CFT and Lattice Correlators Near an RG Domain Wall between Minimal Models

Cameron V. Cogburna111Email<EMAIL_ADDRESS>A. Liam Fitzpatricka222Email:
<EMAIL_ADDRESS>Hao Gengb333Email<EMAIL_ADDRESS>
aDepartment of Physics, Boston University, Boston, MA 02215, USA
bJefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

Abstract

Conformal interfaces separating two conformal field theories (CFTs) provide
maps between different CFTs, and naturally exist in nature as domain walls
between different phases. One particularly interesting construction of a
conformal interface is the renormalization group (RG) domain wall between
CFTs. For a given Virasoro minimal model $\mathcal{M}_{k+3,k+2}$, an RG domain
wall can be generated by a specific deformation which triggers an RG flow
towards its adjacent Virasoro minimal model $\mathcal{M}_{k+2,k+1}$ with the
deformation turned on over part of the space. An algebraic construction of
this domain wall was proposed by Gaiotto in Gaiotto:2012np . In this paper, we
will provide a study of this RG domain wall for the minimal case $k=2$, which
can be thought of as a nonperturbative check of the construction. In this case
the wall is separating the Tricritical Ising Model (TIM) CFT and the Ising
Model (IM) CFT. We will check the analytical results of correlation functions
from the RG brane construction with the numerical density matrix
renormalization group (DMRG) calculation using a lattice model proposed in
Grover:2012bm ; Grover:2013rc , and find a perfect agreement. We comment on
possible experimental realizations of this RG domain wall.

###### Contents

  1. 1 Introduction and Summary
  2. 2 RG Brane Lattice Construction
    1. 2.1 General Strategy
    2. 2.2 Lattice Model for Tricritical Ising to Ising RG Brane
  3. 3 The Computation of Correlators In the Appearance of the RG Brane
    1. 3.1 The General Strategy
    2. 3.2 $\langle\epsilon^{TIM}\epsilon^{TIM}\rangle_{RG}$
    3. 3.3 $\langle\epsilon^{IM}\epsilon^{IM}\rangle_{RG}$
    4. 3.4 Branching Rule and Two-point Function for $\epsilon^{\prime TIM}$
  4. 4 Comparison of Lattice and CFT Calculations
    1. 4.1 Mapping of Operators
    2. 4.2 Numeric Comparisons
  5. 5 Experimental Realizations
    1. 5.1 Engineering the RG Domain Wall on the Boundary of a Topological Superconductor
    2. 5.2 Engineering the RG Domain Wall in a Rydberg Chain
  6. A Algebraic Construction for Minimal Models and Their Tensor Product
    1. A.1 Coset Construction of Unitary Virasoro Minimal Models
    2. A.2 Hidden Symmetry in the Tensor Product of Consecutive Minimal Models
      1. A.2.1 The Hidden Symmetry
      2. A.2.2 Another Representation of the $\widetilde{\mathcal{B}}$ with an Explicit Supersymmetry
  7. B Review of Gaiotto’s Proposal For RG Brane
    1. B.1 Useful Properties of Topological Defects
    2. B.2 RG Flow of the Topological Defect $\mathcal{D}_{r,1}^{(k+2)}$
    3. B.3 Extended Symmetry Algebra on RG Domain Wall from Topological Defect
    4. B.4 Explicit Construction of the RG Domain Wall
    5. B.5 Computation of One-Point Functions
  8. C Topological Superconductors and Majorana Fermions
    1. C.1 A Toy Hamiltonian and Its Topological Properties
    2. C.2 Localized Majorana Modes on the Boundary– The Ising Model

##  1 Introduction and Summary

A useful concept in the study of Quantum Field Theory (QFT) is the idea of a
‘space of QFTs’ Douglas:2010ic . If we view QFTs as Renormalization Group (RG)
flows between Conformal Field Theory (CFT) fixed points, then the space of
QFTs can be envisioned as a network of CFT points connected to each other by
paths along which a family of quantum field theories are defined interpolating
between an ultraviolet (UV) CFT and an infrared (IR) CFT. These QFTs are
parameterized by the energy scale along the RG flow. In general, the dynamics
along such flows is complicated and does not benefit from the relative
rigidity of its CFT endpoints. Instead of studying the full RG flows
themselves, an appealing construction is that of ‘RG branes’ (aka RG Domain
Walls), which capture much of the information of an RG flow but in a simpler
setting. RG branes are constructed by taking a relevant deformation that
triggers the RG flow from the UV CFT to the IR CFT and turning it on over only
part of space, so that in the infrared regime one obtains the ‘IR CFT’ in one
spatial region and the ‘UV CFT’ everywhere else. The boundary between these
two regions is the RG brane, which thereby collapses the entire RG flow to the
geometric action of moving across this boundary. Moreover, symmetric choices
of the boundary can preserve a subset of the conformal symmetries of the CFT
endpoints.

Another advantage of RG branes is that they are relatively easy to engineer in
practice. In the context of numerical simulations, for instance with a
computation on a lattice, one simply has to choose parameters in the
underlying theory to pick out the UV CFT on half of the space and the IR CFT
on the rest of the space. In fact, this approach will be one of our main tools
for studying RG branes in this paper. Moreover, this technique translates into
a well-defined protocol for creating an RG brane experimentally, assuming one
has the flexibility to tune to a critical point over only part of space.

Our main goal in this paper will be to compute physical observables in an
interesting class of RG branes. We will focus on two-point correlation
functions, partly for simplicity but also partly with an eye towards the
potential connection to experimental measurements in the future. In CFTs, two-
point functions in the presence of a boundary are comparable in complexity to
four-point functions in its absence, so the theoretical calculation of them
provides the opportunity to predict a fairly complicated set of observables
that might be measurable in practice.

We will focus on a specific RG brane scenario that brings together two
remarkable pieces of work, one from a CFT perspective and one from an
underlying microscopic perspective. The first of these is a recent proposal
Grover:2012bm ; Grover:2013rc  for an experimental realization, as well as an
explicit lattice Hamiltonian describing it, of a supersymmetric quantum
critical point that can be obtained by tuning only a single parameter. The
specific instance of this proposal that we will use produces a phase diagram
with two distinct phases, one gapped and one gapless, separated by a critical
point. Moreover, the gapless phase is described in the IR by the 2d Ising
Model CFT, and the critical point between the phases is decribed by the
Tricritical Ising Model CFT. This lattice Hamiltonian can therefore be used to
construct an RG brane separating the 2d Ising Model (IM) CFT from the
Tricritical Ising Model (TIM) CFT, and will give us the first numeric handle
for computing observables numerically.

The second result we will use Gaiotto:2012np  is a proposal for the RG brane
between consecutive Virasoro minimal models, directly within the CFT
description.444See Poghosyan:2014jia ; Poghosyan:2022mfw ; Poghosyan:2023brb
for the extension of the construction to other 2d CFTs and Nguyen:2022lie  for
an interesting construction of a domain wall between different symmetry-
protected topological (SPT) phases in the same spirit. The first two Virasoro
minimal models are exactly the Ising Model CFT and the Tricritical Ising Model
CFT, with central charges $c={1\over 2}$ and $c={7\over 10}$ respectively.
This proposal will give us a second handle for computing observables, this
time directly in the IR limit. In particular, we will compute two-point
functions of operators in the presence of the RG brane using the CFT proposal
of Gaiotto:2012np , and compare to computations using a DRMG analysis of the
lattice Hamiltonian from Grover:2013rc . We will find remarkably good
agreement.

The paper is organized as follows. In section 2 we spell out how the RG brane
in our paper in constructed on the lattice. In section 3, we compute several
energy-energy correlators near the RG domain wall between the Tricritical
Ising Model CFT and the Ising Model CFT. In section 4, we compare our
analytical results in 3 with the numerical results obtained from the lattice
construction in 2. In section 5, we discuss potential experimental
consequences and other future directions. We review many technical details of
the RG brane proposal in Gaiotto:2012np  in a series of appendices: in section
A we review the coset (Sugawara) construction of minimal Virasoro models, and
in section B, we review the construction of the RG brane itself. In section C
we review some useful properties of topological superconductors relevant to
our experimental proposal.

##  2 RG Brane Lattice Construction

###  2.1 General Strategy

We are interested in RG branes connecting a “UV” CFT on one side to an “IR”
CFT on the other side. To construct such an RG brane in a lattice model, we
need to be able to dial the parameters of the lattice Hamiltonian such that
for some parameters, the low energy limit of the lattice model is described by
the UV CFT, and for other parameters the low energy limit is described by the
IR CFT. The “UV” and “IR” labels of the two CFTs in this context are solely
relative to each other, as both of them describe the physics at infrared
scales compared to the underlying lattice spacing. In order for the UV CFT to
be able to have an RG flow to the IR CFT, there must exist a relevant
deformation of the UV CFT that triggers this RG flow. In the space of lattice
parameters, this means that points describing the UV CFT require tuning (at
least) one parameter to a critical point (or surface). Moreover, it is then
possible to detune away from this point in some direction such that the low
energy limit is described by the IR CFT. We can choose to parameterize this
direction as a coupling $h$ in our Hamiltonian:

$H=H_{1}+hH_{2},$ (2.1)

such that $h=h_{*}$ is the critical value that produces the UV CFT at low
energies, and $h>h_{*}$ produces the IR CFT at low energies. This setup is
depicted in Fig. 1.

Given a Hamiltonian with these properties, it is then straightforward to
construct the RG brane in terms of the lattice theory: one simply has to tune
$h=h_{*}$ on part of the space, and take $h>h_{*}$ on the rest of the
space,555One should not choose $h$ to be too close to or too far from $h_{*}$.
If $h-h_{*}$ is too small, then this corresponds to setting a very tiny
coefficient in front of the relevant deformation of the UV CFT, so that the IR
CFT is only reached at very long distances. For an infinite sized lattice,
this would not present any problem, but in a finite-sized system it would
prevent one from reaching the IR CFT. On the other hand, if $h-h_{*}$ is taken
to be too large, then it is no longer clear that detuning $h$ away from
$h_{*}$ can be described as a local relevant deformation of the UV CFT.  and
the RG brane lies on the boundary separating these two regions.

Figure 1: Phase diagram required for the construction of the RG brane. At a
critical value $h_{*}$ of the coupling, the low-energy limit is described by
the “UV” CFT, whereas for $h>h_{*}$ it is described by the “IR” CFT. The RG
brane is constructed by tuning $h$ to $h_{*}$ on half of the space, and taking
$h$ slightly greater than $h_{*}$ on the other half of the space. The specific
behavior at $h<h_{*}$ is irrelevant for our discussion.

###  2.2 Lattice Model for Tricritical Ising to Ising RG Brane

Although the general setup described in the previous subsection should be
possible to be realized in many different examples, in this work we will focus
mostly on the specific lattice system proposed in Grover:2012bm ;
Grover:2013rc . Following their notation, the Hamiltonian is $H=H_{1}+hH_{2}$
with

$\displaystyle H_{1}$
$\displaystyle=\sum_{i}\left[\sigma_{i}^{z}\sigma_{i+1}^{z}+\sigma_{i}^{x}+{\mathcal{J}}\left(\mu_{i,a}^{z}\mu_{i,b}^{z}+\mu_{i,b}^{z}\mu_{i+1,a}^{z}\right)+g\left(\sigma_{i}^{x}\mu_{i,a}^{z}-\sigma_{i}^{z}\sigma_{i+1}^{z}\mu_{i,b}^{z}\right)\right],$
(2.2) $\displaystyle hH_{2}$
$\displaystyle=-h\sum_{i}(\mu_{i,a}^{x}+\mu_{i,b}^{x}),$

where $\mu_{i,a},\mu_{i,b}$ and $\sigma_{i}$ are all Pauli matrices for
spin-1/2 spins. The virtue of this model is that, due to an underlying
symmetry of the theory, only one tuning is necessary in order to reach the
(supersymmetric) Tricritical Ising Model fixed point.

To roughly understand why, as explained in Grover:2013rc , first note that if
we neglect the terms proportional to ${\mathcal{J}},g$, and $h$, the
Hamiltonian is just that of the critical Ising Model,

$H_{*}=\sum_{i}\sigma_{i}^{z}\sigma_{i+1}^{z}+\sigma_{i}^{x},$ (2.3)

which describes a gapless free fermion. The coupling between $\sigma$ and
$\mu$ in general could gap out the system, but does not as long as $h$ is
sufficiently large. The reason is that at large (positive) $h$, the $\mu$
spins do not order along the $z$ direction, in which case the vanishing mass
gap is protected by an unbroken $\mathbb{Z}_{2}$ symmetry that flips the sign
of $\mu^{z}$ ($\mu^{z}_{i,a}\rightarrow-\mu^{z}_{i,b}$ and
$\mu^{z}_{i,b}\rightarrow-\mu^{z}_{i+1,a}$) and acts nontrivially
(essentially, as the high-temperature/low-temperature duality map) on
$\sigma_{i}$ ($\sigma^{z}_{i}\sigma^{z}_{i+1}\leftrightarrow\sigma^{x}_{i}$).
As $h$ is decreased, eventually there is a phase transition to the ordered
phase where $\langle\mu^{z}\rangle\neq 0$. At this critical point, the order
parameter behaves as a massless scalar degree of freedom, allowing for a
supersymmetric spectrum. For smaller values of $h$, the theory is gapped.

The upshot is that for any value of model parameters ${\mathcal{J}}$ and $g$,
there is a critical value of $h$ where the low-energy limit is described by
the Tricritical Ising Model, and for larger values of $h$ the low energy limit
is described by the critical Ising Model CFT. The authors of Grover:2012bm
found that for the specific choice of ${\mathcal{J}}=1,g={1\over 2}$, this
critical value is approximately $h_{*}=1.62$.

In field theory language, the theory can be described as a single chiral
superfield $\Phi$ with a superpotential given by

$W(\Phi)=\kappa\Phi+r\Phi^{3}.$ (2.4)

Supersymmetry is broken spontaneously if $W^{\prime}(\Phi)\neq 0$.
Semiclassically, at ${r\over\kappa}<0$ the vacuum has $\langle\Phi\rangle\neq
0$ and $W^{\prime}(\Phi)=0$, so supersymmetry is preserved but a
$\mathbb{Z}_{2}$ symmetry under which $\Phi$ changes sign is spontaneously
broken and the theory is gapped; by contrast, at ${r\over\kappa}>0$,
$\langle\Phi\rangle=0$ and $W^{\prime}(\Phi)\neq 0$, so the supersymmetry of
the theory is broken spontaneously, producing a massless fermionic Goldstino,
which is the massless Majorana fermion of the $c=1/2$ critical Ising Model.
This semiclassical picture continues to hold at the quantum level (see e.g.
Fitzpatrick:2019cif ), and the critical point between the two phases is again
described by the Tricritical Ising Model CFT.

We numerically solve for the RG brane using a Density Matrix Renormalization
Group (DMRG) method based on the iTensor library itensor ; itensor-r0.3 . This
DMRG algorithm is controlled by a few key parameters, as explained in the
documentation. For our simulations we found the parameter values of `nsweeps`
= 15, `maxdim` $=[10,20,100,175,250]$, and `cutoff` = 1E -12 were sufficient
for even large (eg. $N=300$) lattices. In setting up the lattice, periodic
boundary conditions are enforced. Therefore the geometry is such that we have
a tube, the boundary of which is a circle. On this circle the RG brane is
placed at 0 and 180 degrees, with the Ising model ($h=2$) on one side and the
Tricritical Ising Model ($h=1.62$) on the other.

Discretizing a system on a lattice inherently introduces lattice spacing and
finite volume effects. In Sec. 4.1 we expand the $\mu$ lattice operator in
terms of the CFT operators, the coefficients of which are found by computing
the finite volume two-point function when the lattice is entirely in either
the Ising CFT phase or Tricritical Ising CFT phase. These coefficients
precisely determine the overall normalization between the numerical lattice
calculation and the analytic CFT calculation, the result of which is shown in
Sec. 4.2.

##  3 The Computation of Correlators In the Appearance of the RG Brane

In this section, we extend the study of the RG domain wall in Gaiotto:2012np
to the computation of specific four-point functions (in the unfolded picture).
These four-point functions reduce to two-point functions of composite
operators in the folded description. We firstly spell out our general strategy
in computing the relevant four-point functions using Gaiotto’s construction of
the RG domain wall. We then explicitly consider the case of the RG domain wall
of the Tricitical Ising Model. We compute two such four-point functions
following our strategy, compare the results with numerical calculation by DMRG
and find precise agreement.

A priori, it is not at all obvious that one should be able to do analytic
computations of correlation functions in the presence of the RG brane. In
general, even if the UV and IR CFTs are rational, so that all their
correlators can be computed analytically, it will not necessarily be the case
that the RG brane between them is also rational, and one might have to accept
that the resulting correlators cannot be computed based on the algebra alone.
A class of interfaces that do preserve the individual algebras of the UV and
IR CFTs are called rational interfaces. From this point of view, what is
special about rational interfaces is that they satisfy boundary conditions
that glue the chiral sector of the algebra of the theory
$\overline{\mathcal{T}}_{L}\times\mathcal{T}_{R}$ to the anti-chiral sector by
an automorphism (i.e. the map preserves the commutation relations between the
symmetry generators). With the automorphism specified the rational interfaces
can be classified by Cardy’s algebraic description. However, there is an
inherent tension between preserving the UV and IR CFT algebras and
simultaneously coupling them together in a nontrivial way. The reason is that
the UV and IR CFT algebras strongly constrain the allowed form of the
correlators of the theory, and in fact the reason why rational CFTs are
analytically tractable is that the algebra almost completely determines their
correlators. Consequently, if the algebra completely factorizes into the UV
and IR CFT algebras, then their correlators also factorize into UV and IR CFT
correlators. In more technical terms, the automorphism that maps the chiral
and anti-chiral algebras into each other across the RG brane usually doesn’t
mix the $L$ and $R$ sectors of the algebra. In order to mix them in a
nontrivial way, there must be a hidden symmetry that relates the $L$ and $R$
sectors. Naively, our setup has no hidden symmetry and therefore one might
expect that the RG brane is not related to any rational interface.
Fortunately, this expectation is too pessimistic, and Gaiotto Gaiotto:2012np
has proposed a remarkably elegant construction of RG interfaces between
consecutive 2d Virasoro minimal model CFTs using rational interfaces by
identifying the RG branes as the rational interfaces of a different 2d CFT
than $\overline{\mathcal{T}}_{L}\times\mathcal{T}_{R}$ which contains a larger
symmetry algebra. This construction is based on the observation that the
larger symmetry algebra is $\widetilde{\mathcal{B}}$ (that we review in detail
in Sec. A.2) and it extensively uses results reviewed in App. A. The basic
observation that leads to the realization of $\widetilde{\mathcal{B}}$ as the
proper algebra is from the considerations of topological defects
Frohlich:2004ef  and the perturbative results of the RG flow
Zamolodchikov:1987ti ; Fredenhagen:2009tn . We also review Gaiotto’s algebraic
proposal for the RG brane in detail in App. B. The proposal can be roughly
summarized as the statement that the RG brane is simply the product of a
conformal interface that maps operators from the tensor product theory to the
‘B’ theory with the enhanced $\widetilde{\mathcal{B}}$ chiral algebra, times a
Cardy boundary condition with respect to this algebra.

###  3.1 The General Strategy

Our main goal is to compute correlation functions of local operators in the
presence of the RG brane between nearest neighbor minimal models,
$\mathcal{M}_{k+3,k+2}$ and $\mathcal{M}_{k+2,k+1}$. We will mostly focus on
the case of the RG brane between TIM and Ising (i.e. $k=2$), though the
generalization to other minimal models is conceptually straightforward. The
simplest kind of two-point function one might try to consider is that of two
operators in, say, $\mathcal{M}_{k+2,k+1}$:666The individual operators in TIM
and Ising sit in the Kac table as follows:
$\begin{split}&1^{IM}=\phi^{IM}_{(1,1)}=\phi^{IM}_{(2,3)}\,,\quad\epsilon^{IM}=\phi^{IM}_{(2,1)}=\phi^{IM}_{(1,3)}\,,\quad
1^{TIM}=\phi^{TIM}_{(1,1)}=\phi^{TIM}_{(3,4)}\,,\\\
&\epsilon^{TIM}=\phi^{TIM}_{(1,2)}=\phi^{TIM}_{(3,3)}\,,\quad\epsilon^{\prime
TIM}=\phi^{TIM}_{(1,3)}=\phi^{TIM}_{(3,2)}\,,\quad\epsilon^{\prime\prime
TIM}=\phi^{TIM}_{(1,4)}=\phi^{TIM}_{(3,1)}\,.\end{split}$ (3.1)

$\langle\phi_{r,s}^{(k+1)}(x_{1},\overline{x_{1}})\phi_{r,s}^{(k+1)}(x_{2},\overline{x_{2}})\rangle_{RG}.$
(3.2)

However, it will be useful to instead start with correlators where each local
operator is a product of one operator from $\mathcal{M}_{k+2,k+1}$ and one
from $\mathcal{M}_{k+3,k+2}$:

$\langle(\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t})(x_{1},\overline{x_{1}})(\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t})(x_{2},\overline{x_{2}})\rangle_{RG}\,.$
(3.3)

When $(s,t)=(1,1)$, this reduces to the former type of two-point function (and
similarly if $(r,s)=(1,1)$). However, our strategy for computing correlators
will be first to map operators from the tensor product, or “$\mathcal{A}$”,
theory $\mathcal{M}_{k+2,k+1}\times\mathcal{M}_{k+3,k+2}$ (which arises from
the folded description) into operators in the “$\widetilde{\mathcal{B}}$”
theory, which can be described in terms of the Ising model times the
$(k-1)$-th supersymmetric minimal model $\mathcal{SM}_{k+1,k+3}$. For $k=2$,
both descriptions are (loosely speaking) TIM $\times$ Ising, but the mapping
of operators is still nontrivial.

We will work out in detail the two-point function of $\epsilon$ from Ising and
the two-point function of $\epsilon$ from TIM. These operators are represented
in the $\widetilde{\mathcal{B}}$ theory as follows Crnkovic:1989ug :

$(\epsilon^{TIM})_{A}\propto(\sigma^{IM}\sigma^{TIM})_{B},\qquad(\epsilon^{IM})_{A}\propto(\sigma^{IM}\sigma^{\prime
TIM})_{B}.$ (3.4)

More generally, the strategy of computing such two-point function is the
following:

  1. 1.

Use the branching rule to map the composite operator
$(\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t})(x_{1},\overline{x_{1}})$ from the
$\mathcal{M}_{k+2,k+1}\times\mathcal{M}_{k+3,k+2}$ description to
$[r,d^{\prime};t]\otimes[d,\widetilde{d};d^{\prime}]$ in the
$\mathcal{SM}_{k+3,k+1}\times\mathcal{M}_{4,3}$ description.777This mapping,
though nontrivial, is constrained by preserving total conformal weight and the
fact that $r+t$ even lives in the NS sector of the $\mathcal{SM}_{k+3,k+1}$
and odd in the R sector; in subsection 3.4 we demonstrate how to use this
constraint to work out the branching map for $\epsilon^{\prime TIM}$.

  2. 2.

Do the bulk OPE expansion for $\mathcal{SM}_{k+3,k+1}$ and $\mathcal{M}_{4,3}$
separately to have
$[r,d^{\prime};t]\times[r,d^{\prime};t]\rightarrow[\underline{r},\underline{d^{\prime}};\underline{t}]$
and
$[d,\widetilde{d};d^{\prime}]\otimes[d,\widetilde{d};d^{\prime}]\rightarrow[\underline{d},\underline{\widetilde{d}};\underline{d^{\prime}}]$
with known conformal block and OPE coefficients.

  3. 3.

Project the
$[\underline{r},\underline{d^{\prime}};\underline{t}]\otimes[\underline{d},\underline{\widetilde{d}};\underline{d^{\prime}}]$’s
which can not be mapped back to the OPE spectrum of
$\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t}$ with itself in the
$\mathcal{M}_{k+2,k+1}\times\mathcal{M}_{k+3,k+2}$ description.

  4. 4.

Look for the boundary operator expansion (BOE) coefficient of
$[\underline{r},\underline{d^{\prime}};\underline{t}]\otimes[\underline{d},\underline{\widetilde{d}};\underline{d^{\prime}}]$
from the exact Cardy state Equ. (B.7).

  5. 5.

Multiply the BOE coefficients with the corresponding bulk conformal blocks and
bulk OPE coefficients, and sum over all possible
$\underline{r},\underline{d},\underline{d^{\prime}},\underline{t}$ and
$\underline{\widetilde{d}}$.

This procedure fixes the final correlator up to an overall proportionality
constant. This constant comes from the mapping in the first step, and can be
fixed by the one-point function calculated using the method in Sec.
B.5.888Note that there are operators for which the overall constant cannot be
fixed by the last step. These are operators
$\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t}$ whose one-point function is zero (for
example $r+t$ is odd). Alternatively, the overall coefficient can be
determined simply by fixing the leading OPE singularity of the two-point
function.

In general, the prediction for the correlators in the presence of the RG brane
will take the form

$\displaystyle\langle(\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t})(x_{1},\overline{x_{1}})(\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t})(x_{2},\overline{x_{2}})\rangle_{RG}$
(3.5)
$\displaystyle=\sum^{\prime}_{i,j}\langle(\phi_{i}^{(k+1)}\phi_{j}^{(k+2)})_{B}|RG\rangle
C^{(k+1)}_{(r,s),(r,s),i}C_{(s,t),(s,t),j}^{(k+2)}G^{(k+1)}_{(r,s),(r,s),i}(z)G_{(s,t),(s,t),j}^{(k+2)}(z),$

where the $C$s and $G$s are OPE coefficients and conformal blocks in the ‘B’
theory, respectively, the conformal cross ratio $z$ is defined below, and the
prime on the sum indicates that certain terms in the sum are discarded.

$\crossproduct$$O(x_{1})$$\crossproduct$$O(x_{2})$ Figure 2: An illustration
of the calculation where $O(x)\equiv 1^{IM}\times\epsilon^{TIM}(x)$ and we do
the bulk OPE first then compute the resulting one point function due to the
appearance of the Cardy boundary.

###  3.2 $\langle\epsilon^{TIM}\epsilon^{TIM}\rangle_{RG}$

First, we will calculate the two-point function of the TIM $\epsilon$ operator
in the presence of the RG brane, using the branching rule we have
$(1^{IM}\epsilon^{TIM})_{A}\propto(\sigma^{IM}\sigma^{TIM})_{B}$ from the
$\mathcal{A}$ to the $\widetilde{\mathcal{B}}$ theory. Therefore, using the
folded description, our task is to compute the following correlator in the
$\widetilde{\mathcal{B}}$-theory

$\displaystyle\langle(\sigma^{IM}\sigma^{TIM})(x_{1},\overline{x}_{1})(\sigma^{IM}\sigma^{TIM})(x_{2},\overline{x}_{2})\rangle_{RG}$
(3.6)
$\displaystyle=\langle(\sigma^{IM}\sigma^{TIM})(x_{1})(\sigma^{IM}\sigma^{TIM})(\overline{x}_{1})(\sigma^{IM}\sigma^{TIM})(x_{2})(\sigma^{IM}\sigma^{TIM})(\overline{x}_{2})\rangle\,.$

The correlator in the unfolded description is a sum over four-point conformal
blocks, where the four external operators are $(\sigma^{IM}\sigma^{TIM})$ as
written above. These conformal blocks are simply products of the conformal
blocks of $\sigma^{IM}$ and $\sigma^{TIM}$ in the Ising model and TIM,
respectively, which are known in closed form (see below). To compute the
coefficient of each conformal block, we first perform the operator production
expansion (OPE) in the bulk, which is inherited from the individual Ising and
TIM factors:

$[\sigma^{IM}]\times[\sigma^{IM}]=[1^{IM}]+[\epsilon^{IM}]\,,\quad[\sigma^{TIM}]\times[\sigma^{TIM}]=[1^{TIM}]+[\epsilon^{TIM}]+[\epsilon^{\prime
TIM}]+[\epsilon^{\prime\prime TIM}]\,.$ (3.7)

The OPE of the products of chiral components of the fields in the $B$ theory
is, roughly speaking, a sort of “square root” of the products of OPE of the
non-chiral fields from the Ising Model and the Tricritical Ising Model, as
explained in Crnkovic:1989ug  and described in more detail below. Finally, to
obtain the coefficient of the conformal block, we perform the bulk-to-boundary
expansion of each operator in the bulk OPE, which simply contributes an
additional factor of the bulk operator’s one-point function. This OPE,

$\langle\mathcal{O}(x)\times\mathcal{O}(y)\rangle_{RG}=\sum_{i}C_{\mathcal{O}\mathcal{O}i}(x,y)\langle\mathcal{O}_{i}(y)\rangle_{RG},$
(3.8)

is depicted in Fig.2. In principle, we could evaluate the above expression by
doing a brute-force numerical sum over all operators $\mathcal{O}_{i}$ in the
bulk OPE of $\epsilon^{TIM}\times\epsilon^{TIM}$ in the tensor product theory.
The advantage of using the full chiral algebra of the
$\widetilde{\mathcal{B}}$ theory is that there are only a finite number of
primary operators, and so by computing the full conformal blocks we reduce the
sum to a finite number of terms. A useful consistency check that we will
perform of our method is that it reproduces the prediction we get for these
terms purely from using the standard bulk OPE for $\epsilon^{TIM}$ in TIM,
times the RG brane one-point functions, and in fact we will check enough terms
this way to independently fix all the coefficients of the
$\widetilde{\mathcal{B}}$ theory conformal blocks.

To evaluate the correlator (3.6), we need to use the OPE of the chiral
components $(\sigma^{IM}\sigma^{TIM})$ in the $\widetilde{\mathcal{B}}$
theory, in order to sum over the corresponding conformal blocks. As explained
in Crnkovic:1989ug ,999See for instance, their equation (3.6) and surrounding
text. this OPE is not just a simple product of the OPEs of the scalar
$\sigma^{IM}\times\sigma^{IM}$ and $\sigma^{TIM}\times\sigma^{TIM}$ OPEs in
the IM and TIM theories, but instead involves both a projecting-out of certain
cross-products between the two OPEs as well as taking a square root of the
non-holomorphic OPE coefficients. Of the eight operators one would naively get
in the product of the two OPEs in (3.7), four get projected out. A convenient
way to infer this is that the OPE must be consistent with the $\mathcal{A}$
theory OPE $\epsilon^{TIM}\times\epsilon^{TIM}\sim 1^{TIM}+\epsilon^{\prime
TIM}+\textrm{descendants}$, and therefore only products of operators where the
total dimension of the product is equal that of $1^{TIM}$ or $\epsilon^{\prime
TIM}$ plus integers can appear. As a result, only conformal blocks
corresponding to the appearance of $1^{IM}1^{TIM},1^{IM}\epsilon^{\prime
TIM},\epsilon^{IM}\epsilon^{TIM}\text{ and
}\epsilon^{IM}\epsilon^{\prime\prime TIM}$ in the
$\sigma^{IM}\sigma^{TIM}\times\sigma^{IM}\sigma^{TIM}$ OPE survive. The
correlator is given by the following formula

$\displaystyle\langle(\sigma^{IM}\sigma^{TIM})(x_{1},\overline{x}_{1})(\sigma^{IM}\sigma^{TIM})(x_{2},\overline{x}_{2})\rangle_{RG}={1\over|x_{1}-x_{2}|^{2/5}}G^{TIM}_{\epsilon\epsilon}\left(\left|{x_{1}-x_{2}\over
x_{1}-\overline{x}_{2}}\right|^{2}\right),$ (3.9) $\displaystyle\quad
G^{TIM}_{\epsilon\epsilon}(z)\equiv
z^{1/5}\sum^{\prime}_{\phi_{i}=1,\epsilon\atop\phi_{j}=1,\epsilon,\epsilon^{\prime},\epsilon^{\prime\prime}}\langle(\phi_{i}^{IM}\phi_{j}^{TIM})_{B}|RG\rangle
C_{\sigma\sigma\phi_{i}}^{IM}C_{\sigma\sigma\phi_{j}}^{TIM}G^{IM}_{\sigma\sigma\phi_{i}}(z)G^{TIM}_{\sigma\sigma\phi_{j}}(z).$

where the prime on the sum indicates that only the products of conformal
blocks described above are included. The Ising model conformal blocks
$G^{IM}_{\sigma\sigma\phi_{i}}(z)$ are

$\displaystyle G_{\sigma\sigma
1}^{IM}(z)={1\over\sqrt{2}(z(1-z))^{1/8}}\sqrt{1+\sqrt{1-z}},\qquad
G_{\sigma\sigma\epsilon}^{IM}(z)={\sqrt{2}\over(z(1-z))^{1/8}}\sqrt{1-\sqrt{1-z}},$
(3.10)

and the TIM conformal blocks $G^{TIM}_{\sigma\sigma\phi_{i}}(z)$ are
Qiu:1986if

$\displaystyle G_{\sigma\sigma 1}^{TIM}(z)={1\over
z^{3/40}}\Big{(}{\sqrt{\sqrt{1-z}+1}\,_{2}F_{1}\left(-{2\over 5},{1\over
5};{2\over
5};z\right)\over\sqrt{2}(1-z)^{3/40}}+{z(1-z)^{1/40}\,_{2}F_{1}\left({1\over
5},{4\over 5};{7\over 5};z\right)\over 2\sqrt{2}\sqrt{\sqrt{1-z}+1}}\Big{)},$
(3.11) $\displaystyle G_{\sigma\sigma\epsilon}^{TIM}(z)=z^{{1\over
40}}\left({\sqrt{\sqrt{1-z}+1}(1-z)^{1/40}\,_{2}F_{1}\left(-{1\over 5},{2\over
5};{3\over
5};z\right)\over\sqrt{2}}+{\sqrt{2}z(1-z)^{21/40}\,_{2}F_{1}\left({4\over
5},{7\over 5};{8\over 5};z\right)\over 3\sqrt{\sqrt{1-z}+1}}\right)$
$\displaystyle G_{\sigma\sigma\epsilon^{\prime}}^{TIM}(z)=z^{{21\over
40}}\left(2\sqrt{2}\sqrt{\sqrt{1-z}+1}(1-z)^{21/40}\,_{2}F_{1}\left({4\over
5},{7\over 5};{8\over
5};z\right)-{3\sqrt{2}(1-z)^{1/40}\,_{2}F_{1}\left(-{1\over 5},{2\over
5};{3\over 5};z\right)\over\sqrt{\sqrt{1-z}+1}}\right)$ $\displaystyle
G_{\sigma\sigma\epsilon^{\prime\prime}}^{TIM}(z)=z^{17/40}\left({28\sqrt{2}\,_{2}F_{1}\left(-{2\over
5},{1\over 5};{2\over
5};z\right)\over\sqrt{\sqrt{1-z}+1}(1-z)^{3/40}}-14\sqrt{2}\sqrt{\sqrt{1-z}+1}(1-z)^{1/40}\,_{2}F_{1}\left({1\over
5},{4\over 5};{7\over 5};z\right)\right).$

The relevant OPE coefficients are Dotsenko:1984ad ; Dotsenko:1984nm ;
Esterlis:2016psv

$\begin{split}C^{IM}_{\sigma\sigma 1}&=1\,,\\\
C^{IM}_{\sigma\sigma\epsilon}&={1\over 2}\,,\\\ C^{TIM}_{\sigma\sigma
1}&=1\,,\\\ C^{TIM}_{\sigma\sigma\epsilon}&={5\sqrt{21\Gamma\left(-{7\over
4}\right)}\Gamma\left({3\over 4}\right)^{2}\Gamma\left({6\over
5}\right)\left({\Gamma\left({7\over 10}\right)\Gamma\left({5\over
4}\right)\over\Gamma\left({3\over 10}\right)}\right)^{3/2}\over
2^{3/10}\pi^{2}\Gamma\left({4\over 5}\right)}\,,\\\
C^{TIM}_{\sigma\sigma\epsilon^{\prime}}&={5\over 6}\left({\Gamma\left({2\over
5}\right)\over\Gamma\left({1\over 5}\right)\Gamma\left({3\over
5}\right)}\right)^{3/2}\sqrt{\Gamma\left({4\over 5}\right)}\Gamma\left({6\over
5}\right)\,,\\\
C^{TIM}_{\sigma\sigma\epsilon^{\prime\prime}}&={\sqrt{{\Gamma\left(-{7\over
4}\right)\over 21}}\Gamma\left({3\over 4}\right)\Gamma\left({5\over
4}\right)^{3/2}\Gamma\left({7\over 4}\right)\over 2\pi^{2}}\,,\end{split}$
(3.12)

and the relevant one-point functions (or the boundary operator expansion
coefficients) can be found in Equ. (B.7) and Equ. (B.8):

$\begin{split}\langle(1^{IM}1^{TIM})_{B}|RG\rangle\rangle={1\over
2}\sqrt{S^{(1)}_{0,0}S^{(3)}_{0,0}}=\omega_{-}\,,\qquad\quad&\langle(1^{IM}\epsilon^{\prime
TIM})_{B}|RG\rangle\rangle={1\over
2}\sqrt{S^{(1)}_{0,0}S^{(3)}_{0,2}}=\omega_{+}\,,\\\
\langle(\epsilon^{IM}\epsilon^{TIM})_{B}|RG\rangle\rangle=-{1\over
2}\sqrt{S^{(1)}_{0,0}S^{(3)}_{0,2}}=-\omega_{+}\,,\qquad&\langle(\epsilon^{IM}\epsilon^{\prime\prime
TIM})_{B}|RG\rangle\rangle=-{1\over
2}\sqrt{S^{(1)}_{0,0}S^{(3)}_{0,0}}=-\omega_{-}\,,\\\ \textrm{where
}\omega_{\pm}={1\over 2}\left({5\pm\sqrt{5}\over 40}\right)^{{1\over
4}}\,,\quad&\text{and }S^{(k)}_{r,s}=\sqrt{{2\over
k+2}}\sin({\pi(r+1)(s+1)\over k+2})\,.\end{split}$ (3.13)

As mentioned above, a fully independent calculation of the coefficients of the
conformal blocks in (3.9) can be obtained by instead matching its short-
distance expansion to a calculation of (3.8) using the standard TIM bulk OPE
coefficients together with the RG brane one-point functions. There are four
independent coefficients to be fixed, one for each conformal block appearing
in the sum. One of these is simply a normalization which we can factor out.
The small $z$ expansion of $G^{TIM}(z)$ is

$G^{TIM}(z)\propto z^{-1/5}\left(1-{\sqrt{2}(11+5\sqrt{5})^{{1\over
10}}\pi^{{3\over 4}}\Gamma\left({2\over 5}\right)^{{3\over 2}}\over
3\sqrt{\Gamma\left({1\over 5}\right)}\Gamma\left({3\over 10}\right)^{{3\over
2}}\Gamma\left({4\over 5}\right)}z^{3/5}(1+{3z\over 10})+{3\over
280}z^{2}+\dots\right)$ (3.14)

The coefficient of $z^{3/5}$ should be exactly
$C_{\epsilon\epsilon\epsilon^{\prime}}$ times ${\langle(\epsilon^{\prime
TIM})_{A}\rangle_{RG}\over\langle(1)_{A}\rangle_{RG}}$. The OPE coefficient
$C_{\epsilon\epsilon\epsilon^{\prime}}$ is given in equation (3.30). The one-
point functions $\langle(\epsilon^{\prime TIM})_{A}\rangle_{RG}$ and
$\langle(1)_{A}\rangle_{RG}$ are in Gaiotto:2012np  equation (2.56), giving
${\langle(\epsilon^{\prime
TIM})_{A}\rangle_{RG}\over\langle(1)_{A}\rangle_{RG}}=-{1\over
2}\left({{5\over 8}+{\sqrt{5}\over 8}\over{5\over 8}-{\sqrt{5}\over
8}}\right)^{1/4}$, in agreement with the coefficient of $z^{3/5}$ above. The
coefficient of $z^{2}$ follows from the OPE coefficient and one-point function
for the stress tensor. It can easily be calculated using the usual Virasoro
conformal blocks, where the coefficient of $z^{2}$ is ${2h^{2}\over c}$. In
this case, there is an additional suppression for the one-point function of
$T$ in the presence of the RG brane, and this suppression can be read off from
(A.19) where we see that $T^{TIM}$s overlap with itself reflected across the
RG brane picks up a factor of ${3\over k(k+2)}={3\over 8}$. Therefore, the
prediction for the coefficient of $z^{2}$ is ${3\over 8}{2h^{2}\over
c}={3\over 280}$ since $h=h_{\rm\epsilon}^{TIM}={1\over 10}$ and
$c=c^{TIM}={7\over 10}$. Again this agrees with the coefficient of $z^{2}$
above. Finally, we can fix one more coefficient by using the long-distance
limit
$\langle\epsilon(x_{1},\overline{x}_{1})\epsilon(x_{2},\overline{x}_{2})\rangle_{RG}\sim\langle\epsilon(x_{1},\overline{x}_{1})\rangle_{RG}\langle\epsilon(x_{2},\overline{x}_{2})\rangle_{RG}=0$
at $x_{1}-x_{2}\rightarrow\infty$. And indeed, there is a nontrivial
cancellation among the four conformal blocks in $G^{TIM}$ so that the leading
large $z$ term vanishes. This condition completes the independent derivation
of all the coefficients in $G^{TIM}$.

To get the correlator at finite volume, we can conformally map the correlator
from the upper half plane (UHP) to the strip. Without loss of generality, we
can choose units where the volume is $2\pi$, and reintroduce an arbitrary
volume later by scaling. The conformal mapping from the UHP to the strip is
($t_{E}$ being Euclidean time)

$x=e^{t_{E}+i\theta},$ (3.15)

so that the RG brane located at ${\rm Im}(x)=0$ gets mapped to $\theta=0,\pi$.
In the folded description, both TIM and Ising are in the UHP, and get mapped
to the range $0<\theta<\pi$, whereas in the unfolded description, TIM is in,
say, the UHP whereas Ising is in the lower half-plane (LHP), so that they get
mapped to $0<\theta<\pi$ and $\pi<\theta<2\pi$, respectively. When we set up
our DMRG calculation, we will choose periodic boundary conditions, in order to
match the unfolded description in finite volume. It is straightforward to
evaluate the CFT expression for the correlator at arbitrary times, but we will
focus on the case of equal-time correlators and set $t_{E}=0$. Then, in terms
of the function $G^{TIM}_{\epsilon\epsilon}(z)$ from (3.9), the finite-volume
two-point function of $\epsilon^{TIM}$ in the presence of the RG brane is

$\langle\epsilon^{TIM}(\theta_{1})\epsilon^{TIM}(\theta_{2})\rangle_{RG}\propto\left({1\over\sin^{2}\left({\theta_{1}-\theta_{2}\over
2}\right)}\right)^{{1\over
5}}G^{TIM}_{\epsilon\epsilon}\left({\sin^{2}\left({\theta_{1}-\theta_{2}\over
2}\right)\over\sin^{2}\left({\theta_{1}+\theta_{2}\over 2}\right)}\right).$
(3.16)

The proportionality constant in the above equation depends on the
normalization of $\epsilon^{TIM}$ and can easily be fixed by matching the OPE
singularity at $\theta_{1}\sim\theta_{2}$.

###  3.3 $\langle\epsilon^{IM}\epsilon^{IM}\rangle_{RG}$

The calculation of the two-point function of the $\epsilon$ operator on the
Ising model side of the RG brane closely follows the derivation of that on the
TIM side in the previous subsection. The difference is that $\epsilon^{IM}$
maps in the $\widetilde{\mathcal{B}}$ theory to the operator
$(\sigma^{IM}\sigma^{\prime TIM})_{B}$, and so we need to use the TIM
$\sigma^{\prime}$ OPE coefficients and conformal blocks. The TIM
$\sigma^{\prime}\times\sigma^{\prime}$ OPE only contains two primary
operators, $1$ and $\epsilon^{\prime\prime}$. Because $\sigma^{\prime}$ has a
null descendant at level 2, its conformal blocks can easily be calculated by
standard methods. They are

$\displaystyle G_{\sigma^{\prime}\sigma^{\prime}1}^{TIM}(z)$
$\displaystyle={(1-z)^{5/8}\,_{2}F_{1}\left(-{1\over 4},{5\over 4};-{1\over
2};z\right)\over z^{7/8}},\quad
G_{\sigma^{\prime}\sigma^{\prime}\epsilon^{\prime\prime}}^{TIM}(z)$
$\displaystyle=(1-z)^{5/8}z^{5/8}\,_{2}F_{1}\left({5\over 4},{11\over
4};{5\over 2};z\right).$ (3.17)

The corresponding OPE coefficients are

$C_{\sigma^{\prime}\sigma^{\prime}1}^{TIM}=1,\quad
C_{\sigma^{\prime}\sigma^{\prime}\epsilon^{\prime\prime}}^{TIM}={7\over 8}.$
(3.18)

As before, the full correlator is a sum over these conformal blocks with
coefficients given by the bulk OPE coefficients times the RG brane one-point
functions:

$\displaystyle\langle(\sigma^{IM}\sigma^{\prime
TIM})(x_{1},\overline{x}_{1})(\sigma^{IM}\sigma^{\prime
TIM})(x_{2},\overline{x}_{2})\rangle_{RG}={1\over|x_{1}-x_{2}|^{2}}G^{IM}_{\epsilon\epsilon}\left(\left|{x_{1}-x_{2}\over
x_{1}-\overline{x}_{2}}\right|^{2}\right),$ (3.19) $\displaystyle\quad
G^{IM}_{\epsilon\epsilon}(z)\equiv
z\sum^{\prime}_{\phi_{i}=1,\epsilon\atop\phi_{j}=1,\epsilon^{\prime\prime}}\langle(\phi_{i}^{IM}\phi_{j}^{TIM})_{B}|RG\rangle
C_{\sigma\sigma\phi_{i}}^{IM}C_{\sigma^{\prime}\sigma^{\prime}\phi_{j}}^{TIM}G^{IM}_{\sigma\sigma\phi_{i}}(z)G^{TIM}_{\sigma^{\prime}\sigma^{\prime}\phi_{j}}(z).$

The sum on $\phi_{i},\phi_{j}$ includes only two terms in total, the
combination $(\phi_{i}=1,\phi_{j}=1)$ and
$(\phi_{i}=\epsilon,\phi_{j}=\epsilon^{\prime\prime})$. The one-point
functions were given explicitly in (3.13). The finite-volume, equal-time
correlator is

$\langle\epsilon^{IM}(\theta_{1})\epsilon^{IM}(\theta_{2})\rangle_{RG}\propto{1\over\sin^{2}\left({\theta_{1}-\theta_{2}\over
2}\right)}G^{IM}_{\epsilon\epsilon}\left({\sin^{2}\left({\theta_{1}-\theta_{2}\over
2}\right)\over\sin^{2}\left({\theta_{1}+\theta_{2}\over 2}\right)}\right).$
(3.20)

###  3.4 Branching Rule and Two-point Function for $\epsilon^{\prime TIM}$

Finally, let us work out the CFT prediction for a slightly more complicated
example, that of the RG brane two-point function of $\epsilon^{\prime TIM}$.
Now, the operator maps to a linear combination of primaries in the
$\widetilde{\mathcal{B}}$ theory, which we can work out using the action of
the chiral algebra in the $\mathcal{A}$ and $\widetilde{\mathcal{B}}$
descriptions. The branching rule tells us that $1^{IM}\epsilon^{\prime
TIM}(x)_{A}=1^{IM}\epsilon^{\prime
TIM}(x)_{B}\oplus\epsilon^{IM}\epsilon^{TIM}(x)_{B}+1^{IM}\epsilon^{\prime
TIM}(x)_{B}\oplus\epsilon^{IM}\epsilon^{TIM}(x)_{B}$. Our task now is to
determine the precise decomposition coefficients $a$ and $b$ for

$1^{IM}\epsilon^{\prime TIM}(x)_{A}=a1^{IM}\epsilon^{\prime
TIM}(x)_{B}+b\epsilon^{IM}\epsilon^{TIM}(x)_{B}.$ (3.21)

This can be done by using the first equation in Equ. (A.19), which tells us
that

$L_{0;A}^{IM}={5\over 8}L_{0;B}^{TIM}+{\sqrt{15}\over 8}(G\psi)_{0}+{1\over
8}L_{0;B}^{IM}\,,$ (3.22)

together with

$L_{0;A}^{IM}1^{IM}\epsilon^{\prime}(x)_{A}=0\,.$ (3.23)

We have

$0=({3a+\sqrt{15}b\sqrt{2h_{\epsilon^{TIM}}}\over
8}1^{IM}\otimes\epsilon^{\prime
TIM}+{5\sqrt{15}\sqrt{2h_{\epsilon^{TIM}}}a+5b\over
40}\epsilon^{IM}\otimes\epsilon^{TIM})\,,$ (3.24)

and so

$\sqrt{3}a+b=0\,.$ (3.25)

Hence we have the decomposition up to an overall constant.

Translating to the B-theory, the correlator that we are interested in is

$\begin{split}&\langle\Big{(}1^{IM}(x_{1})\epsilon^{\prime
TIM}(x_{1})-\sqrt{3}\epsilon^{IM}(x_{1})\epsilon^{TIM}(x_{1})\Big{)}\Big{(}1^{IM}(x_{2})\epsilon^{\prime
TIM}(x_{2})-\sqrt{3}\epsilon^{IM}(x_{2})\epsilon^{TIM}(x_{2})\Big{)}\rangle\,\,\\\
=&\langle 1^{IM}(x_{1})\epsilon^{\prime
TIM}(x_{1})1^{IM}(x_{2})\epsilon^{\prime
TIM}(x_{2})\rangle+3\langle\epsilon^{IM}(x_{1})\epsilon^{TIM}(x_{1})\epsilon^{IM}(x_{2})\epsilon^{TIM}(x_{2})\rangle\\\
&-\sqrt{3}\langle 1^{IM}(x_{1})\epsilon^{\prime
TIM}(x_{1})\epsilon^{IM}(x_{2})\epsilon^{TIM}(x_{2})\rangle-\sqrt{3}\langle\epsilon^{IM}(x_{1})\epsilon^{TIM}(x_{1})1^{IM}(x_{2})\epsilon^{\prime
TIM}(x_{2})\rangle\,,\end{split}$ (3.26)

and we will compute the resulting four correlators separately.

To compute these correlators, we follow the same procedure as for Equ. (3.6)
doing the operator production expansion (OPE) in the bulk first and computing
the resulting one-point functions (as indicated in Fig.2). The OPE is given by
the (tensor product of the) OPE from the Ising model and the Tricritical Ising
Model. The conformal blocks we get are those for chiral four point functions
in the Ising model and tricitical Ising model. These bulk OPE expansions can
be read from

$\begin{split}[\epsilon^{IM}]\times[\epsilon^{IM}]&=[1^{IM}]\,,\quad[\epsilon^{TIM}]\times[\epsilon^{TIM}]=[1^{TIM}]+[\epsilon^{TIM}]\,,\\\
[\epsilon^{\prime TIM}]\times[\epsilon^{\prime
TIM}]&=[1^{TIM}]+[\epsilon^{\prime
TIM}]\,,\quad[\epsilon^{TIM}]\times[\epsilon^{\prime
TIM}]=[\epsilon^{TIM}]+[\epsilon^{\prime\prime TIM}]\,.\end{split}$ (3.27)

The results are given by

$\begin{split}&\langle 1^{IM}(x_{1})\epsilon^{\prime
TIM}(x_{1})1^{IM}(x_{2})\epsilon^{\prime TIM}(x_{2})\rangle\\\ &={z^{{2\over
5}}(1-z)^{{2\over
5}}\Big{(}H^{I}_{11}(z)+H^{I}_{1\epsilon^{\prime}}(z)\Big{)}\over(x_{1}-x_{2})^{{1\over
15}}(x_{1}-\overline{x}_{1})^{{1\over 15}}(x_{1}-\overline{x}_{1})^{{1\over
15}}(x_{2}-\overline{x}_{1})^{{1\over 15}}(x_{1}-\overline{x}_{2})^{{1\over
15}}(x_{2}-\overline{x}_{2})^{{1\over 15}}(x_{2}-\overline{x}_{1})^{{1\over
15}}(\overline{x}_{1}-\overline{x}_{2})^{{1\over 15}}}\,,\\\
&\langle\epsilon^{IM}(x_{1})\epsilon^{TIM}(x_{1})\epsilon^{IM}(x_{2})\epsilon^{TIM}(x_{2})\rangle\\\
&={z^{{2\over 5}}(1-z)^{{2\over
5}}\Big{(}H^{II}_{11}(z)+H^{II}_{1\epsilon^{\prime}}(z)\Big{)}\over(x_{1}-x_{2})^{{1\over
15}}(x_{1}-\overline{x}_{1})^{{1\over 15}}(x_{1}-\overline{x}_{1})^{{1\over
15}}(x_{2}-\overline{x}_{1})^{{1\over 15}}(x_{1}-\overline{x}_{2})^{{1\over
15}}(x_{2}-\overline{x}_{2})^{{1\over 15}}(x_{2}-\overline{x}_{1})^{{1\over
15}}(\overline{x}_{1}-\overline{x}_{2})^{{1\over 15}}}\,,\\\ &\langle
1^{IM}(x_{1})\epsilon^{\prime
TIM}(x_{1})\epsilon^{IM}(x_{2})\epsilon^{TIM}(x_{2})\rangle\\\ &={z^{{2\over
5}}(1-z)^{{2\over
5}}\Big{(}H^{III}_{\epsilon\epsilon}(z)+H^{III}_{\epsilon\epsilon^{\prime\prime}}(z)\Big{)}\over(x_{1}-x_{2})^{{1\over
15}}(x_{1}-\overline{x}_{1})^{{1\over 15}}(x_{1}-\overline{x}_{1})^{{1\over
15}}(x_{2}-\overline{x}_{1})^{{1\over 15}}(x_{1}-\overline{x}_{2})^{{1\over
15}}(x_{2}-\overline{x}_{2})^{{1\over 15}}(x_{2}-\overline{x}_{1})^{{1\over
15}}(\overline{x}_{1}-\overline{x}_{2})^{{1\over 15}}}\,,\\\
&\langle\epsilon^{IM}(x_{1})\epsilon^{TIM}(x_{1})1^{IM}(x_{2})\epsilon^{\prime
TIM}(x_{2})\rangle\\\ &={z^{{2\over 5}}(1-z)^{{2\over
5}}\Big{(}H^{IV}_{\epsilon\epsilon}(z)+H^{IV}_{\epsilon\epsilon^{\prime\prime}}(z)\Big{)}\over(x_{1}-x_{2})^{{1\over
15}}(x_{1}-\overline{x}_{1})^{{1\over 15}}(x_{1}-\overline{x}_{1})^{{1\over
15}}(x_{2}-\overline{x}_{1})^{{1\over 15}}(x_{1}-\overline{x}_{2})^{{1\over
15}}(x_{2}-\overline{x}_{2})^{{1\over 15}}(x_{2}-\overline{x}_{1})^{{1\over
15}}(\overline{x}_{1}-\overline{x}_{2})^{{1\over 15}}}\,,\end{split}$ (3.28)

where the relevant conformal blocks are

$\begin{split}H_{11}^{I}(z)&=\langle(1^{IM}1^{TIM})_{B}|RG\rangle\rangle
C^{TIM}_{\epsilon^{\prime}\epsilon^{\prime}1}{\left(z^{2}-z+1\right)\,_{2}F_{1}\left(-{2\over
5},{1\over 5};{2\over 5};z\right)\over z^{6/5}(1-z)^{6/5}}\,,\\\
H_{1\epsilon^{\prime}}^{I}(z)&=\langle(1^{IM}\epsilon^{\prime
TIM})_{B}|RG\rangle\rangle
C^{TIM}_{\epsilon^{\prime}\epsilon^{\prime}\epsilon^{\prime}}{\left(z^{2}-z+1\right)\,_{2}F_{1}\left({1\over
5},{4\over 5};{8\over 5};z\right)\over z^{3/5}(1-z)^{6/5}}\,,\\\
H_{11}^{II}(z)&=\langle(1^{IM}1^{TIM})_{B}|RG\rangle\rangle
C^{IM}_{\epsilon\epsilon 1}C^{TIM}_{\epsilon\epsilon 1}{1-z+z^{2}\over
z(1-z)}{\,{}_{2}F_{1}\left(-{2\over 5},{1\over 5};{2\over
5};z\right)\over\sqrt[5]{z}\sqrt[5]{1-z}}\,,\\\
H_{1\epsilon^{\prime}}^{II}(z)&=\langle(1^{IM}\epsilon^{\prime
TIM})_{B}|RG\rangle\rangle C^{IM}_{\epsilon\epsilon
1}C^{TIM}_{\epsilon\epsilon\epsilon^{\prime}}{1-z+z^{2}\over
z(1-z)}{\,{}_{2}F_{1}\left({1\over 5},{4\over 5};{8\over 5};z\right)\over
z^{-{2\over 5}}\sqrt[5]{1-z}}\,,\\\
H_{\epsilon\epsilon}^{III}(z)&=\langle(\epsilon^{IM}\epsilon^{TIM})_{B}|RG\rangle\rangle
C^{TIM}_{\epsilon^{\prime}\epsilon\epsilon}{1\over
1-z}{\,{}_{2}F_{1}\left(-{6\over 5},{1\over 5};-{2\over
5};z\right)\over\sqrt[5]{1-z}z^{3/5}}\,,\\\
H_{\epsilon\epsilon^{\prime\prime}}^{III}(z)&=\langle(\epsilon^{IM}\epsilon^{\prime\prime
TIM})_{B}|RG\rangle\rangle
C^{TIM}_{\epsilon^{\prime}\epsilon\epsilon^{\prime\prime}}{1\over
1-z}{z^{4/5}\,_{2}F_{1}\left({1\over 5},{8\over 5};{12\over
5};z\right)\over\sqrt[5]{1-z}}\,,\\\
H_{\epsilon\epsilon}^{IV}(z)&=\langle(\epsilon^{IM}\epsilon^{TIM})_{B}|RG\rangle\rangle
C^{TIM}_{\epsilon^{\prime}\epsilon\epsilon}{1\over
1-z}{\,{}_{2}F_{1}\left(-{6\over 5},{1\over 5};-{2\over
5};z\right)\over\sqrt[5]{1-z}z^{3/5}}\,,\\\
H_{\epsilon\epsilon^{\prime\prime}}^{IV}(z)&=\langle(\epsilon^{IM}\epsilon^{\prime\prime
TIM})_{B}|RG\rangle\rangle
C^{TIM}_{\epsilon^{\prime}\epsilon\epsilon^{\prime\prime}}{1\over
1-z}{z^{4/5}\,_{2}F_{1}\left({1\over 5},{8\over 5};{12\over
5};z\right)\over\sqrt[5]{1-z}}\,.\end{split}$ (3.29)

The OPE coefficients are

$\begin{split}C^{IM}_{\epsilon\epsilon 1}&=1\,,\\\
C^{TIM}_{\epsilon^{\prime}\epsilon^{\prime}1}&=1\,,\\\
C^{TIM}_{\epsilon^{\prime}\epsilon^{\prime}\epsilon^{\prime}}&={\pi^{5/4}\Gamma\left({2\over
5}\right)\Gamma\left({11\over 5}\right)\over\sqrt[5]{2}\Gamma\left({9\over
10}\right)\left(\Gamma\left({1\over 5}\right)\Gamma\left({13\over
10}\right)\right)^{3/2}\sqrt{3\Gamma\left({7\over 5}\right)}}\,,\\\
C^{TIM}_{\epsilon\epsilon 1}&=1\,,\\\
C^{TIM}_{\epsilon\epsilon\epsilon^{\prime}}&={\sqrt{{\Gamma\left(-{3\over
10}\right)\Gamma\left({2\over 5}\right)\Gamma\left({7\over
5}\right)\over\Gamma\left(-{2\over 5}\right)\Gamma\left({3\over
10}\right)\Gamma\left({8\over 5}\right)}}\over 2^{3/5}}\,,\\\
C^{TIM}_{\epsilon^{\prime}\epsilon\epsilon}&={\sqrt{{\Gamma\left(-{3\over
10}\right)\Gamma\left({2\over 5}\right)\Gamma\left({7\over
5}\right)\over\Gamma\left(-{2\over 5}\right)\Gamma\left({3\over
10}\right)\Gamma\left({8\over 5}\right)}}\over 2^{3/5}}\,,\\\
C^{TIM}_{\epsilon^{\prime}\epsilon\epsilon^{\prime\prime}}&={3\over
7}\,,\end{split}$ (3.30)

and the boundary operator expansion coefficients are

$\begin{split}\langle(1^{IM}1^{TIM})_{B}|RG\rangle\rangle&={1\over
2}\sqrt{S^{(1)}_{0,0}S^{(3)}_{0,0}}\,,\\\ \langle(1^{IM}\epsilon^{\prime
TIM})_{B}|RG\rangle\rangle&={1\over 2}\sqrt{S^{(1)}_{0,0}S^{(3)}_{0,2}}\,,\\\
\langle(\epsilon^{IM}\epsilon^{TIM})_{B}|RG\rangle\rangle&=-{1\over
2}\sqrt{S^{(1)}_{0,0}S^{(3)}_{0,2}}\,,\\\
\langle(\epsilon^{IM}\epsilon^{\prime\prime
TIM})_{B}|RG\rangle\rangle&=-{1\over
2}\sqrt{S^{(1)}_{0,0}S^{(3)}_{0,0}}\,.\end{split}$ (3.31)

##  4 Comparison of Lattice and CFT Calculations

In this section, we numerically study the RG brane system using a DMRG
analysis. Following Grover:2013rc , we apply DMRG to the Hamiltonian (2.2).
Recall that for (2.2) when $h=2$ the system is in the IM phase whereas for
$h=1.62$ the system is in the TIM phase, and the RG brane system is realized
on the lattice by tuning $h=2$ on half the lattice sites and $h=1.62$ on the
other half, with periodic boundary conditions taken at the endpoints.

As mentioned in Sec. 2.2, the DMRG algorithm is controlled by a few key
parameters. For the results below we use a lattice with $N=180$ distinct sites
(540 total sites, since each physical site requires a separate site for
$\mu_{i,a},\mu_{i,b}$ and $\sigma_{i}$) unless otherwise noted. First we will
find the map between lattice operators and CFT operators. Then, we numerically
compute one-point and two-point functions in DMRG, and compare them with the
analytic CFT predictions from Sec. 3.

###  4.1 Mapping of Operators

Because the lattice is a microscopic structure versus the continuum field
theory that we aim to compare it to, we need to verify what lattice operators
correspond to what CFT operators. We do this by expanding the lattice
operators in terms of the CFT operators and then determining the expansion
coefficients. In our case, the $\mu$ lattice operator (i.e. $\mu^{z}_{i,b}$
and $\mu^{z}_{i,a}$) in the Ising domain can be expanded as

$\mu^{z}=A_{\epsilon}^{(I)}\,\epsilon+A_{\partial^{2}\epsilon}^{(I)}\,\partial^{2}\epsilon+\cdots$
(4.1)

where $\epsilon$ is the Ising CFT energy density operator, the $A$s are the
expansion coefficients and the dots are higher decedent operators. In the
tricritical Ising domain the expansion is,

$\mu^{z}=A_{\epsilon}^{(T)}\,\epsilon+A_{\epsilon^{\prime}}^{(T)}\,\epsilon^{\prime}+A_{\partial^{2}\epsilon}^{(T)}\,\partial^{2}\epsilon+\cdots$
(4.2)

where $\epsilon$, $\epsilon^{\prime}$, etc., are the tricritical CFT energy
density operators. The fact that the lattice $\mu$ operator expands to energy
operators can be understood as follows. Adding $\sum_{i}\mu^{z}_{i,b}$ (or
$\sum_{i}\mu^{z}_{i,a}$) to the Hamiltonian Equ. (2.2) breaks the
$\mathbb{Z}_{2}$ symmetry which acts on the $\mu$-sector as sending
$\mu^{z}_{i,a}$ to $-\mu^{z}_{i,b}$ and $\mu^{z}_{i,b}$ to $-\mu^{z}_{i+1,a}$
and on the $\sigma$-sector as the high-temperature/low-temperature duality
transform $\sigma^{z}_{i}\sigma^{z}_{i+1}\leftrightarrow\sigma^{x}_{i}$. Hence
when are in the Ising phase this deformation breaks the high-temperature/low-
temperature duality symmetry101010In the free fermion representation this is
the time-reversal symmetry which prevents the system from being gapped. which
maps $\epsilon$ to $-\epsilon$ so this is equivalent at leading order to an
$\epsilon$ deformation. On the other hand, when we are at the critical point
($h=h_{*}$) the low-energy theory is TIM and such a deformation would gap out
the system. Hence this is a relevant deformation. However, in the TIM case a
$\mathbb{Z}_{2}$ symmetry we have is

$\epsilon^{TIM}\rightarrow-\epsilon^{TIM}\,,\quad\epsilon^{\prime
TIM}\rightarrow\epsilon^{\prime TIM},\quad\text{and }\epsilon^{\prime\prime
TIM}\rightarrow-\epsilon^{\prime\prime TIM}\,,$ (4.3)

which suggests that the expansion of the lattice $\mu$ operator is given by
the $\epsilon^{TIM},\epsilon^{\prime\prime TIM}$ and their descendants.

To determine the coefficients we use DMRG to calculate finite-volume two-point
functions on the lattice in the IM and TIM phases. The equal-time finite-
volume two-point function of a primary operator $\mathcal{O}$ is given by

$\langle\mathcal{O}(x_{1})\mathcal{O}(x_{2})\rangle={({\pi\over
L})^{2\Delta_{\mathcal{O}}}\over\sin\left({\pi x_{12}\over
L}\right)^{2\Delta_{\mathcal{O}}}}\sim{1\over
x_{12}^{2\Delta_{\mathcal{O}}}}\,,$ (4.4)

where we have chosen the canonical normalization of $\mathcal{O}$ so that its
short-distance OPE singularity is $x_{12}^{-2\Delta_{\mathcal{O}}}$, as shown.
Therefore, for the case of (4.1), we fit numeric data to a function of the
form

$\langle\mu(x_{1})\mu(x_{2})\rangle_{\text{TIM}}\approx(A_{\epsilon}^{(T)})^{2}{({\pi\over
L})^{2\Delta_{\epsilon}}\over\sin\left({\pi x_{12}\over
L}\right)^{2\Delta_{\epsilon}}}.$ (4.5)

with $\Delta_{\epsilon}={1\over 5}$, and similarly with the Ising case using
$\Delta_{\epsilon}=1$. Doing this we find, using a $N=180$ site lattice,

$\displaystyle A_{\epsilon}^{(I)}=0.81\,,\qquad A_{\epsilon}^{(T)}=0.61\,.$
(4.6)

In Fig. 3, we show a comparison of the form (4.5) to the result of our DMRG
computation for the $\langle\mu_{i}\mu_{j}\rangle$ two-point function on the
TIM and Ising side, verifying a similar comparison in Grover:2013rc .

It is more difficult to obtain very precise results for the coefficients of
the subleading operators in (4.1) and (4.2). However, such contributions
should be parameterically suppressed by $|i-j|^{2}$ compared to the leading
contribution at long distances. Empirically, we find that this suppression is
roughly a factor of

${b\over|i-j|^{2}}$ (4.7)

(relative to the leading contribution) with $b\sim 20$ for Ising and $b\sim 1$
for TIM.111111A possible explanation for the smallness of this value of $b$ in
TIM is as follows. We expect that the coefficient $A_{\epsilon^{\prime}}$
actually vanishes due to the generalization of Kramers-Wannier symmetry to TIM
(i.e., the $\widehat{N}$ Verlinde line Chang:2018iay ), so the only
contribution to $b$ comes from $\partial^{2}\epsilon$ in both Ising and in
TIM. Then, the cross-terms $\langle\epsilon\partial^{2}\epsilon\rangle$ are
parametrically suppressed relative to the leading terms
$\langle\epsilon\epsilon\rangle$ by
${\langle\epsilon\partial^{2}\epsilon\rangle\over\langle\epsilon\epsilon\rangle}\sim\Delta_{\epsilon}^{2}$,
which is 1 in Ising and $1/25$ in TIM.  So we see that the lattice $\mu$
operators flow to a linear combination of the energy density operators, with
the dominant CFT operator being the $\epsilon$ energy density in the
respective phase, as expected. Moreover, lattice corrections from subleading
operators in (4.1) and (4.2) should be small for $|i-j|\gtrsim 5$ for Ising
and $|i-j|\gtrsim 1$ for TIM. Indeed this is consistent with what we will see
in Fig. 4 when we compare the CFT and lattice calculations of the $\mu_{i}$
two-point function in the presence of the RG brane.

Figure 3: DMRG computation of the $\langle\mu_{i}\mu_{j}\rangle$ two-point
function for the lattice model Equ. (2.2) in the Tricritical Ising ($h=1.62$)
and Ising ($h=2$) phases.

###  4.2 Numeric Comparisons

We start by using DMRG to compute one-point functions. In Gaiotto:2012np  one-
point functions were verified using conformal perturbation theory for large
$k$ Virasoro minimal CFTs; using DMRG allows us to check these formulas at
finite $k$ (in our case $k=1$).

To compute the one-point function, we fit the DMRG data of the two-point
function $\langle\mu_{j_{*}+x}\mu_{j_{*}-x}\rangle_{RG}$, where $j_{*}$ is the
location of the RG brane, to the form

$\langle\mu_{j_{*}-x}\mu_{j_{*}+x}\rangle_{RG}=A_{\epsilon}^{(I)}A_{\epsilon}^{(T)}\langle\epsilon^{TIM}(x)\epsilon^{IM}(-x)\rangle_{RG}=A_{\epsilon}^{(I)}A_{\epsilon}^{(T)}B\left({{\pi\over
L}\over\sin({2x\pi\over L})}\right)^{\Delta},$ (4.8)

where $\Delta=\Delta_{\epsilon}^{IM}+\Delta_{\epsilon}^{TIM}=1.2$ and
$A_{\epsilon}$s are taken from the fits in the previous subsection. As usual,
we have performed a conformal map to take into account finite volume effects.
We obtain the result

$B=0.64\pm 0.01,$ (4.9)

where we have estimated the error by performing the fit over different ranges
of lattice points.

Now we want to compare this numerical result with the prediction from the RG
brane construction. The $\mathcal{A}$-theory operator
$\epsilon^{IM}(x)\epsilon^{TIM}(x)$ can be written as
$\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t}$ where $k=2,r=2,s=1$ and $t=2$. This
tells us that $d=2,\widetilde{d}=2$ and $d^{\prime}=1\text{ or }3$. Hence this
$\mathcal{A}$-theory operator translates into the
$\widetilde{\mathcal{B}}$-theory as the linearly combination of the operators
$[2,1;2]\otimes[2,2;1]=\epsilon^{TIM}\epsilon^{IM}(x)$ and
$[2,3;2]\otimes[2,2;3]=G_{-{1\over 2}}\overline{G}_{-{1\over
2}}\epsilon^{TIM}1^{IM}(x)=\epsilon^{\prime TIM}1^{IM}(x)$.121212Here we used
the fact that the $\widetilde{\mathcal{B}}$-theory is the supersymmetric
representation of the Tricritical Ising Model for our case $k=2$. $G_{-{1\over
2}}$ and $\overline{G}_{-{1\over 2}}$ are the first creation operators of the
supersymmetry generators $G$ and $\overline{G}$ in the NS sector. Let’s write

$\epsilon^{IM}\epsilon^{TIM}(x)_{A}=\alpha\epsilon^{TIM}\epsilon^{IM}(x)_{B}+\beta
G_{-{1\over 2}}\epsilon^{TIM}1^{IM}(x)\,,$ (4.10)

where we only focus on the holomorphic part and the normalization of the
operator gives the constraint

$1=\alpha^{2}+2\beta^{2}h_{\epsilon^{TIM}}=\alpha^{2}+{1\over 5}\beta^{2}\,.$
(4.11)

We can find the coefficients $\alpha$ and $\beta$ as in Equ. (3.21) using the
following relation

$L_{0;A}^{IM}={5\over 8}L_{0;B}^{TIM}+{\sqrt{15}\over 8}(G\psi)_{0}+{1\over
8}L_{0;B}^{IM}\,,$ (4.12)

which tells us that

${\sqrt{15}\over 40}\beta={3\over 8}\alpha\,,\quad{1\over
8}\beta={\sqrt{15}\over 8}C^{IM}_{\epsilon\epsilon 1}\alpha\,.$ (4.13)

The fact that $C^{IM}_{\epsilon\epsilon 1}=1$ tells us that this two equations
are consistent. Then combining with Equ. (4.11) we have

$\alpha=\sqrt{{5\over 8}}\,,\beta=5\sqrt{{3\over 8}}\,.$ (4.14)

As a result, we have

$\langle\langle(\epsilon^{IM}\epsilon^{TIM})_{A}|RG\rangle\rangle=(-\alpha^{2}+2h_{\epsilon^{TIM}}\beta^{2}){\sqrt{S^{(1)}_{0,1}S^{(3)}_{0,1}}\over
S^{(2)}_{0,0}}={1\over 2}{\sqrt{S^{(1)}_{0,1}S^{(3)}_{0,1}}\over
S^{(2)}_{0,0}}\,,$ (4.15)

where the minus sign in the first step comes from the fact that the RG brane
maps $\psi$ (i.e. the homolorphic part of $\epsilon^{IM}_{B}$) to
$-\overline{\psi}$ (i.e. the antiholomorphic part of $\epsilon^{IM}_{B}$).
Moreover, there is one more step before we can match the numerical result Equ.
(4.9). We have to take into account of the fact that in numerics the identity
operator is normalized to one and so we have to consider instead

${\langle\langle(\epsilon^{IM}\epsilon^{TIM})_{A}|RG\rangle\rangle\over\langle\langle(1^{IM}1^{TIM})_{A}|RG\rangle\rangle}={1\over
2}{\sqrt{S^{(1)}_{0,1}S^{(3)}_{0,1}}\over
S^{(2)}_{0,0}}{S^{(2)}_{0,0}\over\sqrt{S^{(1)}_{0,0}S^{(2)}_{0,0}}}={1\over
2}({5+\sqrt{5}\over 5-\sqrt{5}})^{{1\over 4}}\sim 0.63601\,,$ (4.16)

which perfectly matches the numerical result Equ. (4.9).

Figure 4: Comparison of $\langle\mu_{i}\mu_{j}\rangle$ in the RG brane lattice
description vs. the equivalent CFT result Equ. (3.9). Top: The IM side.
Bottom: The TIM side.

Finally, we can compare our CFT result Equ. (3.9) for
$\langle\epsilon\epsilon\rangle_{\textrm{RG}}$ in the presence of the RG brane
to the DMRG calculation. Fig. 4 shows the result of this comparison for the
computing $\langle\mu_{i}\mu_{j}\rangle$ on the Ising and Tricritical Ising
sides of the RG brane, respectively, and shows remarkably good agreement
between the CFT and DMRG calculations. This is one of the main results of the
paper, and provides a highly nontrivial check of our methods for computing
correlators in RG brane backgrounds.

##  5 Experimental Realizations

In this section, we comment on potential experimental realizations of the RG
domain wall we studied in this paper, based on some experimental proposals in
the literature for obtaining the Tricritical Ising Model fixed point. There
are two types of systems that we will focus on and they each have their own
pros and cons. The first type of systems we will consider is (2+1)-dimensional
Type IIID topological superconductors Grover:2012bm ; Grover:2013rc ;
crépel2023topological  and the second type of systems we are interested in is
the (1+1)-dimensional Rydberg chain Slagle:2021ene .

###  5.1 Engineering the RG Domain Wall on the Boundary of a Topological
Superconductor

Let’s consider the edge of a (2+1)-dimensional Type III D topological
superconductor with the bulk in the gapped topological phase (a discussion of
the physics in a toy model for such systems is given in Appendix.C).131313We
thank Tarun Grover for discussions of this approach, and suggestions for how
to implement the RG brane in this context. For such a system we have two
Majorana modes $\chi_{L}$ and $\chi_{R}$ on the edge whose gap is protected by
a time-reversal symmetry. The Lagrangian is given by

$L={i\over 2}\int
dx\Big{[}\chi_{R}(x,t)(\partial_{t}+\partial_{x})\chi_{R}(x,t)+\chi_{L}(x,t)(\partial_{t}-\partial_{x})\chi_{L}(x,t)\Big{]}\,,$
(5.1)

which is the (1+1)-d Ising Model conformal field theory. The two Majorana
fermions $\chi_{R}$ and $\chi_{L}$ are respectively of spin down and spin up
and the time-reversal symmetry exchanges these two Majorana fermions and
meanwhile it protects the system from being gapped (see Appendix.C). Hence the
spontaneous breaking of the time-reversal symmetry (or the magnetic ordering)
provides a portal to realize a different phase (the trivial gapped phase) and
the critical point of the phase transition between this new gapped phase the
gapless phase Equ. (5.1) provides a chance to realize a different CFT. Such a
phase transition can be characterized by an order parameter $\phi(x,t)$ which
transforms under the time-reversal symmetry by a minus sign. The dynamics of
such an order parameter is universally controlled by the usual
$\phi^{4}$-model. The full Lagrangian of the system is given by

$\begin{split}L_{tot}=&{i\over 2}\int
dx\Big{[}\chi_{R}(x,t)(\partial_{t}+\partial_{x})\chi_{R}(x,t)+\chi_{L}(x,t)(\partial_{t}-\partial_{x})\chi_{L}(x,t)\Big{]}\\\
+&{1\over 2}\int
dx\Big{[}(\partial_{t}\phi)^{2}(x,t)-(\partial_{x}\phi)^{2}(x,t)+m^{2}\phi^{2}(x,t)-u\phi^{4}(x,t)+ig\phi(x,t)\chi_{R}(x,t)\chi_{L}(x,-t)\Big{]},\end{split}$
(5.2)

For large positive $u$ the order parameter $\phi$ has a zero vev and the time-
reversal symmetry is not spontaneously broken so the low energy dynamics is
controlled by the free massless Majorana fermions which is the Ising Model. By
contrast, for sufficiently small $u$ the order parameter would have a nonzero
vev which signals the spontaneous breaking of the time-reversal symmetry and
the Majorana fermions are gapped by the Yukawa term (the last term in Equ.
(5.2)) and the low energy dynamics is trivial with a nonvanishing gap. Hence
for a fixed value of $g$ there is a critical value $u_{c}$ for $u$ where the
phase transition happens and the physics is captured by a different CFT (see
Fig.5 for the phase diagram). It is shown in Grover:2013rc  that this CFT is
the Tricritical Ising Model whose central charge is $c={7\over 10}$. As a
result, we can use this system to engineer the RG brane that we studied in
this paper. We can tune the parameter $u$ to $u_{c}$ first and then tune it
above $u_{c}$ on half of the space. In practice, one could try to tune $u$ by
putting the material between two plates of a capacitor in order to turn on a
background electric field, which preserves the time-reversal symmetry and so
should still be described by the same phase diagram.

In real materials the time-reversal symmetry breaking is achieved by magnetic
ordering which sets a preferred direction of the electron spin. In the Ising
Model language, this can be realized by taking into account the fermion
interactions, for example dipole-dipole interactions between the atoms in
Cooper pairs. The orientation of the dipole moments can be driven by an
external electric field which could potentially be used to tune to or away
from the point of magnetic ordering. The advantage of this approach is that we
just have to tune a signal parameter to reach the TIM critical point. However,
at the moment there are no clear $(1+1)$-$d$ candidate experimentally
accessible systems. For example the usual systems that realize a topological
superconductor of our interest are $(2+1)$-$d$ boundaries of $(3+1)$-$d$
systms, but because they sit at the extremely low temperature regime (for
example superfluid He3-B) it is likely to be quite challenging to use them to
create $(1+1)$-$d$ boundaries of, say, $(2+1)$-$d$ films. And, despite various
proposals, there are currently no known intrinsically $(2+1)$-$d$ materials
that are widely accepted to be Type DIII topological superconductors. For more
details of the experimental viability and difficulties we refer the readers to
Grover:2013rc .

$u$$u_{c}$$\bullet$Gapless PhaseGapped Phase Figure 5: The phase diagram of
the system Equ. (5.2) for a fixed value of $g$. At large positive $u$ we enter
a gapless phase of two Majorana fermions which is described by the Ising Model
CFT. At large negative $u$ the system is trivially gapped. Thus there is a
critical value $u_{c}$ at which the second order phase transition happens and
the physics is described by the Tricritical Ising Model CFT.

###  5.2 Engineering the RG Domain Wall in a Rydberg Chain

Another system that one might use to engineer the RG domain wall is the
Rydberg atoms chain Fendley_2004 ; Slagle:2021ene . This system consists of a
one-dimensional chain of bosons (neutral atoms) with each boson as a two-level
quantum mechanical system and the two energy levels are coupled to each other
by a resonant laser field with a Rabi frequency $\Omega$. Nevertheless the
interesting part of system is that the Rydberg atoms are interacting if they
are in the excited state and the interaction is from the dipole-dipole
interaction for neaby excited atoms. Hence the whole Hamiltonian of the chain
can be written as (we will follow the notations in Slagle:2021ene )

$H=\sum_{j}\Big{[}{\Omega\over 2}(b_{j}+b_{j}^{\dagger})-\Delta
n_{j}+V_{1}n_{j}n_{j+1}+V_{2}n_{j}n_{j+2}\Big{]}\,,$ (5.3)

where $\Omega$ is the Rabi frequency of the external resonant laser field that
couples the ground state and the excited state of each atom,
$b_{j}^{\dagger}(b_{j})$ is the creation (annihilation) operator for the atom
at site $j$ to go from the ground (excited) state to the excited (ground)
state141414In the single particle Hilbert space at site $j$, we have
$b_{j}=\ket{0}\bra{1}$ and $b^{\dagger}_{j}=\ket{1}\bra{0}$ so the single
particle Hamiltonian $H_{j}={\Omega\over
2}(b_{j}+b^{\dagger}_{j})={\Omega\over 2}(\ket{0}\bra{1}+\ket{1}\bra{0})$ is
indeed the two level coupled Hamiltonian driven by an external electric field
in standard quantum mechanics., $n_{j}=b^{\dagger}_{j}b_{j}$ is the number
operator at site $j$ which can be either zero or one, $\Delta$ describes the
detuning from the Rydberg state, $V_{1}$ denotes the nearest neighbour dipole-
dipole interaction and $V_{2}$ denotes the next-nearest neighbour dipole-
dipole interaction and we have ignored higher neighbour dipole-dipole
interactions which are suppressed. As Fendley_2004  pointed out this system
has a nice phase diagram when $V_{1}=\infty$. The phase diagram is nicely
plotted in Slagle:2021ene  (see Fig.1 in Slagle:2021ene ). Depending on the
values of ${V_{2}\over\Omega}$ and ${\Delta\over\Omega}$ there are two phases
of the system– a disordered phase $\langle n_{j}\rangle=0$ and a
$\mathbb{Z}_{2}$ ordered phase with $\langle n_{j}\rangle$ and $\langle
n_{j+1}\rangle$ alternatively zero and one. There is a critical line
separating the two phases which describes the phase transition between them.
The interesting aspect of this critical line is that it has a tri-critical
point separating the second order phase transition and the first order phase
transition. This tri-critical point is described by the Tricritical Ising
Model and the second order phase transition is described by the Critical Ising
Model. Hence we can construct the RG domain wall in this paper by firstly tune
the Rydberg chain to the tricritical point and then tune half of them away
from the tricritical point to a nearby critical point for the second order
phase transition between the ordered and disordered phases.

The Rydberg chain has been proposed to be a reliable platform to simulate
exotic many-body spin quantum systems due to its stability and the precise
tunability of the parameters Browaeys_2020 ; Morgado_2021  and some of these
Rydberg chains have been realized experimentally Bernien_2017 . Hence the
advantage of this construction is that the tuning can be done very precisely
in real experimental systems of Rydberg chain and a slight disadvantage is
that we have to tune multiple parameters to locate the tricritical point and
then tune multiple parameters simultaneously to drive half of the space away
from it to a nearby critical point for the second phase transition.

## Acknowledgments

We are grateful to Rich Brower, Davide Gaiotto, Tarun Grover, Ami Katz,
Severin Lüst, Rashmish Mishra, Anatoli Polkovnikov, Lisa Randall, Ruben
Verresen, Ashvin Vishwanath, and Juven Wang for discussions. We would like
thank Davide Gaiotto, Ami Katz and Ruben Verresen for comments on the draft.
CVC and ALF are supported by the US Department of Energy Office of Science
under Award Number DE-SC0015845, and the Simons Collaboration on the Non-
Perturbative Bootstrap. HG is supported by the grant (272268) from the Moore
Foundation “Fundamental Physics from Astronomy and Cosmology”.

##  Appendix A Algebraic Construction for Minimal Models and Their Tensor
Product

In this section, we review relevant results in the coset construction of
unitary minimal models and their tensor product. This provides the relevant
background to understand the construction of RG brane by Gaiotto:2012np  which
we review in Sec.B.

###  A.1 Coset Construction of Unitary Virasoro Minimal Models

Let’s firstly review the coset construction of the Virasoro minimal models.
Virasoro minimal models in 2d are the only unitary conformal field theories in
2d with central charge $c$ less than one. They are denoted as
$\mathcal{M}_{k+3,k+2}$ with integers $k\geq 1$ and for a given $k$ the
central charge is

$c=1-{6\over(k+2)(k+3)}\,,$ (A.1)

the operator spectrum is finite and can be uniquely determined by solving the
bootstrap equation.151515In this paper, we only consider minimal models whose
operator spectrum is diagonal modular invariant. We denote such an operator
content of $\mathcal{M}_{k+3,k+2}$ as $\mathcal{T}_{k+2}$. Their unitarity can
be understood as a consequence of existing exact unitary realization of them
using the coset construction of the current algebra (i.e. the gauged Wess-
Zumino-Witten (WZW) model in the Lagrangian description) as

$\mathcal{M}_{k+3,k+2}={su(2)_{k}\times su(2)_{1}\over su(2)_{k+1}}\,.$ (A.2)

Here $su(2)_{k}$ denotes the $su(2)$ Kac-Moody algebra at level $k$ with which
we can construct a Virasoro algebra using the Sugawara construction whose
central charge is

$c_{su(2)_{k}}={3k\over k+2}\,.$ (A.3)

Therefore we can check that the central charge of the coset construction from
Equ. (A.2) is given by

$c={3k\over k+2}+{3\over 1+2}-{3(k+1)\over k+3}=1-{6\over(k+2)(k+3)}\,,$ (A.4)

which exactly matches that in Equ. (A.1).

Now we will spell out the map between the operator spectrums from the minimal
model to that of its coset construction. A primary representation
$\ket{h,l}_{k}$ of the current algebra $su(2)_{k}$ is specified by two quantum
numbers $h$ and $l$ where $h$ is the conformal weight and $l$ denotes the
spin-${l\over 2}$ representation of the $su(2)$ algebra. The structure of the
$su(2)_{k}$ algebra puts a constraint

$0\leq l\leq k\,,$ (A.5)

and the conformal weight $h$ is related to $l$ as

$h={l(l+2)\over 4(k+2)}\,,$ (A.6)

and for later convenience we will denote this representation as $(l)_{k}$. A
primary representation $h_{r,s}(k+2)$ of the minimal model
$\mathcal{M}_{k+3,k+2}$ is specified by two integer quantum numbers $r$ and
$s$ which satisfies the constriants

$1\leq r\leq k+1\,,\quad 1\leq s\leq k+2\,.$ (A.7)

A primary representation of the tensor product current algebra
$su(2)_{k}\times su(2)_{1}$ is a tensor product of the primary representations
of each of them and it can be decomposed as a direct sum of the primary
representations of the current algebra $su(2)_{k+1}$ and the minimal model
$\mathcal{M}_{k+3,k+2}$:

$\begin{split}(r-1)_{k}\otimes(d-1)_{1}=\mathop{\oplus_{0\leq(s-1)\leq
k+1}}_{r+s+d=1\text{ mod2}}\Big{[}(s-1)_{k+1}\otimes
h_{r,s}(k+2)\Big{]}\,.\end{split}$ (A.8)

As a result, we can denote the minimal model representation $h_{r,s}(k+2)$
using the symbols from the current algebra as $[r,d;s]$ where $1\leq r\leq
k+1$, $1\leq s\leq k+2$, $d=1,2$ and $r+s+d=1\text{ mod2}$. The rule Equ.
(A.8) is called the branching rule and can be used to compute the modular
S-matrix of the minimal model in terms of those of the current algebra as

$S_{[r,d;s],[r^{\prime},d^{\prime};s^{\prime}]}=S^{(k)}_{r-1,r^{\prime}-1}S^{(1)}_{d-1,d^{\prime}-1}S^{(k+1)}_{s-1,s^{\prime}-1}\,,$
(A.9)

where $S^{(k)}_{p,p^{\prime}}=\sqrt{{2\over
k+2}}\sin({\pi(p+1)(p^{\prime}+1)\over k+2})$ is the modular S-matrix between
the primary representations $(p)_{k}$ and $(p^{\prime})_{k}$ of the
$su(2)_{k}$ current algebra.

###  A.2 Hidden Symmetry in the Tensor Product of Consecutive Minimal Models

The folding trick description of the RG brane as a conformal interface
necessities the study of the direct sum of two adjacent minimal models
$\mathcal{M}_{k+2,k+1}\times\mathcal{M}_{k+3,k+2}$.161616A conformal interface
between two CFTs can be described as a Cardy boundary of the tensor product of
the two CFTs if we fold the spacetime with respect to the interface. Using the
coset construction Equ. (A.2) we are studying the following cosets of the
current algebras

$\mathcal{M}_{k+2,k+1}\times\mathcal{M}_{k+3,k+2}={su(2)_{k-1}\times
su(2)_{1}\over su(2)_{k}}\times{su(2)_{k}\times su(2)_{1}\over
su(2)_{k+1}}\,,$ (A.10)

where cancelling the common $su(2)_{k}$ factors in the denominator and
numerator we get

$\mathcal{M}_{k+2,k+1}\times\mathcal{M}_{k+3,k+2}={su(2)_{k-1}\times
su(2)_{1}\times su(2)_{1}\over su(2)_{k+1}}\,.$ (A.11)

This is the relation between the apparent symmetry generators. Nevertheless,
since we only consider diagonal modular invariant operator spectrum we have to
be careful about whether the identity of Equ. (A.11) can be established for
the operator spectrum. We denote the diagonal modular invariant spectrum of
${su(2)_{k-1}\times su(2)_{1}\times su(2)_{1}\over su(2)_{k+1}}$ as
$\mathcal{T}_{B}$.

Now let’s understand the difference between their spectrum. Using the
branching rule Equ. (A.8) we can study the decomposition of the primary
representations of the current algebras to those of the minimal models

$\begin{split}(r-1)_{k-1}\otimes(d-1)_{1}\otimes(\widetilde{d}-1)_{1}&=\mathop{\oplus_{0\leq(s-1)\leq
k}}_{r+s+d=1\text{ mod2}}\Big{[}(s-1)_{k}\otimes
h_{r,s}(k+1)\Big{]}\otimes(\widetilde{d}-1)_{1}\,\\\
&=\mathop{\mathop{\mathop{\oplus_{0\leq(s-1)\leq k}}_{r+s+d=1\text{
mod2}}}_{0\leq(t-1)\leq k+1}}_{s+t+\widetilde{d}=1\text{
mod2}}h_{r,s}(k+1)\otimes h_{s,t}(k+2)\otimes(t-1)_{k+1}\,.\end{split}$ (A.12)

Hence we can use the minimal model symbols to denote the primary
representations of the tensor product of the current algebra
${su(2)_{k-1}\times su(2)_{1}\times su(2)_{1}\over su(2)_{k+1}}$:

$[r,d,\widetilde{d};t]=\mathop{\oplus_{s,r+s+d=1\text{
mod2}}}_{s+t+\widetilde{d}=1\text{ mod2}}h_{r,s}(k+1)\otimes h_{s,t}(k+2)\,,$
(A.13)

where we emphasize that $d,\widetilde{d}=1,2$. As a result, the diagonal
modular invariant spectrum $\mathcal{T}_{B}$ of ${su(2)_{k-1}\times
su(2)_{1}\times su(2)_{1}\over su(2)_{k+1}}$ is different from the tensor
product $\mathcal{T}_{k+1}\times\mathcal{T}_{k+2}$ of the diagonal modular
invariant spectrums of $\mathcal{M}_{k+2,k+1}$ and $\mathcal{M}_{k+3,k+2}$.
Moreover, Equ. (A.13) tells us that the (Kac-Moody) primary representations in
$\mathcal{T}_{B}$ is only a subsector of the (Virasoro) primary
representations in $\mathcal{T}_{k+1}\times\mathcal{T}_{k+2}$:

$\mathcal{T}_{B}=\mathop{\mathop{\oplus_{s=1,2,\cdots,k+1}}_{r=1,2\cdots,k}}_{t=1,2,\cdots,k+2}h_{r,s}(k+1)\otimes
h_{s,t}(k+2)\,.$ (A.14)

####  A.2.1 The Hidden Symmetry

We have seen that the representation space $\mathcal{T}_{B}$ is smaller than
$\mathcal{T}_{k+1}\times\mathcal{T}_{k+2}$ and this usually means that the
symmetry in $\mathcal{T}_{B}$ is larger than those of
$\mathcal{T}_{k+1}\times\mathcal{T}_{k+2}$. For a given pair of $(r,t)$ these
additional symmetries will classify $h_{r,s}(k+1)\otimes h_{s,t}(k+2)$ into
$[r,d,\widetilde{d};t]$ for all values of $s$ as in Equ. (A.13). In other
words, they have to classify the quantum number $s$ which takes value in
$\\{1,2,\cdots,k+1\\}$ into four subsets labeled by $d,\widetilde{d}=1,2$ as
$r+s+d=1\text{ mod2}$ together with $s+t+\widetilde{d}=1\text{ mod2}$. This
can be achieved by including the primary operators
$\phi^{(k+1)}_{1,l}\phi^{(k+2)}_{l,1}$ for $\forall\text{ odd
}l\in\\{1,2,\cdots,k+1\\}$ to the current algebra.171717It is easy to see this
using the fusion rules that $s$ is changed by an even integer under this
symmetry algebra so only representations with the same $(d,\widetilde{d})$
transform to each other under this symmetry algebra. Moreover
$\phi^{(k+1)}_{1,l}\phi^{(k+2)}_{l,1}$ is a legitimate current algebra as its
conformal weight can be calculated  $h={(1-l)^{2}\over 4}+{(l-1)^{2}\over
4}+{1^{2}-1\over 4(k+2)}-{l^{2}-1\over 4(k+3)}+{l^{2}-1\over
4(k+3)}-{1^{2}-1\over 4(k+4)}={(1-l)^{2}\over 2}\,,$ (A.15) as an integer for
odd $l$. Hence we see that $\mathcal{T}_{B}$ enjoys this enlarged chiral
algebra which we denote by $\mathcal{B}$.

Interestingly, the enlarged symmetry we just uncovered for $\mathcal{T}_{B}$
can be further enlarged by assembling the two possible $d,\widetilde{d}$
configurations of $[r,d,\widetilde{d};t]$ for a given pair of
$r,t$181818Remember from Equ. (A.13) for a given $r,t$ there are only two
possible configurations of $d,\widetilde{d}$. into a single multiplet $[r;t]$.
This can be done by further including the primary operators
$\phi^{(k+1)}_{1,l}\phi^{(k+2)}_{l,1}$ $\forall\text{ even
}l\in\\{1,2,\cdots,k+1\\}$ to the current algebra. Nevertheless, these new
current algebra generators have half-integral conformal weights which means
that they are fermionic. Using the facts that the conformal weight of the
primary $\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t}$ is

${(r-s)^{2}\over 4}+{(s-t)^{2}\over 4}+{r^{2}-1\over 4(k+2)}-{t^{2}-1\over
4(k+3)}\,,$ (A.16)

and the fermionic currents $\phi^{(k+1)}_{1,l}\phi^{(k+2)}_{l,1}$ (i.e. $l$ is
even) change the quantum number $s$ by odd integers, we can see that the
change in the conformal weight Equ. (A.16) under such an action is an integer
if $r+t$ is odd and a half-integer if $r+t$ is even. We can conclude that the
representation $[r;t]$ is in the NS sector of this further enlarged algebra if
$r+t$ is an even integer and $R$ sector if $r+t$ is an odd integer. We will
denote this new chiral algebra by $\widetilde{\mathcal{B}}$.

Furthermore, we notice that there is an obvious $\mathbb{Z}_{2}$ symmetry in
the algebra $\mathcal{B}$ which exchanges the two $su(2)_{1}$’s in Equ.
(A.11). This symmetry will mix the representations of $\mathcal{M}_{k+2,k+1}$
and $\mathcal{M}_{k+3,k+2}$ as it is obvious from Equ. (A.10) and Equ. (A.11).

####  A.2.2 Another Representation of the $\widetilde{\mathcal{B}}$ with an
Explicit Supersymmetry

To make the $\mathbb{Z}_{2}$ symmetry manifest, we try to combine the two
$su(2)_{1}$’s together in Equ. (A.10) by cancelling the common $su(2)_{k}$
factor and introducing an $su(2)_{2}$ factor and:

$\mathcal{M}_{k+2,k+1}\times\mathcal{M}_{k+3,k+2}={su(2)_{k-1}\times
su(2)_{2}\over su(2)_{k+1}}\times{su(2)_{1}\times su(2)_{1}\over
su(2)_{2}}\,,$ (A.17)

which can be seen as the tensor product of an $\mathcal{N}=1$ supersymmetric
minimal model and the Ising model
$\mathcal{SM}_{k+3,k+1}\times\mathcal{M}_{4,3}$. With this it is obvious that
the $\mathbb{Z}_{2}$ symmetry exchanging the two $su(2)_{1}$’s is the usualy
$\mathbb{Z}_{2}$ symmetry in the Ising model which maps the free Majorana
fermion $\psi$ to $-\psi$. This can be seen by realizing that the fermion
composite $\psi\overline{\psi}$ in the Ising model sector (remember our
operator spectrum is always diagonal modular invariant) is
$\phi^{(k+1)}_{1,2}\phi^{(k+2)}_{2,1}$ which has conformal weight
$h=\overline{h}={1\over 2}$ from Equ. (A.16) and is invariant under the
transformation $d\leftrightarrow\widetilde{d}$. This can also be seen from the
fusion of two $\phi^{(k+1)}_{1,2}\phi^{(k+2)}_{2,1}$ (the result should be
projected to that satisfies the constraint of operators in $\mathcal{T}_{B}$
i.e. of the form $\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t}$) which only contains
two primaries $\phi^{(k+1)}_{1,1}\phi^{(k+2)}_{1,1}$ (of conformal weight
$h=0$) and $\phi^{(k+1)}_{1,3}\phi^{(k+2)}_{3,1}$ (of conformal weight $h=2$).

The representation $[r,d,\widetilde{d};t]$ is decomposed into the
representation $[r,d^{\prime};t]$ in $\mathcal{SM}_{k+3,k+1}$ tensor product
$[d,\widetilde{d};d^{\prime}]$ in the Ising model where $d^{\prime}=1,2,3$,
$d+\widetilde{d}+d^{\prime}=1\text{ mod2}$ and $r+d^{\prime}+t=1\text{ mod2}$.
This tells us that we either have the tensor product between the NS sector of
$\mathcal{SM}_{k+3,k+1}$ with $1,\epsilon$ in the Ising model or the tensor
product between the R sector of $\mathcal{SM}_{k+3,k+1}$ and $\sigma$ in the
Ising model for representations in $\mathcal{T}_{B}$ (remember from the Sec.
A.2.1 that $r+t$ even is the NS sector and odd is the R sector).

Moreover, it is easier to use the stress-energy tensor to figure out how the
representations in one description is transformed to those in another. We are
more interested in the transformation from
$\mathcal{M}_{k+2,k+1}\times\mathcal{M}_{k+3,k+2}$ to
$\mathcal{SM}_{k+3,k+1}\times\mathcal{M}_{4,3}$. For this purpose, we give the
relations between all their chiral currents of conformal weight two:

$\begin{split}T^{(k+1)}&={k+3\over
2k+4}T_{\mathcal{SM}}+{\sqrt{(k+1)(k+3)}\over 2k+4}G\psi+{k-1\over
2k+4}T_{\psi}\\\ T^{(k+2)}&={k+1\over
2k+4}T_{\mathcal{SM}}-{\sqrt{(k+1)(k+3)}\over 2k+4}G\psi+{k+5\over
2k+4}T_{\psi}\\\
\phi^{(k+1)}_{1,3}\phi^{(k+2)}_{3,1}&=(k+1)(k+3)T_{\mathcal{SM}}-3\sqrt{(k+1)(k+3)}G\psi-3(k-1)(k+5)T_{\psi}\,,\end{split}$
(A.18)

where $T_{\mathcal{SM}}$ is the holomorphic stress-energy tensor of
$\mathcal{SM}_{k+3,k+1}$, $T_{\psi}$ is the holomorphic stress-energy tensor
of the Ising model, $G$ is the superconformal current of
$\mathcal{SM}_{k+3,k+1}$. From here we can see that the total stress-energy
tensor $T^{(k+1)}+T^{(k+2)}$ in the
$\mathcal{M}_{k+2,k+1}\times\mathcal{M}_{k+3,k+2}$ equals to the total stress-
energy tensor $T_{\mathcal{SM}}+T_{\psi}$ in
$\mathcal{SM}_{k+3,k+1}\times\mathcal{M}_{4,3}$.

For later convenience, we give the $\mathbb{Z}_{2}$ ($\psi\rightarrow-\psi$)
transform of the two stress-energy tensors of
$\mathcal{M}_{k+2,k+1}\times\mathcal{M}_{k+3,k+2}$:

$\begin{split}T^{(k+1)}&\rightarrow{3\over(k+2)(k+4)}T^{(k+1)}+{(k-1)(k+3)\over
k(k+2)}T^{(k+2)}+{1\over k(k+2)(k+4)}\phi^{(k+1)}_{1,3}\phi^{(k+2)}_{3,1}\\\
T^{(k+2)}&\rightarrow{(k+1)(k+5)\over(k+2)(k+4)}T^{(k+1)}+{3\over
k(k+2)}T^{(k+2)}-{1\over
k(k+2)(k+4)}\phi^{(k+1)}_{1,3}\phi^{(k+2)}_{3,1}\,.\end{split}$ (A.19)

##  Appendix B Review of Gaiotto’s Proposal For RG Brane

###  B.1 Useful Properties of Topological Defects

Toplogical defects in a CFT are totally transmissive interfaces to the
symmetry currents which can hence be arbitrarily deformed (without passing
through any operators) while they are inserted into any correlators of the CFT
operators. Mathematically, we can denote a topological defect as an operator
$\mathcal{D}$ which satisfies

$[\mathcal{D},\mathcal{J}_{n}]=0\,,\quad[\mathcal{D},\overline{\mathcal{J}}_{n}]=0\,,$
(B.1)

where $\mathcal{J}_{n}$ and $\overline{\mathcal{J}}_{n}$ are the holomorphic
and anti-holomorphic modes of any symmetry current of the CFT. Since we only
consider diagonal modular invariant CFT’s, the topological defects allow
Cardy’s algebraic classification Petkova:2009pe  (see Bachas:2008jd  for a
different perspective from string theory).

Hence, in a minimal model $\mathcal{M}_{k+3,k+2}$ a topological defect can be
denoted as $\mathcal{D}_{r,s}^{(k+2)}$ associated to a Cardy’s state
$\ket{r,s}\rangle_{\text{Cardy}}$ and acts as map

$\mathcal{D}_{r,s}^{(k+2)}=\sum_{r^{\prime},s^{\prime}}{S_{[r;s],[r^{\prime};s^{\prime}]}\over
S_{[1;1],[r^{\prime};s^{\prime}]}}\sum_{n}\ket{r^{\prime},s^{\prime};n}\bra{r^{\prime},s^{\prime};n}\,,$
(B.2)

in the diagonal modular invariant Hilbert space and where
$S_{[r;s],[r^{\prime};s^{\prime}]}$ is the modular S-matrix of primary
representations in $\mathcal{M}_{k+3,k+2}$.191919It is easy to see that Equ.
(B.2) satisfies Equ. (B.1) as for example
$\mathcal{J}_{n}=\mathcal{J}^{\dagger}_{-n}$. This description is useful for
closed topological defect on a plane or a topological defect which wraps
around a nontrivial cycle of a cylinder or a torus. In the former case, we get
a map between operators of the (diagonal modular invariant) CFT
$\mathcal{M}_{k+3,k+2}$ which maps the spinless primary operator
$\mathcal{O}_{(r^{\prime},s^{\prime})}$ to itself with a factor
${S_{[r;s],[r^{\prime};s^{\prime}]}\over S_{[1;1],[r^{\prime};s^{\prime}]}}$
multiplied. In the later case, it provides a specific cutting and gluing
prescription along the cycle in the computation of the partition function.

The most useful property to us is that topological defects can end on certain
fields called disorder fields Frohlich:2004ef  or twist fields Gaiotto:2012np
. These fields are representations of the symmetry algebra of the CFT and can
in general have nonzero spin Frohlich:2004ef . The rule is that the
topological defect $\mathcal{D}_{r,s}^{(k+2)}$ can end on such an operator
$\mathcal{O}_{(p,q),(m,n)}$ if $h_{r,s}(k+2)$ appears in the fusion between
$h_{p,q}(k+2)$ and $h_{m,n}(k+2)$. For example, $\mathcal{D}_{r,s}^{(k+2)}$
can end on a chiral disorder operator $\phi^{(k+2)}_{r,s}$. Moreover, this
tells us that we can move a topological defect $\mathcal{D}_{r,s}^{(k+2)}$
across a spinless primary field $\mathcal{O}_{(r^{\prime},s^{\prime})}$ and
end up with a spinless disorder operator
$\phi^{(k+2)}_{r^{\prime},s^{\prime}}$ connected to the topological defect
$\mathcal{D}_{r,s}^{(k+2)}$ by a tail $\mathcal{D}_{p,q}^{(k+2)}$ such that
$h_{p,q}(k+2)$ appears in the fusion between $h_{r,s}(k+2)$ and itself and
also in the fusion between $h_{r^{\prime},s^{\prime}}(k+2)$ and itself (see.
Fig.6).

As we will see this last property is useful for extracting important
nonperturbative results from perturbative calculations and constraining the RG
follow.

$\mathcal{D}_{r,s}$$\mathcal{O}_{r^{\prime},s^{\prime}}$$\phi_{r^{\prime},s^{\prime}}$$\mathcal{D}_{p,q}$$\mathcal{D}_{r,s}$
Figure 6: The demonstration of crossing the topological defect
$\mathcal{D}_{r,s}$ through a spinless primary operator
$\mathcal{O}_{r^{\prime},s^{\prime}}$. This operation transforms
$\mathcal{O}_{r^{\prime},s^{\prime}}$ to a twist field
$\phi_{r^{\prime},s^{\prime}}$ with a topological defect $D_{p,q}$ connecting
it to $D_{r,s}$.

###  B.2 RG Flow of the Topological Defect $\mathcal{D}_{r,1}^{(k+2)}$

The RG flow we are interested in is triggered by the spinless primary operator
$\phi^{(k+2)}_{1,3}$. The transformation of the topological defect
$\mathcal{D}_{r,1}^{(k+2)}$ under this RG flow can be first understood by the
fact that moving $\mathcal{D}_{r,1}^{(k+2)}$ across the spinless primary
operator $\phi^{(k+2)}_{1,3}$ is a trivial operation as the only
representation which appears in both the fusion between $h_{r,1}(k+2)$ with
itself and the fusion of $h_{1,3}(k+2)$ with itself is the trivial
representation $h_{1,1}(k+2)$. Hence moving the the topological defect
$\mathcal{D}_{r,1}^{(k+2)}$ across the spinless primary operator
$\phi^{(k+2)}_{1,3}$ is almost a trivial operation with at most a scalar
factor multiplied. Since moving $\mathcal{D}_{r,1}^{(k+2)}$ across
$\phi^{(k+2)}_{1,3}$ and back is equivalent to doing nothing, the scalar
factor can only be 1 or $-1$. The precise value of this scalar factor can be
determined by comparing a small $D_{r,1}^{(k+2)}$ loop surrounding
$\phi^{(k+2)}_{1,3}$ and a small $D_{r,1}^{(k+2)}$ loop surrounding nothing.
From Equ. (B.2), we are just taking the following ratio

${S_{[r;1],[1;3]}\over S_{[1;1],[1;3]}}/{S_{[r;1],[1;1]}\over
S_{[1;1],[1;1]}}=1\,.$ (B.3)

As a result, $D_{r,1}^{(k+2)}$ is invisible to the spinless primary operator
$\phi^{(k+2)}_{1,3}$ and it should be mapped to another topological defect
under the RG flow. The result can be extracted from a perturbative RG flow
calculation with large $k$ Fredenhagen:2009tn  and it is that
$\mathcal{D}_{r,1}^{(k+2)}$ will flow to $\mathcal{D}_{1,r}^{(k+1)}$ if we
assume that $\mathcal{M}_{k+2,k+3}$ flows to
$\mathcal{M}_{k+1,k+2}$.202020This is also consistent with the large $k$ RG
calculation Fredenhagen:2009tn .

###  B.3 Extended Symmetry Algebra on RG Domain Wall from Topological Defect

When we have the RG domain wall between $\mathcal{M}_{k+3,k+2}$ on the left
and $\mathcal{M}_{k+2,k+1}$ on the right, we can consider a topological defect
$\mathcal{D}_{r,1}^{(k+2)}$ on the left and deforming half of it through the
RG domain wall to the right (see Fig.7). The RG domain wall will transform the
acrossed half to $\mathcal{D}_{1,r}^{(k+1)}$ as a result of the RG transform
and we end up with a topological defect straddling between the two CFTs on the
two sides of the RG brane. Now applying the folding trick, we end up with a
topological defect which we denote as
$\mathcal{D}_{r,1}^{(k+2)}\mathcal{D}_{1,r}^{(k+1)}$ which end on the Cardy
brane (see Fig.8). We can put a chiral disorder field
$\phi^{(k+2)}_{r,1}\phi^{(k+1)}_{1,r}$ at the end of
$\mathcal{D}_{r,1}^{(k+2)}\mathcal{D}_{1,r}^{(k+1)}$ and push it all the way
to the Cardy brane which gives us a boundary operator that doesn’t change the
boundary condition and have integral (for r odd) or half-integeral (for r
even) conformal weight. We can do similar things for antichiral disorder
fields so from the results in Sec. A.2.1 and Sec. A.2.2 we get two copies of
the algebra $\widetilde{B}$ on the Cardy brane.

Gaiotto’s suggestion Gaiotto:2012np  is that the Cardy brane should glue these
two copies of $\widetilde{B}$ and the two copies of
$\mathcal{T}_{k+1}\times\mathcal{T}_{k+2}$ inisde them are mapped to each
other by the $\mathbb{Z}_{2}$ twisting described in Sec. A.2.2. In other
words, the two copies of $\widetilde{B}$’s are glued to each other by the
$\mathbb{Z}_{2}$ automorphism. As a result, the boundary operators should be
the boundary extrapolation of $\mathbb{Z}_{2}$ invariant operators
$\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t}$ with $r+t$ even.212121This can be seen
by remember the $\mathbb{Z}_{2}$ exchanges the two $su(2)_{1}$ algebras and so
it exchanges $d$ and $\widetilde{d}$ on the LHS of Equ. (A.13). As a result
$\mathbb{Z}_{2}$ invariant operators should have $d=\widetilde{d}$ and so we
get $r+t$ even from the RHS. This is consistent with the results from the
perturbative RG calculation Zamolodchikov:1987ti .

$\mathcal{D}_{r,1}^{k+2}$$\mathcal{D}_{r,1}^{(k+2)}$$\mathcal{D}_{1,r}^{(k+1)}$
Figure 7: Moving half of the topological defect $\mathcal{D}_{r,1}^{(k+2)}$ on
the right through the RG brane is equivalent to performing an RG transform of
of it. This gives another half of the resulting topological defect on the left
as $\mathcal{D}_{1,r}^{(k+1)}$.
$\mathcal{D}_{r,1}^{(k+2)}$$\mathcal{D}_{1,r}^{(k+1)}$$\mathcal{D}^{(k+1)}_{1,r}\mathcal{D}^{(k+2)}_{r,1}$
Figure 8: From the unfolded picture (left) to the folded picture (right). The
red wall denotes a Cardy brane in the folded picture. In the folded picture we
have a topological defect $\mathcal{D}^{(k+1)}_{1,r}\mathcal{D}^{(k+2)}_{r,1}$
ending on the Cardy brane.

###  B.4 Explicit Construction of the RG Domain Wall

With the boundary operator spectrum known, a general boundary state can be
written down as a linear combination of the Ishibashi states corresponding to
the boundary operator spectrum under the constrains that the Verlinde formula
should be satisfied by those coefficients. There are in general many such
states but Gaiotto proposed the following one to be the correct one

$\ket{\widetilde{B}}=\sum_{r,t}^{r+t\in
2\mathbb{Z}}\sqrt{S^{(k-1)}_{0,r-1}S^{(k+1)}_{0,t-1}}|r,t;\widetilde{B}\rangle\rangle\,,$
(B.4)

where $|r,t;\widetilde{B}\rangle\rangle$ is the Ishibashi state for the
algebra $\widetilde{B}$ corresponding to the representation $[r;t]$. This
state is simple as it satisfies the Verlinde formula such that all the
multiplicities in the modular dual channel is uniformly one. Moreover, we can
use the $\mathcal{T}_{B}$ language to rewrite the Ishibashi states as

$|r,t;\widetilde{B}\rangle\rangle=\sqrt{S^{(1)}_{0,0}}|r,1,1,t;\mathcal{T}_{B}\rangle\rangle+\sqrt{S^{(1)}_{1,1}}|r,2,2,t;\mathcal{T}_{B}\rangle\rangle\,,$
(B.5)

where $|r,d,\widetilde{d},t;\mathcal{T}_{B}\rangle\rangle$ is the Ishibashi
state for the algebra $\mathcal{T}_{B}$ associated to the representation
$[r,d,\widetilde{d};t]$ and we have used the fact that we could assemble the
$\mathcal{T}_{B}$ representations $[r,d,\widetilde{d};t]$ for given $r$ and
$t$ into the $\widetilde{B}$ representation $[r;t]$. Furthermore, using the
decomposition Equ. (A.17) we can write the Ishibashi state
$|r,d,\widetilde{d},t;\mathcal{T}_{B}\rangle\rangle$ as a superposition of
tensor products of Ishibashi states in the supersymmetric minimal model and
the Ising model. This is associated with the following decompositions of
representations

$\begin{split}[r,1,1;t]&=[r,1;t]\otimes[1,1;1]+[r,3;t]\otimes[1,1;3]\,,\\\
[r,2,2;t]&=[r,1;t]\otimes[2,2;1]+[r,3;t]\otimes[2,2;3]\,.\end{split}$ (B.6)

However, when we translate them into the relationships between Ishibashi
states we have to know the linear superposition coefficients. This can be
fixed by observing that the $\mathbb{Z}_{2}$ automorphism we are using is
localized purely in the Ising model sector. As a result, the Cardy state
$\ket{\widetilde{B}}$ we get should be a tensor product of the identity Cardy
state of the supersymmetric minimal model $\mathcal{SM}_{k+3,k+1}$ and a
nontrivial Cardy state of the Ising model whose Ishibashi components are
$|1,1;1\rangle\rangle$, $|1,1;3\rangle\rangle$, $|2,2;1\rangle\rangle$ and
$|2,2;3\rangle\rangle$ (i.e. $|\epsilon\rangle\rangle$ and $|1\rangle\rangle$
by the branching rule Equ. (A.8)). This Cardy state is
$\ket{\sigma}={1\over\sqrt{2}}(|1\rangle\rangle-|\epsilon\rangle\rangle)$
which indeed implements the $\mathbb{Z}_{2}$ transformation
$\psi\rightarrow-\psi$.222222This can be seen by using the doubling trick,
identifying the Cardy boundary $\ket{\sigma}$ as a topological defect
$\mathcal{D}_{\sigma}$ in the resulting chiral theory and using the fact that
$\mathcal{D}_{\sigma}$ is a group like defect. Hence, the Cardy state Equ.
(B.4) can be written as

$\ket{\widetilde{B}}=\ket{1}_{\text{NS}}\otimes\ket{\sigma}_{\text{Ising}}={1\over\sqrt{2}}\ket{1}_{\text{NS}}\otimes(|1\rangle\rangle-|\epsilon\rangle\rangle)\,,$
(B.7)

where the Cardy state $\ket{1}_{\text{NS}}$ can be written in the Ishibashi
states as

$\begin{split}\ket{1}_{\text{NS}}&=\sum_{r,t}^{r+t\in
2\mathbb{Z}}\sqrt{S_{[r;t],[0,0]}}|r;t\rangle\rangle\,\\\ &=\sum_{r,t}^{r+t\in
2\mathbb{Z}}\sqrt{S^{(k-1)}_{0,r-1}S^{(k+2)}_{0,t-1}S^{(2)}_{0,0}}|r,1;t\rangle\rangle+\sum_{r,t}^{r+t\in
2\mathbb{Z}}\sqrt{S^{(k-1)}_{0,r-1}S^{(k+2)}_{0,t-1}S^{(2)}_{0,2}}|r,3;t\rangle\rangle\,\\\
&={1\over\sqrt{2}}\sum_{r,t}^{r+t\in
2\mathbb{Z}}\sqrt{S^{(k-1)}_{0,r-1}S^{(k+2)}_{0,t-1}}\Big{(}|r,1;t\rangle\rangle+|r,3;t\rangle\rangle\Big{)}\,.\end{split}$
(B.8)

Here we notice that $|r,1;t\rangle\rangle$ and $|r,3;t\rangle\rangle$ (for
$r+t\in 2\mathbb{Z}$) form an NS sector supermultiplet.

###  B.5 Computation of One-Point Functions

So far we have constructed the RG brane between two consecutive minimal models
$\mathcal{M}_{k+2,k+1}$ and $\mathcal{M}_{k+3,k+2}$ as a rational brane by
embedding the diagonal modular invariant representations $h_{r,s}(k+1)\otimes
h_{s,t}(k+2)=[r,d;s]\otimes[s,\widetilde{d};t]$ of the algebra
$\mathcal{M}_{k+2,k+1}\times\mathcal{M}_{k+3,k+2}$ into the diagonal modular
invariant representations $[r,d,\widetilde{d};t]$ of an equivalent algebra
${su(2)_{k-1}\times su(2)_{1}\times su(2)_{1}\over su(2)_{k+1}}$. For
simplicity we will called the former theory $\mathcal{T}_{\mathcal{A}}$ (i.e.
$\mathcal{T}_{k+1}\times\mathcal{T}_{k+2}$) and the latter theory
$\mathcal{T}_{\mathcal{B}}$. The precise value of the one-point functions of
the operators $\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t}$ in
$\mathcal{T}_{\mathcal{A}}$ is of important physical relevance as they tell us
how the operators $\phi^{(k+2)}_{r,s}$ and $\phi^{(k+1)}_{s,t}$ are mixed
under the RG flow. To find the precise value of these one-point functions we
have to map the rational brane Equ. (B.7) we have constructed in
$\mathcal{T}_{\mathcal{B}}$ back to $\mathcal{T}_{\mathcal{A}}$ and then the
one-point functions can be easily computed.

However, the map of the rational brane Equ. (B.7) to
$\mathcal{T}_{\mathcal{A}}$ is not trivial. This can be achieved by firstly
constructing a proper $\mathcal{T}_{\mathcal{A}}$ topological interface (i.e.
totally transmissive for symmetry currents in $\mathcal{T}_{\mathcal{A}}$)
$\mathcal{I}_{1}$ separating $\mathcal{T}_{\mathcal{A}}$ and
$\mathcal{T}_{\mathcal{B}}$ and then fusing it with the
$\mathcal{T}_{\mathcal{B}}$ rational brane Equ. (B.7) (see Fig.9). A proper
$\mathcal{I}_{1}$ should allow the $\mathcal{T}_{\mathcal{A}}$ topological
defects

$\mathcal{D}^{\mathcal{A}}_{[r,d;s]\otimes[s,\widetilde{d};t]}=\sum_{r^{\prime},d^{\prime},s^{\prime},\widetilde{d}^{\prime},t^{\prime}}{S_{[r,d;s]\otimes[s,\widetilde{d};t],[r^{\prime},d^{\prime};s^{\prime}]\otimes[s^{\prime},\widetilde{d}^{\prime},t^{\prime}]}\over
S_{[1,1;1]\otimes[1,1;1],[r^{\prime},d^{\prime};s^{\prime}]\otimes[s^{\prime},\widetilde{d}^{\prime},t^{\prime}]}}\sum_{n}\ket{r^{\prime},d^{\prime},s^{\prime},\widetilde{d^{\prime}},t^{\prime};n}\bra{r^{\prime},d^{\prime},s^{\prime},\widetilde{d^{\prime}},t^{\prime};n}\,,$
(B.9)

to end on it.232323We of course have the following constraints
$\begin{split}r+d+s=&1\text{ mod}2\,,\quad s+\widetilde{d}+t=1\text{
mod}2\,,\\\ r^{\prime}+d^{\prime}+s^{\prime}=&1\text{ mod}2\,,\quad
s^{\prime}+\widetilde{d}^{\prime}+t^{\prime}=1\text{ mod}2\,.\end{split}$
(B.10) This would be ensured if
$D^{\mathcal{A}}_{[r,d;s]\otimes[s,\widetilde{d};t]}$ appears in the fusion of
$\overline{\mathcal{I}_{1}}$ and $\mathcal{I}_{1}$ or in other words if
$\mathcal{I}_{1}\overline{\mathcal{I}_{1}}$ is a direct sum of the topological
defects $D^{\mathcal{A}}_{[r,d;s]\otimes[s,\widetilde{d};t]}$ (see Fig.10).
This is satisfied by the following construction of $\mathcal{I}_{1}$

$\mathcal{I}_{1}=\sum_{r,t,s,d,\widetilde{d}}\sqrt{{S_{[1;1],[r;t]}\over
S_{[1,1;1]\otimes[1,1;1],[r,d;s]\otimes[s,\widetilde{d},t]}}}\sum_{n}\ket{r,d,s,\widetilde{d},t;n}\bra{r,d,s,\widetilde{d},t;n}\,,$
(B.11)

where the operator $\bra{r,d,s,\widetilde{d},t;n}$ are orthonormal
$\mathcal{T}_{\mathcal{A}}$ descendents of the primary states
$\bra{r,d,s,\widetilde{d},t}$. The operator
$\sum_{n}\ket{r,d,s,\widetilde{d},t;n}\bra{r,d,s,\widetilde{d},t;n}$ acts
nontrivially only on the Ishibashi state $|r,t;\widetilde{B}\rangle\rangle$
and maps it to the Ishibashi state
$|r,d,s,\widetilde{d},t;\mathcal{A}\rangle\rangle$.242424This is because
$[r;t]$ can be split into $\mathcal{A}$ representations
$[r,d;s]\otimes[s;\widetilde{d};t]$ with multiplicities as one (see
Sec.A.2.1).

Moreover, it is easy to check that we have

$\begin{split}\mathcal{I}_{1}\overline{\mathcal{I}_{1}}&=\sum_{r,t,s,d,\widetilde{d}}{S_{[1;1],[r;t]}\over
S_{[1,1;1]\otimes[1,1;1],[r,d;s]\otimes[s,\widetilde{d},t]}}\sum_{n}\ket{r,d,s,\widetilde{d},t;n}\bra{r,d,s,\widetilde{d},t;n}\,,\\\
&=\sum_{s^{\prime},d^{\prime},\widetilde{d^{\prime}}}\mathcal{D}^{\mathcal{A}}_{[1,d^{\prime};s^{\prime}]\otimes[s^{\prime},\widetilde{d^{\prime}};1]}\,,\end{split}$
(B.12)

where the last step comes from the fact that

$\sum_{s^{\prime},d^{\prime},\widetilde{d^{\prime}}}S_{[1,d^{\prime};s^{\prime}]\otimes[s^{\prime},\widetilde{d^{\prime}};1],[r,d;s]\otimes[s,\widetilde{d};t]}=S_{[1;1],[r;t]}\,.$
(B.13)

This relation can be obtained from the following considerations. Let’s
consider the modular character $\chi_{[r;t]}(\tau)$ of the $\widetilde{B}$
representation $[r;t]$. Since we know that $[r;t]$ can be split into
$\mathcal{A}$ representations $[r,d;s]\otimes[s;\widetilde{d};t]$ with
multiplicities as one (see Sec.A.2.1) so we have

$\chi_{[r;t]}(\tau)=\sum_{s,d,\widetilde{d}}\chi_{[r,d;s]\otimes[s,\widetilde{d};t]}(\tau)\,.$
(B.14)

Now we consider $r=1,t=1$, do a modular transform

$S:\quad\tau-\rightarrow-{1\over\tau}\,,$ (B.15)

and we will have

$\begin{split}\chi_{[1;1]}(\tau)&=\sum_{r,t}S_{[1;1],[r;t]}\chi_{[r;t]}(-{1\over\tau})=\sum_{r,t,s,d,\widetilde{d}}S_{[1;1],[r;t]}\chi_{[r,d;s]\otimes[s,\widetilde{d};t]}(-{1\over\tau})\\\
&=\sum_{s^{\prime},d^{\prime},\widetilde{d^{\prime}}}\chi_{[1,d^{\prime};s^{\prime}]\otimes[s^{\prime},\widetilde{d^{\prime}};1]}(\tau)=\sum_{s^{\prime},d^{\prime},\widetilde{d^{\prime}}}\sum_{r,t,s,d,\widetilde{d}}S_{[1,d^{\prime};s^{\prime}]\otimes[s^{\prime},\widetilde{d^{\prime}};1],[r,d;s]\otimes[s,\widetilde{d};t]}\chi_{[r,d;s]\otimes[s,\widetilde{d};t]}(-{1\over\tau})\,,\end{split}$
(B.16)

which gives us Equ. (B.13) if we compare the end of the first line and the end
of the second line. Furthermore, Equ. (B.12) tells us that the
$\mathcal{T}_{\mathcal{A}}$ defects
$\mathcal{D}^{\mathcal{A}}_{[1,d,r]\otimes[r,\widetilde{d},1]}=\mathcal{D}^{(k+2)}_{r,1}\mathcal{D}^{(k+1)}_{1,r}$
can end on the defect $\mathcal{I}_{1}$ and hence could end on the Cardy
boundary $\mathcal{I}_{1}|\widetilde{B}\rangle\rangle$ after we fuse
$\mathcal{I}_{1}$ with $|\widetilde{B}\rangle\rangle$ getting a Cardy state
for $\mathcal{T}_{\mathcal{A}}$. This is precisely what we used in the
construction of the RG domain wall as we discussed in Sec.B.3 (see Fig.11).

Now we can use the map Equ. (B.11) to map the boundary state Equ. (B.4) to a
boundary state in the $\mathcal{T}_{\mathcal{A}}$ theory (see Fig.9)

$|\mathcal{A}\rangle\rangle=\mathcal{I}_{1}|\widetilde{B}\rangle\rangle=\sum_{r,t,s,d,\widetilde{d},d^{\prime}}^{r+t\in
2\mathbb{Z}}\alpha_{r,d,s,\widetilde{d},t,d^{\prime}}\sqrt{S^{(k-1)}_{0,r-1}S^{(k+1)}_{0,t-1}S^{(1)}_{0,d-1}S^{(1)}_{0,\widetilde{d}-1}}\sqrt{{S_{[1;1],[r;t]}\over
S_{[1,1;1]\otimes[1,1;1],[r,d;s]\otimes[s,\widetilde{d},t]}}}|r,d,s,\widetilde{d},t\rangle\rangle\,,$
(B.17)

where $|r,d,s,\widetilde{d},t\rangle\rangle$ is the
$\mathcal{T}_{\mathcal{A}}$ Ishibashi state associated with the representation
$[r,d;s]\otimes[s,\widetilde{d};t]$ and here we emphasize that the constrains
$r+d+s=1\text{ mod2}$, $s+\widetilde{d}+t=1\text{ mod2}$ and
$d+\widetilde{d}+d^{\prime}=1\text{ mod2}$ should be satisfied. The
coefficient $\alpha_{r,d,s,\widetilde{d},t,d^{\prime}}$ can be figured out by
normalization of the state and the decomposition Equ. (A.18). For the
$(r,s,t,d,\widetilde{d})$ such that $d^{\prime}$ can be uniquely determined
this factor is just one. This tells us that in these cases the one-point
function of the operator $\phi^{(k+1)}_{r,s}\phi^{(k+2)}_{s,t}$ is given by

$\sqrt{S^{(k-1)}_{0,r-1}S^{(k+1)}_{0,t-1}S^{(1)}_{0,d-1}S^{(1)}_{0,\widetilde{d}-1}}\sqrt{{S_{[1;1],[r;t]}\over
S_{[1,1;1]\otimes[1,1;1],[r,d;s]\otimes[s,\widetilde{d},t]}}}={\sqrt{S^{(k-1)}_{0,r-1}S^{(k+1)}_{0,t-1}}\over
S^{(k)}_{0,s-1}}\delta_{d,\widetilde{d}}\,.$ (B.18)

Examples of more general cases that $d^{\prime}$ is not uniquely fixed can be
found in Equ. (3.21) and Equ. (4.10).

$\mathcal{B}$$\mathcal{A}$$\mathcal{I}_{1}$$|\widetilde{B}\rangle\rangle$
Figure 9: The picture in the folded description of mapping the
$\mathcal{T}_{\mathcal{B}}$ Cardy state to a $\mathcal{A}\subset\mathcal{B}$
theory Cardy state by fusing it with a $\mathcal{T}_{\mathcal{A}}$ topological
defect $\mathcal{I}_{1}$ that separate the $\mathcal{A}$ theory and the
$\mathcal{B}$ theory. The fusion is topologically achieved by pushing
$\mathcal{I}_{1}$ all the way to $|\widetilde{B}\rangle\rangle$.
$\mathcal{B}$$\mathcal{A}$$\mathcal{I}_{1}$$\overline{\mathcal{I}}_{1}$$\mathcal{A}$
Figure 10: The picture in of fusing $\mathcal{I}_{1}$ with
$\overline{\mathcal{I}}_{1}$. The fusion is achieved topologically by pushing
$\mathcal{I}_{1}$ all the way to be coincident with
$\overline{\mathcal{I}}_{1}$. The result is a $\mathcal{T}_{\mathcal{A}}$
topological defect.
$\mathcal{D}^{(k+1)}_{1,r}\mathcal{D}^{(k+2)}_{r,1}$$\mathcal{I}_{1}$$|\widetilde{\mathcal{B}}\rangle\rangle$$\mathcal{B}$$\mathcal{A}$
Figure 11: The demonstration in the folded picture that how we could have
$\mathcal{T}_{\mathcal{A}}$ topological defects
$\mathcal{D}^{(k+1)}_{1,r}\mathcal{D}^{(k+2)}_{r,1}$ ending on the RG brane
which is the fusion of $\mathcal{I}_{1}$ with
$|\widetilde{\mathcal{B}}\rangle\rangle$. The fusion is achieved topological
by pushing $\mathcal{I}_{1}$ all the way to the $\mathcal{T}_{\mathcal{B}}$
Cardy boundary $|\widetilde{B}\rangle\rangle$.

##  Appendix C Topological Superconductors and Majorana Fermions

In this appendix we will illustrate the gist of topological superconductors
for readers with high-energy physics background. We will consider a toy
Hamiltonian for the Type IIID topological superconductor which we will use to
illustrate the relevant physical background of Sec. 5.1.

###  C.1 A Toy Hamiltonian and Its Topological Properties

Let’s consider a two-dimensional electronic system which hosts electrons and
holes. A typical such system is described by a lattice Hamiltonian where we
have a fermionic degree of freedom on each lattice site. The feromions can be
of spin-up and spin-down. We consider such a system in momentum space with a
p-wave superconducting pairing

$\begin{split}H&={1\over 2}\sum_{\vec{p}\in BZ}H_{\vec{p}}\\\ &={1\over
2}\sum_{\vec{p}\in BZ}\Big{[}\psi^{\dagger}_{\vec{p},\uparrow}({p^{2}\over
2m}-\mu)\psi_{\vec{p},\uparrow}+\Delta\psi_{-\vec{p},\uparrow}(p_{x}+ip_{y})\psi_{\vec{p},\uparrow}+\psi^{\dagger}_{\vec{p},\downarrow}({p^{2}\over
2m}-\mu)\psi_{\vec{p},\downarrow}+\Delta\psi_{-\vec{p},\downarrow}(p_{x}-ip_{y})\psi_{\vec{p},\downarrow}+H.C.\Big{]}\,,\end{split}$
(C.1)

where $\psi^{\dagger}_{\vec{p},s}$ is a creation operator for electron with
momentum $\vec{p}$ and spin $s$ ($s=\uparrow,\downarrow$), $\psi_{\vec{p},s}$
is the corresponding creation operator for a hole, $\mu$ is the chemical
potential, $\Delta$ is the p-wave pairing parameter and the momentum $\vec{p}$
lives in the Brillouin zone. This is called the Bogolubov-de Genes Hamiltonian
in condensed matter literature and the way the property of an electronic
material is analyzed is to firstly write down the Bogolubov-de Genes
Hamiltonian for that material and then solve for the eigenenergy and
eigenmodes. These eigenmodes are called quasi-particles. We don’t intend to
solve for the eigenmodes and eigenenergy of the Hamiltonian Equ. (C.1) but we
will analyze some important properties of it that tells us nontrivial
information of the spectrum and phases for the electronic system that is
described by this Hamiltonian.

Firstly, this Hamiltonian transforms under the particle-hole symmetry

$\mathcal{P}:\psi_{\vec{p},s}\rightarrow\psi^{\dagger}_{-\vec{p},-s}\,,\quad\psi^{\dagger}_{\vec{p},s}\rightarrow\psi_{-\vec{p},-s}\,$
(C.2)

as

$PH_{\vec{p}}P=-H_{-\vec{p}}\,.\,\quad PHP=-H\,.$ (C.3)

Hence, the spectrum of this Hamiltonian is paired into pairs of opposite
values

$H=\sum_{E\geq
0}\Big{[}E\gamma^{\dagger}_{E}\gamma_{E}-E\gamma^{\dagger}_{-E}\gamma_{-E}\Big{]}\,,$
(C.4)

where $\gamma^{\dagger}_{E}$ and $\gamma_{E}$ are the creation and
annihilation operators for the quasi-particles. Moreover, the spin-up and spin
down degress of freedom don’t mixed in the Hamiltonian. We can analyze
spectrum of $H_{\vec{p}}$. This can be down by defining

$\Psi^{\dagger}_{\vec{p},\uparrow}=(\psi^{\dagger}_{\vec{p},\uparrow},\psi_{-\vec{p},\uparrow})\,,\quad\Psi^{\dagger}_{\vec{p},\downarrow}=(\psi^{\dagger}_{\vec{p},\downarrow},\psi_{-\vec{p},\downarrow})\,.$
(C.5)

We can see that in both the spin-up and spin-down sector we have eigenvalues
$E_{\pm}(\vec{p})=\pm\sqrt{({p^{2}\over 2m}-\mu)^{2}+\Delta^{2}p^{2}}$. Hence,
there is a degeneracy of each eigenvalue $E$ and this degeneracy is due to the
spin quantum number. As a result, we can see that when $\mu=0$ there is a
gapless zero mode sector for $\vec{p}=0$ and close to this zero mode sector
there is an emergent time-reversal symmetry:

$\mathcal{T}:\psi_{\vec{p},s}\rightarrow\psi_{-\vec{p},-s}\,,\quad\psi^{\dagger}_{\vec{p},s}\rightarrow\psi^{\dagger}_{-\vec{p},-s}\,,\quad\vec{p}\rightarrow-\vec{p}\,,\quad
i\rightarrow-i\,,$ (C.6)

which can be seen from Equ. (C.1) by taking $\mu=0$ and ignoring $p^{2}$ terms
and this symmetry protects the gap.

Secondly, when $\mu\neq 0$, there is no zero modes and there is always a gap
in the spectrum. We want to understand the difference between $\mu>0$ and
$\mu<0$ cases. Since the two spin sectors don’t couple, we can focus on the
spin-up sector for which we can write the Hamiltonian as (for simplicity we
will take $\Delta=1$ hereafter)

$H_{\vec{p},\uparrow}+H_{-\vec{p},\uparrow}=\Psi^{\dagger}_{\vec{p},\uparrow}\begin{pmatrix}{p^{2}\over
2m}-\mu&p_{x}+ip_{y}\\\ p_{x}-ip_{y}&-({p^{2}\over
2m}-\mu)\end{pmatrix}\Psi_{\vec{p},\uparrow}=\Psi^{\dagger}_{\vec{p},\uparrow}\vec{h}(\vec{p})\cdot\vec{\sigma}\Psi_{\vec{p},\uparrow}\,,$
(C.7)

where we have $\vec{h}(\vec{p})=(p_{x},-p_{y},{p^{2}\over 2m}-\mu)$. As long
as $\mu\neq 0$, the vector $\vec{h}(\vec{p})$ is never zero, so we can
normalize it and define

$\widehat{h}(\vec{p})={\vec{h}(\vec{p})\over|\vec{h}(\vec{p})|}\,,$ (C.8)

and consider the Chern-number of the map $\widehat{h}(\vec{p})$ (notice that
$\widehat{h}(\vec{p})\rightarrow(0,0,1)$ as $\vec{p}\rightarrow\infty$)

$C=\int{d^{2}\vec{p}\over
4\pi}\big{[}\widehat{h}(\vec{p})\cdot\big{(}\partial_{p_{x}}\widehat{h}(\vec{p})\times\partial_{p_{y}}\widehat{h}(\vec{p})\big{)}\big{]}\,,$
(C.9)

which determines the winding number of the map $\widehat{h}(\vec{p})$. We can
see that $h^{z}(\vec{p})$ behaves rather differently when $\mu>0$ and $\mu<0$.
As a result, in the former case as $\vec{p}$ goes from $0$ to $\infty$
$\widehat{h}(\vec{p})$ starts with pointing to the South pole and ends up with
pointing to the North pole and in the later case $\widehat{h}^{z}(\vec{p})$
never takes a negative value so it always points to a point on the Northern
hemisphere. That is that in the $\mu>0$ case the winding number $C=1$ and in
the $\mu<0$ case $C=0$.

In summary, the two gapped phases $\mu>0$ and $\mu<0$ are different
topologically and their boundary $\mu=0$ has gapless zero modes degenerate in
spin and enjoy an emergent time-reversal symmetry in the low energy regime.
Moreover, we can see that the $\mu<0$ phase is a trivially gapped phase (this
can be seen by taking $\mu\rightarrow-\infty$) and $\mu>0$ is a weakly gapped
phase which is topological as it has a nontrivial topological number $C=1$.

###  C.2 Localized Majorana Modes on the Boundary– The Ising Model

Now we want to consider, the system described by the Hamiltonian Equ. (C.1) on
a manifold with boundary. For example a finite piece of the electronic
material whose physics is described by Equ. (C.1). We are interested in the
physics of its boundary when we tune its bulk to the topologically nontrivial
phase $\mu>0$. This can be done in experiment by electron doping.

At a fine-grained level we have to specify the precise boundary conditions and
solve for the energy spectrum of the Hamiltonian under this boundary
conditions. Nevertheless, we can grasp the gist using a slightly coarse-
grained model for the boundary. We can think of the environment as in the
trivial phase $\mu<0$ of the Hamiltonian Equ. (C.1). Since the bulk of the
system has been tuned to the topologically nontrivial phase $\mu>0$, the
boundary can be thought of as a thin buffer zone between these two phases
where $\mu=0$. For simplicity we will take the $y-$direction to be the
direction normal to the boundary of the material i.e. the buffer zone is thin
in the $y-$direction. We are interested in the low energy physics in the
buffer zone. In the low energy regime where the spin-up sector Hamiltonian can
be written as

$H_{\vec{p},\uparrow}+H_{-\vec{p},\uparrow}=\Psi^{\dagger}_{\vec{p},\uparrow}\begin{pmatrix}0&p_{x}+ip_{y}\\\
p_{x}-ip_{y}&0\end{pmatrix}\Psi_{\vec{p},\uparrow}.$ (C.10)

The buffer zoom is extremely thin in the $y-$direction so we can set $p_{y}$
to zero in the low energy regime. Expanding the matrix representation, this
gives us the Hamiltonian

$H_{\uparrow}=\sum_{p_{x}\geq
0}p_{x}(\psi^{\dagger}_{p_{x},\uparrow}\psi^{\dagger}_{-p_{x},\uparrow}+\psi_{-p_{x},\uparrow}\psi_{p_{x},\uparrow})\,.$
(C.11)

Let’s do the following redefinition

$\psi_{p_{x},\uparrow}={1\over\sqrt{2}}(\chi^{\dagger}_{p_{x},\uparrow}+\chi_{-p_{x},\uparrow})\,,$
(C.12)

where $\chi_{p_{x},\uparrow}$ is a fermionic annihilation operator satisfies
the standard algebra with the creation operator. This gives us

$\begin{split}H_{\uparrow}&=\sum_{p_{x}\geq
0}p_{x}(\chi_{p_{x},\uparrow}\chi_{-p_{x},\uparrow}+\chi_{p_{x},\uparrow}\chi^{\dagger}_{p_{x},\uparrow}+\chi^{\dagger}_{-p_{x},\uparrow}\chi_{-p_{x},\uparrow}+\chi^{\dagger}_{-p_{x},\uparrow}\chi^{\dagger}_{p_{x},\uparrow})\,\\\
&={1\over
2}\sum_{p_{x}}p_{x}(-\chi_{-p_{x},\uparrow}\chi_{p_{x},\uparrow}+\chi_{p_{x},\uparrow}\chi^{\dagger}_{p_{x},\uparrow}-\chi^{\dagger}_{p_{x},\uparrow}\chi_{p_{x},\uparrow}+\chi^{\dagger}_{-p_{x},\uparrow}\chi^{\dagger}_{p_{x},\uparrow})+const.\,\\\
&={1\over 2}\int
dx\chi_{\uparrow}(x,t)i\partial_{x}\chi_{\uparrow}(x,t)\,,\end{split}$ (C.13)

where we have a Majorana field

$\chi_{\uparrow}(x,t)=\sum_{p_{x}}(\chi_{p_{x},\uparrow}e^{ip_{x}(x+t)}+\chi^{\dagger}_{p_{x},\uparrow}e^{-ip_{x}(x+t)})\,.$
(C.14)

The Majorana field satisfies
$\chi_{\uparrow}(x,t)=\chi^{\dagger}_{\uparrow}(x,t)$. So we have the
Lagrangian

$L_{\uparrow}={i\over 2}\int
dx\chi_{\uparrow}(x,t)(\partial_{t}-\partial_{x})\chi_{\uparrow}(x,t)\,,$
(C.15)

which is chiral. Similarly, in the spin-down sector we have the Lagrangian

$L_{\downarrow}={i\over 2}\int
dx\chi_{\downarrow}(x,t)(\partial_{t}+\partial_{x})\chi_{\downarrow}(x,t)\,,$
(C.16)

where

$\psi_{p_{x},\downarrow}={i\over\sqrt{2}}(\chi^{\dagger}_{p_{x},\downarrow}+\chi_{-p_{x},\downarrow})\,.$
(C.17)

In total we have the Lagrangian

$L={i\over 2}\int
dx\Big{[}\chi_{\downarrow}(x,t)(\partial_{t}+\partial_{x})\chi_{\downarrow}(x,t)+\chi_{\uparrow}(x,t)(\partial_{t}-\partial_{x})\chi_{\uparrow}(x,t)\Big{]}\,,$
(C.18)

which describes two chiral Majorona fermions and the time-reversal symmetry
Equ. (C.19) translates to

$\mathcal{T}:\chi_{\uparrow}(x,t)\rightarrow-i\chi_{\downarrow}(x,-t)\,,\quad\chi_{\downarrow}(x,t)\rightarrow
i\chi_{\uparrow}(x,-t)\,,\quad i\rightarrow-i\,.$ (C.19)

Hence we have

$\mathcal{T}^{2}=-1\,.$ (C.20)

This time reversal symmetry prevents the Majorana ferimons to have Majorana
mass term $im\chi_{\downarrow}(x,t)\chi_{\uparrow}(x,t)$. As a result, we have
a gapless Majorana fermion whose mass is protected by the time-reversal
symmetry Equ. (C.19) and this theory is localized in the thin buffer zone or
equivalent on the boundary of the bulk topological material. This theory is
nothing but the 1+1-dimensional Ising Model. We emphasized that Equ. (C.20) is
an important character of the Type IIID topological superconductor which
ensure the boundary of the nontrivial topological phase to host two anti-
propagating Majorana fermions and hence gives us the Ising Model.

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