# Tensor Network Methods for Extracting CFT Data from Fixed-Point Tensors and
Defect Coarse Graining

Wenhan Guo  C. N. Yang Institute for Theoretical Physics and Department of
Physics and Astronomy, State University of New York at Stony Brook, Stony
Brook, NY 11794-3840, USA  Tzu-Chieh Wei  C. N. Yang Institute for Theoretical
Physics and Department of Physics and Astronomy, State University of New York
at Stony Brook, Stony Brook, NY 11794-3840, USA

###### Abstract

We present a comprehensive study on the extraction of CFT data using tensor
network methods, specially, from the fixed-point tensor of the linearized
tensor renormalization group (lTRG) for the 2D classical Ising model near the
critical temperature. Utilizing two different methods, we extract operator
scaling dimensions and operator-product-expansion (OPE) coefficients by
introducing defects on the lattice and by employing the fixed-point tensor. We
also explore the effects of point-like defects in the lattice on the coarse-
graining process. We find that there is a correspondence between coarse-
grained defect tensors and conformal states obtained from lTRG fixed-point
equation. We also analyze the capabilities and limitations of our proposed
coarse-graining scheme for tensor networks with point-like defects, which
includes graph independent local truncation (GILT) and higher-order tensor
renormalization group (HOTRG). Our results provide a better understanding of
the capacity and limitations of the tenor renormalization group scheme in
coarse-graining defect tensors, and we show that GILT+HOTRG can be used to
give accurate two- and four-point functions under specific conditions. We also
find that employing the minimal canonical form further improves the stability
of the RG flow.

##  I Introduction

The study of the renormalization group (RG) is of central importance in
studying critical phenomena in statistical mechanics and many-body quantum
systems [1, 2]. The basic idea is to remove the short-range information of a
given system while keeping its long-range behavior intact. For critical
systems, there is a similarity for behaviors at different length scales and,
very often, they can be described by a conformal field theory (CFT) [3], which
emerges at fixed points of the RG. By studying the perturbative behavior of
the RG near fixed points, one can extract useful information about the CFT,
such as the central charge, the scaling dimensions of conformal operators, and
the operator product expansion (OPE) coefficients between them.

Tensor network (TN) is a versatile framework for describing strongly
correlated systems by encoding the correlation and/or the entanglement
structure into mutually-connected tensors. It can be used to describe
partition functions of classical statistical models and wave functions of
quantum many-body ground states and topological order [4], as well as toy
models to study conformal field theory and holography [5, 6, 7]. In
particular, tensor network renormalization provides a new perspective of the
RG flow, where the properties of the coarse-grained system is being
parametrized by a tensor. Given the comparison that CFTs are fixed-points of
the RG flow when ‘integrating out’ the UV (short-distance) details, it is a
natural question whether one can build a correspondence or relationship
between the fixed-points tensor and the CFT. One particular direction of the
above question is whether it is possible to determine the CFT data of a
critical lattice system when given the fixed-point tensor from coarse graining
the lattice model. This question has been asked and dealt with previously
using tensor network methods [8, 9, 10, 11, 12, 13].

Tensor-based real-space RG methods have been developed over the past several
decades, offering certain advantages over traditional numerical methods like
Monte Carlo methods. The density matrix renormalization group (DMRG) [14, 15,
16, 17] method, which can be understood in terms of the matrix product state
(MPS) [18, 19, 20, 21] ansatz, has been the most successful numerical method
for one dimensional many-body systems that are gapped. The multiscale-
entanglement-renormalization-ansatz (MERA) [13, 22, 23] is a tree-tensor
network that provides reliable and accurate results for studying one-
dimensional gapless systems.

Project entangled pair states (PEPS) [24, 25, 26], proposed by Cirac and
Verstraete, are a natural extension of the MPS for quantum systems on a
lattice of two or higher dimensions that obey the area entanglement law.
However, contracting the tensor network in two and higher dimensions is a
computationally difficult task, which necessitates approximations to render it
efficient. One of the early coarse-graining methods is the corner transfer
matrix, originally developed by Baxter [27, 28], which has since been applied
to quantum systems [29]. Another approach for the real-space coarse-graining
scheme in two-dimensional systems is the tensor renormalization group (TRG) by
Levin and Nave [30], which prompted several variants and further idea [31, 32,
33, 34, 35], including the higher-order TRG (HOTRG) [33] that we will use in
this work.

Although these methods give an accurate estimation of free energy density and
local observables, they still do not generate a correct RG flow in the tensor
space. Gu and Wen [36] discovered that real-space TN RG methods did not remove
all the short-distance correlations, and local entanglement filtering was
needed to achieve a fixed-point tensor. In particular, during the real-space
RG, the local correlations residing across the boundary of the coarse-graining
blocks survive, which can be illustrated by a toy model called corner double-
line (CDL) tensors. Various methods were proposed to deal with the issue of
the CDL tensors [36, 34, 35, 37]. With those methods, the fixed-point tensors
of the RG flow can be approximately found [38].

Among those entanglement filtering methods, graph independent local truncation
(GILT), proposed by Hauru, Delcamp, and Mizera [37], has the advantage of
simplicity, especially since it is compatible with several coarse-graining
methods, such as TRG and HOTRG. It also offers a clear geometric meaning of
how CDLs are removed in the coarse-graining. By using HOTRG+GILT, as well as a
$Z_{2}$ gauge fixing, Lyu, Xu, and Kawashima demonstrated that the scaling
dimensions can be extracted by linearizing the equation that relates
successive coarse-graining tensors around the fixed point [39], which is a
procedure referred to as the linearized TRG (lTRG) and is analogous to the
traditional RG equation. Scaling dimensions were also extracted from the
transfer matrix at the fixed point [36] or from the eigenvalues of the local-
scale transformation [34, 40, 9].

It was emphasized in Ref. [39] that a good gauge-fixing is essential in making
the notion of fixed-point tensor well-defined and in getting the correct
scaling dimensions from lTRG. Here, in this work, in addition to the $Z_{2}$
gauge fixing employed in Ref. [39], we also employ the minimal canonical form
(MCF), proposed recently by Acuaviva et al. [41], who showed how to identify
gauge-equivalent tensors by an iterative scheme. We adopt the MCF into lTRG
and obtain a more consistent RG flow of coarse-grained tensors, allowing
better characterization of how the system approaches and flows away from the
fixed point. This allows us to study 2D classical Ising model and reproduce
the CFT data of the Ising CFT, including scaling dimensions $\Delta_{\sigma}$,
$\Delta_{\varepsilon}$, and OPE coefficients $C_{\sigma\sigma\varepsilon}$,
from the fixed-point tensor.

Our results of scaling dimensions and OPE coefficients are consistent with
calculations of the two-point and four-point correlation functions. However,
we encounter and resolve issues in computing the latter to achieve the
comparison. Correlation functions calculated from HOTRG suffer from a smearing
effect. Despite removing CDL local correlations, we find that GILT suffers
inaccuracy in evaluating correlation functions at longer distances. We provide
a solution via an averaging procedure (described in detail below).
Furthermore, we compare the coarse-grained lattice-level defect tensors and
the tensors extracted from the lTRG spectrum, confirming that both represent
conformal states. We also explore whether our coarse-graining method can
accommodate excited states in addition to the ground state.

The remaining structure of this paper is as follows. In Sec. II, we review the
Tensor Renormalization Group concept and the coarse-graining techniques that
we employ, which include HOTRG, MCF, and GILT. We also review the concept of
lTRG. In Sec. III, we present our main numerical findings. In Sec. III.1, we
demonstrate that our coarse-graining scheme successfully obtains a fixed-point
tensor, and this allows us to extract the full CFT data of the Ising CFT,
including scaling dimensions $\Delta_{\sigma}$, $\Delta_{\varepsilon}$, and
OPEs $C_{\sigma\sigma\varepsilon}$, respectively in Sec. III.2 and Sec. III.3.
For comparison, in Sec. III.4 and Sec. III.5, we calculate the two-point and
four-point correlation functions to extract the full CFT data. We also compute
the two-point function on a finite-size torus in Sec. III.6. In Sec. III.7 and
Sec. III.8, we compare the coarse-grained lattice-level defect tensors and the
tensors extracted from the lTRG spectrum, confirming that both represent
conformal states. In Sec. IV, we explore whether our coarse-graining method
can accommodate excited states in addition to the ground state. We identify
the primary error from HOTRG and GILT as resulting from the ”smearing” of
point-like defects and suppression of edge modes. To address this, we
recommend sampling points randomly and avoiding placing impurities tensors at
corners and edges of coarse-grained blocks. Moreover, we demonstrate that the
advanced gauge-fixing method MCF improves TRG flow stability and propose a
method for detecting CDL tensors along the RG flow. We also provide evidence
that the coarse-graining introduces an error that breaks conformal symmetry
and ultimately drives the system away from the fixed point. In Sec. V, we
summarize our work and suggest potential research directions. In the Appendix,
we present further details to supplement the main text, such as the
algorithmic procedures that we use in this paper. In addition, we also review
some related CFT backgrounds and our proposed TN approach to relevant CFT
data.

##  II Background

In this section, we review the connection between conventional RG flow and
their tensor network counterpart. We start by introducing the tensor network
and the concept of TRG in Sec. II.1. Then we briefly review the concept of RG
flow, their fixed points, and CFT in Sec. II.2. After that, we describe some
the previous studies on TRG, including HOTRG, local entanglement filtering,
and GILT, in Secs II.2 and II.4, as these are the main tools we will use in
this work. Further details are presented in the Appendix. We also discuss the
gauge redundancy in the tensor network and the newly proposed gauge fixing
method we will use, i.e., MCF in Sec. II.5. We then discuss the linearized
tensor renormalization group proposed in Ref. [39] to extract scaling
dimensions of operators in Sec II.6. Finally, we describe our coarse-graining
scheme, which involves a combination of HOTRG, GILT, MCF, and a residual
$U(\chi)^{\otimes m}$ gauge fixing procedure in Sec. II.7. Additionally, we
describe how we coarse-grain defect tensors, which can be used to compute
correlation functions.

###  II.1 Tensor Network and Tensor Renormalization Group

Translation invariant tensor network is a natural way to describe the
partition function of many statistical models and wave functions of quantum
systems, such as the Ising model on a square lattice [42] and the ground state
wavefunction of the Affleck-Kennedy-Lieb-Tasaki model [43]. By replacing
certain tensors with a “defect tensor”, one can further compute the $n$-point
expectation value or correlation function. This is illustrated in Eq. (1),
which shows the tensor network representation of the vacuum partition function
(or free energy $F=-\frac{1}{\beta}\log Z$) and the 2-point expectation value,

$Z=\vbox{\hbox{\includegraphics[page=1]{TRGCFTGraphics.pdf}}},\expectationvalue{\mathcal{O}_{x_{1}}\mathcal{O}_{x_{2}}}=\frac{1}{Z}\vbox{\hbox{\includegraphics[page=2]{TRGCFTGraphics.pdf}}}$
(1)

We denote the “virtual” Hilbert space as $\mathcal{H}_{v}=\mathbb{C}^{\chi}$
associated with tensor indices on one leg, where $\chi$ is the corresponding
bond dimension of the tensor leg. A tensor $T$ on the lattice has $2m$ legs,
where $m$ is the dimension of the lattice used to describe the system. For
example, $m=2$ in the classical Ising model on the square lattice. Thus, the
vector space of all the $2m$-legged tensors is $T\in\mathcal{H}_{v}^{\otimes
2m}$.

To evaluate the contraction of the tensor network in Eq. (1) on an infinitely
large lattice, which yields the partition function of the system at the
continuum limit, one iteratively transforms the tensor network into a new
tensor network on a coarse-grained lattice until the boundary effect and
finite-size effect is negligible. This is generally called the real-space
tensor renormalization procedure, as illustrated below.

$\vbox{\hbox{\includegraphics[page=3]{TRGCFTGraphics.pdf}}}\approx\vbox{\hbox{\includegraphics[page=4]{TRGCFTGraphics.pdf}}}$
(2)

The bond dimension of the coarse-grained tensor under exact merging will grow
exponentially. To make it feasible for numerical computation, one needs to
truncate the Hilbert space of the virtual degrees of freedom (i.e., bond
dimensions) in each leg. Various numerical methods such as TRG [39], HOTRG
[33], TEFR [36], TNR [34, 40], SRG [31], and GILT [37] are proposed to
properly truncate the Hilbert space in the coarse-graining process.

Truncation to finite bond dimension. When coarse-graining the tensor towards a
fixed point, scale invariance requires that
$\mathcal{H}_{v}\otimes\mathcal{H}_{v}\cong\mathcal{H}_{v}$, which implies
that $\text{dim}\mathcal{H}_{v}=1$ or $\infty$ strictly. So in principle, one
needs infinitely many iterations so that the coarse-grained tensor network can
reach either a trivial or an infinite-dimensional tensor. In practice, we
assume the system approximately reaches the thermodynamic limit after a finite
but large number of steps, and the higher-index components of the coarse-
grained tensors are negligible, and we truncate the Hilbert space on each leg
to its most significant $\chi$-dimensional subspace.

One has to bear in mind that the truncation error introduces an artificial
length scale to the coarse-grained system, that breaks the conformal symmetry.
This issue will be demonstrated in Sec. III.2; for example, see Fig. 16. In
this paper, we focus on a combination of methods using HOTRG and GILT [39].
The description of our coarse-graining scheme is in Sec. II.7.

###  II.2 Renormalization Group Flow and Conformal Field Theory

The behavior of statistical and quantum systems at different length scales can
be described using the renormalization group flow in the system parameter
space, as illustrated in Fig. 1.

Figure 1: Illustration of the renormalization group flow (top) and the changes
in tensors during the RG process.

At criticality, the system flows into a certain type of fixed point, which
sometimes can be described by a CFT, where there is an emergence of scale
invariance at long distances. As a result, the two-point functions obey a
power law

$\expectationvalue{\mathcal{O}_{x_{1}}\mathcal{O}_{x_{2}}}\propto\frac{1}{(x_{1}-x_{2})^{2\Delta_{\mathcal{O}}}},$
(3)

where $\Delta_{\mathcal{O}}$ is the scaling dimension of the operator
$\mathcal{O}$. Further information about CFT will be summarized in Appendix. B
and Appendix. C.

When the system is near the critical manifold of the system parameters, it can
firstly converge to someplace near the critical manifold, and then deviate
from it exponentially, moving towards perhaps another fixed point. On the
logarithmic scale, the deviation after each RG step follows a V-shaped curve
in terms of the change of the tensor and the coarse-graining step, as
illustrated in Fig. 1.

In TRG, the difference between coarse-grained tensors from successive RG steps
also follows a curve schematically shown in Fig. 1, which was previously
reported in Refs. [44, 34, 39] and can be used as a method to determine how
close the flow is to the criticality; see also Fig. 3 from our numerical
simulations. We expect the change of the system after each RG transformation
to behave as a V-shaped curve at the logarithmic scale. For example, the
location $B$ in Fig. 1 is the closest to a nearby critical manifold, and for
length scales where the system is at the proximity of B, the system behaves
like the CFT at the corresponding fixed point. As a result, the system can
behave like different CFTs at corresponding length scales, dependent on the
detailed RG flow.

A CFT can be uniquely determined by its CFT data. The CFT data consists of the
list of primary operators $\\{\mathcal{O}_{i}\\}$, their conformal dimensions
$\\{\Delta_{i}\\}$, and OPE coefficients $C_{ijk}$ between them. With them,
one can compute all the n-point correlation functions. In the context of
tensor networks, the conformal dimensions describe how a defect tensor scales
under coarse graining, and the OPE coefficients describe how two defect
tensors fuse into another defect tensor during this process.

###  II.3 Higher Order Tensor Renormalization Group

Now we describe the key coarse-graining procedure that we shall use, i.e., the
HOTRG. The idea is to find a coarse-grained tensor $T^{\prime}$ and isometries
$w$: $\mathcal{H}_{v}\rightarrow\mathcal{H}_{v}\otimes\mathcal{H}_{v}$, which
specifies the subspace to keep after one step of coarse-graining, as
illustrated in the equation below.

$\vbox{\hbox{\includegraphics[page=20]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=21]{TRGCFTGraphics.pdf}}}$
(4)
$\vbox{\hbox{\includegraphics[page=22]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=23]{TRGCFTGraphics.pdf}}}$
(5)

One can repeat the coarse-graining procedure horizontally and vertically at
alternative steps.

The strategy of choosing the isometry tensor $w$ is to make the coarse-grained
tensor be able to mimic the original tensor sub-network as much as possible,
as follows.

$\vbox{\hbox{\includegraphics[page=9]{TRGCFTGraphics.pdf}}}\approx\vbox{\hbox{\includegraphics[page=10]{TRGCFTGraphics.pdf}}}$
(6)

The detailed prescription of computing $w$ is described in Algorithm. A.1.

###  II.4 Local Entanglement Filtering and Graph Independent Local Truncation

Figure 2: Local Entanglement resides on the boundary and Long Range
Entanglement in the bulk between two rectangular blocks of lattice sites.

Real-space RG methods suffer from accumulating the microscopic entanglement
between the pair of sites adjacent to the boundary of coarse-grained regions.
As illustrated in Fig. 2, the entanglement between two blocks, which
determines the necessary bond dimension required for faithfully representing
the system, arises from the contribution of short-distance pairs across the
boundary of the two blocks.

On a square lattice, local entanglement can be parametrized by corner-double-
line tensors [36],

$\vbox{\hbox{\includegraphics[page=11]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=12]{TRGCFTGraphics.pdf}}}$
(7)

To effectively capture the long-wavelength behavior of the system using a
tensor with a finite bond dimension, local entanglement removal must be done.
The basic idea of local entanglement removal involves gluing two tensors
together and subsequently slicing them apart again.

CDL tensors can be removed by the Graph Independent Local Truncation [37]. The
idea of GILT is to modify a subset of a tensor network such as a plaquette, so
that the modified subnetwork still looks the same from external legs, but the
entanglement through a certain internal leg is reduced,

$\vbox{\hbox{\includegraphics[page=13]{TRGCFTGraphics.pdf}}}\approx\vbox{\hbox{\includegraphics[page=14]{TRGCFTGraphics.pdf}}}\stackrel{{\scriptstyle\text{SVD}}}{{=}}\vbox{\hbox{\includegraphics[page=15]{TRGCFTGraphics.pdf}}}$
(8)

###  II.5 Gauge Fixing and Minimal Canonical Form

The “virtual” Hillbert space $\mathcal{H}_{v}$ on each leg is unphysical,
suffering from a $GL(\chi)^{\otimes m}$ gauge-redundancy [41], as shown in Eq.
(9). We recall that $\chi$ is the bond dimension and $m$ is the ‘spacetime’
dimension of the lattice. It is essential to fix the gauge in order to compare
the tensors from different RG steps componentwise.

$\vbox{\hbox{\includegraphics[page=5]{TRGCFTGraphics.pdf}}}\sim\vbox{\hbox{\includegraphics[page=6]{TRGCFTGraphics.pdf}}}$
(9)

Ref. [39] introduced a rudimentary gauge-fixing procedure which is based on
the eigenvalue decomposition during the standard HOTRG procedure Eq. (33).
Their method successfully yields a V-shape curve. 111 The Figure.9 in [39]
does not show a clear V-shape curve, indicating that the gauge is not being
completely fixed when comparing tensors at successive RG steps. However, the
github code they provides yields a clear V-shape curve, as shown in Fig. 13.

Recently, it has been proposed in Ref. [41] that the tensor $T$ can be
iteratively gauge-transformed to a “minimal canonical form” (MCF) [41], which
is unique among gauge-equivalent tensors up to a $U(\chi)^{\otimes m}$ gauge
ambiguity,

$T_{min}=\operatorname{argmin}\\{||\tilde{T}||_{2}:\tilde{T}\in\overline{G\cdot
T}\\},$ (10)

where $G\cdot T=\\{g_{1}\otimes g_{2}\otimes g_{1}^{-1}\otimes g_{2}^{-1}\cdot
T|g_{1},g_{2}\in GL(\chi)\\}$. The overline means we take the closure of the
gauge orbit, i.e., including the limits of any sequence of group actions. (The
consideration of closure is for mathematical rigor and does not need to be
explicitly considered in our scenario.)

We summarize prescription of obtaining the MFC in Algorithm. A.3. The residual
$U(\chi)^{\otimes m}$ gauge can be robustly fixed using an improved method
based on the preliminary method in Ref. [39]. The method is summarized in
Algorithm. A.4. With MCF implement, we obtain a more consistent gauge-fixing
when comparing coarse-grained tensors between successive RG steps, and it
yields a V-shape curve, as shown in Fig. 13.

###  II.6 Linearized Tensor Renormalization Group

The lTRG [39] studies how the perturbation in the input tensor changes the
flow of the coarse-grained tensor. It is defined as follows,

$M^{i^{\prime}j^{\prime}k^{\prime}l^{\prime}}_{ijkl}=\frac{\partial
T^{\prime}_{i^{\prime}j^{\prime}k^{\prime}l^{\prime}}}{\partial T_{ijkl}},$
(11)

where $T$ and $T^{\prime}$ are tensors before and after coarse-graining. At
the critical point, the eigenvalues $\lambda_{\alpha}$ of $M$ give the
conformal dimensions $\Delta_{\mathcal{O}_{\alpha}}$ of the CFT operators,

$M\ket{\mathcal{O}_{\alpha}}=\lambda_{\alpha}\ket{\mathcal{O}_{\alpha}},$ (12)

where

$\frac{\lambda_{\alpha}}{\lambda_{0}}=s^{-\Delta_{\mathcal{O}_{\alpha}}},$
(13)

and $s=2$ is the linear scaling of the system after one RG step (by merging
two sites). Note that proper normalization is required, see Appendix. G.

Ref. [39] demonstrates that one can obtain the scaling dimensions of the first
few operators in the Ising CFT from the linearized HOTRG scheme with GILT and
a preliminary $Z_{2}$ ($U(1)$ for the complex case) gauge-fixing procedure.
One can also compare the result from lTRG with the scaling dimensions obtained
from the transfer-matrix method [36], as well as known CFT results [3]. For
example, in Ref. [40], the first 101 scaling dimensions of local and non-local
operators in the Ising CFT are calculated using the transfer matrix method on
the coarse-grained tensor obtained from the TNR method, which confirms well
with the expected analytical result.

In this paper, we will further demonstrate that the eigenvectors
$\ket{\mathcal{O}_{\alpha}}$ of lTRG in Eq. (11) have a clear physical
meaning. They are the conformal states associated with the corresponding
operators $\mathcal{O}_{\alpha}$. Conformal states are obtained by inserting
conformal primaries and descendants into the vacuum state (see Appendix. B).
In this paper, the states are defined on the boundary of a coarse-graining
block. Thus we place the operator at the center of the block. On the lattice
level, those states can be achieved by coarse-graining the tensor network Eq.
(1) with the operator insertion as defect tensors:

$\ket{\mathcal{O}_{\alpha}}=\lim_{r\rightarrow
0}\mathcal{O}_{\alpha}(r)\ket{0}\approx\sigma(x_{1})\sigma(x_{2})...\ket{0}_{\text{lattice}}.$
(14)

By also calculating the two-point correlation function of the Ising model on
the square lattice using the TRG method, we find that the position of the
operator in the coarse-grained defect tensor $T_{\sigma_{x_{1}}}$ is encoded
by its projections on the primaries and descendants tensors or lTRG.

###  II.7 Coarse Graining using HOTRG, GILT and MCF

Here, we integrate GILT and MCF into the HOTRG steps,

$\vbox{\hbox{\includegraphics[page=7]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=8]{TRGCFTGraphics.pdf}}}$
(15)

In the above, at each coarse-graining layer, $w$, $w^{\dagger}$ are the
isometry tensors from HOTRG, and $g_{12}$, $g_{13}$, $g_{22}$, and $g_{23}$
are the entanglement truncation matrices from GILT. They filter out certain
local correlations which cannot be removed by $w$ and $w^{\dagger}$. Moreover,
$h_{0}$ and $h_{2}$ are the gauge-fixing matrices, determined by MCF, and the
residual $U(\chi)^{\otimes m}$ gauge redundancy is fixed by the sign-fixing
procedure in Ref. [39].

Adapting GILT to HOTRG. In GILT, one has to repeat the process on all the
possible plaquette-leg combinations to remove all the local entanglements. In
practice, as suggested by [39], it is sufficient only to apply GILT on the
following plaquette-legs:

$\vbox{\hbox{\includegraphics[page=16]{TRGCFTGraphics.pdf}}}\approx\vbox{\hbox{\includegraphics[page=17]{TRGCFTGraphics.pdf}}}$
(16)

The tensor in the gray shade is to be coarse-grained, with circle-shaped GILT
tensors being splitted using the SVD. We only truncate half of the CDL loops
across the legs to be coarse-grained.

###  II.8 Coarse-graining the Defect Tensors

We use the same set of coarse-graining tensors $w$, $g$, $h$ from coarse-
graining vacuum tensors, to coarse-grain the network with insertion of defect
tensors. Similarly, when calculating the linearized TRG Eq. (11), we keep the
coarse-graining tensors $w$, $g$, $h$ as constants, i.e., their variation with
respect to the change of the input tensor is discarded. As a result, the
coarse-grained tensor is quad-linear in the four input tensors. Eq. (17) shows
the coarse-graining of a defect tensor. Eq. (18) shows the variation of the
coarse-grained tensor under the change of the input tensor, which is used in
obtaining the lTRG equation Eq. (11). In both equations, the isometries $w$
and truncations $g$ are omitted for the simplicity of the diagrams.

$\vbox{\hbox{\includegraphics[page=51]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=52]{TRGCFTGraphics.pdf}}}$
(17)

$\vbox{\hbox{\includegraphics[page=18]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=19]{TRGCFTGraphics.pdf}}}+\text{
3 other terms}.$ (18)

As shown below in Sec. IV, one can obtain faithful scaling dimensions from the
linearized TRG with fixed $g$ and $w$. However, in terms of $n$-point
correlation functions, one sometimes gets undesirable results if GILT is
applied, which will be carefully discussed later in Sec. IV.4.

##  III Main Results

Here we present our results of the CFT data from the fixed-point tensor of the
linearized Tensor Renormalization Group [39], using the improved gauging
fixing via MCF. In particular, we separately extract the CFT data, i.e.,
scaling dimension and OPE coefficients, from two different methods: one is by
introducing defects on the lattice, and another only depends on the fixed-
point tensor, with no knowledge about the lattice. We also compare the tensors
corresponding to conformal operators extracted from the two methods.

We first follow the work of Lyu et al. [39], which extracted the conformal
dimension spectrum from the eigenvalues of lTRG. Our first contribution is the
improving of the stability of the RG flow by introducing MCF for gauge fixing.
The results of RG flow and scaling dimensions are summarized in Sections III.1
and III.2.

Next, we go beyond reproducing scaling dimensions and propose and implement
how to extract the OPE coefficients, such as $C_{\sigma\sigma\varepsilon}$,
through the fusion of eigenvectors the of lTRG fixed point equations (note
that these eigenvectors are themselves tensors). The results agree with known
values and are summarized in Section III.3.

Additionally, we verify that eigenvectors of lTRG serve as a useful
representation of conformal states at low energy sectors. To do this,
specifically, we also construct these conformal states by placing lattice-
level operators on the lattice (see Table 1) and then coarse-graining them. We
compare these with the eigenvectors of lTRG and find that the latter aligns
well with the coarse-grained tensors in the low-energy (or small scaling-
dimension) subspace, as demonstrated in Section III.7. Moreover, by expanding
the coarse-grained tensor using the basis of primary and descendant states, it
becomes evident that the position of the primary operator inserted in the
lattice is encoded in the components of the descendant basis, similar to the
standard Taylor expansion used for operators. This observation is discussed in
Section III.8.

However, it remains unclear whether coarse graining introduces significant
truncation errors in such states with higher excited levels. To characterize
the quality of the coarse-graining on defect tensors, we calculate their two-
point and four-point correlation functions using HOTRG+GILT. They agree well
with the analytical result, and we can also use them to extract the CFT data
$\Delta_{\sigma}$, $\Delta_{\varepsilon}$ and $C_{\sigma\sigma\varepsilon}$ by
fitting the correlation functions. But the error is noticeably larger than the
one we obtain from the fusion method. Additionally, we calculate the two-point
function on a finite-size torus and find that the numerical result agrees well
with the analytical one and that the correlation function respects the
periodic boundary condition of the torus geometry. These results are
summarized in sections III.4, III.5, and III.6.

By studying the $N$-point correlation functions, we can better understand the
capability and limitations of our real-space RG scheme in coarse-graining
defect tensors. A technical discussion is provided in section IV. With this
knowledge, we are confident that the coarse-grained tensor can serve as a
reference for conformal states.

In summary, we demonstrate two different methods of extracting CFT data from a
critical lattice model. The first method involves extracting operator scaling
dimensions from two-point correlation functions and OPE coefficients from
four-point correlation functions by introducing defects on the lattice. The
second method utilizes the fixed-point tensor to extract the CFT data without
relying on lattice-level knowledge (i.e., only from the fixed-point tensor and
equation). Specifically, we obtain the scaling dimension from the eigenvalues
of lTRG and the OPE coefficient from the fusion rules of the eigenvectors of
lTRG, which correspond to conformal states. Our findings demonstrate good
agreement between the conformal states obtained from lTRG and those obtained
from coarse-graining lattice-level defects.

###  III.1 Convergence onto the Conformal Fixed Point in Higher-order Tensor
Renormalization Group

After applying GILT [37] to remove local entanglement and performing the MCF
[41] for gauge-fixing, we can identify a nontrivial fixed point in HOTRG [33],
where the tensor remains almost the same after coarse-graining. This fixed
point from the 2D classical Ising model corresponds to the Ising CFT
$\mathcal{M}(4/3)$, with a central charge of $c=1/2$ [3].

Figure 3: Difference of coarse-grained tensor before and after every two
layers of HOTRG coarse-graining $||T^{(l)}-T^{(l-2)}||$ for the 2D square
lattice Ising model near the critical temperature $T_{c}$. Three temperatures
are considered: $T\approx T_{c}$, $T=T_{c}-5\times 10^{-5}$ and
$T=T_{c}+5\times 10^{-5}$. Bond dimension $\chi=24$, GILT is applied with
$\epsilon_{\text{GILT}}=8\times 10^{-7}$, $\text{nIter}_{\text{GILT}}=1$. The
critical temperature $T_{c}=2.26920066107$ is numerically determined with
respect to the fixed bond dimension. Figure 4: The first few components of
fixed-point tensors $T_{ijkl}^{(fix)}$ for the a) high temperature, b) low
temperature and c) critical fixed point of the RG flow demonstrated in Fig. 3.
The same parameters are used as in Fig. 3. The components $T_{ijkl}^{(fix)}$’s
are shown as pixels at row $4\times i+j$ and at column $4\times k+l$. The
brightness of the pixel stands for the absolute value of the component, and
the color of the pixel stands for the argument of the component (i.e., red is
positive, blue is negative, and black is zero). One can see that a) is the
trivial fixed point of HOTRG, b) is the tensor product between two trivial
tensors, indicating an unbroken $Z_{2}$ symmetry and c) is a more complicated
fixed point tensor. The same parameters are used as in Fig. 3.

In Fig. 3, we present the tensor difference $||T^{(l)}-T^{(l-2)}||$ of the
coarse-grained Ising model on a 2D square lattice as a function of the HOTRG
iterations near the critical temperature $T_{c}$ (see green dots in the
figure). As the system approaches the critical point around step 30, the
coarse-grained tensor converges to a fixed point approximately, where the
tensor remains almost the same after the coarse-graining steps (with the
change in the tensors being less than $3\times 10^{-4}$). After step 30, the
tensor deviates exponentially, because the critical fixed point is an unstable
fixed point, and any error from numerical error, truncation error, or
fluctuation of initial values, will result in a spontaneous symmetry broken
after RG iterations.

This behavior of first approaching and then deviating from the fixed point is
a hallmark of the RG flow near a critical point that can be identified from a
V-shape curve in the logarithmic plot. Fig. 3 also illustrates how the system
evolves into distinct fixed points as the temperature is perturbed slightly.
When starting at a temperature slightly away from $T_{c}$ (above and below),
the system does not converge as closely to the critical fixed point as the one
at the critical temperature. The flows in both cases approach their
“perihelions” of the fixed point around step 20.

In the low-temperature phase, the system first flows to a degenerate low-
temperature fixed point (with the only two nonzero elements are
$T_{0,0,0,0}=T_{1,1,1,1}$) before eventually flowing to a tensor of the form
$T_{0,0,0,0}=0$, due to a spontaneous symmetry breaking caused by
perturbations from numerical errors, as we did not enforce the $Z_{2}$
symmetry during the coarse graining. In the high-temperature phase, the system
converges directly to the high-temperature fixed point, with one nonzero
element being $T_{+,+,+,+}$ (whose tensor form is nevertheless equivalent to
$T_{0,0,0,0}$ by a basis transformation in the bond degrees of freedom Eq.
(9)). The tensor components at the three fixed points are shown in Fig. 4.

###  III.2 Scaling Dimension from Linearized Tensor Renormalization group

In Table. 1, we compare the results of scaling dimensions obtained by two
different methods: (1) diagonalizing the transfer matrix on a cylinder and (2)
solving eigenvalues of lTRG Eq. (11). We also compare for each case how the
results depend on whether GILT is used or not. First, we confirm that GILT
generally improves the numerical results for scaling dimensions [37],
especially for the higher values. However, we observe that the transfer matrix
spectrum achieves more accurate information about higher conformal dimensions
compared to the lTRG method (see their values for operators having $\Delta=3$
or higher).

|  | with GILT | without GILT | Ref.[39]
---|---|---|---|---
exact | $\Delta$ | $\Delta_{\text{cyl}}$ | $\Delta_{\text{RG}}$ | $\Delta_{\text{cyl}}$ | $\Delta_{\text{RG}}$ | $\Delta_{\text{cyl}}$ | $\Delta_{\text{RG}}$
$\mathbb{1}$ | 0 |  |  |  |  |  |
$\sigma$ | $\frac{1}{8}$ | 0.1256 | 0.1275 | 0.1217 | 0.1266 | 0.125 | 0.127
$\varepsilon$ | 1 | 1.0059 | 1.0027 | 1.0085 | 1.0836 | 1.002 | 1.009
$\partial_{x}\sigma$, | $1+\frac{1}{8}$ | 1.1320 | 1.1269 | 1.1422 | 1.1857 | 1.128 | 1.125
$\partial_{y}\sigma$ |  | 1.1320 | 1.1448 | 1.1422 | 1.1911 | 1.128 | 1.128
$\partial_{x}\varepsilon$, | $2$ | 2.0146 | 1.9985 | 2.0496 | 2.0098 | 2.014 | 2.002
$\partial_{y}\varepsilon$, |  | 2.0151 | 2.0004 | 2.0525 | 2.0936 | 2.014 | 2.004
$T_{xx}-T_{yy}$, |  | 2.0165 | 2.0042 | 2.0701 | 2.1000 | 2.016 | 2.068
$T_{xy}+T_{yx}$, |  | 2.0166 | 2.0434 | 2.0701 | 2.1085 | 2.016 | 2.073
$\partial_{xx}\sigma$, | $2+\frac{1}{8}$ | 2.1411 | 2.1215 | 2.1795 | 2.1085 |  |
$\partial_{xy}\sigma$, |  | 2.1463 | 2.1981 | 2.2202 | 2.1140 |  |
$\partial_{yy}\sigma$, |  | 2.1471 | 2.2049 | 2.2752 | 2.1140 |  |
$\partial_{xx}\varepsilon$ | 3 | 3.0289 | 2.5006 | 2.9658 | 2.1165 |  |
$\partial_{xy}\varepsilon$ |  | 3.0372 | 2.6803 | 2.9680 | 2.1165 |  |
$\partial_{yy}\varepsilon$ |  | 3.0375 | 2.6884 | 2.9680 | 2.1251 |  |
$\partial_{x}T_{xx}$ |  | 3.0377 | 2.7914 | 2.9707 | 2.1251 |  |
$\partial_{y}T_{yy}$ |  | 3.0421 | 2.8247 | 3.1250 | 2.1430 |  |
… | $3+\frac{1}{8}$ | 3.1534 | 2.8255 | 3.1269 | 2.1430 |  |
Table 1: The central charge and the first few scaling dimensions at RG step 20
obtained using two different methods: (1) transfer matrix on a cylinder[36]
with scaling dimensions denoted as $\Delta_{\text{cyl}}$, and (2) linearized
Tensor Renormalization Group with scaling dimensions denoted as
$\Delta_{\text{RG}}$. The Bond dimension is set to $\chi=24$, and GILT is
applied with $\epsilon_{\text{GILT}}=8\times 10^{-7}$ and
$\text{nIter}_{\text{GILT}}=1$. Note that Ref. [39] reports results at step=22
and with $\chi=30$. Moreover, the conformal operators in each group are
degenerate.

The transfer matrix spectrum, $\Delta_{cyl}$, represents the quality of the
coarse-grained tensor. The method with GILT yields the correct scaling
dimensions and their degeneracy number up to $\Delta=3+\frac{1}{8}$. Without
GILT, the number of degeneracy is incorrect starting from $\Delta=3$, and the
error in the scaling dimensions is much larger.

The lTRG scaling dimension spectrum, $\Delta_{\text{RG}}$, is of greater
interest to us, as we shall extend the approach to OPE coefficients. The
method with GILT fails to provide the correct scaling dimensions from
$\Delta=3$ and up, but it does show a gap between $\Delta=2+\frac{1}{8}$ and
$\Delta=3$, with the correct number of degeneracy at $\Delta=2+\frac{1}{8}$.
On the other hand, the case without GILT has the last meaningful gap between
$\Delta=1+\frac{1}{8}$ and $\Delta=2$. Additionally, the GILT case has fewer
errors at $\Delta=1,2$, although it performs slightly worse at
$\Delta=1+\frac{1}{8}$.

For comparison, we present the number of degeneracy at each scaling dimension
in Table. 4. The error of scaling dimensions compared to the analytic results
at each RG step is further discussed in IV.3, Fig. 16.

###  III.3 OPE Coefficient from Linearized Tensor Renormalization Group

Despite that lTRG performs worse than the transfer matrix in scaling
dimensions, we demonstrate in this subsection that using lTRG one can extract
the OPE coefficients $C_{ijk}$ only from its fixed-point tensor
$T_{\mathbb{1}}$, without any knowledge about the lattice-level details. In
particular, what one needs is the coarse-graining $\mathcal{M}:T_{1}\otimes
T_{2}\otimes T_{3}\otimes T_{4}\mapsto T^{\prime}$, and the conformal states
$T_{\mathbb{1}}$, $T_{\sigma}$, and $T_{\varepsilon}$, which are the first few
eigenvectors of the lTRG transformation $M:T\mapsto T\otimes T\otimes T\otimes
T\mapsto T^{\prime}$.

We recall the OPE expansion and its coefficients in terms of the conformal
states rather than operators,

$\ket{\mathcal{O}_{2}\mathcal{O}_{3}}=\frac{C_{231}}{r_{23}^{\Delta_{2}+\Delta_{3}-\Delta_{1}}}\ket{\mathcal{O}_{1}}+...,$
(19)

where $...$ denotes states orthogonal to $\ket{\mathcal{O}_{1}}$. We summarize
in Appendix G the steps to estimate the OPE coefficient from Eq. (19), which
involve the rescaling of fixed-point tensors and the renormalization on the
length scale. The result of a few OPE coefficients is shown in Table. 2. The
scaling dimensions and the fusion between two $\sigma$’s are fairly accurate;
however, the values that incur slightly larger errors are
$C_{\sigma\varepsilon\sigma}$ and $C_{\varepsilon\varepsilon\varepsilon}$.
This is because as an operator of higher scaling dimensions, $\epsilon$,
suffers more error than $\sigma$ from the coarse-graining process. We remark
that OPE coefficients (and scaling dimensions) can also be extracted from
four-point correlation functions, as will be discussed below in Sec. III.5.
However, the results there are not as accurate.

| exact | lTRG
---|---|---
$\Delta_{\sigma}$ | 1/8 | 0.128
$\Delta_{\varepsilon}$ | 1 | 1.003
$C_{\sigma\sigma\sigma}$ | 0 | $-1.1\times 10^{-9}$
$C_{\sigma\sigma\varepsilon}$ | 1/2 | 0.511
$C_{\varepsilon\varepsilon\sigma}$ | 0 | $-1.7\times 10^{-10}$
$C_{\varepsilon\varepsilon\varepsilon}$ | 0 | 0.162
$C_{\sigma\varepsilon\sigma}$ | 1/2 | 0.408
$C_{\sigma\varepsilon\varepsilon}$ | 0 | $-4.3\times 10^{-10}$
Table 2: The CFT data extracted from the fusion rules of lTRG following the
procedure detailed in Appendix. G.

###  III.4 Scaling Dimension from Two-point Correlators

It is well-known that the two-point correlation function
$\langle\sigma(0)\sigma(r)\rangle$ at criticality scales as $1/r^{d-2+\eta}$,
where the anomalous dimension $\eta$ is related to the scaling dimension
$\Delta$, via $d-2+\eta=2\Delta$. Thus, it provides an alternative approach to
extracting scaling dimensions. Fig. 5 shows the two-point correlation function
$\langle\sigma(x_{0},y_{0})\sigma(x_{1},y_{1})\rangle$ of the square lattice
Ising model near the critical temperature, where

Figure 5: The two-point correlator
$\langle\sigma(x_{0},y_{0})\sigma(x_{1},y_{1})\rangle$ of the Ising model on a
$2^{30}\crossproduct 2^{30}$ square lattice obtained from coarse-graining
defect tensors using HOTRG. Three temperatures are considered: $T\approx
T_{c}$, $T=T_{c}-5\times 10^{-5}$ and $T=T_{c}+5\times 10^{-5}$. The locations
of $(x_{0},y_{0})$ and $(x_{1},y_{1})$ were chosen randomly. The bond
dimension used is $\chi=24$ and GILT is applied with
$\epsilon_{\text{GILT}}=8\times 10^{-7}$, $\text{nIter}_{\text{GILT}}=1$. The
critical temperature $T_{c}=2.26920066107$ is numerically determined with
respect to the bond dimension, not from the exact solution.

the points $(x_{0},y_{0})$ and $(x_{1},y_{1})$ were randomly sampled on the
xy-plane. The correlation function is determined by the distance between
$(x_{0},y_{0})$ and $(x_{1},y_{1})$, irrespective of the relative orientation
or the absolute positions for most cases. This indicates that the
translational symmetry is preserved and the $SO(2)$ rotational symmetry has
emerged at large distances.

We use three different temperatures. (1) When $T<T_{c}$, the correlation
function saturates at a finite distance, indicating a long-range order. (2)
When $T\approx T_{c}$, the correlation function follows a power-law behavior,
with a slope in the log-log scale determined by the scaling dimension,
indicating critical behavior, discussed in more detail below. (3) When
$T>T_{c}$, the correlation function decays at a finite distance, indicating
the absence of long-range order.

We now analyze the results of the two-point correlations in more detail and
aim to extract the scaling dimension. To do this, we fit the correlation
functions to the following ansatzes, in the respective temperature ranges,

$\displaystyle\langle\sigma(x_{0},y_{0})\sigma(x_{1},y_{1})\rangle$
$\displaystyle=A\left(\frac{1}{\zeta^{1/B}}+\frac{1}{x^{1/B}}\right)^{2\Delta
B},\,T<T_{c},$ (20)
$\displaystyle\langle\sigma(x_{0},y_{0})\sigma(x_{1},y_{1})\rangle$
$\displaystyle=A\frac{1}{x^{2\Delta}},\,T=T_{c},$ (21)
$\displaystyle\langle\sigma(x_{0},y_{0})\sigma(x_{1},y_{1})\rangle$
$\displaystyle=A\frac{e^{-x/\zeta}}{x^{2\Delta}},\,T>T_{c},$ (22)

where $\Delta$ is the scaling dimension of the magnetization operator
$\sigma$, $\zeta$ is the correlation length, $m_{0}$ is the contribution of
1-point function. We discuss the form of the low-temperature ansatz further in
Appendix. I. In the fitting process, we use Huber loss [46] with
$\epsilon=0.1$ to remove outliers.

When $T\approx T_{c}$, we obtain $\Delta=0.1293$; when $T=T_{c}-5\times
10^{-5}$, we obtain $\Delta=0.1273$, $\zeta=2.24\times 10^{3}$ and $B=0.9$;
when $T=T_{c}+5\times 10^{-5}$, we obtain $\Delta=0.1330$ and
$\zeta=1.95\times 10^{4}$. The three conformal dimensions thus obtained agree
well with one another and with the analytic result $\Delta=0.125$.

We remark that in order to get the correct scaling dimension, it is necessary
to average over the position of the correlated pairs, due to an issue in the
coarse-graining of impurity operators. Also, Huber loss is used to filter out
the outliers. We elaborate on these issues later in Sec. IV.

###  III.5 OPE Coefficient from Four-point Correlators

Figure 6: The four-point correlator
$\langle\sigma(x_{1},y_{1})\sigma(x_{2},y_{2})\sigma(x_{3},y_{3})\sigma(x_{4},y_{4})\rangle$
of the Ising model on a $1024\crossproduct 1024$ square lattice at the
critical temperature $T\approx T_{c}$ for different choices of operator
locations. The horizontal axis indicates the analytical value from Eq. (65)
using the known CFT data $\Delta_{\sigma}=0.125$, $\Delta_{\varepsilon}=1$,
$C_{\sigma\sigma\varepsilon}=0.5$. The vertical axis indicates our numerical
results. The four points were chosen randomly. The parameters used here are
the same as in Fig. 5.

The essential data for the Ising CFT are the scaling dimensions
$\Delta_{\sigma}$, $\Delta_{\varepsilon}$, and the OPE coefficient
$C_{\sigma\sigma\varepsilon}$. In 2D CFT, the four-point function can be
uniquely determined by the CFT data according to Eq. (65). We thus compute the
four-point correlation function
$\langle\sigma(x_{1},y_{1})\sigma(x_{2},y_{2})\sigma(x_{3},y_{3})\sigma(x_{4},y_{4})\rangle$
of the square lattice Ising model near the critical temperature. To confirm
that we can extract the OPE coefficients, we first compare our numerical
values with the analytical CFT prediction, shown in Fig. 6. The horizontal
axis is the analytical value and the vertical is the numerical value obtained
from our TN procedure. One can see that, although the data is noisy, the trend
indicates that they are proportional to each other up to some normalization
factor, resulting from the normalization of the spin operators.

With the above comparison, we further use Eq. (65) for the fitting and thus
extract the CFT data from our numerical results. We use Huber loss
$\epsilon=0.1$ at the logarithm space and convert the input as the distances
between pairs of points. The fitted values (vs. known values) are
$\Delta_{\sigma}=0.1120\,({\rm vs.}\,0.125)$,
$\Delta_{\varepsilon}=0.9729\,({\rm vs.}\,1)$, and
$C_{\sigma\sigma\varepsilon}=0.4564\,({\rm vs.}\,0.5)$, respectively. Thus,
four-point functions provide a crude estimation of all the essential data of
Ising CF and give another confirmation of our previous results using lTRG.

###  III.6 Two-point Correlator on a Torus

We also compute the two-point correlation function
$\langle\sigma(0,0)\sigma(x,0)\rangle$ on a $1024\times 1024$ torus using the
HOTRG method. The results are shown in Figure 7.

Figure 7: The two-point correlator $\langle\sigma(0,0)\sigma(x,0)\rangle$ of
the Ising model at $T\approx T_{c}$ on a $1024\times 1024$ torus. GILT is not
used. The bond dimension is $\chi=24$.  Figure 8: The two-point correlator
$\langle\sigma(0,0)\sigma(x,0)\rangle$ of the Ising model at $T\approx T_{c}$
on a $2^{30}\times 2^{30}$ torus. The data is plotted in linear and loglog
space. Here, same parameters are used as in Fig. 7.

The result confirms well with the spin-spin correlation of the Ising CFT on a
torus [47] (12.108):

$\expectationvalue{\sigma(0,0)\sigma(x,y)}\propto\left|\frac{\partial_{z}\theta_{1}(0|\tau)}{\theta_{1}(z|\tau)}\right|^{\frac{1}{4}}\frac{\sum_{\nu=1}^{4}|\theta_{\nu}(z/2|\tau)|}{\sum_{\nu=2}^{4}|\theta_{\nu}(0|\tau)|},$
(23)

where $z=\frac{1}{L_{x}}(x+iy)$ is the complex coordinate, $L_{x}$, $L_{y}$
are the size of the lattice, $\tau=iL_{y}/L_{x}$, and $\theta_{\nu}$’s are the
elliptic theta functions, following the convention in [48]. We only examine
the case with $y=0$ here.

Fig. 8 shows the two-point function on a much larger torus, corresponding to
60 layers in the coarse graining. The correlator looks like the high-
temperature correlation function as in Fig. 5 and Eq. (22) from both sides.
This is because the coarse-grained tensors we use had already flowed away from
the conformal fixed point at the length scale $2^{15}$, as shown in Fig. 3 and
Fig. 5.

As expected by the periodic boundary condition, the curves in Fig. 7 and Fig.
8 both exhibit a peak when $x\rightarrow L_{x}$. However, the sharp peak on
$x\rightarrow L_{x}$ contradicts the purpose of coarse-graining in HOTRG:
removing the microscopic details. The peak is a result of correlations between
two point-like defects. However, in the case $x\rightarrow L_{x}$, the two
defects meet at the very last step of the coarse-graining process. So what was
computed instead is the correlation function between two “smeared” operators,
which gives a much smoother curve compared to that of the “bare” operators.
Further discussions will be presented in Sec. IV.

The fact that one can get a sharp peak when $x\rightarrow L_{x}$ indicates
that the truncated Hilbert space of the coarse-grained tensor still captures
the modes where the excitation is sharply localized on the boundary of the
coarse-grained regions, as described in Sec. IV. Fig. 8 further illustrates
that those point-like localized modes are being kept throughout the whole RG
flow.

###  III.7 Comparison between Eigenvectors of linearized Tensor
Renormalization Group with Coarse-Grained Defect Tensors

As demonstrated in Sec. III.2 that lTRG via HOTRG+GILT+MCF gives good
estimates of scaling dimensions, we now propose that additionally the
eigenvectors of the lTRG coarse-graining equation (Eq. (11)) at the fixed
point correspond to physical states of conformal primaries and their
descendants. This claim is supported by a direct comparison between the
coarse-grained tensors with operator insertion
$T_{\sigma_{x_{1}}\sigma_{x_{2}}...}$ and the first few eigenvectors $v_{i}$
of the lTRG Eq. (11) near the conformal point, as shown in Fig. 9.

Figure 9: Projection of coarse-grained tensors with operator insertion onto
the first few eigenvectors eigenvectors of lTRG at the RG step 30. The color
indicates the argument $\theta$ of the overlapping
$\innerproduct{T_{\text{lattice}}}{T_{\text{lTRG}}}=re^{i\theta}$: red
indicates positive values $\theta=0$ and blue indicates negative values
$\theta=\pi$. The bond dimension used is $\chi=24$, and GILT is applied with
$\epsilon_{\text{GILT}}=8\times 10^{-7}$, $\text{nIter}_{\text{GILT}}=1$. The
results suggest that the eigenvectors correspond to states of conformal
primaries and descendants.

From there we observe that the coarse-grained vacuum tensor $T_{\mathbb{1}}$
lies on the eigenspace with conformal dimension 0, which corresponds to the
identity operator $\mathbb{1}$. The coarse-grained tensor with one operator
insertion $T_{\sigma_{x}}$ lies on the eigenspace of conformal dimension
0.125, which corresponds to the spin operator $\sigma$. However, as can be
seen in Fig. 9, there are also significant projections on higher eigenvectors,
which may be due to numerical errors and/or the operator insertion position
not being exactly in the middle. Next, the coarse-grained tensor with two
operator insertions $T_{\sigma_{x_{1}}\sigma_{x_{2}}}$ lies on the joint
subspace of $\Delta=0$ and $\Delta=1$. This is consistent with the fusion rule
of the Ising CFT: $\sigma\times\sigma=\mathbb{1}+\varepsilon$, where
$\varepsilon$ is the energy density operator with $\Delta_{\varepsilon}=1$.
Lastly, the coarse-grained tensor with three operator insertions
$T_{\sigma_{x_{1}}\sigma_{x_{2}}\sigma_{x_{3}}}$ further confirms the fusion
rule $\varepsilon\times\sigma=\sigma$ and $\mathbb{1}\times\sigma=\sigma$.

However, we note that there are also undesired projections on eigenspaces with
$\Delta\gtrsim 2$. This is likely due to limitations in the numerical
precision or the fact that the eigenspaces are not strictly orthogonal. We
also note that the above comparison is done in the subspace spanned by the
first few eigenvectors of lTRG, in order to avoid the UV details from lattice
implementation and the “junk” part [35] that corresponds to the higher indices
of the tensor.

We further compare the higher eigenvectors with the conformal descendants
obtained by finite-differencing the coarse-grained tensors with different
operator insertion positions. The scheme is elaborated in Table. 3. The
procedure consists of inserting operators at different positions in the
original lattice and then coarse-graining the resultant tensors. Finally,
finite differences are taken between the coarse-grained tensors with operator
insertions to obtain the lattice version of conformal descendants.

Coarse-grained tensor | $\Delta$ |  Inserted operator at lattice level
---|---|---
$\mathbb{1}$ | 0 | $\mathbb{1}$
$\sigma$ | 1/8 | $\sigma_{0,0}$
$\sigma\times\sigma=\mathbb{1}+\varepsilon$ | 0,1 | $\sigma_{l,0}\,\sigma_{-l,0}$
$\sigma\times\sigma\times\sigma=\sigma$ | 1/8 | $\sigma_{l,0}\,\sigma_{-\frac{1}{2}l,\frac{\sqrt{3}}{2}l}\,\sigma_{-\frac{1}{2}l,-\frac{\sqrt{3}}{2}l}$
$\partial_{x}\sigma$ | 1+1/8 | $(\sigma_{d,0}-\sigma_{-d,0})$
$\partial_{y}\sigma$ | 1+1/8 | $(\sigma_{0,d}-\sigma_{0,-d})$
$\partial_{x}\varepsilon$ | 2 | $(\sigma_{d,l}\,\sigma_{d,-l}-\sigma_{-d,l}\,\sigma_{-d,-l})$
$\partial_{y}\varepsilon$ | 2 | $(\sigma_{l,d}\,\sigma_{-l,d}-\sigma_{l,-d}\,\sigma_{-l,-d})$
$T_{\times}$ | 2 | $(\sigma_{l,l}\,\sigma_{-l,-l}-\sigma_{l,-l}\,\sigma_{-l,l})$
$T_{+}$ | 2 | $(\sigma_{l,0}\,\sigma_{-l,0}-\sigma_{0,l}\,\sigma_{0,-l})$
$\partial^{2}_{x}\sigma$ | 2+1/8 | $(\sigma_{d,0}-\sigma_{d^{\prime},0}-\sigma_{-d^{\prime},0}+\sigma_{-d,0})$
$\partial^{2}_{y}\sigma$ | 2+1/8 | $(\sigma_{0,d}-\sigma_{0,d^{\prime}}-\sigma_{0,-d^{\prime}}+\sigma_{0,-d})$
$\partial_{x}\partial_{y}\sigma$ | 2+1/8 | $(\sigma_{d,d}-\sigma_{-d,d}-\sigma_{d,-d}+\sigma_{-d,-d})s$
Table 3: Coordinates and patterns of operator insertion for Fig. 9 and Fig.
10. The stress-energy tensor $T$ has spin 2, which is the descendent of
$\mathbb{1}$, which is decomposed into its quadratic momentum components
$T_{+}$ and $T_{\crossproduct}$. Figure 10: Finite difference of coarse-
grained tensors onto eigenvectors of linearized TRG. The color indicates the
argument $\theta$ of the overlapping
$\innerproduct{T_{\text{lattice}}}{T_{\text{lTRG}}}=re^{i\theta}$: red
indicates positive values $\theta=0$ and blue indicates negative values
$\theta=\pi$. Here, same parameters are used as in Fig. 9.

Fig. 10 displays the projection of the lattice-realized conformal descendants
onto the eigenvectors of lTRG. We observe that $\partial_{x}\sigma$ and
$\partial_{y}\sigma$ belong to a nearly degenerate 2-dimensional subspace with
a conformal dimension of approximately 1.125. Similarly, the other descendants
such as $T_{\times}$, $T_{+}$, $\partial_{x}\varepsilon$, and
$\partial_{y}\varepsilon$ have their maximal overlap near conformal dimension
2. Meanwhile, the descendants $\partial_{x}^{2}\sigma$,
$\partial_{x}\partial_{y}\sigma$, and $\partial_{y}^{2}\sigma$ appear near the
subspace corresponding to $\Delta=2.125$.

We note, however, that the order of the eigenvalues of the most significantly
overlapped eigenvectors between $\partial_{x}\varepsilon$ and
$\partial_{x}^{2}\sigma$ $\partial_{y}^{2}\sigma$ are not correct. The most
significant overlap of $\partial_{x}\varepsilon$ is at 2.119, which shows a
big discrepancy with the overlap of $\partial_{y}\varepsilon$ between 1.998
and 2.004. Moreover, the most significant overlap of the next two operators,
$\partial_{x}^{2}\sigma$ and $\partial_{y}^{2}\sigma$, is at scaling dimension
2.044, which is smaller than 2.121. So the order of the scaling dimensions of
some operators are wrong, and operators at different degeneracy subspace are
being mixed up due to numerical error.

We also note that the pair $T_{\crossproduct}$, $\partial_{y}\varepsilon$, and
the pair $\partial_{x}^{2}\sigma$, $\partial_{y}^{2}\sigma$ are not very
distinguishabie at the first few eigenvectors. This indicates that the
information to distinguish those states is lost during the coarse-graining.
This results in the unexpected behavior of $\partial_{y}^{2}\sigma$ in Fig.
11, which is to be discussed in the next subsection.

###  III.8 Expansion of One-point Operators into the Conformal Family

Expansion on flat space. Another way to show the tensors such as
$\partial_{x}\sigma$ and $\partial_{x}\partial_{x}\sigma$ obtained from lTRG
eigenvectors in the previous subsection have the physical meaning just as
their name suggests, i.e., one can check if the one-point tensor
$T_{\sigma_{x,0}}$ can be Taylor expanded into the conformal family as
follows,

$\displaystyle\sigma(x,y)=\sigma(0,0)+x\partial_{x}\sigma(0,0)+y\partial_{y}\sigma(0,0)$
(24)
$\displaystyle\,\,+\frac{1}{2}(x^{2}\partial_{x}^{2}\sigma(0,0)+xy\partial_{x}\partial_{y}\sigma(0,0)+y^{2}\partial_{y}^{2}\sigma(0,0))+...$

We thus compute operators on both sides and compare if the above expansion
holds. Fig. 11 shows the decomposition of $T_{\sigma_{x,0}}$ onto the
conformal tensors $\sigma$, $\partial\sigma$, $\partial^{2}\sigma$.

Figure 11: Projection of coarse-grained tensor with different operator
insertion positions onto the conformal state obtained from linearized Tensor
Renormalization Group. $L=1024$ is the lattice size. Same parameter as Fig. 9

From Fig. 11 we see that, as the operator $\sigma(x)$ moves along the x-axis,
its projection on $T_{\partial_{x}\sigma}$ changes linearly, and its
projection on $T_{\partial_{x}^{2}\sigma}$ changes quadratically. Also its
projection on $T_{\sigma}$ remains almost constant, except near the edge. This
confirms the Taylor expansion shown in Eq. (24). However, the unexpected
behavior of projections onto $T_{\partial_{y}^{2}\sigma}$ might be a result of
numerical error. From Fig. 10, one can see that the lattice tensors of
$\partial_{x}^{2}\sigma$ and $\partial_{y}^{2}\sigma$ are not quite
distinguishable from the projection onto the first few eigenvectors of lTRG.
So one can conclude that the coarse-graining process erroneously removes the
information which is crucial in distinguishing the two operators. This results
in the unexpected $\partial_{y}^{2}\sigma$ value in Fig. 10.

The suppression of $T_{\sigma}$ near the edges of the coarse-grained block
$x\rightarrow 0$ and $x\rightarrow 1023$ may be a result that GILT
artificially suppresses the defects near the coarse-grained block edges and
corners, which will be discussed in Sec. IV.7.

We do not directly compare $T_{\sigma_{x,0}}$ with the coarse-grained tensors
$T_{\sigma}$, $T_{\partial\sigma}$… It is because, as suggested by [35], the
higher-index components of the coarse-grained tensors, or equivalently, the
subspace spanned by less significant eigenvectors, suffer substantial from the
unphysical ’noise’ of bond-dimension truncation and numerical errors. So we
only did the comparison in the subspace spanned by the most significant
eigenvectors.

Expansion on a cylinder. In parallel with the eigenvalues of lTRG, it is well-
known that one can also extract scaling dimensions from the eigenvalues of the
transfer matrix. In particular, we consider the periodic boundary condition
where the transfer matrix is a chain of $N=2$ coarse-grained tensors:

(25)

At the radial quantization, one would expect the following expansion,

$\sigma(r_{0},\theta)\sim\sigma+e^{i\theta}\partial\sigma+e^{-i\theta}\bar{\partial}\sigma+...,$
(26)

where $\sim$ indicates omitting the coefficient at each term.

Figure 12: Projection $\bra{O_{i}}\tr(TT)_{\sigma_{(r_{0},\theta)}}\ket{0}$ of
a coarse-grained tensor with different operator insertion positions onto the
conformal state obtained from transfer matrix method. $\bra{O_{i}}$’s are the
eigenvectors of the transfer matrix on a cylinder Eq. (25), $\ket{0}$ is the
ground state of the transfer matrix, and $\tr(TT)_{\sigma_{(r_{0},\theta)}}$
is a cyclic chain of two coarse-grained tensors, with a defect on one of the
tensor, depending on the position of $\theta$. The eigenvectors
$\mathcal{O}_{i}$’s are labeled by their estimated scaling dimensions.

Fig. 12 compares the eigenvectors of the transfer matrix Eq. (25) and the
lattice state associated with placing a defect at $(r_{0},\theta)$. The
lattice state is obtained from applying a chain of $N=2$ coarse-grained
tensors $\tr(T_{\sigma_{(r_{0},\theta)}}T_{0})$ (or
$\tr(T_{0}T_{\sigma_{(r_{0},\theta)}})$) to the ground state $\ket{0}$ of the
transfer matrix. The size of the coarse-grained block is $L/2$. At
$\theta<\pi$, the defect is on the first tensor. At $\theta\geq\pi$, the
defect is on the second tensor.

In Fig. 12, one can see that the eigenvector of scaling dimension
$\Delta\approx 0.125$, which corresponds to $\sigma$, is approximately
constant in the expansion Eq. (26); the eigenvectors of scaling dimension
$\Delta\approx 1.125$, which correspond to $\partial\sigma$ and
$\bar{\partial}\sigma$, have a sine and cosine coefficient in the expansion;
similarly, eigenvectors associated with $\Delta\approx 2.125$ have a
$\sin(2\theta)$ and $\cos(2\theta)$ coefficient. This observation confirms
that, like in lTRG, the eigenvectors of the transfer matrix can also be seen
as conformal states.

##  IV Further discussions

In this section, we evaluate the capabilities and limitations of our adopted
coarse-graining scheme for tensor networks on a lattice with point-like
defects, consisting of HOTRG coarse-graining, GILT local entanglement
filtering, and MCF gauge fixing as outlined in Sec. II.7.

Initially, we focused on improving our numerical method without the presence
of defects, as having a good vacuum tensor could facilitate further
computations with defects. Following the work of [39], we have analyzed the
per-component tensor differences (see the result presented earlier in Fig. 3)
and scaling dimension spectrum (see Table 1). Here our goal is to confirm the
crucial role of GILT in achieving a stable RG flow and show that the MCF
gauge-fixing method further enhances the stability by showing the per-
component tensor differences, scaling dimension spectrum, and the central
charge (extracted via the transfer matrix method) throughout the RG flow in
Sec. IV.1.

We also demonstrate how GILT effectively suppresses the accumulation of CDL
tensors by comparing the transfer matrix spectrum between even and odd RG
steps in Sec. IV.2. (Note in our HOTRG procedure, we define each step of the
RG is to coarse-graining two tensors either horizontally to vertically,
reducing the effective number of sites by half). However, in Sec. IV.3, we
show that even with the GILT truncation implemented, the error in the scaling
dimensions from the spectrum of the transfer matrix and the spectrum of lTRG
continues to accumulate along the coarse-graining flow. This eventually
results in the system flowing away from the conformal fixed point.

With a good vacuum tensor at hand, a natural next step is to study the two-
point function. However, our initial “on-paper” discussion in Sec. IV.4
suggests that GILT might introduce a significant error when calculating
correlation functions with more than one defect points, compared to the
vanilla HOTRG. Surprisingly, our actual numerical calculations in Sec. IV.5
show that GILT can achieve results in two-point functions as well as the
vanilla HOTRG, as long as the two points are randomly sampled within the
coarse-graining region, and outliers being properly treated. However, if the
points are located at the corner or edge of the coarse-graining region, GILT
introduces a significantly larger error.

Further evidence shows that corners and edges are special places that can be
seen in the two-point function on a torus as discussed in Sec. IV.7. Only
without GILT and when the two operators are located at the corners of the
coarse-grained region can the two distantly separated defects give a strong
correlation value when they are reunited after the region is wrapped as a
torus.

The above clue strongly suggests the smearing of point-like operators due to
coarse-graining, which is a consequence of removing short-distance details in
traditional RG techniques. For a pair of smeared operators, when close to each
other, they yield a much lower and smoother peak compared to their unsmeared
counterparts. In the vanilla HOTRG, the modes at the edges and the corners
experience less smearing due to the algorithm’s preference for retaining more
information about the CDL tensors. This explains why it produces a significant
correlation function when two corner defects reunite after the region is
wrapped as a torus. In contrast, GILT introduces some suppression of the
boundary modes, which explains why it yields a much smaller correlation
function when the two points are located on the boundary of the coarse-
graining region.

Suppose that the smearing radius is correlated with the size of the coarse-
graining block, when the operators are randomly sampled within the region, it
is less likely for them to be within each other’s smearing radius. This
results in a more reliable correlation function calculation.

To verify our conjecture, in Sec. IV.6, we compare the correlation function
between points with different degrees of smearing. The more coarse-graining
two defect points have experienced before they are fused into one tensor, the
more smearing we believe they have suffered from. We find that the correlation
for points at the same coarse-graining level before fusion aligns well with
the smeared two-point function between two Gaussian profiles. The outlier in
our data confirms our conjecture that GILT and non-GILT treat boundary modes
differently.

We further investigate the relationship between the smearing radius and the
size of the coarse-graining blocks. Unfortunately, we find that the smearing
radius grows faster than the size of the coarse-graining blocks. This implies
that as the coarse-graining process continues, the smearing radius eventually
overtakes the block size, leading to an increasing loss of information in the
correlation functions.

In summary, we conclude that it is possible to evaluate $N$-point functions
using HOTRG+GILT. To avoid the issue of operator smearing, it is necessary to
sample the points randomly in the bulk, avoid the edges and corners of the
coarse-grained regions, and filter out the outliers, e.g., using Huber loss.
GILT introduces some artificial suppression for operators at the boundary, but
this minor flaw can be ameliorated as discussed above and is compensated by
the method’s ability to stabilize the RG flow and facilitate the extraction of
the fixed-point tensor.

###  IV.1 Improved Stability of RG Flow with GILT

Figure 13: Comparison of HOTRG with and without GILT via the per-component
difference of coarse-grained tensors from successive RG steps.[l+2]. The red
line is the result by running the code of [39] at $\chi=24$, adjusting the
number of RG steps and tensor normalization to our definition. The bond
dimension we used here is $\chi=24$, and some GILT parameters are:
$\epsilon_{\text{GILT}}=8\times 10^{-7}$ and $\text{nIter}_{\text{GILT}}=1$.
Figure 14: Comparison of scaling dimensions and the central charge obtained
from diagonalizing the transfer matrix, as well as their difference compared
to analytic results at the conformal point, via HOTRG with and without GILT.
The parameters are the same parameter as in Fig. 13.

The use of GILT, as discussed in Ref. [37] and briefly reviewed in Sec. II.4,
is critical in ensuring the stability of the RG flow. As depicted in Fig. 13,
GILT enables the coarse-grained tensor to converge to a fixed-point tensor,
with the change in coarse-grained tensors decreasing exponentially after each
RG iteration. Conversely, without GILT, the change in coarse-grained tensors
remains constant, indicating either continued system evolution during the RG
steps or the failure of the gauge-fixing procedure due to degeneracy in the
spectrum decomposition (see Appendix. F). Compared to Ref. [39], the stability
of our RG flow has been slightly improved, which can be attributed to the
additional introduction of the MCF gauge-fixing method.

Furthermore, Fig. 14 illustrates the scaling dimensions and central charge
extracted from the spectrum of the transfer matrix of the coarse-grained
tensors. With GILT, the scaling dimensions and the central charge remain
consistent throughout most of the RG flow. In contrast, without GILT,
information about the scaling dimension is lost at the beginning of the
coarse-graining process, and the central charge is less stable during the
flow.

###  IV.2 Removal of CDL Tensors

As suggested in Ref. [37], the accumulating error during coarse graining
without GILT may be due to the accumulation of CDL tensors during the RG
process (i.e., after vertical and horizontal graining steps, respectively),
which consume all the bond dimension resources at the end. To investigate
this, we compare the spectrum $s^{(l)}_{i}$ of the transfer matrix between
even and odd RG steps, normalized by the largest eigenvalue $s^{(l)}_{0}$. At
odd RG steps, the coarse-grained region of each tensor is a 2:1 rectangle,
compared to the square at even RG steps. Therefore, we expect
$(s^{(odd)}_{i})^{2}\approx s^{(even)}_{i}$ near the fixed point. However, for
the fixed point of CDL tensor network, we expect $s^{(odd)}_{i}\approx
s^{(even)}_{i}$. Fig. 15 shows the oscillation of eigenvalues of the transfer
matrix between odd and even RG steps. Without GILT, the oscillation decays
during the coarse-graining process, indicating that the tensor network behaves
more like CDL tensors after coarse-graining. This result confirms GILT’s
capability to remove CDL tensors.

Figure 15: The eigenvalues $s^{(l)}_{i}$ of the transfer matrix of coarse-
grained tensors with and without GILT, normalized by the largest eigenvalue
$s^{(l)}_{0}$. At even RG steps, the coarse-grained region is a square, and at
odd RG steps, the coarse-grained region is a 2:1 rectangle. This results in an
oscillation of eigenvalues of the transfer matrix, as shown in the graph.
Without GILT, the oscillation decays after a few RG steps, indicating that the
tensor network behaves like CDL tensors. The parameters used here are the same
as those in Fig. 13.

###  IV.3 Accumulation of Error from Truncations

Figure 16: The flow of scaling dimensions obtained from the transfer matrix
and lTRG methods, and their absolute error compared to the Ising Conformal
Field Theory $\mathcal{M}(4/3)$. The grey lines (in the background) indicate
the analytic results. The bond dimension is $\chi=24$, and GILT is applied
with $\epsilon_{\text{GILT}}=8\times 10^{-7}$ and
$\text{nIter}_{\text{GILT}}=1$. The critical temperature
$T_{c}=0.44068381309509275$ is numerically determined with respect to the
current bond dimension.

The error of scaling dimensions obtained from the transfer matrix and from the
lTRG method are summarized in Fig. 16. The error in the transfer matrix
spectrum increases monotonically because, in principle, one can extract the
precise scaling dimension from the initial tensor before coarse-graining if
the chain length is infinitely long. Thus, when the chain length is long
enough to suppress finite-size effects, the only source of error is the
truncation error after each RG step, which accumulates throughout the RG flow.
This accumulation starts far before the system reaches its proximity to the
conformal fixed point (around step 30).

On the other hand, the error in the linearized Tensor Renormalization Group
generally improves as the system approaches the conformal fixed point. This
improvement is due to the scaling invariance that lTRG equation is based on,
which emerges at the continuous limit. Thus, as the system becomes more
conformal, the lTRG method becomes more accurate, leading to a reduced error
in the calculated scaling dimensions.

In Fig. 3, one can see that at approximately the critical temperature, the
coarse-grained tensor will still flow away from the fixed point tensor. This
can be due to the floating-point errors or the imprecision of the critical
temperature we found. Note due to finite bond dimensions, we do not just use
the exact $T_{c}$ value, but optimize the temperature using the method
described in Ref. [39] to find the corresponding effective $T_{c}$. We also
want to emphasize that the truncation from HOTRG and GILT introduces an
artificial length scale that breaks conformal symmetry. This might be the
reason for the instability of the critical fixed point tensor and the shift of
the critical temperature.

###  IV.4 Theoretical Insufficiency of HOTRG and GILT in Coarse-graining the
Lattice with Defects

Most prior use cases on HOTRG and GILT have been focused on translational
invariant tensor networks, where the same tensor is placed at all the lattice
sites. These studies typically consider the partition function/free energy
without any defects/impurity operators inserted. The coarse-graining tensor
network, with $w$, $g$, and $h$ insertions, can be seen as a tree-like tensor
network ansatz that tries to approximate the ground state wavefunction ($g$,
$h$ are omitted) or the partition function:

$\vbox{\hbox{\includegraphics[page=53]{TRGCFTGraphics.pdf}}}\rightarrow\vbox{\hbox{\includegraphics[page=54]{TRGCFTGraphics.pdf}}}$
(27)

In this section, we discuss how the vanilla HOTRG and HOTRG+GILT may respond
if we replace certain tensors in the MERA-like coarse-graining scheme with
defect tensors, i.e., we change a few of the tensors $T$ in Eq. (27) to defect
tensors such as $T_{\sigma}$, but keep the $w$, $g$, $h$ invariant. HOTRG can
produce good fidelity of one-point functions as long as the bond dimension is
sufficiently large for the fidelity of the ‘vacuum’ partition function (or
free energy). Note that the HOTRG equation Eq. (6) holds if both $T_{1}$ and
$T_{2}$ are vacuum tensors. So, the environment $E$ of a defect tensor
$T_{\mathcal{O}}$, which is the contraction of the tensor network with
$T_{\mathcal{O}}$ removed and only contains vacuum tensors, can be faithfully
computed using HOTRG coarse-graining method.

What about two-point functions and higher? The environment can still be
obtained from coarse-graining the remaining vacuum tensor. The source of error
comes from the insertion of $w^{\dagger}w$ between two coarse-grained defect
tensors. Note that only one side of $w^{\dagger}w\simeq\mathbb{1}$ is
connected to a vacuum tensor, sufficient for the HOTRG equation to hold.

The GILT equation Eq. (8) holds only when the tensors $ABCD$ in the subnetwork
Eq. (8) are all vacuum tensors. One may question whether the Hilbert space
that GILT truncates is the subspace corresponding to defect tensors. This
conjecture is supported by the results shown in Fig. 20 that the HOTRG+GILT
gives a lower estimation of the two-point correlation function.

In calculating the two-point function, the HOTRG equation only fails when the
two coarse-grained defects meet each other in the coarse-graining. In
contrast, the GILT equation introduces some errors at each coarse-grained
level. This suggests that GILT introduces more errors in two-point functions.
This is consistent with what we have observed. Nevertheless, we will
demonstrate that it is still possible to make HOTRG+GILT yield as good results
as the vanilla HOTRG for correlation functions.

###  IV.5 Comparison of Two-point Correlation Functions with and without GILT

Figure 17: The two-point correlation function
$\expectationvalue{\sigma(x_{0},y_{0})\sigma(x_{1},y_{1})}$ can be calculated
using different strategies. One option is whether to use GILT, while another
is whether to take the correlation function averaged over different starting
points $(x_{0},y_{0})$ or to start from the corner $(0,0)$ of the coarse-
grained block $0\leq x<L_{x},0\leq y<L_{y}$. When using GILT and a fixed
starting point, the correlation function gives an incorrect scaling factor. On
the other hand, without GILT and with a fixed starting point, the correlation
is slightly overestimated. However, by taking the average starting points
$(x_{0},y_{0})$ in the block, both cases with or without GILT agree well with
the analytic result.

Fig. 17 shows the two-point correlation function
$\expectationvalue{\sigma(x_{0},y_{0})\sigma(x_{1},y_{1})}$ calculated by
different methods. When GILT is used, the correlator gives the wrong scaling
dimension when one of the defects is localized at a corner of the coarse-
grained block. The reason seems to be that the corner modes are being
artificially suppressed by GILT, as discussed in Sec. IV.7.

The overall best results are given when GILT is used, but the correlation
function averaged over the positions of the defects. GILT gives a more
reliable vacuum tensor, and the averaging helps avoid corners of coarse-
grained blocks at various levels. We have used this strategy for the earlier
part of our results.

###  IV.6 Smearing of Point Defects

In the coarse-graining process, a point-like defect is expected to be
“smeared” into a “particle cloud”. When the two smeared defects move toward
each other, one might expect a regularized two-point correlation function,
with a smoother, lower peak compared to the correlation function between two
“bare” defects. Eq. (28) below shows a type of regularized two-point function,
where the point-like defect is being replaced by a Gaussian profile Eq. (29),
as illustrated in Fig. 18. In the log-log plot, Eq. (28) features a flat line,
which indicates the smeared tip of the correlation function, followed by a
negative slope, indicating the unsmeared part of the correlation function,
which follows the power law Eq. (30).

Figure 18: a) In a smeared two-point function, the two-point-like defects are
replaced by a Gaussian profile with a smearing radius $r_{s}$. b) The shape of
a smeared two-point function Eq. (28) in a log-log plot. The vertical grey
line indicates the smearing radius $r_{s}$.

$\displaystyle f_{\text{smeared}}(x,y)$ $\displaystyle=\int\int\int\int
dx_{1}dy_{1}dx_{2}dy_{2}$ (28) $\displaystyle G(x-x_{1},y-y_{1})$
$\displaystyle G(x-x_{2},y-y_{2})P(x_{1}-x_{2},y_{1}-y_{2}),$

$G(x,y)=\frac{1}{2\pi
r_{s}^{2}}e^{-\frac{1}{2}\frac{x^{2}+y^{2}}{r_{s}^{2}}},$ (29)

$P(x,y)=\frac{1}{(x^{2}+y^{2})^{\Delta}}.$ (30)

Figure 19: a) The sites on a $8\times 8$ lattice are grouped into coarse-
graining blocks at various levels $l$. The coordinate system is originated at
the lower left corner of the biggest block $l=6$. The above illustration shows
the coarse-graining blocks of a lattice that contains two defects at $(2,2)$
and $(4,6)$. b) At level $1\leq l\leq 5$, there are two defect tensors at each
level, each contained at a separate block. At level $l=6$, the two defect
tensors are fused into one.

In HOTRG, the fusion of two defects happens when two coarse-grained tensors,
each has one defect in its region, are being coarse-grained into a larger
tensor. As illustrated in Fig. 19, the two point defects $(2,2)$ and $(4,6)$
meets at a $8\times 8$ block at level $l=6$. Naturally, one would expect that
the amount of smearing the two defects suffer is related to the block size of
the largest coarse-grained tensor before the two defect meets.

Figure 20: The two-point correlation function evaluated at different coarse-
graining blocks. The color indicates the size of the biggest block before the
two defects are fused into one block. The black line is fitted at the
log2-log2 scale by a one-sided Huber loss with $\epsilon=0.1$ for $r<256$ with
scaling dimension $\Delta_{\text{No GILt}}=0.118$ and
$\Delta_{\text{GILT}}=0.131$. The points are sampled in a fine-tuned
probability distribution that prioritizes the outliers below the main trend.
For the trend of different block sizes, see Fig. 21-Fig. 24 Figure 21: The
two-point correlation function evaluated at different coarse-graining blocks.
GILT is not used. This figure represents the same data as in Fig. 20, but
separated in different frames by different block sizes. Each subfigure
represents a certain size $l_{x}\times l_{y}$ of the biggest block before the
two defects are fused into one block. The two black lines are the fitted main
trend and the lower bound, respectively. The vertical grey dashed line
indicates the block size $l_{x},l_{y}$. The vertical grey solid line indicates
the smearing radius $r_{s}$. Figure 22: The two-point correlation function
evaluated at different coarse-graining blocks. GILT is used. This figure
represents the same data as Fig. 20, but separated in different frames by
different block size. Each subfigure represents a certain size $l_{x}\times
l_{y}$ of the biggest block before fusing the two defects into one block. The
two black lines are the fitted main trend and the lower bound, respectively.
The vertical grey dashed line indicates the block size $l_{x},l_{y}$. The
vertical grey solid line indicates the smearing radius $r_{s}$.

Fig. 20 shows how the coarse-grained block size $l_{x}\times l_{y}$ right
before two defects merge regularizes the short-distance behavior of two-point
correlators. The color indicates the size of the biggest block before the two
defects are fused into one block. The black line is fitted at the log2-log2
scale by a one-sided Huber loss with $\epsilon=0.1$ for $r<256$ with
respective fitted scaling dimensions $\Delta_{\text{No GILt}}=0.118$ and
$\Delta_{\text{GILT}}=0.131$. The points are sampled with a fine-tuned
probability distribution that prioritizes the outliers below the main trend so
as to illustrate the smmearing effect.

As mentioned earlier, the size of the coarse-grained block $l_{x}\times l_{y}$
implies the amount of smearing regularization a two-point correlator suffers.
For correlators with the same separation $r$, the bigger the coarse-grained
block, the lower the regularized correlator is. Also note that the amount of
smearing is not only determined by the size of the block but also the relative
position of a defect inside a block, which has not been examined carefully
yet.

Fig. 21 and Fig. 22 further demonstrate the trend of the smeared correlation
function for a given coarse-graining size $l_{x}\times l_{y}$. There are two
main trends in the data. On the log-log plot, the first one is a straight line
with a negative slope, implying the unsmeared, exact correlation function. The
second trend is a flat line, indicating the smeared two-point function. The
two trends together confirm the smearing picture described in Eq. (28). Note
that there are also data scattered between the two trends, as we have already
seen in Fig. 20.

The first trend resembles the exact, unsmeared correlation function. In Fig.
20, the first trend of data with various block sizes almost overlaps with each
other, showing consistency between different levels of coarse-graining. It
might indicate that certain pairs of defects do not suffer from the smearing,
or the smearing radius is far smaller than the separation $r$.

The second trend resembles the smeared correlation function, which is a flat
line, clipping the tip of the first trend at short distances. The flat line
indicates the correlation function is almost invariant when the separation of
the smearing operator changes. It might be explained by the picture that the
separation is less than the smearing radius $l_{s}$. The smearing radius
$l_{s}$ is estimated by the intersection of the first and the second trend.

Figure 23: The smearing radius $l_{s}$ in Fig. 21 and the size of the coarse-
graining block. GILT is not used here. Figure 24: The smearing radius $l_{s}$
in Fig. 22 and the size of the coarse-graining block. GILT is used here.

The smearing radius $l_{s}$ increases with the increase of the coarse-graining
block size ($l_{x}\times l_{y}$) before two defect tenors merge, as shown in
Fig. 23 and Fig. 24. However, the smearing radius grows faster than the growth
of block size: $l_{s}\propto l_{x}^{\delta_{\text{LS}}}$, with
$\delta_{\text{LS, No GILT}}=1.20$ and $\delta_{\text{LS, GILT}}=1.68$. This
indicates that as the coarse-graining iteration proceeds, the smearing radius
will catch up the lattice size, and may resulting errors in further coarse-
graining.

Comparing the cases with and without GILT. When GILT is applied, most of the
short-distance correlation points on the first, unsmeared trend are moved to
the second, smeared trend. It means GILT successfully removes most of the
short-distance details of the system, compared to the case without GILT.
However, at the long-distance end, the data without GILT follows more
precisely to the fitted line, while the data with GILT has a more significant
error with a much lower value. It means that GILT also removes some of the
information about the long-range behavior of the defects.

The data analysis in this subsection confirms that the defect is being smeared
after coarse-graining. Without GILT, defects at certain places might not be
smeared as much. With GILT, most defects are being smeared at approximately
the same degree, but GILT also substantially removes some of the long-range
information about the defects, resulting in an error that lowers the
calculated two-point function value.

We also find the positioning of the defects plays a crucial role in
determining whether the defect are being missed from smearing in the no GILT
case, and the defects are being over-suppressed in the GILT case. There is a
sign that the corner of the largest unfused block is the abovementioned
special place, but further numerical evidence is required to support this
conjecture.

Finally, we provide an explanation for why the naive averaging over different
places when sampling the two-point function in Fig. 5 yields the best result
for the scaling dimension when GILT is used. As shown in Fig. 22, to obtain
the unsmeared two-point function, we require $r>r_{s}$. When points are
randomly sampled, it is most probable that the size of the largest block
$l_{x}\times l_{y}$ before merging is comparable to the distance between the
two points $r$, and $l_{x}>r_{s}$. The only situation where $l_{x}\gg r$ is
when $r$ is located in the corner or edge of the largest unfused block, which
is a very unlikely scenario. Thus, in Fig. 22, we intentionally increase the
likelihood that $r$ is inside the smearing radius, so we can observe the
reduction of correlation function due to smearing, and the fitted scaling
dimension has a larger error.

However, in Fig. 5, when points are randomly sampled, most of them satisfy
$r>r_{s}$, which guarantees that the smearing only occurs at a shorter scale
compared to $r$, thus ensuring the correctness of the two-point function.
Therefore, the naive averaging over different places yields the best scaling
dimension result in this case.

###  IV.7 Removal of Localized Edge Modes

When GILT is not applied, the dominance of the CDL tensor suggests that the
truncated Hilbert space of the coarse-grained tensor is mainly capturing the
exciting modes near the boundaries and corners of the coarse-grained block.
This can be shown from the observation that there is no smearing at the corner
of the coarse-grained block when GILT is turned off.

Figure 25: The two point correlation function
$\expectationvalue{\sigma(0,y_{0})\sigma(x,y_{0})}$ on a torus $x\sim
x+L_{x},y\sim y+L_{y}$ calculated using different methods. One option is
whether to use GILT, while another is whether the start point is at the corner
$(0,0)$ of the coarse-grained block $0\leq x<L_{x},0\leq y<L_{y}$, or near the
center $(0,L_{y}/2)$ of the block edge. Figure 26: On a torus geometry, the
two operators will experience coarse graining before being fused together. a)
When the operator is at the center of the edge, it will be “smeared” into a
“particle cloud”, resulting in a lower correlation function. b) When the
operator is at the corner, and GILT is not applied, smearing will not happen,
resulting in a sharp peak at the end of the two-point function in Fig. 25.
Figure 27: The two-point correlation function
$\expectationvalue{\sigma(0,y_{0})\sigma(x,y_{0})}$ on a torus. In a) and c)
the color denotes the values of $y_{0}$, where red indicates $y_{0}=L_{y}/2$
and blue indicates $y_{0}=0$. In b) and d), $x=L_{x}-1$.

In particular, we compare the torus correlation function at $x\rightarrow 0$
and $x\rightarrow L_{x}$. As shown in Fig. 26, the correlator at $x\rightarrow
L_{x}$ is being evaluated by contracting opposite edges of a $L_{x}\times
L_{y}$ block. So, the largest coarse-grained block containing a single defect
is of size $L_{x}/2$. One would expect a very large smearing effect, resulting
in a vanishing peak at $x\rightarrow L_{x}$. However, Fig. 25 shows that the
peak at $x\rightarrow L_{x}$ is as sharp as the bare correlation function
evaluated at smaller scales at $x\rightarrow 0$, in the case that GILT is not
used and the two defects are localized at the corner of the block.

It suggests that the defect is not being smeared after the coarse-graining
iterations. In other words, the truncated Hilbert space contains point-like
excitations at the corners of the block. To further verify it, we plot the
height of the peak when $y_{0}$ moves from the corner to the midpoint of the
block edge. As shown in Fig. 27c,d, the smearing effect is stronger when the
two defects are on the midpoint of the oppositing edges. Giving the finiteness
of the dimension of the truncated Hilbert space, It suggests that the defects
are being smeared into a one-dimensional “cloud” along the edge.

In the case of GILT, one would expect that the edge modes are being
suppressed, and there is no peak at $x\rightarrow L_{x}$, as confirmed in Fig.
27a,b. Note that there are some inconsistencies in the correlation function
when the defects are located at the corners. It suggests that GILT introduces
some artificial suppression for the corner excitation.

##  V Conclusion

In this work, we have integrated the MCF gauge-fixing method [41], with the
tensor renormalization scheme [39] that combines the local entanglement
filtering method GILT [37] and the HOTRG coarse-graining method [33]. The
combined method enhances the stability of the RG flow, which is manifested
visually in the tensor component space. Additionally, we have demonstrated the
detection of CDL-like effects from the oscillation of the transfer matrix
spectrum between horizontal and vertical coarse-graining steps.

Following [39], we have calculated the scaling dimensions for conformal states
with $\Delta<3$ in the critical square lattice Ising model using two different
methods: the transfer matrix spectrum [36] and the lTRG approach [39]. Both
methods exhibit good agreement with analytical results. We have further
calculated the OPE coefficient $C_{\sigma\sigma\varepsilon}$ from lTRG and its
eigenvectors.

Important quantities to calculate in TN include correlation functions. We have
computed the two-point spin-spin correlation function for the square lattice
Ising model at temperatures near the critical point (at, above, and below).
The extracted exponent is in good agreement with the scaling dimension. We
have also examined the correlation function on a finite-size torus and
obtained results consistent with analytical results. In addition, we have also
computed four-point correlation functions, and from those we were able to
extract the scaling dimensions and the OPE coefficient, although the quality
was not as good as other approaches we have used.

We have extended the use of lTRG beyond the computation of scaling dimensions
and compared the eigenvectors of lTRG fixed-point equation with coarse-grained
defect tensors, discovering that inserting operators on the lattice allows for
the construction of conformal states/operators, which corroborate well with
those obtained from lTRG. We have also found that the inserted operator’s
position is reflected in its decomposition onto the conformal states
numerically.

Our work has explored the effects of coarse-graining (HOTRG) and local
entanglement filtering (GILT) on lattice defects/impurities. We have observed
that point-like defects become “smeared” into “particle clouds” after coarse-
graining and that the degree of smearing varies depending on the defect’s
position in the coarse-graining region, particularly when it is located at the
corner or edge of the region. GILT also exhibits an artificial suppression for
edge modes; despite this, we were able to mitigate this and demonstrate that a
good-quality two-point function on the plane or torus can still be retrieved
if the smearing effect and the edge mode suppression are properly
circumvented.

We now conclude this work by discussing possible future directions. The HOTRG
and GILT algorithms we used in this do not account for lattice defects when
determining truncation and filtering strategies. It may be possible to modify
these coarse-graining algorithms to be sensitive to lattice defects. It is
therefore interesting to explore whether this modification leads to more
robust coarse-graining on $n$-point functions and whether it can improve the
overall quality of fixed-point tensors. We also expect that by studying the
tensor-space RG flow, oscillation of CDL tensors, and smearing of point-like
operators, one can troubleshoot the current limitations of coarse-graining and
local entanglement filtering methods, and develop more precise and efficient
computational methods for tensor-based RG approaches. Furthermore, we note
that our methods can be extended to other systems, including quantum spin
models, such as the transverse-field Ising model, possibly to 3D classical
systems. It may also be interesting to study the tensor-based RG flow of
extended defect objects, such as defect lines in 2D and defect planes in 3D.
Finally, it might be useful to investigate the reverse. In this paper, we
extracted the CFT data from a fixed-point tensor. Then, a natural question is,
where there exist some prescriptions for constructing a fixed-point tensor
from the given CFT data, which might pave a new way of conformal bootstrapping
via studying the fixed-points of the tensor renormalization group in the
tensor space.

Note added: Close to the completion of our initial manuscript, we became aware
of a recent work by Atsuhi Ueda and Masaki Oshikawa on “finite-size and finite
bond dimension effects of tensor network renormalization” [49].

##  VI Acknowledgements

We are grateful for many useful and valuables discussion with Ning Bao, Nikko
Pomata, Andreas Weichselbaum, and Alexander B. Zamolodchikov. We acknowledge
partial support by the Materials Science and Engineering Divisions, Office of
Basic Energy Sciences of the U.S. Department of Energy under Contract No.
DESC0012704 (in particular, for developing TN algorithms) and by the National
Science Foundation under Grant No. PHY 1915165 (in particular, for
applications in CFT).

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##  Appendix A List of Algorithms in TRG

###  A.1 Higher-Order Tensor Renormalization Group

Here, we recapitulate the essence of the HOTRG coarse-graining procedure, as
schematically shown in Algorithm A.1. We define one step of HOTRG as merging
two tensors into one. The first (or odd number of) step involves finding the
isometry $w$ to merge two tensors along the horizontal directions, resulting
in a rectangular shape of the whole system. The second (or even number of)
step involves merging two resultant tensors from the previous step into one,
restoring the whole system to a square shape.  Algorithm 1 Higher-Order Tensor
Renormalization Group [33]

$\vbox{\hbox{\includegraphics[page=20]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=21]{TRGCFTGraphics.pdf}}}$
(31)

$\vbox{\hbox{\includegraphics[page=22]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=23]{TRGCFTGraphics.pdf}}}$
(32)

Inputs: A translational invariant tensor network on a lattice. The tensor on
each site is $T$.

Outputs: The coarse-grained tensor $T^{\prime}$ by grouping every two adjacent
tensors in the chosen spatial dimension, and an isometry $w$ which truncates
the bond dimension of the coarse-grained tensor from $\chi\times\chi$ to
$\chi$.

Steps:

To minimize the truncation error:

$\left\|\,\vbox{\hbox{\includegraphics[page=24]{TRGCFTGraphics.pdf}}}-\vbox{\hbox{\includegraphics[page=25]{TRGCFTGraphics.pdf}}}\,\right\|^{2},$
(33)

the isometry $w$ can be estimated from the singular value decomposition of the
two-site reduced density matrix $M$:

$\vbox{\hbox{\includegraphics[page=26]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=27]{TRGCFTGraphics.pdf}}}\stackrel{{\scriptstyle\text{SVD}}}{{=}}\vbox{\hbox{\includegraphics[page=28]{TRGCFTGraphics.pdf}}}$
(34)

On a 2D square lattice, one does the vertical and horizontal coarse-graining
step alternatively, as shown in Eq. (31) and Eq. (32).

###  A.2 Graph Independent Local Truncation

Next, we describe the GILT procedure [37] that aims to remove CDL-like tensors
and it is shown in Algorithm A.2. It involves a local truncation $R$ that
interrupts loop-like correlations.

Algorithm 2 Graph Independent Local Truncation [37]

$\vbox{\hbox{\includegraphics[page=29]{TRGCFTGraphics.pdf}}}\approx\vbox{\hbox{\includegraphics[page=30]{TRGCFTGraphics.pdf}}}\stackrel{{\scriptstyle\text{SVD}}}{{=}}\vbox{\hbox{\includegraphics[page=31]{TRGCFTGraphics.pdf}}}$
(35)

Inputs: a subgraph of a tensor network, and an edge in the subgraph to
truncate.

Outputs: a truncation matrix $R$ to insert in the chosen edge, or
equivalently, a pair of linear transformations $g_{1}$ and $g_{2}$,
$R=g_{1}g_{2}$ to apply to the legs of the adjacent tensors, that partially
remove the local entanglements passing through the edge.

Steps:

$\vbox{\hbox{\includegraphics[page=32]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=33]{TRGCFTGraphics.pdf}}}\stackrel{{\scriptstyle\text{SVD}}}{{=}}\vbox{\hbox{\includegraphics[page=34]{TRGCFTGraphics.pdf}}}$
(36)

The environment $E$ of an edge is defined by removing that edge from the
tensor network. $E$ is interpreted as an isometry from the two ends of broken
edges to the external edges.

$\vbox{\hbox{\includegraphics[page=35]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=36]{TRGCFTGraphics.pdf}}}$
(37)

By spectral decomposing $E$, one can see that the subspace of variations of
$R\in\text{Mat}_{\chi\times\chi}$ which does not significantly affect the
tensor output, is spanned by the singular vectors corresponding to smaller
singular values.

In particular, we choose $R$ in the following way:

$\vbox{\hbox{\includegraphics[page=37]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=38]{TRGCFTGraphics.pdf}}}$
(38)
$t^{\prime}_{i}=t_{i}\frac{\hat{S}_{i}^{2}}{\hat{S}_{i}^{2}+\epsilon^{2}},\hat{S}_{i}=\frac{S_{i}}{\max(S)},\vbox{\hbox{\includegraphics[page=39]{TRGCFTGraphics.pdf}}}\;=\vbox{\hbox{\includegraphics[page=40]{TRGCFTGraphics.pdf}}}$
(39)

Split $R$ into $g_{1}$ and $g_{2}$ by SVD, and absorb them into the adjacent
tensors, as in Eq. (8). One can further truncate the local entanglement by
repeating the process on edge between $g_{1}$ and $g_{2}$.

###  A.3 Minimal Canonical Form

In order to compare tensors in different coarse-graining steps and determine
the change in the RG flow, it is necessary to convert them into certain
canonical forms. One key tool that we use is the recently proposed MCF [41].
It fixes the redundancy under the group $GL(\chi)^{\otimes m}$ up to a residue
$U(\chi)^{\otimes m}$, where $m$ is the bond dimension of each leg. MCF uses
an iterative procedure, described in Algorithm A.3.

Algorithm 3 Minimal Canonical Form[41]

$\vbox{\hbox{\includegraphics[page=41]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=42]{TRGCFTGraphics.pdf}}}$
(40)
$\vbox{\hbox{\includegraphics[page=43]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=44]{TRGCFTGraphics.pdf}}}$
(41)

Inputs: A translationally invariant tensor network on a lattice. The tensor on
each site is $T$.

Outputs: A linear transformation $(g_{1},g_{2})\in GL(\chi)^{\otimes m}$,
which gauge transforms the tensor into its minimal canonical form:

$T_{min}=\operatorname{argmin}\\{||\tilde{T}||_{2}:\tilde{T}\in\overline{G\cdot
T}\\},$ (42)

where $m$ is the spatial dimension of the lattice, $\chi$ is the bond
dimension, and $\overline{G\cdot T}$ is the closure of the gauge orbit
generated by the gauge transform Eq. (41).

Steps:

The object to minimize is:

$||T||_{2}=\tr\rho=\tr\ket{T}\bra{T}=\vbox{\hbox{\includegraphics[page=45]{TRGCFTGraphics.pdf}}}$
(43)

Its first-order variation with respect to gauge transformation is:

$\partial_{t=0}||(e^{tX_{1}},e^{tX_{2}})\cdot T||_{2}^{2}=2\sum_{k=1}^{m}\tr
X_{k}(\rho_{k,1}-\rho_{k,2}^{T}),$ (44)

where the reduced density matrices $\rho_{k,1}$,$\rho_{k,2}$ are:

$\begin{split}&\rho_{1,1}=\vbox{\hbox{\includegraphics[page=46]{TRGCFTGraphics.pdf}}},\rho_{1,2}^{T}=\vbox{\hbox{\includegraphics[page=47]{TRGCFTGraphics.pdf}}},\\\
&\rho_{2,1}=\vbox{\hbox{\includegraphics[page=48]{TRGCFTGraphics.pdf}}},\rho_{2,2}^{T}=\vbox{\hbox{\includegraphics[page=49]{TRGCFTGraphics.pdf}}},\end{split}$
(45)

According to [41], the global minimum can be reached by gradient descent. We
apply the following gauge transform by the gradient in Eq. (44) and the step
length suggested in [41],

$\vbox{\hbox{\includegraphics[page=50]{TRGCFTGraphics.pdf}}}=e^{-\frac{1}{4m}\frac{1}{tr\rho}(\rho_{k,1}-\rho_{k,2}^{T})}$
(46)

and repeat the whole process until converges.

###  A.4 Preliminary procedure of fixing the $U(\chi)$ gauge

After fixing the $GL(\chi)^{\otimes m}$ gauge using MCF, there is a remaining
$U(\chi)^{\otimes m}$ gauge. Ref. [39] states that one only needs to fix the
residue $U(1)^{\otimes m\chi}$ subgroup, which reduces to the sign ambiguity
at each index $Z_{2}^{\otimes m\chi}$ for the case of real tensors. However,
we found that, after fixing the signs using an improved method modified from
Ref. [39], one can further minimize the difference of $T$ from the reference
tensor $\tilde{T}$ by applying a standard local minimization procedure for the
two gauging matrices $g_{1},g_{2}$ by singular value decomposing the
corresponding environment tensors iteratively. This is because the argument in
Ref. [39] relies on that there is no level crossing between eigenvalues when
one obtains $T$ through HOTRG method Eq. (34). The potential level crossing
mixes two such indices and can be fixed by the $U(\chi)^{\otimes m}$ gauge.

Here shows our procedure to fix the $U(\chi)^{\otimes m}$ gauge in Algorithm
A.4.

Algorithm 4 Procedure of fixing the $U(\chi)^{\otimes m}$ gauge [39]

Input: $T_{ijkl}$, the tensor to be gauge-fixed. $\tilde{T}_{ijkl}$, the
reference tensor for the gauge choice

Output: $T_{ijkl}$, the gauge fixed tensor

Steps:

for repeat a few steps do

for $\tilde{\chi}\leftarrow 1$ to $\chi$ do

$\rho_{1,i}\leftarrow\sum_{j,k,l\leq\tilde{\chi}}T_{ijkl}\tilde{T}_{ijkl}$

$d_{1,i}\leftarrow\operatorname{sgn}{\rho_{1,i}}$

$T_{ijkl}\leftarrow T_{ijkl}d_{1,i}d_{1,j}$

$\rho_{2,k}\leftarrow\sum_{i,j,l\leq\tilde{\chi}}T_{ijkl}\tilde{T}_{ijkl}$

$d_{2,k}\leftarrow\operatorname{sgn}{\rho_{2,k}}$

$T_{ijkl}\leftarrow T_{ijkl}d_{2,k}d_{2,l}$

end for

for repeat a few steps do

loss$\leftarrow\bra{\tilde{T}}\ket{(g_{1},g_{2})\cdot T}$

optimize $g_{1}$ by svd the environment tensor of $g_{1}$

optimize $g_{2}$ by svd the environment tensor of $g_{2}$

$T\leftarrow(g_{1},g_{2})\cdot T$

end for

end for

We note that in Algorithm A.4, the overlap is evaluated as

$\bra{\tilde{T}}\ket{(g_{1},g_{2})\cdot
T}=\vbox{\hbox{\includegraphics[page=55]{TRGCFTGraphics.pdf}}},$

where the environment tensor of $\bar{g_{1}}$ is the tensor subnetwork after
removing $g_{1}$ from the network, and the standard iteratively optimization
method for unitary $g_{1}$ is: $g_{1}\leftarrow(UV^{\dagger})^{*}$, with $U$
and $V$ obtained from SVD the environment tensor: $\bar{g_{1}}=UsV^{\dagger}$.
The tensor $g_{2}$ can be similarly updated.

##  Appendix B Counting the number of conformal states

This section discusses how to obtain the number of conformal states at each
conformal level in Ising CFT $\mathcal{M}(3/4)$. It is a brief summary of some
relevant formulas from Chapters 7 and 8 in Ref. [47]. For comparison, we also
list the number of degeneracy states of the first 12 scaling dimensions of
Ising CFT in Table. 4.

$\Delta$ | Total | $\mathbb{1}$ | $\sigma$ | $\varepsilon$
---|---|---|---|---
0 | 1 | 1 |  |
$\frac{1}{8}$ | 1 |  | 1 |
1 | 1 | 0 |  | 1
$1+\frac{1}{8}$ | 2 |  | 2 |
2 | 4 | 2 |  | 2
$2+\frac{1}{8}$ | 3 |  | 3 |
3 | 5 | 2 |  | 3
$3+\frac{1}{8}$ | 6 |  | 6 |
4 | 9 | 5 |  | 4
$4+\frac{1}{8}$ | 9 |  | 9 |
5 | 13 | 6 |  | 7
$5+\frac{1}{8}$ | 14 |  | 14 |
Table 4: Number of conformal states at each scaling dimension.  level | $h_{1,1}=0$ | $h_{2,1}=1/16$ | $h_{1,2}=1/2$
---|---|---|---
0 | 1 | 1 | 1
1 | 0 | 1 | 1
2 | 1 | 1 | 1
3 | 1 | 2 | 1
4 | 2 | 2 | 2
5 | 2 | 3 | 2
6 | 3 | 4 | 3
7 | 3 | 5 | 4
8 | 5 | 6 | 5
9 | 5 | 8 | 6
10 | 7 | 10 | 8
Table 5: Number of conformal states for different chiral representation of
Ising CFT

In Ising CFT $\mathcal{M}(3/4)$, there are three primary operators (states):
$\mathbb{1}$, $\sigma$, $\varepsilon$. Their conformal dimensions are
$\Delta_{\mathbb{1}}=0$, $\Delta_{\sigma}=\frac{1}{8}$,
$\Delta_{\varepsilon}=1$. They are the tensor product of the left and right
chiral representations of the Virasoro algebra:

$\displaystyle\ket{\mathbb{1}}$
$\displaystyle=\ket{h=0}\otimes\ket{\bar{h}=0},$
$\displaystyle\ket{\mathbb{\sigma}}$
$\displaystyle=\ket{h=1/16}\otimes\ket{\bar{h}=1/16},$
$\displaystyle\ket{\mathbb{\varepsilon}}$
$\displaystyle=\ket{h=1/2}\otimes\ket{\bar{h}=1/2}.$

The corresponding descendant states can be achieved by applying left and right
Virasoro operators on those highest-weight states:

$L_{-m_{1}}L_{-m_{2}}L_{-m_{3}}...\ket{h}\otimes\bar{L}_{-\bar{m}_{1}}\bar{L}_{-\bar{m}_{2}}\bar{L}_{-\bar{m}_{3}}...\ket{\bar{h}}.$
(47)

The conformal dimension of a state is the sum of the left and right conformal
spin:

$\Delta=h+m_{1}+m_{2}+...+\bar{h}+\bar{m}_{1}+\bar{m}_{2}+...$ (48)

The operators $L_{m}$,$\bar{L}_{m}$ satisfies the Virasoro Algebra:

$[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}(m-1)m(m+1)\delta_{m+n,0},$ (49)

where

$L_{-m}=L_{m}^{\dagger},$ (50)

and the primary state $\ket{h}$ satisfies:

$L_{0}\ket{h}=h,\qquad L_{m>0}\ket{h}=0.$ (51)

Given $c$ and $h$, the inner product between states of the form
$L_{-m_{1}}...\ket{h}$ is uniquely defined and can be computed by moving all
the annihilation operators $L_{m>0}$ to the right and creation operators
$L_{m<0}$ to the left using commutator relationship Eq. (49). For example, the
inner product between $L_{-2}\ket{h}$ and $L_{-2}\ket{h}$ is:

$\bra{h}L_{2}L_{-2}\ket{h}=\bra{h}4L_{0}+\frac{c}{2}+L_{-2}L_{2}\ket{h}=4h+\frac{c}{2}.$
(52)

Note that different descendent states of the same primary state might be
linearly dependent if they are at the same level. It can be seen from the
inner product of different states. In particular, define the Kac matrix:

$M^{(l)}_{m,n}=\bra{m}\ket{n},$ (53)

where $\ket{m}$, $\ket{n}$ are the descendent states of the primary state
$\ket{h}$ and have the level $l$.

For example, the explicit expression of $M^{(2)}$ of given $c$ and $h$:

$\begin{pmatrix}\frac{c}{2}+4h&6h\\\ 6h&4h+8h^{2}\end{pmatrix},$ (54)

which represents the inner products between $L_{-2}\ket{h}$ and
$L_{-1}L_{-1}\ket{h}$.

The rank of $M^{(l)}$ determines the number of linearly independent
descendants of $\ket{h}$ at level $l$. To compute the degeneracy, one might
write down the Kac Matrix manually or use symbolic packages such as [50].

Instead, the general expression for degeneracy can also be read out from the
character of irreducible Verma Module $M_{r,s}$ :

$\displaystyle\chi_{h}(q)$ $\displaystyle:=$
$\displaystyle\tr_{M_{r,s}}(q^{L_{0}})$ $\displaystyle=$ $\displaystyle
q^{h-c/24}\sum_{n=0}^{\infty}(\text{degeneracy at level n})*q^{n}.$

The expression of $\chi_{h}$ is found as follows (see 8.17 in [47]):

$\chi_{h_{r,s}}(q)=K^{p,p^{\prime}}_{r,s}(q)-K^{p,p^{\prime}}_{r,-s}(q),$ (56)

where $K$ is given as

$K^{(p,p^{\prime})}_{r,s}(q)=\frac{q^{-1/24}}{\varphi(q)}\sum_{n\in\mathbb{Z}}q^{(2pp^{\prime}n+pr-p^{\prime}s)^{2}/4pp^{\prime}},$
(57)

and the $\varphi(q)$ is defined via

$\displaystyle\frac{1}{\varphi(q)}$ $\displaystyle=$
$\displaystyle\frac{1}{\prod_{n\geq 1}(1-q^{n})}$ $\displaystyle=$
$\displaystyle 1+q+2q^{2}+3q^{3}+5q^{4}+7q^{5}+11q^{6}+...,$

where $p,p^{\prime}$ labels the minimal model $M(p^{\prime}/p)$. Moreover,
$r,s$ labels the highest weight state $\ket{h_{r,s}}$. The central charge $c$
is given as follows,

$c=1-6\frac{(p-p^{\prime})^{2}}{pp^{\prime}},$ (59)

and the scaling dimension is given by

$h_{r,s}=\frac{(pr-p^{\prime}s)^{2}-(p-p^{\prime})^{2}}{4pp^{\prime}}.$ (60)

The degeneracy of the Kac matrix for $c=1/2$ at the first few levels is
summarized at Table. 5. By collecting the tensor product of left and right-
hand states by the total conformal dimension, we reached Table. 4.

##  Appendix C The OPE Coefficient of Ising Model

The three-point function in CFT has a general form,

$\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}=\frac{C_{123}}{r_{12}^{\Delta_{1}+\Delta_{2}-\Delta_{3}}r_{23}^{\Delta_{2}+\Delta_{3}-\Delta_{1}}r_{31}^{\Delta_{3}+\Delta_{1}-\Delta_{2}}},$
(61)

where $r_{ij}=\left|r_{i}-r_{j}\right|$ is the distance between operators and
the scaling behavior $a,b,c$ is determined by the scaling dimensions of the
operators.

The OPE coefficient $C_{123}$ is the only thing not fixed by conformal
symmetry. In 2D Ising CFT $\mathcal{M}(3/4)$, the OPE coefficients are:

$\displaystyle
C_{\sigma\sigma\mathbb{1}}=C_{\varepsilon\varepsilon\mathbb{1}}=1,\
C_{\sigma\sigma\varepsilon}=\frac{1}{2},\
{C_{\varepsilon\sigma\sigma}}=\frac{1}{2}.$ (62)

Note that the first line indicates field operators $\sigma$ and $\varepsilon$
need to be properly rescaled such that:

$\displaystyle\expectationvalue{\sigma(r_{1})\sigma(r_{2})}$
$\displaystyle=\frac{1}{\left|r_{1}-r_{2}\right|^{\frac{1}{4}}},$ (63)
$\displaystyle\expectationvalue{\varepsilon(r_{1})\varepsilon(r_{2})}$
$\displaystyle=\frac{1}{\left|r_{1}-r_{2}\right|^{2}}.$

In terms of state-operator correspondence, the OPE coefficient can also be
written as:

$\ket{\mathcal{O}_{2}\mathcal{O}_{3}}=\frac{C_{231}}{r_{23}^{\Delta_{2}+\Delta_{3}-\Delta_{1}}}\ket{\mathcal{O}_{1}}+...,$
(64)

where $\innerproduct{\mathcal{O}_{i}}{\mathcal{O}_{i}}$ is normalized to 1,
and … denotes operators other than $\ket{\mathcal{O}_{1}}$, such as the
descendent states of $\mathcal{O}_{1}$ like $\partial\ket{\mathcal{O}_{1}}$
and the other operators $\mathcal{O}_{j}$ which also has a non-vanishing three
point function with $\mathcal{O}_{2}\mathcal{O}_{3}$.

##  Appendix D Four-point Function of 2D Ising CFT

Here we briefly conclude the calculation of the four-point function of Ising
CFT, following the lecture notes in Ref. [51]. For simplicity, we only
consider the CFT on a plane.

In a CFT, $N$-point correlation functions can be completely determined from
the “CFT data”, which are the scaling dimensions $\Delta_{\mathcal{O}_{i}}$
and the OPE coefficients $C_{\mathcal{O}_{i}\mathcal{O}_{j}\mathcal{O}_{k}}$.
In particular, one uses Eq. (64) to replace the product of two operators into
the sum of single operators and iteratively convert the problem to the problem
of calculating $(N-1)$-point functions.

By fusing the first and second pair of operators, the four-spin correlation
function in 2D Ising CFT can be written as:

$\expectationvalue{\sigma(x_{1})\sigma(x_{2})\sigma(x_{3})\sigma(x_{4})}=\sum_{\mathcal{O}=\mathbb{1},\varepsilon}\frac{C_{\sigma\sigma\mathcal{O}}^{2}}{x_{12}^{2\Delta_{\sigma}}x_{34}^{2\Delta_{\sigma}}}g_{\Delta_{\mathcal{O}},l_{\mathcal{O}}}(u,v),$
(65)

where $\mathcal{O}=\mathbb{1},\varepsilon$ are the two families (primary and
its descendants) of operators that appear in the OPE of
$\sigma\crossproduct\sigma$, and $C_{\sigma\sigma\mathcal{O}}$ is the
corresponding OPE coefficient. In the above, $u$ and $v$ are the cross ratios
of the four points $x_{1}$,$x_{2}$,$x_{3}$,$x_{4}$, which captures the degree
of freedom for the points which is invariant under conformal transformation.
They can be further summarized by a single complex variable $z$, which is the
position of the fourth point when one transforms the first three points to
$0$, $1$, and $\infty$:

$\displaystyle
u=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}},v=\frac{x_{23}^{2}x_{14}^{2}}{x_{13}^{2}x_{24}^{2}},$
(66) $\displaystyle u=z\bar{z},v=(1-z)(1-\bar{z}),$ (67)

where $g_{\Delta_{\mathcal{O}},l_{\mathcal{O}}}(u,v)$ is the conformal block
for the operator family $\mathcal{O}$. $\Delta$ and $l$ are the scaling
dimension and the spin, respectively. ($l_{\mathbb{1}}$, $l_{\varepsilon}=0$).
$g_{\Delta_{\mathcal{O}},l_{\mathcal{O}}}(u,v)$ captures the correlation
function between the primaries and descendants in the OPE of the first and
second pair of operators onto family $\mathcal{O}$. In 2D, the conformal block
can be conveniently calculated using the hypergeometric function
${}_{2}F_{1}$:

$\displaystyle
g_{\Delta,l}(u,v)=k_{\Delta+l}(z)k_{\Delta-l}(\bar{z})+k_{\Delta-l}(z)k_{\Delta+l}(\bar{z}),$
(68) $\displaystyle
k_{\beta}(x)=x^{\beta/2}{}_{2}F_{1}(\beta/2,\beta/2,\beta,x).$ (69)

##  Appendix E Scaling Dimension from Transfer Matrix

For a cylinder of shape $W\times H$, when conformally mapped to an annulus of
inner radius $r_{1}=1$, the outer radius is:

$r_{2}=e^{\frac{2\pi H}{W}}.$ (70)

One can extract the scaling dimension of each eigenvector of the transfer
matrix as follow, where $\lambda_{i}$ are the eigenvalues of the transfer
matrix,

$\frac{\lambda_{i}}{\lambda_{0}}=\left(\frac{r_{2}}{r_{1}}\right)^{-\Delta_{i}}.$
(71)

##  Appendix F Fixing the degeneracy of conformal states

From linearized TRG $M$ we obtain its ’raw’ eigenvectors and eigenvalues:

$M\ket{u_{a}}=\lambda_{a}\ket{u_{a}}.$ (72)

Ideally, they should correspond to the conformal states:

$\ket{\mathcal{O}_{\alpha}}\in\\{\ket{\mathbb{1}},\ket{\sigma},\ket{\varepsilon},\ket{\partial\sigma},...\\}$
(73)

However, only from conformal dimensions one cannot different states which
share the same conformal dimension, such as $\partial_{x}\sigma$ and
$\partial_{y}\sigma$. Moreover, in practice, the truncation error from finite
bond dimensions, the anisotropy introduced by our HOTRG scheme, and the
truncation from GILT, had introduced various numerical errors. We have
observed that higher-level states with different conformal dimensions will
also mix with each other. Furthermore, if the lTRG matrix is not strictly
hermitian, the eigenvectors $\ket{u_{\alpha}}$ might not be orthogonal to each
other.

To calibrate the above issues, we first orthonormalize the eigenvectors using
the standard Gram-Schmidt process, where the eigenvectors with lower conformal
dimensions have higher priorities. The next step is to rotate the basis
$\ket{u_{a}}$ to build the conformal states $\ket{\mathcal{O}_{\alpha}}$:

$\ket{\mathcal{O}_{\alpha}}=\sum_{\lambda_{a}=\Delta}S_{\alpha a}\ket{u_{a}}.$
(74)

We use the scheme described in Table. 3 to build the lattice implementation of
conformal operators using the finite-difference method and operator fusion.
Note that $\ket{T_{\alpha}}$’s are the corresponding coarse-grained tensors
and they contain higher states in the conformal tower of
$\mathcal{O}_{\alpha}$. For example:

$\displaystyle\ket{T_{\partial_{x}\sigma(x_{0})}}$
$\displaystyle\sim\frac{1}{2\delta}(\sigma(x_{0}+\delta)-\sigma(x_{0}-\delta))\ket{0}$
(75)
$\displaystyle\sim\ket{\partial_{x}\sigma(x_{0})}+\frac{1}{6}\ket{\partial^{3}_{x}\sigma(x_{0})}+...$

The Gram matrix $S^{\prime}$ is the dot product between $\ket{u_{a}}$ and
$\ket{T_{\alpha}}$:

$S^{\prime}_{a\alpha}=\bra{u_{a}}\ket{T_{\alpha}}.$ (76)

We expect the block-diagonal part of $S^{\prime}$ will cancel the rotation
matrix, and the off-block-diagonal part is the higher terms that correspond to
finite-difference error like Eq. (75):

$S^{(\Delta)}=S^{\prime(\Delta),-1},$ (77)

where $(\Delta)$ means restricted on eigenspace
$\lambda_{\alpha},\lambda_{a}\approx\Delta$.

However, in practice, some states with different scaling dimensions, such as
$\Delta=2$ and $\Delta=2.125$, are mixed into each other, was well as into
eigenvectors of higher conformal dimensions. It is shown in Fig. 10. An easy
fix is to use a bigger block that both contains $\Delta=2$ and $\Delta=2.125$.
One might also include more eigenvectors and replace the submatrix inverse
with the submatrix pseudo-inverse. We find that we can assume all the
eigenvectors are somehow mixed and do the pseudo-inverse for the whole matrix
regardless of the theoretical degeneracy, will give the best result. So,
rather than “fixing the degeneracy of eigenvectors of lTRG using lattice
operators”, what we actually did is more like projecting the lattice operators
onto the low-energy (conformal dimension) subspace of lTRG.

##  Appendix G Renormalization of fixed-point tensors and extraction of OPE
coefficient

To extract the CFT data from the fusion rules of coarse-grained tensors, it is
tempting to identify the coarse-grained tensors with the ket vectors in Eq.
(64). From it, one may extract the OPE coefficient by comparing the coarse-
grained tensor of two defect tensors with the third one. However, proper
renormalization is required to get the correct result. It is due to the
following reasons.

1\. We want the coarse-graining operation, which maps the states on a square
of length $2a_{0}$ to states on the square of length $a_{0}$, to have the same
behavior as the scaling operator in CFT, when acting on defects, as
illustrated in Eq. (78) and Eq. (79). It requires the vacuum tensor
$T_{\mathbb{1}}$ to be properly normalized.

2\. The size of the coarse-grained block $a_{0}$ is related to the lattice
detail which we should be agnostic of when only the fixed-point tensor is
provided. One needs to reassign $a_{0}$ with a finite number (e.g. $a_{0}=1$)