phi_{\alpha\beta}=-\phi_{\beta\alpha}$ and thus has $N_{\rm
b}\equiv N(N-1)/2$ number of independent components. The bulk action is given
by

$\displaystyle S_{\rm b}$
$\displaystyle=\frac{1}{2}\int\mathrm{d}^{3}x\left[\sum_{\alpha<\beta}(\partial_{\mu}\phi_{\alpha\beta})^{2}+\lambda(x)\left(\sum_{\alpha<\beta}\phi_{\alpha\beta}^{2}-\frac{N_{\rm
b}}{g}\right)\right],$ (35)

which has an emergent $O(N(N-1)/2)$ symmetry containing the original $O(N)$ as
a subgroup. The large-$N$ limit of this action has been explained in Section
III and we will be using the same notations. As before, we take $N$ to be odd.

For the spin impurity, we again introduce $N+1$ Majorana operators
$\gamma_{i}\leavevmode\nobreak\ (i=0,\dots,N)$ acting on a
$2^{(N+1)/2}$-dimensional extended Hilbert space and then project onto one of
the two fermion-number parity sectors, say
$P_{f}=(-\mathrm{i})^{(N+1)/2}\gamma_{0}\gamma_{1}\cdots\gamma_{N}=1$. The
impurity and the coupling terms in the action take the form

$\displaystyle S_{\rm imp}+S_{\rm cp}$
$\displaystyle=\frac{1}{4}\int\mathrm{d}\tau\sum_{i=0}^{N}\gamma_{i}\partial_{\tau}\gamma_{i}$
$\displaystyle+J\int\mathrm{d}\tau\sum_{\alpha<\beta}(\mathrm{i}\gamma_{\alpha}\gamma_{\beta})\phi_{\alpha\beta}(\tau,{\bm{r}}=0),$
(36)

where we have ignored $\gamma_{0}$ which completely decouples from other
fields. The sign of $J$ is irrelevant as before due to the $\phi\mapsto-\phi$
bulk symmetry. Similar to Section III, there is no need to actually worry
about the fermion-number parity projection: the operator $\gamma_{0}$ serves
as a unitary transformation that flips the fermion-number parity but commutes
with the Hamiltonian, which guarantees all correlation functions that we will
be interested in take the same value in the two fermion-number parity sectors.
We will therefore again consider the extended problem where the fermion
operators are physical.

Now we analyze the large-$N$ limit of the coupled system. Integrating out all
the $\phi$ fields in the bulk, then decoupling the resulting four-fermion
interaction term into $\gamma_{\alpha}\gamma_{\alpha}$ channels by an
antisymmetric bilocal field $Q(\tau,\tau^{\prime})$, and finally integrating
out all fermion fields, we obtain the following impurity part of the action:

$\displaystyle S^{\prime}$
$\displaystyle=-N\ln\operatorname{Pf}\left[\frac{1}{2}\partial_{\tau}\delta(\tau-\tau^{\prime})-Q(\tau,\tau^{\prime})\right]$
$\displaystyle+\frac{1}{4J^{2}}\int\mathrm{d}\tau\mathrm{d}\tau^{\prime}Q^{2}(\tau,\tau^{\prime})(g^{\lambda}_{\phi,0}(\tau,\tau^{\prime}))^{-1}.$
(37)

We now choose $J^{2}$ to scale as $J^{2}_{0}/N$ and find that $S^{\prime}$
contains an overall factor of $N$. Moreover, the bulk effective action that we
did not show explicitly has an $N_{\rm b}$ factor, thus as
$N\rightarrow\infty$, both $\lambda(x)$ and $Q(\tau,\tau^{\prime})$ become
classical and their expectation values can be obtained from the saddle-point
equations. Since $N_{\rm b}$ is of order $N^{2}$, the classical solution of
$\lambda$ is the same as that in the absence of impurity up to order $1/N$
corrections. As a result, the classical solution for $Q(\tau,\tau^{\prime})$
can be obtained by setting $\lambda(x)=m^{2}$. Let
$G_{\gamma}(\tau)=\left\langle
T\gamma_{\alpha}(\tau)\gamma_{\alpha}(0)\right\rangle$ with an arbitrary
$\alpha=1,\cdots,N$ be the Majorana Green’s function. In the large-$N$ limit
where $Q(\tau,\tau^{\prime})$ is pinned to its saddle-point solution, we have

$\displaystyle
G^{-1}_{\gamma}(\tau,\tau^{\prime})=\frac{1}{2}\partial_{\tau}\delta(\tau-\tau^{\prime})-Q(\tau,\tau^{\prime}).$
(38)

The saddle-point equation for $Q(\tau,\tau^{\prime})$ as $N\rightarrow\infty$
is

$\displaystyle
Q(\tau,\tau^{\prime})=J^{2}_{0}G_{\gamma}(\tau-\tau^{\prime})g_{\phi,0}(\tau-\tau^{\prime}),$
(39)

where we have used

$\displaystyle\frac{\delta\ln\operatorname{Pf}G^{-1}_{\gamma}}{\delta
Q(\tau,\tau^{\prime})}=\frac{1}{2}\operatorname{Tr}\left(G_{\gamma}\frac{\delta
G^{-1}_{\gamma}}{\delta
Q(\tau,\tau^{\prime})}\right)=G_{\gamma}(\tau-\tau^{\prime}).$ (40)

###  IV.1 Impurity spin correlation function

The impurity spin Green’s function $G_{S}(\tau)=\left\langle
TS_{\alpha\beta}(\tau)S_{\alpha\beta}(0)\right\rangle$ (no sum over $\alpha$,
$\beta$) with the definition
$S_{\alpha\beta}=-\frac{1}{2}\mathrm{i}\gamma_{\alpha}\gamma_{\beta}$ can be
computed as $G_{S}(\tau)=\frac{1}{4}G^{2}_{\gamma}(\tau)$ for $N\to\infty$.
Our task now is to solve the self-consistent equations (38) and (39). We focus
on the bulk critical point $g=g_{c}$. First consider the situation $T=0$ and
$\tau^{-1}\ll\Lambda_{\rm UV}$. Dropping the first term on the right hand side
of Eq. (38) and using $g_{\phi,0}(\tau)=1/(4\pi|\tau|)$, the self-consistent
equations are solved with the ansatz

$\displaystyle
G_{\gamma}(\tau)=\frac{A_{\gamma}}{|\tau|^{2\Delta}}\operatorname{sgn}(\tau)$
$\displaystyle\Rightarrow\leavevmode\nobreak\ $ $\displaystyle
G_{\gamma}(\mathrm{i}\omega)=\int_{-\infty}^{\infty}\mathrm{d}\tau\mathrm{e}^{\mathrm{i}\omega\tau}G_{\gamma}(\tau)$
$\displaystyle=2\mathrm{i}A_{\gamma}\cos(\pi\Delta)\Gamma(1-2\Delta)|\omega|^{2\Delta-1}\operatorname{sgn}(\omega).$
(41)

when $\Delta=1/4$ and $A_{\gamma}^{2}=1/J_{0}^{2}$. This scaling dimension of
the Majorana fermions is the same as that in the SYK model with $q=4$
Maldacena and Stanford (2016) a point that we will return to in Section VII.2
below.

It is easy to see from the definition111Recall that we are already considering
an extended problem where fermion operators are physical.  that for all
temperature, $G_{\gamma}(\tau)\geq 0$ when $\tau\geq 0$, thus
$A_{\gamma}=1/|J_{0}|$. Notice that since $Q(\omega)\sim\sqrt{\omega}$, the
term we dropped in Eq. (38) is indeed negligible. The spin two-point function
is determined as

$\displaystyle G_{S}(\tau)=\frac{1}{4J_{0}^{2}|\tau|}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (T=0,\leavevmode\nobreak\
\tau^{-1}\ll\Lambda_{\rm UV}).$ (42)

Figure 3:
$\operatorname{sgn}(\tilde{\omega}_{n})\operatorname{sgn}(J_{0})\operatorname{Im}\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})$
as a function of $1/\sqrt{|\tilde{\omega}_{n}|}$ for $100$ values of
$|\tilde{\omega}_{n}|$. At large $|n|$,
$\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})$ approaches the zero-temperature
form
$\mathrm{i}\operatorname{sgn}(J_{0})\operatorname{sgn}(\tilde{\omega}_{n})\sqrt{2\pi}|\tilde{\omega}_{n}|^{-1/2}$,
indicated by the red dashed line.

Next, consider nonzero temperature with $T,\tau^{-1}\ll\Lambda_{\rm UV}$.
Recall when $T>0$, a nonzero mass $m=\mu T$ is generated for the bulk $\phi$
bosons. We can write $G_{\gamma}(\tau)$ in the following scaling form:

$\displaystyle J_{0}G_{\gamma}(\tau)=\sqrt{T}\Phi_{\gamma}(\tau T),$ (43)

where $\Phi_{\gamma}$ is some universal function. In the frequency space, this
becomes

$\displaystyle
J_{0}G_{\gamma}(\mathrm{i}\omega_{n})=\frac{1}{\sqrt{T}}\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n}),$
(44)

where we have used dimensionless frequencies
$\tilde{\omega}_{n}=\omega_{n}/T=(2n-1)\pi$ and the corresponding Fourier
transform
$\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})=\int_{0}^{1}\mathrm{d}(\tau
T)\mathrm{e}^{\mathrm{i}\tilde{\omega}_{n}(\tau T)}\Phi_{\gamma}(\tau T)$.
$G_{\gamma}(\tau)$ being real and antisymmetric implies
$G_{\gamma}(-\mathrm{i}\omega_{n})=G_{\gamma}(\mathrm{i}\omega_{n})^{*}$ as
well as $G_{\gamma}(-\mathrm{i}\omega_{n})=-G_{\gamma}(\mathrm{i}\omega_{n})$,
thus $\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})$ is purely imaginary and odd
in frequency. In terms of
$\operatorname{Im}\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})$, the self-
consistent equations are equivalent to

$\displaystyle\frac{1}{\operatorname{Im}\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})}$
$\displaystyle=-\frac{1}{4\pi}\sum_{\tilde{\nu}_{k}>0}\operatorname{Im}\Phi_{\gamma}(\mathrm{i}\tilde{\nu}_{k})\ln\left[\frac{(\tilde{\omega}_{n}-\tilde{\nu}_{k})^{2}+\mu^{2}}{(\tilde{\omega}_{n}+\tilde{\nu}_{k})^{2}+\mu^{2}}\right],$
(45)

where we have used

$\displaystyle
g_{\phi,0}(\mathrm{i}\omega_{n})=\frac{1}{4\pi}\ln\left(\frac{\Lambda^{2}}{\omega_{n}^{2}+m^{2}}\right)$
(46)

with $\Lambda$ \- the UV cutoff of the bulk momentum. This equation can be
solved numerically. In practice, to accurately estimate
$\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})$ up to some $\tilde{\omega}_{\rm
max}=(2n_{\rm max}-1)\pi$, we need to truncate the frequency space at a larger
$\tilde{\omega}_{\rm ext}=(2n_{\rm ext}-1)\pi$. As $|n|\rightarrow\infty$, we
expect $\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})$ to approach the zero-
temperature form
$\mathrm{i}\operatorname{sgn}(J_{0})\operatorname{sgn}(\tilde{\omega}_{n})\sqrt{2\pi}|\tilde{\omega}_{n}|^{-1/2}$.
Using this asymptotic behavior, one can show that if we send both
$\tilde{\omega}_{\rm max}$ and $\tilde{\omega}_{\rm ext}$ to infinity but have
$\tilde{\omega}_{\rm max}/\tilde{\omega}_{\rm ext}\rightarrow 0$, then the
relative error of $\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{\rm max})$ goes to
zero. Therefore, we took $n_{\rm ext}=n_{\rm max}^{2}$ in our actual
calculation. The result for $\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})$ is
shown in Fig. 3. It is clear that as $|n|$ gets large,
$\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})$ indeed approaches the expected
zero-temperature form.

###  IV.2 Impurity susceptibility

Next, we compute the impurity susceptibility. We again generalize the magnetic
field to an antisymmetric matrix $H_{\alpha\beta}$ with
$\alpha,\beta=1,\cdots,N$ which reduces to a vector field in the $SO(3)$ case
by $H_{\alpha}=\frac{1}{2}\epsilon^{\alpha\beta\gamma}H_{\beta\gamma}$. Recall
$\phi_{\alpha\beta}$ also embeds into an antisymmetric matrix $\phi$ by
$\phi_{\alpha\beta}=-\phi_{\beta\alpha}$. The coupling of $H_{\alpha\beta}$ to
the impurity site is

$\displaystyle\Delta\mathcal{L}_{\rm
imp}=\frac{1}{2}\sum_{\alpha<\beta}H_{\alpha\beta}(\mathrm{i}\gamma_{\alpha}\gamma_{\beta}),$
(47)

and the coupling to the bulk is

$\displaystyle\partial_{\tau}\phi_{\alpha\beta}\mapsto\partial_{\tau}\phi_{\alpha\beta}-\mathrm{i}(\phi_{\alpha\gamma}H_{\gamma\beta}-H_{\alpha\gamma}\phi_{\gamma\beta})$
(48)

which reduces to Eq. (26) as one can easily check. As before, to compute the
susceptibility $\chi$, we take $H_{12}(x)=-H_{21}(x)=h$ to be a constant and
turn off other components of $H_{\alpha\beta}$. Then $\chi$ is given by

$\displaystyle\chi=T\left.\frac{\partial^{2}\ln Z}{\partial
h^{2}}\right|_{h=0}=T\frac{1}{Z}\left.\frac{\partial^{2}Z}{\partial
h^{2}}\right|_{h=0}.$ (49)

The additional terms in the Lagrangian now take the form

$\displaystyle\Delta\mathcal{L}_{\rm
b}=h\sum_{\alpha>2}2(\mathrm{i}\partial_{\tau}\phi_{1\alpha})\phi_{2\alpha}-\frac{1}{2}h^{2}\sum_{\alpha>2}\left(\phi^{2}_{1\alpha}+\phi^{2}_{2\alpha}\right),$
$\displaystyle\Delta\mathcal{L}_{\rm
imp}=\frac{1}{2}h(\mathrm{i}\gamma_{1}\gamma_{2}).$ (50)

In the decoupled limit $J_{0}=0$, we still have $\chi_{\rm
imp}=1/(4T)\equiv\mathcal{C}_{\rm free}/T$ for all $N$. For the interacting
system, it turns out that the low temperature behavior of $\chi_{\rm imp}$ is
dominated by $\chi_{\rm b,b}$ (cf. Eq. (31)), thus we will focus on this term
in the following.

Figure 4: (a) Feynman diagrams relevant in the large-$N$ limit for $\chi_{\rm
b,b}$, the bulk-bulk part of the impurity susceptibility to a uniform magnetic
field. Single circles are source terms of the form $(\partial_{\tau}\phi)\phi$
which couples linearly to $h$, and double circles are sources terms of the
form $\phi^{2}$ which couples to $h^{2}$ (cf. Eq. (50)). The solid lines are
Majorana fermion propagators and the dashed lines are $\phi$ boson
propagators. (b) The shaded bubble which appears in the first panel. The wavy
line represents the propagator of the Lagrange multiplier field $\lambda$ and
is of order $1/N^{2}$.

Fig. 4a presents all the diagrams of order $N^{0}$ for $\chi_{\rm b,b}$, where
the shaded bubble is explained in panel b. We note that the two terms
contributing to the shaded bubble again have a remarkable divergence
cancellation. Computing all the diagrams in Fig. 4, we found that

$\displaystyle\chi_{\rm imp}=\frac{\mathcal{C}_{\rm cp}}{T}.$ (51)

There is no anomalous dimension in $T$. The analytic expression for
$\mathcal{C}_{\rm cp}$ is given in Appendix A. Using the numerical solution
for $\Phi_{\gamma}$, we found

$\displaystyle\mathcal{C}_{\rm cp}=0.2109\pm 0.0018=(0.844\pm
0.007)\mathcal{C}_{\rm free}.$ (52)

One may recall that in Section III (Model 1), the coupling to the bulk gives
an order $1/N$ correction to the impurity Curie coefficient, in contrast to
the order $1$ correction here. In fact, we now have an order $N^{2}$ number of
fields in the bulk, and the bulk susceptibility scales as $N$ (in contrast to
Model 1 where the bulk susceptibility is of $O(1)$). Thus, the relative order
in $N$ of boundary susceptibility correction to the bulk susceptibility is the
same in Models 1 and 2. Notice that $\mathcal{C}_{\rm cp}$ in Eq. (52) is
numerically quite close to the $N=3$ result in Section III, which is expected
because the two models reduce to the same one at $N=3$. As in section III,
there is a similar symmetry fractionalization argument for the impurity being
not fully screened. Let us briefly explain our numerical computation. Given a
numerical solution of $\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})$ up to
$|\tilde{\omega}_{n}|\leq\tilde{\omega}_{\rm max}$, we can further extrapolate
this solution by its known asymptotic form at larger frequencies. By
increasing the number of frequency modes in the summation, we asymptotically
converge to the susceptibility result corresponding to the infinite sum over
frequency. We collect two sources of error for the susceptibility obtained
this way: one is from the extrapolation fitting, and the other is from the
inaccuracy of $\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})$ with
$|\tilde{\omega}_{n}|\leq\tilde{\omega}_{\rm max}$. To estimate the latter
error, we reduce $n_{\rm max}$ by half, compute the susceptibility in the same
way and then evaluate the discrepancy. We regard the two kinds of errors as
independent and the total error is the root mean square of them. For the
numbers given above, we used $n_{\rm max}=100$, and the error is dominated by
the extrapolation.

##  V Antisymmetric Tensor Impurity with Tensor Bulk Fields

It is natural to generalize the previous analysis to impurities in other
representations. For example, in the $SO(3)$ case, one may ask whether there
is an even-odd effect; what will happen if we replace the spin-$1/2$ impurity
(spinor representation) by spin-$1$ (vector representation)? For the $SO(N)$
case, we consider a totally antisymmetric tensor representation as explained
below.

We introduce complex fermion modes $c_{\alpha}$ for $\alpha=1,2,\cdots,N$, and
make no assumption on whether $N$ is even or odd. The impurity spin operators
$S_{\alpha\beta}$ are
$-\mathrm{i}(c^{\dagger}_{\alpha}c_{\beta}-c^{\dagger}_{\beta}c_{\alpha})$. We
also fix the total fermion number as
$\sum_{\alpha}c^{\dagger}_{\alpha}c_{\alpha}=\nu N$, namely the impurity
Hilbert space is spanned by the states
$c^{\dagger}_{\alpha_{1}}c^{\dagger}_{\alpha_{2}}\cdots
c^{\dagger}_{\alpha_{\nu N}}\ket{0}$. When $\nu=1/N$, the impurity forms a
vector representation of $SO(N)$. For a generic $\nu$, this is the
antisymmetric $\nu N$-tensor representation. In the following, we will hold
$\nu$ as a constant when taking the large-$N$ limit. We will still use a
Majorana representation that manifest the analogy with the previous section:

$\displaystyle\gamma_{A\alpha}=c_{\alpha}+c^{\dagger}_{\alpha},\leavevmode\nobreak\
\gamma_{B\alpha}=-\mathrm{i}(c_{\alpha}-c^{\dagger}_{\alpha}).$ (53)

The bulk consists of tensor fields $\phi_{\alpha\beta}$ with the action in Eq.
(35). The impurity and coupling terms in the action now take the form

$\displaystyle S_{\rm imp}$
$\displaystyle=\frac{1}{4}\int\mathrm{d}\tau\sum_{\alpha}\sum_{I=A,B}\gamma_{I\alpha}\partial_{\tau}\gamma_{I\alpha}$
(54) $\displaystyle+\int\mathrm{d}\tau\leavevmode\nobreak\
\xi(\tau)\left[\sum_{\alpha}\mathrm{i}\gamma_{A\alpha}\gamma_{B\alpha}-(2\nu-1)N\right],$
(55) $\displaystyle S_{\rm cp}$
$\displaystyle=\sum_{I=A,B}\frac{J_{0}}{\sqrt{N}}\int\mathrm{d}\tau\sum_{\alpha<\beta}(\mathrm{i}\gamma_{I\alpha}\gamma_{I\beta})\phi_{\alpha\beta}(\tau,\bm{r}=0),$
(56)

where $\xi(\tau)$ is a Lagrange multiplier field imposing the fermion-number
constraint
$\sum_{\alpha}\mathrm{i}\gamma_{A\alpha}\gamma_{B\alpha}=(2\nu-1)N$. We define
the imaginary-time Green’s functions222Strictly speaking, these correlation
functions are ill-defined since the Majorana operators are fictitious. They
should really be thought of as some bilocal fields in the path integral
language and we are interested in their saddle point values from which we can
compute physical correlation functions.
$G_{IJ}(\tau,\tau^{\prime})=\left\langle
T\gamma_{I\alpha}(\tau)\gamma_{J\alpha}(\tau^{\prime})\right\rangle$ where
$I,J\in\\{A,B\\}$ and $\alpha$ is arbitrary. It directly follows from the
definition that $G_{BA}(\tau,\tau^{\prime})=-G_{AB}(\tau^{\prime},\tau)$. For
convenience, we will denote $G_{AA}$ and $G_{BB}$ by $G_{A}$ and $G_{B}$,
respectively. We found the following self-consistent equations in the
$N\rightarrow\infty$ limit:

$\displaystyle Q_{I}(\tau,\tau^{\prime})$
$\displaystyle=J_{0}^{2}g_{\phi,0}(\tau-\tau^{\prime})G_{I}(\tau,\tau^{\prime}),$
(57) $\displaystyle R(\tau,\tau^{\prime})$
$\displaystyle=J_{0}^{2}g_{\phi,0}(\tau-\tau^{\prime})G_{AB}(\tau,\tau^{\prime}),$
(58) $\displaystyle\mathrm{i}G_{AB}(\tau,\tau)$ $\displaystyle=2\nu-1,$ (59)

In terms of the bilocal fields $Q_{I}$ and $R$, the Green’s function matrix
$G\equiv(G_{IJ})$ is given by

$\displaystyle
G^{-1}(\tau,\tau^{\prime})=\begin{pmatrix}\frac{1}{2}\partial_{\tau}\delta(\tau-\tau^{\prime})-Q_{A}(\tau,\tau^{\prime})&\mathrm{i}\xi(\tau)\delta(\tau-\tau^{\prime})-R(\tau,\tau^{\prime})\\\
-\mathrm{i}\xi(\tau)\delta(\tau-\tau^{\prime})+R(\tau^{\prime},\tau)&\frac{1}{2}\partial_{\tau}\delta(\tau-\tau^{\prime})-Q_{B}(\tau,\tau^{\prime})\end{pmatrix}\equiv
M_{\gamma}(\tau,\tau^{\prime}).$ (60)

In this expression, $G^{-1}$ is the inverse on both the $\\{A,B\\}$ and time
indices. In other words,
$\sum_{J}\int\mathrm{d}\tau^{\prime}G_{IJ}(\tau,\tau^{\prime})M_{\gamma}(\tau^{\prime},\tau^{\prime\prime})_{JK}=\delta_{IK}\delta(\tau-\tau^{\prime\prime})$.
$Q_{A}$ and $Q_{B}$ are by definition antisymmetric, but $R$ has no such
restriction, i.e. $R(\tau,\tau^{\prime})$ are all independent in the path
integral. From these self-consistent equations, we look for solutions to the
classical fields $Q_{I}$, $R$ and $\xi$. Our approach presented below is
largely inspired by Parcollet  _et al._ (1998).

###  V.1 Scaling ansatz

Before writing down the explicit ansatz for the Green’s functions, let us
first examine the general constraints they need to satisfy. The problem has a
$U(1)$ _gauge_ symmetry generated by
$\sum_{\alpha}\mathrm{i}\gamma_{A\alpha}\gamma_{B\alpha}$ which acts on
Majorana operators as

$\displaystyle\begin{pmatrix}\gamma_{A\alpha}\\\
\gamma_{B\alpha}\end{pmatrix}\mapsto\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}\gamma_{A\alpha}\\\
\gamma_{B\alpha}\end{pmatrix}.$ (61)

Equivalently, the action is $c_{\alpha}\mapsto
e^{\mathrm{i}\theta}c_{\alpha}$. For now let us restrict to saddle point
solutions which preserve this gauge symmetry and we will comment on other
possibilities later. Taking $\theta=\pi/2$, we find
$G_{A}(\tau,\tau^{\prime})=G_{B}(\tau,\tau^{\prime})$ and
$G_{AB}(\tau,\tau^{\prime})=-G_{BA}(\tau,\tau^{\prime})\equiv
G_{AB}(\tau^{\prime},\tau)$. This implies $G_{AB}$ is symmetric. In contrast,
$G_{A}$ is antisymmetric directly from the definition. We assume the time
translation symmetry, thus $G(\tau,\tau^{\prime})=G(\tau-\tau^{\prime})$ and
$\xi$ in Eq. (60) is time-independent.

If we regard the $\gamma$’s as actual fermion operators, there will be more
constraints on the Green’s functions. On the one hand,
$G_{A}(\tau)\in\mathbb{R}$ and $G_{A}(\tau)\geq 0$ when $\tau>0$. On the other
hand, $G_{AB}(\tau)^{*}=G_{BA}(\tau)=-G_{AB}(\tau)$, thus $G_{AB}$ is purely
imaginary. It then follows from the expression for $G^{-1}$ that the saddle-
point value of $\xi$ is real, thus it effectively sets a chemical potential.
In the following, we will assume an ansatz with these properties. Of course
there could be other solutions but there should be at least one solution
satisfying these properties. This is because we can imagine really solving a
problem of fermions on the impurity site with a chemical potential determined
by $\xi$. As $N$ goes to infinity, we expect that the total fermion number
does not fluctuate and therefore a solution for this problem also serves as a
valid solution to our original problem a with fermion number projection.

Define a self-energy matrix

$\displaystyle Q(\tau)=\begin{pmatrix}Q_{A}(\tau)&R(\tau)\\\
-R(\tau)&Q_{A}(\tau)\end{pmatrix},$ (62)

then the self-consistent equations can be compactly written as

$\displaystyle[G(\mathrm{i}\omega_{n})]^{-1}$
$\displaystyle=-\frac{1}{2}\mathrm{i}\omega_{n}-\xi\sigma_{y}-Q(\mathrm{i}\omega_{n}),$
(63) $\displaystyle Q(\tau)$ $\displaystyle=J_{0}^{2}g_{\phi,0}(\tau)G(\tau),$
(64)

where $[G(\mathrm{i}\omega_{n})]^{-1}$ denotes the $2\times 2$ matrix inverse
and the symmetry constraints mentioned above have been used. The first
equation is written in the frequency space with a nonzero-temperature form.
When considering zero temperature, one just needs to replace $\omega_{n}$ by a
continuous variable $\omega$.

Now let us state our ansatz for the Green’s functions at $T=0$. We emphasize
that we only attempt to solve the self-consistent equations in the regime
$\tau^{-1}\ll\Lambda$. In this long time limit, we expect the leading behavior
of $G_{A}$ and $G_{AB}$ to be the following power-law form:

$\displaystyle G_{A}(\tau)$
$\displaystyle=\frac{A}{|\tau|^{2\Delta}}\operatorname{sgn}(\tau)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (A\geq 0),$ (65) $\displaystyle
G_{AB}(\tau)$
$\displaystyle=\frac{\mathrm{i}B}{|\tau|^{2\Delta}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (B\in\mathbb{R}),$ (66)

where $A$ and $B$ are constants to be determined later. In terms of the
complex fermion operators,

$\displaystyle\left\langle
Tc_{\alpha}(\tau)c^{\dagger}_{\alpha}(0)\right\rangle=\frac{A\operatorname{sgn}(\tau)+B}{2|\tau|^{2\Delta}}\quad(\text{no
sum over $\alpha$}).$ (67)

We assume $g_{\phi,0}$ at long time takes the form:

$\displaystyle
g_{\phi,0}(\tau)=\frac{C_{\phi}}{|\tau|^{2\delta}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (\frac{1}{2}\leq\delta<1),$ (68)

where a convenient range for the $\phi$-field scaling dimension $\delta$ is
imposed. Physically, $\delta=1/2$. Equations (57) and (58) then imply that at
$\tau^{-1}\ll\Lambda$,

$\displaystyle Q_{A}(\tau)$
$\displaystyle=J_{0}^{2}C_{\phi}A\frac{\operatorname{sgn}(\tau)}{|\tau|^{2(\Delta+\delta)}},$
(69) $\displaystyle R(\tau)$
$\displaystyle=\mathrm{i}J_{0}^{2}C_{\phi}B\frac{1}{|\tau|^{2(\Delta+\delta)}}.$
(70)

In order to solve (63), Green’s functions in the frequency space are needed.
Assuming $0<2\Delta<1$, we find that

$\displaystyle G_{A}(\mathrm{i}\omega)$
$\displaystyle=A\left[2\mathrm{i}\cos(\pi\Delta)\Gamma(1-2\Delta)\operatorname{sgn}(\omega)|\omega|^{2\Delta-1}\right],$
(71) $\displaystyle G_{AB}(\mathrm{i}\omega)$
$\displaystyle=\mathrm{i}B\left[2\sin(\pi\Delta)\Gamma(1-2\Delta)|\omega|^{2\Delta-1}\right].$
(72)

These can be obtained by directly Fourier transforming the power-law ansatz as
if it was valid for all $\tau$. However, as the leading behavior at small
$\omega$, the above results are more generally correct; with our assumptions
for $\Delta$, both $G_{A}(\mathrm{i}\omega)$ and $G_{AB}(\mathrm{i}\omega)$
have power-law divergences at $\omega\rightarrow 0$, and the leading terms are
completely determined by the slowly decaying long-time behavior of
$G_{A}(\tau)$ and $G_{AB}(\tau)$. The self-energy functions $Q_{I}$ and $R$
are trickier to deal with. We assume in addition that $1<2(\Delta+\delta)<2$,
then the Fourier transform of $Q_{A}(\tau)$ can be directly obtained using the
ansatz as well as principal value integral near $\omega=0$:

$\displaystyle
Q_{A}(\mathrm{i}\omega)=J_{0}^{2}C_{\phi}A\left[2\mathrm{i}\cos(\pi(\Delta+\delta))\Gamma(1-2(\Delta+\delta))\operatorname{sgn}(\omega)|\omega|^{2(\Delta+\delta)-1}\right].$
(73)

As $\omega\rightarrow 0$, the above expression has no divergence, thus one may
wonder whether this converging power-law behavior is still only determined by
the long-time limit of $Q_{A}(\tau)$. We note that $\partial
Q_{A}(\mathrm{i}\omega)/\partial\omega$ does have a divergence at $\omega=0$
whose leading term is determined by the slowly decaying tail of $\tau
Q_{A}(\tau)$. Therefore, _modulo a constant term_ , this expression is indeed
the leading behavior of $Q_{A}(\mathrm{i}\omega)$ near $\omega=0$ which
depends only on the long-time form of $Q_{A}(\tau)$. Moreover, the oddness of
$Q_{A}(\mathrm{i}\omega)$ guarantees that the constant term
$Q_{A}(\mathrm{i}\omega=0)$ actually vanishes.

For the Fourier transform of $R(\tau)$, we assume

$\displaystyle R(\mathrm{i}\omega=0)$ $\displaystyle=\mathrm{i}\xi,$ (74)
$\displaystyle R(\mathrm{i}\omega)-R(\mathrm{i}\omega=0)$
$\displaystyle=\mathrm{i}J_{0}^{2}C_{\phi}B\int_{-\infty}^{\infty}\mathrm{d}\tau\left[\frac{\mathrm{e}^{\mathrm{i}\omega\tau}}{|\tau|^{2(\Delta+\delta)}}-\frac{1}{|\tau|^{2(\Delta+\delta)}}\right]$
$\displaystyle=\mathrm{i}J_{0}^{2}C_{\phi}B\left[2\sin(\pi(\Delta+\delta))\Gamma(1-2(\Delta+\delta))|\omega|^{2(\Delta+\delta)-1}\right].$
(75)

In the first integral expression of
$R(\mathrm{i}\omega)-R(\mathrm{i}\omega=0)$, we did not mean that
$R(\mathrm{i}\omega)$ is really given by
$\mathrm{i}J_{0}^{2}C_{\phi}B|\tau|^{-2(\Delta+\delta)}$ for all $\tau$ (which
does not have a Fourier transform); we are just using a convenient
regularization. It is not hard to prove that, again, the final result only
relies on $R(\tau\gg\Lambda^{-1})$. Plugging our ansatz into Eq. (63), we
found that

$\displaystyle\Delta=\frac{1}{2}(1-\delta),$ (76) $\displaystyle
A^{2}\sin^{2}\left(\frac{\pi\delta}{2}\right)+B^{2}\cos^{2}\left(\frac{\pi\delta}{2}\right)=\frac{\delta\sin(\pi\delta)}{4\pi
J_{0}^{2}C_{\phi}}.$ (77)

Notice that our previous assumptions on the scaling dimensions are met self-
consistently. The ratio $A/B$ can only be determined using the fermion number
constraint (59). This is not an easy task; here we have only obtained the
correlation functions in the long-time limit, but (59) is a UV equation.
Therefore, we need to find a relation similar to the Luttinger theorem in a
Fermi liquid which translates this UV constraint to an IR one, and this is
done in Appendix B. Just quoting the result, we found

$\displaystyle{B}/{A}={\tan(\theta_{0+})}/{\tan(\pi\Delta)},$ (78)

where $\theta_{0+}$ is the unique solution within $[-\pi\Delta,\pi\Delta]$ to
the equation

$\displaystyle\frac{\sin(2\theta_{0+})}{\sin(2\pi\Delta)}=\frac{1}{2\Delta-1}\left(2\nu-1+\frac{2}{\pi}\theta_{0+}\right).$
(79)

The impurity spin operator is now
$S_{\alpha\beta}=-\frac{1}{2}\mathrm{i}(\gamma_{A\alpha}\gamma_{A\beta}+\gamma_{B\alpha}\gamma_{B\beta})$
and its two-point function is
$G_{S}(\tau)=\frac{1}{2}(A^{2}-B^{2})/|\tau|^{4\Delta}$.

Plugging in the values $\delta=1/2$ and $C_{\phi}=1/(4\pi)$, we have
$\Delta=1/4$, $A^{2}+B^{2}=1/J_{0}^{2}$ and $B/A=\tan\theta_{0+}$. Recalling
$A\geq 0$, these yield

$\displaystyle A=\frac{\cos\theta_{0+}}{|J_{0}|},\quad
B=\frac{\sin\theta_{0+}}{|J_{0}|},$ (80)

and $\theta_{0+}$ is determined by (79) which simplifies to

$\displaystyle\sin(2\theta_{0+})=-\frac{4}{\pi}\theta_{0+}-4\nu+2.$ (81)

We plot both sides of this equation as well as the solution for $B/A$ as a
function of $\nu$ in Fig. 5. The particle-hole symmetry $\nu\to 1-\nu$ is
clear from the plot.

Figure 5: (a) Both sides of (81) as functions of $\theta_{0+}$. The blue solid
line represents the left-hand side. The black dashed lines represent the
right-hand side which is a linear function with intercept $b_{\nu}=-4\nu+2$,
plotted for a few different values of $\nu$ as indicated in the figure. (b)
Solution for $B/A$ as a function of $\nu$ in the case $\Delta=1/4$.

The impurity spin correlator is

$\displaystyle G_{S}(\tau)=\frac{\cos(2\theta_{0+})}{2J_{0}^{2}|\tau|}.$ (82)

We conclude this section by noting that an alternative ansatz

$\displaystyle G_{A}=G_{B}=0,\quad
G_{AB}(\tau)=\pm\frac{\operatorname{sgn}(\tau)}{J_{0}\sqrt{|\tau|}},$ (83)

which breaks the $U(1)$ gauge symmetry also solves the _long time limit_ of
the self-consistent equations. However, we believe this is not a physical
solution because it gives the same nonzero spin correlation function for all
filling $\nu$, including $\nu=0$ where we expect the correlator to vanish.

###  V.2 Correlator at nonzero temperature and susceptibility

In this subsection, we compute the susceptibility to a uniform external
magnetic field as we did in the spinor cases.

After eliminating the $Q$ matrix from (63) and (64), the self-consistent
equations become

$\displaystyle[G(\mathrm{i}\omega_{n})]^{-1}=-\xi\sigma_{y}-\frac{J_{0}^{2}}{\beta}\sum_{\nu_{n}}g_{\phi,0}(\mathrm{i}\omega_{n}-\mathrm{i}\nu_{n})G(\mathrm{i}\nu_{n}),$
(84)

where we have ignored the $-\mathrm{i}\omega_{n}/2$ term. Similar to the case
of a spinor impurity coupled to tensor bulk fields, we define

$\displaystyle
J_{0}G(\mathrm{i}\omega_{n})=\frac{1}{\sqrt{T}}\Phi(\mathrm{i}\tilde{\omega}_{n}).$
(85)

$\Phi(\mathrm{i}\tilde{\omega}_{n})$ takes the form
$\Phi_{A}(\mathrm{i}\tilde{\omega}_{n})\mathbbm{1}+\Phi_{AB}(\mathrm{i}\tilde{\omega}_{n})\mathrm{i}\sigma_{y}$
and is purely imaginary. We can then rewrite the equation as

$\displaystyle[\operatorname{Im}\Phi(\mathrm{i}\tilde{\omega}_{n})]^{-1}$
$\displaystyle=-\frac{\xi}{J_{0}\sqrt{T}}\mathrm{i}\sigma_{y}$
$\displaystyle+\sum_{\nu_{n}}g_{\phi,0}(\mathrm{i}\omega_{n}-\mathrm{i}\nu_{n})\operatorname{Im}\Phi(\mathrm{i}\tilde{\nu}_{n}).$
(86)

Using an isomorphism of (algebraic) fields:

$\displaystyle\\{a\mathbbm{1}+b(\mathrm{i}\sigma_{y})|a,b\in\mathbb{R}\\}$
$\displaystyle\cong\mathbb{C}.$ (87) $\displaystyle
a\mathbbm{1}+b(\mathrm{i}\sigma_{y})$ $\displaystyle\mapsto a+b\mathrm{i}$

the $2\times 2$ matrix function
$\operatorname{Im}\Phi(\mathrm{i}\tilde{\omega}_{n})$ can be conveniently
represented by a complex number function
$F=\operatorname{Im}\Phi_{A}+\mathrm{i}\operatorname{Im}\Phi_{AB}\equiv
F_{1}+\mathrm{i}F_{2}$, satisfying

$\displaystyle
F^{-1}(\mathrm{i}\tilde{\omega}_{n})=-\frac{\mathrm{i}\xi}{J_{0}\sqrt{T}}+\sum_{\nu_{n}}g_{\phi,0}(\mathrm{i}\omega_{n}-\mathrm{i}\nu_{n})F(\mathrm{i}\tilde{\nu}_{n}).$
(88)

$F_{1}$ is antisymmetric, thus the real part of the above equation becomes

$\displaystyle\frac{F_{1}(\mathrm{i}\tilde{\omega}_{n})}{|F(\mathrm{i}\tilde{\omega}_{n})|^{2}}=\frac{1}{4\pi}\sum_{\tilde{\nu}_{n}>0}\ln\left[\frac{(\tilde{\omega}_{n}+\tilde{\nu}_{n})^{2}+\mu^{2}}{(\tilde{\omega}_{n}-\tilde{\nu}_{n})^{2}+\mu^{2}}\right]F_{1}(\mathrm{i}\tilde{\nu}_{n}).$
(89)

For the imaginary part of the equation, we eliminate $\xi$ from a specific
frequency $\tilde{\omega}_{n}=\tilde{\lambda}$, leading to

$\displaystyle-\frac{F_{2}(\mathrm{i}\tilde{\omega}_{n})}{|F(\mathrm{i}\tilde{\omega}_{n})|^{2}}+\frac{F_{2}(\mathrm{i}\tilde{\lambda})}{|F(\mathrm{i}\tilde{\lambda})|^{2}}+\frac{1}{4\pi}\sum_{\tilde{\nu}_{n}>0}\left\\{\ln\left[\frac{(\tilde{\omega}_{n}-\tilde{\nu}_{n})^{2}+\mu^{2}}{(\tilde{\lambda}-\tilde{\nu}_{n})^{2}+\mu^{2}}\right]\right.$
$\displaystyle\left.+\ln\left[\frac{(\tilde{\omega}_{n}+\tilde{\nu}_{n})^{2}+\mu^{2}}{(\tilde{\lambda}+\tilde{\nu}_{n})^{2}+\mu^{2}}\right]\right\\}F_{2}(\mathrm{i}\tilde{\nu}_{n})=0.$
(90)

For example, one may take $\tilde{\lambda}=\pi$. We still need one more
equation to effectively implement the fermion number constraint. In a
numerical calculation where the frequency space is truncated, this is also
necessary for matching the number of unknowns. From the zero temperature
limit, we know that as $n\rightarrow\infty$,
$F_{2}(\mathrm{i}\tilde{\omega}_{n})/F_{1}(\mathrm{i}\tilde{\omega}_{n})$
should approach the $B/A$ ratio. However, this is hard to impose in practice
as we are not able to directly access the $n\rightarrow\infty$ limit. In our
calculation, we constrained the ratio
$F_{2}(\mathrm{i}\tilde{\omega}_{n})/F_{1}(\mathrm{i}\tilde{\omega}_{n})$ for
one certain $\tilde{\omega}_{n}$, and then determined
$F_{2}(\mathrm{i}\infty)/F_{1}(\mathrm{i}\infty)$ by an extrapolation
procedure, which then gave an estimate of the filling fraction $\nu$. As
before, when computing $F(\mathrm{i}\tilde{\omega}_{n})$ up to some
$\tilde{\omega}_{\rm max}$, we truncated the frequency space at a larger
$\tilde{\omega}_{\rm ext}$ with $n_{\rm ext}=n_{\rm max}^{2}$. In Fig. 6, we
present an example of the numerical solution to
$F(\mathrm{i}\tilde{\omega}_{n})$ where $\nu$ is closed to $1/3$.

Figure 6:
$\operatorname{sgn}(J_{0})\operatorname{Im}\Phi_{IJ}(\mathrm{i}\tilde{\omega}_{n})$
as a function of $1/\sqrt{\tilde{\omega}_{n}}$ for $60$ positive values of
$\tilde{\omega}_{n}$. Blue squares and black circles represent $IJ=AA$ and
$IJ=AB$, respectively. The filling fraction is estimated to be $\nu=0.3354\pm
0.0001$ by a linear fitting of
$F_{2}(\mathrm{i}\tilde{\omega}_{n})/F_{1}(\mathrm{i}\tilde{\omega}_{n})$ with
respect to $1/\tilde{\omega}_{n}$ using the $13$ smallest positive frequency
values. Red dashed lines and purple dotted dashed lines indicate the
asymptotic (zero-temperature) behavior for $\nu=1/3$.

Computation of the susceptibility parallels the previous case. When the
impurity is decoupled, we found

$\displaystyle\text{(free imp.)}\quad\chi_{\rm
imp}=\frac{2\nu(1-\nu)N}{N-1}\frac{1}{T}\stackrel{{\scriptstyle
N\rightarrow\infty}}{{\longrightarrow}}2\nu(1-\nu)/T.$ (91)

For example, when $N=3,\leavevmode\nobreak\ \nu=1/3$, the result coincides
with $S(S+1)/(3T)$ for $S=1$, as expected. After turning on the coupling, with
the same Feynman diagrams as in Fig. 4, we find $\chi_{\rm
imp}={\mathcal{C}}_{\rm cp}/T$. Again there is no anomalous dimension in $T$.
Our numerical result for $\mathcal{C}_{\rm cp}$ is shown in Fig. 7.

Figure 7: (a) $\mathcal{C}_{\rm cp}$ for an antisymmetric tensor impurity as a
function of $\nu$ for $\nu\leq 1/2$. Error bars are indicated. Note that
$\chi_{\rm imp}$ for the filling $\nu$ is the same as that for the filling
$1-\nu$ due to the particle-hole symmetry. (b) $\mathcal{C}_{\rm
cp}/\mathcal{C}_{\rm free}$ as a function of $\nu$.

The computation strategy is similar to that in Section IV. For the special
filing $\nu=1/2$, $F_{2}=0$ due to the particle-hole symmetry and we simply
multiplied the result in the spinor case by a factor of $2$. For other values
of $\nu$, we solved $F(\mathrm{i}\tilde{\omega}_{n})$ for $n_{\rm max}=60$,
extended the solution using the known asymptotic form, and then obtained an
extrapolated result for the susceptibility at infinite frequency cutoff. We
collected three sources of error: one from the extrapolation fitting for the
susceptibility, another one from varying $n_{\rm max}$ and a new one from the
estimation of $\nu$ (another extrapolation) as explained earlier.

From our result for the susceptibility, at least at this particular fixed
point, the antisymmetric tensor impurity is not fully screened by the bulk.
This may sound surprising because the impurity now transforms under a regular
representation of $SO(N)$. In fact, there is still a secret symmetry
fractionalization happening. Suppose $N$ is even, the $\mathbb{Z}_{2}$ center
of $SO(N)$ (generated by $-\mathbbm{1}_{N\times N}$) has a trivial action on
the bulk fields $\phi_{\alpha\beta}$, i.e. the low-energy symmetry in the bulk
is truly $SO(N)/\mathbb{Z}_{2}$. In contrast, this $\mathbb{Z}_{2}$ center
symmetry acts nontrivially on the impurity Hilbert space when $\nu N$ is odd.
This argument does not apply to odd $N$ or even $\nu N$, but for large $N$ the
continuity of our solution in $N$ and $\nu$ implies the existence of a fixed
point with only partial screening in these cases also. In Section VI, we will
consider an impurity problem free of similar symmetry fractionalization and
there we, indeed, find a fully screened fixed point.

##  VI A $\mathbb{Z}_{2}$ symmetric impurity in the $O(3)$ model

Figure 8: Bilayer antiferromagnet and an impurity (marked as a red dot)
symmetric under the $\mathbb{Z}_{2}$ layer interchange symmetry.

As our final model, we come back to a spin $S$ impurity in the $O(3)$ model,
Eq. (2). However, we now require that the impurity respect the unitary
$\phi_{\alpha}\to-\phi_{\alpha}$ symmetry. For instance, we can consider the
$O(3)$ transition in a bilayer antiferromagnet.Wang  _et al._ (2006) The layer
exchange symmetry acts as $\phi_{\alpha}\to-\phi_{\alpha}$. We place the
impurity midway along a rung of the bilayer and require it to couple in a way
that respects the layer exchange symmetry. See Fig. 8 for an illustration. In
this setup the coupling $L_{\rm cp}\sim S_{\alpha}\phi_{\alpha}$ of the
impurity spin operator $S_{\alpha}$ to the bulk fields is prohibited. The next
most relevant coupling is

$\displaystyle S_{\rm cp}=-J\int
d\tau\,O_{\alpha\beta}(\tau)t_{\alpha\beta}(\tau,\bm{r}=0)$ (92)

where

$\displaystyle O_{\alpha\beta}$
$\displaystyle=\frac{1}{2}(S_{\alpha}S_{\beta}+S_{\beta}S_{\alpha})-\frac{\delta_{\alpha\beta}}{3}S(S+1)$
(93) $\displaystyle t_{\alpha\beta}$
$\displaystyle\sim\phi_{\alpha}\phi_{\beta}-\frac{1}{3}\delta_{\alpha\beta}(\vec{\phi}^{2})$
(94)

We note that $O_{\alpha\beta}=0$ for spin $S=1/2$, thus, we only consider
$S\geq 1$ below.

The traceless symmetric tensor $t_{\alpha\beta}$ operator in the $O(3)$ model
is known to have a scaling dimension $\Delta_{t}=1.20954(32)$.Chester  _et
al._ (2020) Thus, the impurity coupling $J$ in Eq. (92) is slightly
irrelevant, $dJ/d\ell=-(\Delta_{t}-1)J$. (Here and below $x\sim e^{\ell}$).
This means that for small enough $J$ the impurity decouples from the bulk,
forming an essentially free spin.

We next perform a perturbative renormalization group analysis in $J$ searching
for non-trivial weak coupling fixed points, using $\Delta_{t}-1\approx 0.21$
as an expansion parameter. We note that the next most relevant impurity
coupling is $\delta S\sim\lambda\int d\tau s(\tau,0)$, where $s(x)$ is the
leading $SO(3)$ scalar. The scaling dimension $\Delta_{s}=1.59488(81)$ (see
Chester  _et al._ (2020)), thus, this coupling is somewhat more irrelevant
than $J$ and will not be included in our analysis.

Applying standard perturbation theory in $J$ to second order we obtain (see
appendix C):

$\displaystyle\frac{dJ}{d\ell}=-(\Delta_{t}-1)J+aJ^{2}$ (95)

where the coefficient $a$ is given by

$\displaystyle
a=\frac{\sqrt{2}}{3}\left(S(S+1)-\frac{15}{4}\right)\lambda_{ttt}$ (96)

and $\lambda_{ttt}\approx 1.499$ is the coefficient of $t$ in the $t\times t$
OPE (see appendix C for our normalization convention333We thank Shai Chester
for explaining the normalization conventions in Ref. Chester  _et al._ , 2020
to us).Chester  _et al._ (2020) Intriguingly, $a<0$ for $S=1$, $a=0$ for
$S=3/2$, and $a>0$ for $S>3/2$. This means that for $S=1$ we have an IR
unstable fixed point at ${\it negative}$ $J_{c}=-(\Delta_{t}-1)/|a|$, and for
$J<-|J_{c}|$, $J$ runs away to $-\infty$. As we discuss below, we expect this
runaway flow is towards a fully screened impurity phase. On the other hand,
for $S\geq 2$ we have an unstable fixed point at positive
$J_{c}=(\Delta_{t}-1)/a$, and for $J>J_{c}$ $J$ runs away to $+\infty$. We do
not currently know the nature of the $J=+\infty$ fixed point, but we don’t
expect the impurity to be fully screened here. (Indeed, this flow occurs both
for integer and half-integer $S\geq 2$, and for half-integer $S$ we don’t
expect full screening to occur.) Finally, for $S=3/2$, $a=0$ and any
perturbatively accessible fixed points will be controlled by the $O(J^{3})$
term in $dJ/dl$. Below, we exclude $S=3/2$ from our discussion.

From the $\beta$-function (95) we conclude that the correlation length
exponent $\nu$ at the critical point $J_{c}$ ($\xi\sim|J-J_{c}|^{-\nu}$) is
given by

$\displaystyle\nu^{-1}=\Delta_{t}-1\approx 0.21$ (97)

both for $S=1$ and for $S\geq 2$. We also compute perturbatively the scaling
dimension of the impurity spin $S^{a}$ at the critical point $J=J_{c}$ (see
appendix C):

$\displaystyle\Delta_{S^{a}}$ $\displaystyle\approx 2(S(S+1)-3/4)J^{2}$
$\displaystyle\approx\frac{9(\Delta_{t}-1)^{2}}{\lambda_{ttt}^{2}}\frac{S(S+1)-3/4}{(S(S+1)-15/4)^{2}}$
$\displaystyle\approx 0.18\frac{S(S+1)-3/4}{(S(S+1)-15/4)^{2}}$ (98)

We now say a few words about the nature of the $J=-\infty$ phase that appears
to be realized for $S=1$ and $J<-|J_{c}|$. We believe this is a fully screened
phase. There are several indications that this is the case. Most
heuristically, let’s begin by freezing the direction of the bulk order
parameter $\phi_{\alpha}$ (e.g. along the $z$ direction). The impurity
coupling (92) is then $\delta H\sim-JS^{2}_{z}$. For $J<0$ this coupling
favors the state with minimal $S_{z}$. For integer $S$, this is the unique
state $S_{z}=0$. Now, turning back on slow fluctuations of $\phi_{\alpha}$,
the impurity spin will be in the instantaneous eigenstate of
$\phi_{\alpha}S_{\alpha}$ with zero eigenvalue, which produces no Berry phase.
This absence of Berry phase makes us conclude that for $J\to-\infty$ the
magnetic impurity essentially acts as a non-magnetic one - i.e. full screening
occurs.

Another indication that the $J=-\infty$ fixed point for $S=1$ is fully
screened comes from considering the large $N$ generalization of the current
problem that we discuss below. This generalization is quite similar to the
$SU(N)$ impurity model studied in Ref. Florens  _et al._ (2006).

###  VI.1 Large $N$ generalization

We consider a vector $O(N)$ model in the bulk, Eq. (6). The impurity is taken
to transform in the traceless symmetric tensor representation of $SO(N)$ with
$N_{b}$ indices. There is a unique traceless symmetric two-index tensor
operator $O_{\alpha\beta}$ acting on the impurity Hilbert space, which can be
taken as

$\displaystyle
O_{\alpha\beta}=-\frac{1}{2}(S_{\alpha\gamma}S_{\beta\gamma}+S_{\beta\gamma}S_{\alpha\gamma})+\frac{2}{N}\delta_{\alpha\beta}C$
(99)

with $S_{\alpha\beta}$ the generators of $SO(N)$ in the representation
considered and $C=\frac{1}{2}S_{\alpha\beta}S_{\alpha\beta}=N_{b}(N_{b}+N-2)$
\- the quadratic Casimir.444We’ve normalized $O_{\alpha\beta}$ so that it
reduces to (93) for $N=3$. We consider the impurity coupling:

$\displaystyle S_{\rm cp}=-J\int\,d\tau
O_{\alpha\beta}(\tau)t_{\alpha\beta}(\tau,\bm{r}=0)$ (100)

with $t_{\alpha\beta}$ \- the bulk symmetric tensor

$\displaystyle
t_{\alpha\beta}\sim\phi_{\alpha}\phi_{\beta}-\frac{\delta_{\alpha\beta}}{N}(\vec{\phi}^{2})$
(101)

The resulting model is similar to the large-$N$ model considered in Ref.
Florens  _et al._ , 2006, except the symmetry group is $SO(N)$ instead of
$SU(N)$. At large $N$ the scaling dimension $\Delta_{t}$ is given byGracey
(2002)

$\displaystyle\Delta_{t}=1+\frac{32}{3\pi^{2}N}+O(1/N^{2})$ (102)

Thus, the impurity coupling $J$ is slightly irrelevant in the large-$N$ limit,
so the impurity forms a free spin for sufficiently small $J$.

We again perform a perturbative RG analysis in $J$ utilizing the parametric
smallness of $\Delta_{t}-1$ (see appendix C). We obtain a flow equation as in
(95) but with the coefficient $a$ now given by

$\displaystyle a$
$\displaystyle=-\frac{\sqrt{2}\lambda_{ttt}}{4N}(4(N-4)N^{2}_{b}+4(N^{2}-6N+8)N_{b}$
$\displaystyle+N(N^{2}-4N+8))$ (103)

where the OPE coefficient $\lambda_{ttt}\to 2$ for $N\to\infty$. Note that
$a<0$ for all integer $N>3$ independent of $N_{b}$. Thus, we expect that the
critical point with $J_{c}>0$ found for $S\geq 2$ in the $N=3$ case does not
exist for $N\geq 4$ (or at least is not perturbatively acessible). Instead,
for $N\geq 4$, the critical coupling $J_{c}=(\Delta_{t}-1)/a$ is negative, and
for $J<-|J_{c}|$ we have a runaway flow to $J=-\infty$ as in the $S=1$, $N=3$
case. In particular, taking the limit $N\to\infty$ with $\nu=N_{b}/N$-fixed,
we have

$\displaystyle
a=-2\sqrt{2}N^{2}(\nu+1/2)^{2},J_{c}=-\frac{8\sqrt{2}}{3\pi^{2}N^{3}(\nu+1/2)^{2}}$
(104)

One may ask whether perturbation theory in $J$ continues to work for the
coupling $J_{c}\sim O(1/N^{3})$. As we explain below, the answer is yes.

####  VI.1.1 Beyond perturbation theory

Next, to understand the nature of the flow $J\to-\infty$ for $J<J_{c}$ we set-
up a non-perturbative large-$N$ calculation. We can represent our impurity in
terms of canonical complex boson modes $b_{\alpha}$. The constraint of $N_{b}$
boxes in the symmetric $SO(N)$ tensor representation translates to
$b^{\dagger}_{\alpha}b_{\alpha}|\psi\rangle=N_{b}|\psi\rangle$, where
$|\psi\rangle$ is a state in the physical subspace $V_{phys}$. The traceless
condition translates to $b_{\alpha}b_{\alpha}|\psi\rangle=0$. The action of
the operator $O_{\alpha\beta}$ on $|\psi\rangle\in V_{phys}$ is

$\displaystyle O_{\alpha\beta}$
$\displaystyle\to-\frac{1}{2}(N+2N_{b}-4)\left(b^{\dagger}_{\alpha}b_{\beta}+b^{\dagger}_{\beta}b_{\alpha}-2\frac{N_{b}\delta_{\alpha\beta}}{N}\right)$
$\displaystyle+(b^{\dagger}_{\gamma}b^{\dagger}_{\gamma})b_{\alpha}b_{\beta}$
(105)

In particular, when calculating matrix elements of $O_{\alpha\beta}$ within
$V_{phys}$ we may drop the last term in Eq. (105). Thus, we consider the
action

$\displaystyle S_{\rm imp}$ $\displaystyle=\int\mathrm{d}\tau
b^{\dagger}_{\alpha}\partial_{\tau}b_{\alpha}+\int\mathrm{d}\tau\leavevmode\nobreak\
\xi(\tau)\left(b^{\dagger}_{\alpha}b_{\alpha}-\nu N\right)$
$\displaystyle+V\int d\tau
b^{\dagger}_{\alpha}b^{\dagger}_{\alpha}b_{\beta}b_{\beta},$ (106)
$\displaystyle S_{\rm cp}$
$\displaystyle=\frac{2\pi\sqrt{2}J_{0}}{N}\int\mathrm{d}\tau
b^{\dagger}_{\alpha}b_{\beta}\phi_{\alpha}\phi_{\beta}(\bm{r}=0).$ (107)

Here $\xi(\tau)$ is a Lagrange multiplier enforcing
$b^{\dagger}_{\alpha}b_{\alpha}|\psi\rangle=N_{b}|\psi\rangle$. We enforce the
constraint $b_{\alpha}b_{\alpha}|\psi\rangle=0$ energetically by sending the
energy $V\to\infty$. The coupling $J_{0}$ is related to $J$ in Eq. (100) via
$J_{0}=N(N+2N_{b}-4)J$ and the factor $2\pi\sqrt{2}$ in (107) is introduced to
match the normalization of $t_{\alpha\beta}$ in the large-$N$ limit (see
appendix C).

We can decouple the quartic terms in (106), (107):

$\displaystyle S_{\rm imp}$ $\displaystyle=\int\mathrm{d}\tau
b^{\dagger}_{\alpha}\partial_{\tau}b_{\alpha}+\int\mathrm{d}\tau\leavevmode\nobreak\
\xi(\tau)\left(b^{\dagger}_{\alpha}b_{\alpha}-\nu N\right)$
$\displaystyle+i\int
d\tau\,\left(ub^{\dagger}_{\alpha}b^{\dagger}_{\alpha}+u^{*}b_{\alpha}b_{\alpha}\right)+\frac{1}{V}\int
d\tau u^{*}u$ (108) $\displaystyle S_{\rm cp}$ $\displaystyle=\int
d\tau\left(sb^{\dagger}_{\alpha}\phi_{\alpha}(\bm{r}=0)+s^{*}b_{\alpha}\phi_{\alpha}(\bm{r}=0)\right)$
$\displaystyle-\frac{N}{2\pi\sqrt{2}J_{0}}\int d\tau s^{*}s$ (109)

$s(\tau)$ and $u(\tau)$ are auxiliary fields. The limit $V\to\infty$ can now
be taken by simply dropping the last term in (108). It is now clear that the
theory possesses a large-$N$ limit with fixed $J_{0}$, $\nu$ where
fluctuations of $\xi$, $s$ and $u$ are suppressed. Note that taking $J_{0}\ll
1$ only suppresses fluctuations of $s$ further - in this regime we can perform
combined expansion in $J_{0}$ and $1/N$, as we have done in the previous
section ($J_{0}\ll 1$ translates to $J\ll 1/N^{2}$ and our $J_{c}$ in Eq.
(104) lies in this range). Below, we focus on $J_{0}\sim O(N^{0})$ in order to
understand the phase diagram of our model.

As already noted, when $N=\infty$ the fields $\xi$, $s$, $u$ (and
$\lambda(x)$) become frozen at their saddle-point value. The saddle-point
equations are:

$\displaystyle\langle b^{\dagger}_{\alpha}b_{\alpha}\rangle=\nu N,\quad\langle
b_{\alpha}b_{\alpha}\rangle=0$ (110) $\displaystyle\langle
b_{\alpha}\phi_{\alpha}(\bm{r}=0)\rangle=\frac{N}{2\pi\sqrt{2}J_{0}}s,\quad\langle
b^{\dagger}_{\alpha}\phi_{\alpha}(\bm{r}=0)\rangle=\frac{N}{2\pi\sqrt{2}J_{0}}s^{*}$
(111) $\displaystyle\langle\phi_{\alpha}\phi_{\alpha}(x)\rangle=\frac{N}{g}$
(112)

One saddle-point is $s=s^{*}=u=u^{*}=0$. Here the impurity and the bulk
decouple. Working at finite temperature $T$ we then find the familiar saddle-
point $\lambda(x)=m^{2}=\mu T$ for the bulk, whereas
$\xi(\tau)=T\log(\nu^{-1}+1)$, so that the $b$ propagator $G_{b}(\tau)=\langle
b_{\alpha}(\tau)b^{\dagger}_{\alpha}(0)\rangle=\frac{1}{\beta}\sum_{\omega_{n}}\frac{1}{-i\omega_{n}+\xi}$.
It is instructive to compute the $s$ propagator at this decoupled fixed point.
Letting $G_{s}(i\omega_{n})=\int d\tau\langle s(\tau)s^{*}(0)\rangle
e^{i\omega_{n}\tau}$, we obtain

$\displaystyle G_{s}(i\omega_{n})$
$\displaystyle=\frac{1}{N}\left(-(2\pi\sqrt{2}J_{0})^{-1}+\Pi_{s}(i\omega_{n})\right)^{-1}$
(113) $\displaystyle\Pi_{s}(i\omega_{n})$ $\displaystyle=-\int d\tau
G_{b}(\tau)g_{\phi,0}(\tau)e^{i\omega_{n}\tau}$ (114)

with $g_{\phi,0}$ given by Eq. (21). From the small $\tau$ behavior of
$G_{b}(\tau)$, $g_{\phi,0}(\tau)$, the UV divergent part of
$\Pi_{s}(i\omega_{n})\stackrel{{\scriptstyle
UV}}{{=}}-\frac{1}{4\pi}(2\nu+1)\log(\Lambda/|\omega_{n}|)$, leading to the RG
flow

$\displaystyle\frac{dJ_{0}}{d\ell}=-\sqrt{2}(\nu+\tfrac{1}{2})J^{2}_{0}$ (115)

which agrees with our perturbative result in Eqs. (95), (104). Thus, at
$N=\infty$, $J_{0}>0$ runs logarithmically to zero and $J_{0}<0$ gives a run-
away flow $J_{0}\to-\infty$. We also now have a hint of the nature of this
run-away flow. For $J_{0}<0$ we have

$\displaystyle
G_{s}(i\omega_{n}=0)^{-1}=\frac{N}{2\pi\sqrt{2}}\left(|J_{0}|^{-1}-\sqrt{2}(\nu+\tfrac{1}{2})\log(\Lambda/T)\right)$
(116)

Thus, $G_{s}^{-1}(i\omega_{n}=0)$ switches sign and becomes negative for
$T<T_{0}=\Lambda\exp\left[-\left((\nu+\tfrac{1}{2})\sqrt{2}|J_{0}|\right)^{-1}\right]$.
This implies that the $s=0$ saddle-point becomes unstable for $T<T_{0}$ and
suggests that the true saddle-point in this regime should have $\langle
s\rangle\neq 0$. (Note that once $1/N$ corrections are included, we expect no
true phase transition at $T_{0}$, but rather a crossover.)

Thus, we expect the $T=0$ saddle point for $J_{0}<0$ has $s\sim\langle
b_{\alpha}\phi_{\alpha}\rangle\neq 0$ (and also by symmetry $u\neq 0$ and
$u^{*}\neq 0$). We will not attempt to solve the saddle-point equations (112)
in this regime. (At such a saddle $\lambda(x)$ acquires spatial dependence and
has to be solved for self-consistently.) We expect that this saddle-point
describes a fully screened impurity: the $b$ propagator will be non-singular
near $\omega=0$, so that the impurity modes can be safely integrated out.

##  VII Comparison with Related Results

###  VII.1 Impurity problems at $N=3$

It is interesting to compare our $1/N$-expansion results with the results of
different approaches to related problems. The $\epsilon$-expansion with
$\epsilon=4-D$ Vojta  _et al._ (2000) and quantum Monte Carlo (QMC) studies
Höglund  _et al._ (2007); Höglund and Sandvik (2007) have been applied to the
$SO(3)$ (or $Spin(3)$) symmetric impurity problem where a spin-$S$ impurity is
coupled to the 2+1D $O(3)$ critical bulk.

First consider the case of a spin-$1/2$ impurity for which the full action is
given in Eqs. 1-4. Model 1 (Section III, Eqs. 5-8) and Model 2A (Section IV,
Eqs. 35 and 36) are both $SO(N)$ generalizations of it. In Table 2, we list
the results from different approaches on (1) the scaling dimension of the spin
operator $S_{\alpha}$ or its $SO(N)$ analogs and (2) ${\mathcal{C}}_{\rm
cp}/{\mathcal{C}}_{\rm free}$, the ratio between the renormalized and free
Curie coefficients. See the table caption for more details such as the
expansion orders. Notice that there are two inequivalent generalizations of
the spin operators to $N>3$, one proportional to
$\mathrm{i}\gamma_{0}\gamma_{\alpha}$ and the other proportional to
$\mathrm{i}\gamma_{\alpha}\gamma_{\beta}$, appearing in both Model 1 and 2A.
There is no natural way to prefer one or the other in comparing to $N=3$
results, thus the scaling dimensions for both sets of operators are given in
the table, denoted as $\Delta_{\rm vec}$ and $\Delta_{\rm adj}$, respectively.
Let us also quote the analytic formulas from the $\epsilon$-expansion:

$\displaystyle[S_{\alpha}]=\frac{1}{2}\epsilon-\epsilon^{2}\left(\frac{5}{484}+\frac{\pi^{2}}{11}[S(S+1)-1/3]\right)+\cdots$
(117) $\displaystyle{\mathcal{C}}_{\rm cp}/{\mathcal{C}}_{\rm
free}=1+\sqrt{\frac{33\epsilon}{40}}-\frac{7\epsilon}{4}+\cdots$ (118)

Note that ${\mathcal{C}}_{\rm cp}/{\mathcal{C}}_{\rm free}$ is $S$-independent
to this order in $\epsilon$. We emphasize that when using the above
expressions to obtain concrete numbers, we have directly plugged $\epsilon=1$
and the value of $S$ into the power series; usually better results can be
obtained using Padé approximants or other resummation techniques. In both
large-$N$ models, $\Delta_{\rm vec}$ and $\Delta_{\rm adj}$ differ a lot from
each other, which indicates a sensitive dependence on $N$ of at least one of
the two scaling dimensions. Nonetheless, $\Delta_{\rm vec}$ of Model 2A does
roughly agrees with the QMC result. Also, both large-$N$ models predict that
the impurity operator which directly couples to the bulk field has a scaling
dimension $1/2$, coinciding with the 1-loop $\epsilon$-expansion. Our
predictions for the Curie coefficient are numerically much closer to the QMC
finding of ${\mathcal{C}}_{\rm cp}/{\mathcal{C}}_{\rm free}\sim 1.04$ than the
$\epsilon$-expansion, but there is still a qualitative difference: in QMC the
impurity susceptibility is slightly enhanced by the bulk instead of being
suppressed.

Methods | $\bm{\Delta_{\rm vec}}$ | $\bm{\Delta_{\rm adj}}$ | $\bm{{\mathcal{C}}_{\rm cp}/{\mathcal{C}}_{\rm free}}$
---|---|---|---
Model 1 | $1/2$ | $0$ | $0.860$
Model 2A | $1/4$ | $1/2$ | $0.844(7)$
$\epsilon$-Exp. | $0.116$ | $\leftarrow$ | $0.158$
QMC | $0.20(1)$ | $\leftarrow$ | $1.048(8)$
Table 2: Comparison between different approaches for the problem of a
spin-$1/2$ impurity coupled to the 2+1D $O(3)$ critical bulk. For the two
large-$N$ models (Model 1 and 2A) considered in this paper, we display both
the scaling dimension $\Delta_{\rm vec}$ of
$S_{\alpha}\propto\mathrm{i}\gamma_{0}\gamma_{\alpha}$ that transforms as an
$SO(N)$ vector and the scaling dimension $\Delta_{\rm adj}$ of
$S_{\alpha\beta}\propto\mathrm{i}\gamma_{\alpha}\gamma_{\beta}$ that
transforms in the adjoint representation. For the actual $SO(3)$ problem
($N=3$) studied previously by both the $\epsilon$-expansion and QMC methods,
these two set of operators are equivalent. The last column displays
${\mathcal{C}}_{\rm cp}/{\mathcal{C}}_{\rm free}$. The $\epsilon$-expansion
results are of 2-loop order. The numerical error for ${\mathcal{C}}_{\rm
cp}/{\mathcal{C}}_{\rm free}$ of Model 1 is not shown since it is smaller than
the last digit retained here.

Next consider the case of a spin-$1$ impurity where less data is available.
Model 2B (Section V) with either $\nu=1/N$ or $\nu=1/3$ is an $SO(N)$
generalization of this problem. The $\epsilon$-expansion gives
$[S_{\alpha}]=1/2$ at 1-loop order and a negative number at 2-loop order. Once
again, our prediction $[S_{\alpha\beta}]=1/2$ matches the 1-loop result. We
are not aware of any QMC study of this scaling dimension. Regarding the
susceptibility, 2-loop $\epsilon$-expansion gives the same ratio
${\mathcal{C}}_{\rm cp}/{\mathcal{C}}_{\rm free}=0.158$ and QMC gives
${\mathcal{C}}_{\rm cp}/{\mathcal{C}}_{\rm free}=0.995(3)$. These should be
compared with Fig. 7b either at $\nu=1/3$ or in the $\nu\rightarrow 0$ limit.
It seems that the large-$N$ value of the susceptibility ratio
${\mathcal{C}}_{\rm cp}/{\mathcal{C}}_{\rm free}$ is closed to one for all
$\nu$ and thus compatible with the QMC prediction (although we are less
certain about the $\nu\rightarrow 0$ limit due to the huge error bar).

###  VII.2 Sachdev-Ye-Kitaev models

The SYK models Kitaev (2015a, b); Maldacena and Stanford (2016); Sachdev
(2015); Gu  _et al._ (2020) are 0+1D quantum models which have been
extensively studied in the large-$N$ limit, that exhibit a conformal
structure. We can compare correlators in the SYK models with correlators of
our 0+1D quantum impurity, which is immersed in a 2+1D conformal bulk.

Models 2A (Section IV) and 2B (Section V) have strong similarities to the real
(Majorana) and complex SYK models, respectively, at least at zero temperature.
We will elaborate on this connection in this subsection. For our purpose here,
we will regard the fermion operators in these two models as physical555We were
in fact already doing so when looking for large-$N$ solutions to these two
models earlier., i.e. we do not apply the fermion-number parity projection in
Model 2A and treat $\xi$ as a real constant in Model 2B instead of a Lagrange
multiplier.

Zero Temperature Correlators: In Model 2A, there are $N$ Majorana operators
$\gamma_{\alpha}$ coupled to the bulk fields. We have found that
$[\gamma_{\alpha}]=1/4$, same as that in the real SYK model Kitaev (2015a, b);
Maldacena and Stanford (2016), therefore, up to an overall factor, the
Majorana Green’s function in Model 2A also coincides with that in the SYK
model in the long-time limit at $T=0$. We may also identify this equivalence
from the self-consistent equations. In the real SYK model, the self-energy
$Q(\tau)$ is related to the Green’s function $G_{\gamma}(\tau)$ by
$Q(\tau)\propto G_{\gamma}(\tau)^{3}$, while we have $Q(\tau)\propto
g_{\phi,0}(\tau)G_{\gamma}(\tau)$ in Model 2A. These are not of the same form
but $g_{\phi,0}(\tau)$ happens to be proportional to $G_{\gamma}(\tau)^{2}$ in
the long-time limit at $T=0$, thus they lead to the same solution. We note
that, interestingly, it appears from QMC numerics that the large-$N$ scaling
dimension $\gamma_{\alpha}=1/4$ in Model 2A does not change much as we go down
to $N=3$. Recall from the previous subsection, QMC found that in the $N=3$
version of Model 2A, the spin operator666In Model 2A, the operator
$S_{\alpha\beta}\propto\mathrm{i}\gamma_{\alpha}\gamma_{\beta}$ couples
directly to the bulk $\phi$ fields. When $N=3$, $S_{\alpha}$ is defined by
$S_{\alpha}=\frac{1}{2}\epsilon^{\alpha\beta\gamma}S_{\beta\gamma}$ and is
proportional to $\mathrm{i}\gamma_{0}\gamma_{\alpha}$ in either of the
fermion-number parity sectors.
$S_{\alpha}\propto\mathrm{i}\gamma_{0}\gamma_{\alpha}$ has scaling dimension
0.20. This quantity does not depend on whether we impose the fermion-number
parity projection or not as explained in Section IV. Noticing that
$\gamma_{0}$ completely decouples from all other degrees of freedom, we
conclude $[\gamma_{\alpha}]=0.20$ when $N=3$, quite closed to its large-$N$
limit $1/4$.

The equivalence between Model 2B and the complex SYK model Sachdev (2015); Gu
_et al._ (2020) is even more remarkable. The complex fermion Green’s function
in Model 2B is given by

$\displaystyle\left\langle
Tc_{\alpha}(\tau)c^{\dagger}_{\alpha}(0)\right\rangle=\frac{A\operatorname{sgn}(\tau)+B}{2|\tau|^{1/2}}\quad(\tau\gg\Lambda_{\rm
UV}^{-1},\leavevmode\nobreak\ T=0).$ (119)

Apart from the fermion scaling dimension $[c_{\alpha}]=1/4$ and the overall
normalization, this two-point function contains one more parameter $B/A$
characterizing the particle-hole asymmetry, which is determined by the
Luttinger theorem introduced in Section V.1. Let us quote the result again:

$\displaystyle\nu-\frac{1}{2}=-\frac{\theta}{\pi}+\left(\Delta-\frac{1}{2}\right)\frac{\sin(2\theta)}{\sin(2\pi\Delta)},$
(120)

where $\Delta=1/4$ is the fermion scaling dimension and $B/A$ is related to
$\theta$ by $B/A=\tan(\theta)/\tan(\pi\Delta)$. It turns out that not only do
the fermion scaling dimensions match but the above Luttinger theorem also
coincides exactly with the charge formula in the complex SYK model Georges
_et al._ (2001); Sachdev (2015); Gu  _et al._ (2020). This is a rather
nontrivial result since the Luttinger theorem presented here contains an
anomalous term, the second term on the right-hand side, which comes from
certain singularities at $\omega=0$ as detailed in Appendix B. For example,
such an anomalous term does not exist in the context of the multichannel
$SU(N)$ Kondo problem Parcollet  _et al._ (1998). A deeper understanding of
the unexpected similarities between these models awaits future investigation.

Finite Temperature: The equivalence between Model 2A/2B with the real/complex
SYK model in the long-time limit does not seem to hold when $T>0$. In the SYK
models, the Green’s functions at $T>0$ can be obtained from the zero-
temperature ones by a simple conformal transformation. However, in our
impurity problems, we have an intrinsically 2+1D bulk CFT such that the
correlation functions at different temperatures are not simply related by
conformal transformations as in 1+1D or 0+1D (nearly) CFTs.

##  VIII Conclusion

In this paper, we investigated four $SO(N)$ symmetric models of a quantum
impurity coupled to a 2+1D critical bulk using $1/N$ expansions. Models 1, 2A
and 2B are large-$N$ generalizations of $SO(3)$ symmetric models. More
specifically, Models 1 and 2A reduce to the problem of a spin-$\frac{1}{2}$
impurity coupled to the $O(3)$ Wilson-Fisher bulk CFT when $N=3$. On the other
hand, Model 2B together with $\nu=1/N$ reduces to the problem of a spin-$1$
impurity coupled to the same bulk. Model 3 describes the $O(3)$ Wilson-Fisher
bulk coupled to a spin $S>1/2$ impurity that respects the
$\mathbb{Z}_{2}:\phi_{\alpha}\to-\phi_{\alpha}$ symmetry of the bulk.

For Models 1, 2A and 2B, we found that the impurity is not fully screened by
the bulk and exhibits a Curie form static susceptibility with a renormalized
Curie coefficient. The absence of screening can be understood using symmetry
fractionalization arguments. Model 3 has a rich phase diagram: it possesses a
free spin phase for all $S$. In addition, for $S=1$ it has a fully screened
phase separated from the free spin phase by a transition that we describe. For
$S\geq 2$ we find a transition from the free spin phase to a yet undetermined
phase.

Let us think about the implications for $N=3$. Extrapolating to $N=3$, our
results for Models 1 and 2A both imply that a spin-$\frac{1}{2}$ impurity is
not fully screened by the $O(3)$ critical bulk. How about a spin-$1$ impurity?
From Model 2B, we have found that in the large-$N$ limit, either a vector
impurity ($\nu\rightarrow 0$) or an impurity with $\nu=1/3$ has a nonzero
$\mathcal{C}_{\rm cp}/\mathcal{C}_{\rm free}$ and is therefore also not fully
screened by the $O(N(N-1)/2)$ critical bulk generated by the tensor fields
$\phi_{\alpha\beta}$. Both cases reduce to $S=1$ when $N=3$. The existence of
a stable not fully screened fixed point for $S=1$ was also confirmed using QMC
and $\epsilon$-expansion, which we discussed in Section VII.1. In principle,
for integer spin $S$ there should also exist a fully screened impurity fixed
point, however, it is not accessible in our large-$N$ treatment of Model 2B.
In contrast, the transition to what we believe to be a fully screened phase is
accessible for $S=1$ in Model 3 and also in its large-$N$ generalizations.

Let us now comment on other possible future directions. An obvious open
problem is to upgrade the impurity to 1+1D, i.e. to couple a 1+1D gapless spin
chain to the boundary of a 2+1D critical bulk. Indeed this problem was one of
the motivations for this work. This problem was recently investigated in Refs.
Jian  _et al._ (2020); Metlitski (2020).

We have found that Models 2A and 2B are closely related to the real and
complex SYK models, respectively. In particular, the Luttinger theorem we
proved in Model 2B takes exactly the same form as that in the complex SYK
model. It would be interesting to further explore the connections between our
impurity problems and the SYK models. For example, it is presently unclear if
the impurity residual entropy in Models 2A and 2B matches with the SYK
results. A beautiful formula

$\displaystyle\frac{\mathrm{d}S_{\rm res}}{\mathrm{d}\nu}=2\pi\mathcal{E}$
(121)

which relates the entropy $S_{\rm res}$, charge $\nu$ and an “electric field”
parameter $\mathcal{E}$ characterizing particle-hole asymmetry has been proved
in both the multichannel $SU(N)$ Kondo problem Parcollet  _et al._ (1998) and
the complex SYK model Sachdev (2015); Gu  _et al._ (2020), although the
relation between $\mathcal{E}$ and $\nu$ is different in these two cases. We
wonder if the same holds in Model 2B where
$\mathcal{E}=\ln\left[(1+B/A)/(1-B/A)\right]$; so far we were not able to make
progress on this question due to the lack of analytical control of the finite-
temperature Green’s function.

## Acknowledgements

We would like to thank Ruihua Fan, Subir Sachdev and Matthias Vojta for
insightful discussions. HS, SL and AV were supported by a Simons Investigator
award (AV) and by the Simons Collaboration on Ultra-Quantum Matter, which is a
grant from the Simons Foundation (651440, AV). MM is supported by the National
Science Foundation under grant number DMR-1847861.

##  Appendix A Analytic Expressions for the Static Susceptibility

In this section, we explicitly spell out the analytic expressions for the
interacting Curie coefficients mentioned in the main text.

First consider the case of a spinor impurity coupled to vector bulk fields
studied in Section III, recall there is the decomposition:

$\displaystyle\chi_{\rm imp}=\frac{\mathcal{C}_{\rm cp}}{T}=\chi_{\rm
b,b}+2\chi_{\rm b,imp}+\chi_{\rm imp,imp}.$ (122)

We have the following expressions (cf. Fig. 2) for the order-$1/N$
contribution to $\chi_{\rm*}$, denoted as $\chi^{(1)}_{\rm*}$.

$\displaystyle T\chi^{(1)}_{b,b}$ $\displaystyle=I_{1}+I_{2}+I_{3},$ (123)
$\displaystyle I_{1}+I_{2}$ $\displaystyle=-\frac{1}{\pi}\sum_{\omega}B_{\rm
reg}(\mathrm{i}\omega_{n})\frac{\tilde{\omega}_{n}^{2}}{(\tilde{\omega}_{n}^{2}+\mu^{2})^{2}},$
(124) $\displaystyle I_{3}$ $\displaystyle=\frac{1}{2\pi}\sum_{\omega}B_{\rm
reg}(\mathrm{i}\omega_{n})\frac{1}{\tilde{\omega}_{n}^{2}+\mu^{2}}=0,$ (125)

where $B_{\rm reg}(\mathrm{i}\omega_{n})$ represents the shaded bubble and is
given by

$\displaystyle B_{\rm reg}(\mathrm{i}\omega_{n})$
$\displaystyle=\frac{1}{N}\frac{\sum_{\lambda,\nu}(\tilde{\lambda}_{n}^{2}+\mu^{2})^{-1}\Phi(\mathrm{i}\tilde{\nu}_{n})[\Phi_{0}(\mathrm{i}\tilde{\omega}_{n}-\mathrm{i}\tilde{\nu}_{n})-\Phi_{0}(\mathrm{i}\tilde{\lambda}_{n}-\mathrm{i}\tilde{\nu}_{n})]}{\sum_{\lambda}(\tilde{\lambda}_{n}^{2}+\mu^{2})^{-1}}.$
(126)

In deducing the above result, the following leading-order expression for the
$\lambda$ field propagator has been used Moshe and Zinn-Justin (2003):

$\displaystyle\Delta_{\lambda}(\mathrm{i}\nu_{n},\bm{q})$
$\displaystyle=-\frac{2}{N}\left[\frac{1}{\beta}\sum_{\omega_{n}}\int\frac{\mathrm{d}^{2}p}{(2\pi)^{2}}g_{\phi}(\mathrm{i}\omega_{n},\bm{p})g_{\phi}(\mathrm{i}\nu_{n}-\mathrm{i}\omega_{n},\bm{q}-\bm{p})\right]^{-1}$
(127)
$\displaystyle=\begin{cases}-\frac{16}{N}\sqrt{\nu_{n}^{2}+q^{2}}&(m=0),\\\
-\frac{16\pi}{\sqrt{5}N}m&(\nu_{n}=0,\leavevmode\nobreak\ \bm{q}=0),\\\
\cdots&\end{cases}$ (128)

where the minus sign reveals the fact that $\lambda$ has imaginary fluctuation
around the saddle point solution. For $\chi_{\rm b,imp}$ we have

$\displaystyle 2T\chi^{(1)}_{\rm b,imp}$ $\displaystyle=\frac{2}{\pi
N}\sum_{\omega,\nu}\frac{\tilde{\omega}_{n}+\tilde{\nu}_{n}}{(\tilde{\omega}_{n}+\tilde{\nu}_{n})^{2}+\mu^{2}}\frac{1}{\tilde{\omega}_{n}^{2}}\operatorname{Im}\Phi_{0}(\mathrm{i}\tilde{\nu}_{n}).$
(129)

For $\chi_{\rm imp,imp}$, we have

$\displaystyle T\chi^{(1)}_{\rm imp,imp}=\frac{2}{\pi
N}\sum_{\omega>0}\sum_{\nu>0}\frac{1}{\tilde{\omega}_{n}^{3}}\operatorname{Im}\Phi_{0}(\mathrm{i}\tilde{\nu}_{n})\ln\left[\frac{(\tilde{\omega}_{n}-\tilde{\nu}_{n})^{2}+\mu^{2}}{(\tilde{\omega}_{n}+\tilde{\nu}_{n})^{2}+\mu^{2}}\right].$
(130)

As we mentioned in the main text, there is a divergence cancellation between
$\chi^{(1)}_{\rm imp,imp}$ and $2\chi^{(1)}_{\rm b,imp}$, which is not obvious
from the analytic expressions above.

For the case of a spinor impurity coupled to tensor bulk fields studied in
Section IV, the analytic expression for $\mathcal{C}_{\rm cp}$ is (cf. Fig. 4)

$\displaystyle\mathcal{C}_{\rm cp}$ $\displaystyle=I_{1}+I_{2}+I_{3}+I_{4},$
(131) $\displaystyle I_{1}+I_{2}$
$\displaystyle=-\frac{1}{\pi}\sum_{\omega}[NB_{\rm
reg}(\mathrm{i}\omega_{n})]\frac{\tilde{\omega}_{n}^{2}}{(\tilde{\omega}_{n}^{2}+\mu^{2})^{2}},$
(132) $\displaystyle I_{3}$
$\displaystyle=\frac{1}{4\pi^{2}}\sum_{\omega,\lambda,\nu}\frac{(\tilde{\nu}_{n}-\tilde{\omega}_{n})(\tilde{\nu}_{n}-\tilde{\lambda}_{n})}{[(\tilde{\nu}_{n}-\tilde{\omega}_{n})^{2}+\mu^{2}][(\tilde{\nu}_{n}-\tilde{\lambda}_{n})^{2}+\mu^{2}]}\Phi^{2}_{\gamma}(\mathrm{i}\tilde{\nu}_{n})\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n})\Phi_{\gamma}(\mathrm{i}\tilde{\lambda}_{n}),$
(133) $\displaystyle I_{4}$ $\displaystyle=\frac{1}{2\pi}\sum_{\omega}[NB_{\rm
reg}(\mathrm{i}\omega_{n})]\frac{1}{\tilde{\omega}_{n}^{2}+\mu^{2}}=0,$ (134)

where $B_{\rm reg}(\mathrm{i}\omega_{n})$ represents the shaded bubble and is
given by

$\displaystyle NB_{\rm reg}(\mathrm{i}\omega_{n})$
$\displaystyle=\frac{\sum_{\lambda,\nu}(\tilde{\lambda}_{n}^{2}+\mu^{2})^{-1}\Phi_{\gamma}(\mathrm{i}\tilde{\nu}_{n})[\Phi_{\gamma}(\mathrm{i}\tilde{\omega}_{n}-\mathrm{i}\tilde{\nu}_{n})-\Phi_{\gamma}(\mathrm{i}\tilde{\lambda}_{n}-\mathrm{i}\tilde{\nu}_{n})]}{\sum_{\lambda}(\tilde{\lambda}_{n}^{2}+\mu^{2})^{-1}}.$
(135)

For the case of an antisymmetric $\nu N$-tensor impurity coupled to tensor
bulk fields, we have

$\displaystyle\mathcal{C}_{\rm cp}$ $\displaystyle=I_{1}+I_{2}+I_{3}+I_{4},$
(136) $\displaystyle I_{1}+I_{2}$
$\displaystyle=-\frac{1}{\pi}\sum_{\omega}[NB_{\rm
reg}(\mathrm{i}\omega_{n})]\frac{\tilde{\omega}_{n}^{2}}{(\tilde{\omega}_{n}^{2}+\mu^{2})^{2}},$
(137) $\displaystyle I_{3}$
$\displaystyle=\frac{1}{2\pi^{2}}\sum_{\omega,\lambda,\nu}\frac{(\tilde{\nu}_{n}-\tilde{\omega}_{n})(\tilde{\nu}_{n}-\tilde{\lambda}_{n})}{[(\tilde{\nu}_{n}-\tilde{\omega}_{n})^{2}+\mu^{2}][(\tilde{\nu}_{n}-\tilde{\lambda}_{n})^{2}+\mu^{2}]}\operatorname{Re}\left[F^{2}(\mathrm{i}\tilde{\nu}_{n})F(\mathrm{i}\tilde{\omega}_{n})F(\mathrm{i}\tilde{\lambda}_{n})\right],$
(138) $\displaystyle I_{4}$ $\displaystyle=\frac{1}{2\pi}\sum_{\omega}[NB_{\rm
reg}(\mathrm{i}\omega_{n})]\frac{1}{\tilde{\omega}_{n}^{2}+\mu^{2}}=0,$ (139)

where

$\displaystyle NB_{\rm reg}(\mathrm{i}\omega_{n})$
$\displaystyle=\frac{2\sum_{\lambda,\nu}(\tilde{\lambda}_{n}^{2}+\mu^{2})^{-1}\operatorname{Re}\\{F(\mathrm{i}\tilde{\nu}_{n})[F(\mathrm{i}\tilde{\nu}_{n}-\mathrm{i}\tilde{\omega}_{n})-F(\mathrm{i}\tilde{\nu}_{n}-\mathrm{i}\tilde{\lambda}_{n})]\\}}{\sum_{\lambda}(\tilde{\lambda}_{n}^{2}+\mu^{2})^{-1}}.$
(140)

##  Appendix B A Luttinger Theorem for the Impurity Problem

Here we derive a Luttinger-like theorem which relates the fermion number
constraint (59) at UV to the IR behavior of the Green’s functions. The
derivation here is inspired by Parcollet  _et al._ (1998).

We first introduce a convenient alternative representation of the Green’s
function and the self-energy function. Note that $G_{A}(\tau)$ being real
antisymmetric, $G_{AB}(\tau)$ being imaginary symmetric and $g_{\phi,0}(\tau)$
being real symmetric altogether imply that both $G(\mathrm{i}\omega_{n})$ and
$Q(\mathrm{i}\omega_{n})$ are purely imaginary. Moreover,
$\operatorname{Im}G(\mathrm{i}\omega_{n})$ and
$\operatorname{Im}Q(\mathrm{i}\omega_{n})$ both have the form
$a\mathbbm{1}+b(\mathrm{i}\sigma_{y})$ for $a,b\in\mathbb{R}$. There is an
isomorphism of (algebraic) fields:

$\displaystyle\\{a\mathbbm{1}+b(\mathrm{i}\sigma_{y})|a,b\in\mathbb{R}\\}$
$\displaystyle\cong\mathbb{C}.$ (141) $\displaystyle
a\mathbbm{1}+b(\mathrm{i}\sigma_{y})$ $\displaystyle\mapsto a+b\mathrm{i}$

Therefore, we can define complex valued functions

$\displaystyle D(\mathrm{i}\omega_{n})$
$\displaystyle=D_{1}+\mathrm{i}D_{2}=\operatorname{Im}G_{A}(\mathrm{i}\omega_{n})+\mathrm{i}\operatorname{Im}G_{AB}(\mathrm{i}\omega_{n}),$
(142) $\displaystyle\Sigma(\mathrm{i}\omega_{n})$
$\displaystyle=\Sigma_{1}+\mathrm{i}\Sigma_{2}=\operatorname{Im}Q_{A}(\mathrm{i}\omega_{n})+\mathrm{i}\operatorname{Im}Q_{AB}(\mathrm{i}\omega_{n}),$
(143)

and they satisfy the same algebra as the two-by-two matrices
$\operatorname{Im}G(\mathrm{i}\omega_{n})$ and
$\operatorname{Im}Q(\mathrm{i}\omega_{n})$ (but one need to be careful when
taking traces). In particular, the self-consistent equations for them have the
same form.

Let us now derive the Luttinger-like theorem. We consider $T=0$, and the
fermion number constraint can be written as

$\displaystyle-\int\frac{\mathrm{d}\omega}{2\pi}\operatorname{Im}D(\mathrm{i}\omega)=2\nu-1.$
(144)

We do not know the full functional form of $D(\mathrm{i}\omega)$, so our goal
is to evaluate the left-hand side only using the behavior of
$D(\mathrm{i}\omega)$ at $\omega\rightarrow 0\pm$ and
$\omega\rightarrow\pm\infty$. The self-consistent equations can be written as

$\displaystyle D^{-1}(\mathrm{i}\omega)$
$\displaystyle=\frac{1}{2}\omega-\mathrm{i}\xi+\Sigma(\mathrm{i}\omega),$
(145) $\displaystyle\Sigma(\mathrm{i}\omega)$
$\displaystyle=J_{0}^{2}\int\frac{\mathrm{d}\nu}{2\pi}g_{\phi,0}(\mathrm{i}\nu)D(\mathrm{i}\omega-\mathrm{i}\nu).$
(146)

Taking $\omega$ derivative on the first equation, we found

$\displaystyle D=-2\left(D^{-1}\frac{\partial
D}{\partial\omega}+D\frac{\partial\Sigma}{\partial\omega}\right).$ (147)

Eq. (144) can now be written as

$\displaystyle
2\int\frac{\mathrm{d}\omega}{2\pi}\operatorname{Im}\left(D^{-1}\frac{\partial
D}{\partial\omega}+D\frac{\partial\Sigma}{\partial\omega}\right)=2\nu-1.$
(148)

We separate the left-hand side into two integrals. In doing so, we actually
need to specify how to deal with the singularities of both integrands at
$\omega=0$. We choose777One can also regularize both $D(\mathrm{i}\omega)$ and
$\Sigma(\mathrm{i}\omega)$ near $\omega=0$ in a way compatible with (145) and
that both integrands become smooth. As a result, values of the two
integrations will be different from our choice, although the sum of them is
still be same.  to break
$\int_{-\infty}^{\infty}\mathrm{d}\omega\mapsto\int_{-\infty}^{0-}\mathrm{d}\omega+\int_{0+}^{\infty}\mathrm{d}\omega$.
The first integral is straightforward:

$\displaystyle\iota_{1}$
$\displaystyle=2\left(\int_{-\infty}^{0-}+\int_{0+}^{\infty}\right)\frac{\mathrm{d}\omega}{2\pi}\operatorname{Im}\left(D^{-1}\frac{\partial
D}{\partial\omega}\right)=\frac{1}{\pi}\left(\int_{-\infty}^{0-}+\int_{0+}^{\infty}\right)\mathrm{d}\omega\frac{\partial}{\partial\omega}\operatorname{Im}\ln
D$
$\displaystyle=\frac{1}{\pi}(\theta_{\infty}-\theta_{0+}+\theta_{0-}-\theta_{-\infty}),$
(149)

where $\theta_{\omega}$ is the phase angle of $D(\mathrm{i}\omega)$. By the
symmetries of the Green’s functions,
$D(-\mathrm{i}\omega)=-D(\mathrm{i}\omega)^{*}$, thus we can take
$\theta_{-\omega}=\pi-\theta_{\omega}$ and the above simplifies to
$\iota_{1}=(2/\pi)(\theta_{\infty}-\theta_{0+})$. (145) implies that as
$|\omega|\rightarrow\infty$, $D(\mathrm{i}\omega)\rightarrow 2/\omega$. We
thus have $\iota_{1}=(-2/\pi)\theta_{0+}$ by taking $\theta_{\infty}=0$. At
small frequency, we have

$\displaystyle D(\mathrm{i}\omega)\rightarrow
2[A\cos(\pi\Delta)\operatorname{sgn}(\omega)+\mathrm{i}B\sin(\pi\Delta)]\Gamma(1-2\Delta)|\omega|^{2\Delta-1},$
(150)

hence,

$\displaystyle(\cos\theta_{0+},\sin\theta_{0+})=\frac{1}{\sqrt{A^{2}\cos^{2}(\pi\Delta)+B^{2}\sin^{2}(\pi\Delta)}}(A\cos(\pi\Delta),B\sin(\pi\Delta)).$
(151)

From our previous result,
$A^{2}\cos^{2}(\pi\Delta)+B^{2}\sin^{2}(\pi\Delta)=\delta\sin(\pi\delta)/(4\pi
J_{0}^{2}C_{\phi})$ is a constant independent of $\nu$. Also recall that we
assume $1/2\leq\delta<1$, implying $0<\Delta\leq 1/4$.

Next, let us analyze the second integral

$\displaystyle\iota_{2}=2\left(\int_{-\infty}^{0-}+\int_{0+}^{\infty}\right)\frac{\mathrm{d}\omega}{2\pi}\operatorname{Im}\left(D\frac{\partial\Sigma}{\partial\omega}\right),$
(152)

which turns out to be tricky. We will first describe an naïve approach leading
to a wrong result, and then explain how to fix it. Define a Luttinger-Ward
functional

$\displaystyle\Phi_{\rm LW}=\int\mathrm{d}\tau
g_{\phi,0}(\tau)\operatorname{Tr}[\sigma_{y}G(-\tau)G(\tau)]=-2\mathrm{i}\int\frac{\mathrm{d}\nu\mathrm{d}\omega}{(2\pi)^{2}}g_{\phi,0}(\mathrm{i}\nu)\operatorname{Im}[D(\mathrm{i}\omega)D(\mathrm{i}\omega-\mathrm{i}\nu)].$
(153)

$\Phi_{\rm LW}$ is a finite number; it is in fact zero due to the symmetries
of the Green’s functions. $\Phi_{\rm LW}$ should be invariant by
$\omega\mapsto\omega+\epsilon$, which implies

$\displaystyle\int\frac{\mathrm{d}\nu\mathrm{d}\omega}{(2\pi)^{2}}g_{\phi,0}(\mathrm{i}\nu)\operatorname{Im}\left[\frac{\partial
D(\mathrm{i}\omega)}{\partial\omega}D(\mathrm{i}\omega-\mathrm{i}\nu)+D(\mathrm{i}\omega)\frac{\partial
D(\mathrm{i}\omega-\mathrm{i}\nu)}{\partial\omega}\right]=0.$ (154)

Applying the change of variable $\omega=\omega^{\prime}+\nu$ to the second
term in the square bracket, and noticing Eq. (146), the above reduces to

$\displaystyle\int\frac{\mathrm{d}\omega}{2\pi}\operatorname{Im}\left[\frac{\partial
D(\mathrm{i}\omega)}{\partial\omega}\Sigma(\mathrm{i}\omega)\right]=0\quad\Rightarrow\quad\int\frac{\mathrm{d}\omega}{2\pi}\operatorname{Im}\left[D(\mathrm{i}\omega)\frac{\partial\Sigma(\mathrm{i}\omega)}{\partial\omega}\right]=0,$
(155)

where integration by parts was used in the last step. Does the above
calculation imply $\iota_{2}=0$? Suppose this is true, we conclude that
$\theta_{0+}=(1/2-\nu)\pi$. As a result, $A=0$ for $\nu=0$ and the impurity
spin correlation function (at long time)
$G_{S}(\tau)=\frac{1}{2}(A^{2}-B^{2})/|\tau|^{4\Delta}$ becomes negative in
this case, which is not possible given that $S_{\alpha\beta}$ is a bosonic
hermitian operator. Therefore, the integration in the above result, whatever
it means, is not equivalent to that in $\iota_{2}$. In fact, for this
calculation to be valid, we need to regularize the singularity of
$D(\mathrm{i}\omega)$ near $\omega=0$, which also modifies
$\Sigma(\mathrm{i}\omega)$ through (146). The regularization leaves a
contribution and makes $\iota_{2}$ nonzero. Notice that although both
$G_{A}\partial R/\partial\omega$ and $G_{AB}\partial Q_{A}/\partial\omega$
contain $\mathcal{O}(|\omega|^{-1})$ terms, these terms cancel in
$\operatorname{Im}(D\partial\Sigma/\partial\omega)$ and the integral in
$\iota_{2}$ has no logarithmic divergence near $\omega=0$. Similar
cancellation may no longer exist for the regularization; suppose we regularize
$D(\mathrm{i}\omega)$ within $|\omega|\lesssim\epsilon$, then there is a
contribution of order $\epsilon\times(1/\epsilon)\sim 1$ to
$\int\mathrm{d}\omega\operatorname{Im}(D\partial\Sigma/\partial\omega)$, hence
$\iota_{2}$ should be of order $1$. According to this argument, in order to
get rid of the effect of regularization, we need to somehow reduce the small-
frequency divergence in $D$ or $\partial\Sigma/\partial\omega$. We can achieve
this goal by utilizing the $D$ and $\Sigma$ functions at the special filling
$\nu=0$, denoted by $D_{0}$ and $\Sigma_{0}$. Let $A_{0},B_{0}$ be the $A,B$
coefficients at $\nu=0$. For a general filling, we define

$\displaystyle\tilde{D}$
$\displaystyle=\frac{A}{A_{0}}\operatorname{Re}D_{0}+\mathrm{i}\frac{B}{B_{0}}\operatorname{Im}D_{0},$
(156) $\displaystyle\tilde{\Sigma}$
$\displaystyle=\frac{A}{A_{0}}\operatorname{Re}\Sigma_{0}+\mathrm{i}\frac{B}{B_{0}}\operatorname{Im}\Sigma_{0},$
(157)

Then $D-\tilde{D}$ as well as $\partial(\Sigma-\tilde{\Sigma})/\partial\omega$
no longer have $\mathcal{O}(|\omega|^{-1/2})$ terms. Now we can perform a
similar calculation. Define a new Luttinger-Ward functional:

$\displaystyle\Phi^{\prime}_{\rm
LW}=\int\frac{\mathrm{d}\nu\mathrm{d}\omega}{(2\pi)^{2}}g_{\phi,0}(\mathrm{i}\nu)\operatorname{Im}[(D+\tilde{D})(\mathrm{i}\omega)(D-\tilde{D})(\mathrm{i}\omega-\mathrm{i}\nu)].$
(158)

By the invariance under $\omega\mapsto\omega+\epsilon$, we obtain the sum rule

$\displaystyle\int\frac{\mathrm{d}\omega}{2\pi}\operatorname{Im}\left[(D-\tilde{D})\frac{\partial(\Sigma+\tilde{\Sigma})}{\partial\omega}+(D+\tilde{D})\frac{\partial(\Sigma-\tilde{\Sigma})}{\partial\omega}\right]=0.$
(159)

Here we can substitute $\int_{-\infty}^{\infty}\mathrm{d}\omega$ by
$\int_{-\infty}^{0-}\mathrm{d}\omega+\int_{0+}^{\infty}\mathrm{d}\omega$, and
the above simplifies to

$\displaystyle\left(\int_{-\infty}^{0-}+\int_{0+}^{\infty}\right)\frac{\mathrm{d}\omega}{2\pi}\operatorname{Im}\left[D\frac{\partial\Sigma}{\partial\omega}-\tilde{D}\frac{\partial\tilde{\Sigma}}{\partial\omega}\right]=0.$
(160)

Noticing

$\displaystyle\operatorname{Im}\left(\tilde{D}\frac{\partial\tilde{\Sigma}}{\partial\omega}\right)=\frac{AB}{A_{0}B_{0}}\operatorname{Im}\left(D_{0}\frac{\partial\Sigma_{0}}{\partial\omega}\right),$
(161)

we have

$\displaystyle\iota_{2}(\nu)=\frac{AB}{A_{0}B_{0}}\iota_{2}(\nu=0).$ (162)

For $\nu=0$, the impurity Hilbert space is trivial, thus $G_{S}(\tau)$ should
be identically zero, implying $A_{0}^{2}=B_{0}^{2}\neq 0$. We know that
$A_{0}>0$, but the sign of $B_{0}$ can be arbitrary. Let us assume $B_{0}>0$,
which may follow from the assumption that $\operatorname{Im}G_{AB}(\tau)$ does
not change sign at zero temperature, or can be checked numerically, then
$\theta_{0+}(\nu=0)=\pi\Delta+2n\pi$ for some integer $n$. Using
$\iota_{1}+\iota_{2}=2\nu-1$, we obtain $\iota_{2}(\nu=0)=4n+2\Delta-1$.
Combining all these results, it follows that

$\displaystyle-\frac{2}{\pi}\theta_{0+}+\frac{\sin(2\theta_{0+})}{\sin(2\pi\Delta)}(4n+2\Delta-1)=2\nu-1,$
(163)

where $AB/(A_{0}B_{0})=\sin(2\theta_{0+})/\sin(2\pi\Delta)$ has been used. The
system has a particle-hole symmetry:
$\gamma_{A}\mapsto\gamma_{A},\gamma_{B}\mapsto-\gamma_{B},\xi\mapsto-\xi$ and
$\nu\mapsto 1-\nu$, which implies $A(\nu)=A(1-\nu)$ and $B(\nu)=-B(1-\nu)$.
Assuming the continuity of $\theta_{0+}$ as a function of $\nu$, this symmetry
and $A$ being nonnegative imply that
$-\pi\Delta+2n\pi\leq\theta_{0+}\leq\pi\Delta+2n\pi$ as $\nu$ varies. In
particular, $\theta_{0+}(\nu=1/2)=2n\pi$. Plugging this into the above
equation yields $n=0$. We finally conclude that $\theta_{0+}$ satisfies the
following transcendental equation:

$\displaystyle\frac{\sin(2\theta_{0+})}{\sin(2\pi\Delta)}=\frac{1}{2\Delta-1}\left(2\nu-1+\frac{2}{\pi}\theta_{0+}\right),$
(164)

which then determines the ratio $B/A=\tan(\theta_{0+})/\tan(\pi\Delta)$.

##  Appendix C Perturbative calculations for the $\mathbb{Z}_{2}$ symmetric
impurity

We start with the $O(N)$ symmetric model in Eq. (100). We normalize the $O(N)$
traceless symmetric operator $t_{\alpha_{1}\alpha_{2}}$ as

$\displaystyle\langle
t_{\alpha_{1}\alpha_{2}}(x_{1})t_{\beta_{1}\beta_{2}}(x_{2})\rangle$
$\displaystyle=\frac{1}{x^{2\Delta_{t}}_{12}}\frac{1}{2}\left(\delta_{\alpha_{1}\beta_{1}}\delta_{\alpha_{2}\beta_{2}}+\delta_{\alpha_{1}\beta_{2}}\delta_{\alpha_{2}\beta_{1}}-\frac{2}{N}\delta_{\alpha_{1}\alpha_{2}}\delta_{\beta_{1}\beta_{2}}\right)$