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

The translation invariant tensor network is a natural way to describe the partition function of many statistical models and the wave functions of quantum systems, such as the Ising model on a square lattice [42] and the ground state wave function 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.

Eq. (1) below shows the tensor network representation of the vacuum partition function (or, in turn, the free energy via $F=-\frac{1}{\beta}\log Z$) of a lattice model on a square lattice with periodic boundary condition,

$$Z=\tr\vbox{\hbox{\includegraphics[page=1]{TRGCFTGraphics.pdf}}},$$

(1)

Eq. (2) below shows how to evaluate the two-point correlator,

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

(2)

where $\tr T=\vbox{\hbox{\includegraphics[page=57,scale={0.5}]{TRGCFTGraphics.pdf}}}$ means contracting opposite pairs of edges, according to the periodic boundary condition.

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 spacetime 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 the finite-size effect are negligible. This is generally called the real-space tensor renormalization procedure, as illustrated below. (We note that in the infinite system instead of the value of the partition function, we obtain the free-energy density instead.)

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

(3)

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.

One has to bear in mind that the truncation error introduces an artificial length scale to the coarse-grained system, which breaks the conformal symmetry. This issue will be demonstrated in Sec. III.2; for example, see Fig. 17. In this paper, we focus on a combination of methods using HOTRG and GILT [39]. The description of our coarse-grain scheme is given in Sec. II.7.

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.

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

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

(4)

where $\Delta_{\mathcal{O}}$ is the scaling dimension of the operator $\mathcal{O}$. Further information about CFT will be summarized in App. B and App. 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 change in the tensor is represented by a V-shaped curve after each RG step, as demonstrated 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 ¹¹1See 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 on 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 a CFT at the corresponding fixed point. As a result, the system can behave like different CFTs at corresponding length scales, depending 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 the OPE coefficients $C_{ijk}$ between them. With these, 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.

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 isometry $w$: $\mathcal{H}_{v}\rightarrow\mathcal{H}_{v}\otimes\mathcal{H}_{v}$, which specifies the subspace to keep after one step of coarse-graining, as illustrated in the equations below.

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

(5)

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

(6)

One can repeat the coarse-graining procedure horizontally and then 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 subnetwork as much as possible, as follows.

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

(7)

The detailed prescription of computing $w$ is described in Alg. A.1 (App. A.1).

Real-space RG methods suffer from accumulating 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 to faithfully represent 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 parameterized by corner-double-line tensors [36],

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

(8)

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 then slicing them apart again.

CDL tensors can be removed by 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 sub-network still looks the same from external legs, but the entanglement through a certain internal leg is reduced,

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

(9)

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. (10). We recall that $\chi$ is the bond dimension and $m$ is the ‘spacetime’ dimension of the lattice. It is essential to fix the gauge to compare the tensors of different RG steps component-wise; a gauge transformation on the four legs is illustrated in the equation below.

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

(10)

Ref. [39] introduced a rudimentary gauge-fixing procedure, which is based on the eigenvalue decomposition during the standard HOTRG procedure Eq. (38). Their method successfully yields a V-shaped curve ²²2 The Figure 9 in [39] does not show a clear V-shaped curve, indicating that the gauge is not completely fixed when comparing tensors at successive RG steps. However, the GitHub code they provide yields a clear V-shaped 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,

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

(11)

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 that 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 the prescription for obtaining the MFC in Alg. A.3 (App. 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 Alg. A.4 (App. A.4). With the implementation of MCF, we obtain a more consistent gauge fixing when comparing coarse-grained tensors between successive RG steps, and it produces a V-shaped curve; see the discussion in Sec. IV.

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

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

(12)

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,

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

(13)

where

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

(14)

and $s=2$ is the linear scaling of the system after two RG steps (which merge four sites by a vertical and a horizontal coarse-graining steps). Note that proper normalization is required; see App. G.

Ref. [39] shows that the scaling dimensions of the first few operators in the Ising CFT can be obtained 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 lTRG result with the scaling dimensions obtained from the transfer matrix method [36], as well as the known analytical results from conformal field theory [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. (12) have a clear physical meaning. They are the conformal states associated with the corresponding operators $\mathcal{O}_{\alpha}$. The conformal states are obtained by inserting conformal primaries and descendants into the vacuum state (see App. 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. At the lattice level, these states can be achieved by coarse-graining the tensor network Eq. (1) with the operator insertion as defect tensors:

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

(15)

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 onto the primary and descendant tensors or lTRG.

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

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 that 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].

To compute the $n$-point correlation function in Eq. (2), we iteratively apply the coarse-graining step Eq. (16) to each pair of tensors in all coarse-graining blocks, as shown in Eq. (18) below:

$$\tr\vbox{\hbox{\includegraphics[page=59]{TRGCFTGraphics.pdf}}}=\tr\vbox{\hbox{\includegraphics[page=60]{TRGCFTGraphics.pdf}}}=\tr\vbox{\hbox{\includegraphics[page=61]{TRGCFTGraphics.pdf}}}=\tr\vbox{\hbox{\includegraphics[page=62]{TRGCFTGraphics.pdf}}}=\tr\vbox{\hbox{\includegraphics[page=63]{TRGCFTGraphics.pdf}}},$$

(18)

where $\sigma_{1}$, $\sigma_{2}$ are two defect operators, and $:\sigma_{1}\sigma_{2}:$ means the two defects are fused together.

The coarse-graining of tensors is performed according to Eq. (16), in which we use the same set of coarse-graining tensors $w$, $g$, $h$ as when coarse-graining vacuum tensors. The symbol $\tr$ indicates contracting opposite pairs of edges, according to periodic boundary conditions. Note that the coarse-grained tensor is dependent on not only the type of defect but also the position of the defect relative to the coarse-graining block.

Similarly, when calculating the linearized TRG Eq. (12), on the other hand, we also keep the coarse-graining tensors $w$, $g$, $h$ as constants, that is, 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:

$$\vbox{\hbox{\includegraphics[page=18]{TRGCFTGraphics.pdf}}}=\vbox{\hbox{\includegraphics[page=64]{TRGCFTGraphics.pdf}}}+\vbox{\hbox{\includegraphics[page=65]{TRGCFTGraphics.pdf}}}+\vbox{\hbox{\includegraphics[page=66]{TRGCFTGraphics.pdf}}}+\vbox{\hbox{\includegraphics[page=67]{TRGCFTGraphics.pdf}}}$$

(19)

Eq. (19) above shows the variation of the coarse-grained tensor with the change in the input tensor, which is used to obtain the lTRG equation Eq. (12).

Faithful scaling dimensions can be computed from the linearized TRG with fixed $g$ and $w$, as shown in Sec. IV. However, in terms of $n$-point correlation functions, sometimes undesirable results are obtained if GILT is applied, which will be discussed in more detail later in Sec. IV.5.

After applying GILT [37] to remove local entanglement and performing the MCF [41] for gauge-fixing, we can identify a non-trivial 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].

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 purple circles in the figure). As the system approaches the critical point around step 30, the coarse-grained tensor approximately converges to a fixed point, 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, such as the 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-shaped curve in the logarithmic plot. Figure 3 also illustrates how the system evolves into distinct fixed points as the temperature is slightly perturbed. 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 being $T_{0,0,0,0}=T_{1,1,1,1}$) before eventually flowing to a tensor of the form $T_{0,0,0,0}=0$, due to spontaneous symmetry breaking caused by perturbations from numerical errors, since we did not enforce the $Z_{2}$ symmetry during 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 virtual indices Eq. (10)). The tensor components at the three fixed points are shown in Fig. 4.

The numerical scheme for the critical temperature $T_{c}(\chi)$ is obtained using a bisection algorithm, as suggested in [39]. We assume that there is an abrupt change in the coarse-grained tensors across the phase transition. The critical point can be identified by comparing the gauge-fixed coarse-grained tensors $T^{(l_{M})}$. The detailed procedure is described in Alg. J in App. J.

The curves in Fig. 3 exhibits more fluctuation and discontinuities in the range where $|T^{\prime}-T|/|T|$ is large. It is because when two tensors are more different from each other, it is more likely that the MCF fails to find the correct gauge to compare the two tensors. More specifically, a gauge fixing scheme could introduce discontinuities; that is, two physically nearly identical tensors are separated by the “cut” of a gauge choice, resulting in an O(1) inconsistent in their gauge-fixed form. This problem might be diagnosed by inspecting the level crossing incident in the SVD spectrum.

In Table 1, we compare the results of the scaling dimensions obtained by two different methods: (1) diagonalizing the transfer matrix on a cylinder and (2) solving the eigenvalues of lTRG Eq. (12). 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 higher values. However, we observe that the transfer matrix spectrum achieves more accurate information on higher conformal dimensions compared to the lTRG method (see their values for operators having $\Delta=3$ or higher).

The spectrum of the transfer matrix, $\Delta_{cyl}$, can be used to represent the quality of the coarse-grained tensor. The method with GILT enabled produces the correct scaling dimensions and their degeneracy number up to $\Delta=3$. Without GILT, the number of degeneracy errors starts from $\Delta=2$, and the error in the scaling dimensions is much larger.

The lTRG scaling dimension spectrum, $\Delta_{\text{RG}}$, is of greater interest to us, since we shall extend the approach to OPE coefficients. The method with GILT enabled does not provide the correct scaling dimensions from $\Delta=3$ and above, 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 $\Delta=1$ and $\Delta=1+\frac{1}{8}$. Furthermore, the GILT case has fewer errors at $\Delta=\frac{1}{8},1,1+\frac{1}{8}$.

For further comparison, we present the number of degeneracy in 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.4,

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,

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

(20)

where $...$ denotes states orthogonal to $\ket{\mathcal{O}_{1}}$.

From our coarse-graining scheme, we have that

$$\bra{T_{\varepsilon}}\mathcal{M}\ket{\begin{array}[]{cc}T_{\sigma}&T_{\sigma}\\ T_{\mathbb{1}}&T_{\mathbb{1}}\end{array}}=\frac{C_{\sigma\sigma\varepsilon}}{a_{0}^{2\Delta_{\sigma}-\Delta_{\varepsilon}}},$$

(21)

where $\ket{\begin{array}[]{cc}T_{\sigma}&T_{\sigma}\\ T_{\mathbb{1}}&T_{\mathbb{1}}\end{array}}$ is the 8-legged tensor from contracting the corresponding tensor sub-network, $\mathcal{M}$ is the coarse-graining procedure Eq. (16) that converts the 8-legged tensor to the 4-legged coarse-grained tensor. The inner product $\bra{T_{2}}\ket{T_{1}}=\vbox{\hbox{\includegraphics[page=58,scale={0.7}]{TRGCFTGraphics.pdf}}}$. $\ket{T_{\mathbb{1}}}$ and $\ket{T_{\sigma}}$ are renormalized so that:

$$\mathcal{M}\ket{\begin{array}[]{cc}T_{\mathbb{1}}&T_{\mathbb{1}}\\ T_{\mathbb{1}}&T_{\mathbb{1}}\end{array}}=\ket{T_{\mathbb{1}}}.$$

(22)

$$\mathcal{M}\ket{\begin{array}[]{cc}T_{\sigma}&T_{\sigma}\\ T_{\mathbb{1}}&T_{\mathbb{1}}\end{array}}=\frac{C_{\sigma\sigma\mathbb{1}}}{a_{0}^{2\Delta_{\sigma}}}\ket{T_{\mathbb{1}}}+...$$

(23)

The detailed scheme is further explained in Appendix G. Note that in practice, we place defect tensors on different combinations of sites in the coarse-graining block and average over the resulting renormalization coefficients and the OPE coefficients.

The result of the OPE coefficients of Ising CFT 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 in Sec. III.5. However, the results there are not as accurate.

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. Figure 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.

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 absolute positions for most cases. This indicates that the translational symmetry is preserved and the $SO(2)$ rotational symmetry has emerged at large distances.

To calculate the two-point function using coarse-grain tensors. We place two defect tensors $T_{\sigma(x_{1},y_{1})}^{(0)}$, $T_{\sigma(x_{2},y_{2})}^{(0)}$ on a $L_{x}\times L_{y}$ square lattice tensor network with periodic boundary conditions, where the tensors at the other sites are the vacuum tensor $T^{(0)}$. Then we coarse-grain every adjacent pair of tensors as described in Sec. II.8. Then the coarse-grained vacuum tensor $T^{(l+1)}$ and the coarse-grained defect tensors $T^{(l+1)}_{\sigma(x_{1},y_{1})}$, $T^{(l+1)}_{\sigma(x_{2},y_{2})}$ are obtained. Furthermore, when two defects are met in the same coarse grain block, they fuse into one defect tensor $T^{(l+1)}_{\sigma(x_{1},y_{1})\sigma(x_{2},y_{2})}$.

At the final coarse-graining step, all defects fuse into one tensor $T^{(l_{M})}_{\sigma(x_{1},y_{1})\sigma(x_{2},y_{2})}$. The correlation function is computed using the following

$$\expectationvalue{\sigma(x_{1},y_{1})\sigma(x_{2},y_{2})}=\frac{\tr T^{(l_{M})}_{\sigma(x_{1},y_{1})\sigma(x_{2},y_{2})}}{\tr T^{(l_{M})}},$$

(24)

where $\tr T=\sum_{i,k}T_{iikk}=\vbox{\hbox{\includegraphics[page=57,scale={0.5}]{TRGCFTGraphics.pdf}}}$ indicates contracting the opposite pairs of legs of the tensor.

We used 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 the 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,

	$\displaystyle\langle\sigma(x_{0},y_{0})\sigma(x_{1},y_{1})\rangle$	$\displaystyle=A\frac{e^{-x/\zeta}}{x^{2\Delta^{\prime}}}+m_{0}^{2},\,T<T_{c},$		(25)
	$\displaystyle\langle\sigma(x_{0},y_{0})\sigma(x_{1},y_{1})\rangle$	$\displaystyle=A\frac{1}{x^{2\Delta}},\,T=T_{c},$		(26)
	$\displaystyle\langle\sigma(x_{0},y_{0})\sigma(x_{1},y_{1})\rangle$	$\displaystyle=A\frac{e^{-x/\zeta}}{x^{2\Delta}},\,T>T_{c},$		(27)

where $\Delta(T)$ is the scaling dimension of the magnetization operator fitted at the corresponding temperature. $\zeta(T)$ is the correlation length, $m_{0}=\expectationvalue{\sigma}$ is the magnetization in the ordered phase, $\Delta^{\prime}(T)$ is the “scaling dimension” of $\sigma-\expectationvalue{\sigma}$, where $\expectationvalue{\sigma}=(L_{x}L_{y})^{-1}\sum_{x,y}\expectationvalue{\sigma(x,y)}$ is the one-point function averaged at every lattice site.

Regarding the outliers in the data, we adopt Huber loss Eq. (98) (explained in App. I) with $\epsilon=10^{-2}$ (in log-log space) to suppress their excessive effects. Since the fitting is done in the log-log space, we also give the data points a weight that is proportional to the magnitude of its correlation.

When $T\approx T_{c}$, we obtain $\Delta=0.1266$; when $T=T_{c}+5\times 10^{-5}$, we obtain $\Delta=0.1293$ and $\zeta=1.70\times 10^{4}$; when $T=T_{c}-5\times 10^{-5}$, we obtain $\Delta^{\prime}=0.1652$, $\zeta=1.62\times 10^{3}$, and $m_{0}=0.313245$.

The three correlation functions behave in a similar way at short distance $x<10^{3}$. By fitting with Eq. (26), we get $\Delta=0.1251,0.1272,0.1300$ for low, critical, and high temperatures, respectively. The above results agree well with each other 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 positions of the correlated pairs, due to an issue in the coarse-graining of impurity operators. Also, Huber loss is used to take into account the outliers. We elaborate on these issues later in Sec. IV.

The essential data for 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. (74). Thus, we 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 of the four-point correlation function 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. (74) for the fitting and thus extract the CFT data from our numerical results. We use the Huber loss Eq. (98) with $\epsilon=0.1$ to fit the curve in the logarithmic space, where the input data are 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 CFT and give another confirmation of our previous results using lTRG.

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.

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

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

(28)

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. (27) from both sides. This is because the coarse-grained tensors we use had already flowed away from the conformal fixed point on 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 in $x\rightarrow L_{x}$ seemingly contradicts the purpose of the coarse graining in HOTRG: removing microscopic details. The peak is the result of correlations between two point-like defects. However, in the case $x\rightarrow L_{x}$, the two defects meet in 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 a sharp peak can be obtained when $x\rightarrow L_{x}$ indicates that the truncated Hilbert space of the coarse-grained tensor still captures the modes in which the excitation is sharply localized on the boundary of the coarse-grained regions, as described in Sec. IV. Figure 8 further illustrates that those point-like localized modes are kept throughout the entire RG flow.

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. (12)) at the fixed point correspond to the 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. (12) near the conformal point, as shown in Fig. 9.

From there we observe that the coarse-grained vacuum tensor $T_{\mathbb{1}}$ lies in the eigenspace with conformal dimension 0, which corresponds to the identity operator $\mathbb{1}$. The coarse-grained tensor with an insertion of one operator $T_{\sigma_{x}}$ lies in the 0.125 conformal dimension eigenspace, 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 in 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 to 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.

Figure 10 displays the projection of the conformal descendants realized in the lattice 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 $\Delta=2.125$.

However, we note that the order of the eigenvalues of the eigenvectors with the most significant overlap 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 2.120, which shows a great discrepancy with the overlap of $\partial_{y}\varepsilon$ between 1.998 and 2.003. Moreover, the most significant overlaps of the next two operators, $\partial_{x}^{2}\sigma$ and $\partial_{y}^{2}\sigma$, are at the scaling dimension 2.045, which is smaller than 2.120. So the order of the scaling dimensions of some operators is wrong, and operators at different degeneracy subspaces 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 distinguishable 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 will be discussed in the next subsection.

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. (29). However, the unexpected behavior of projections in $T_{\partial_{y}^{2}\sigma}$ might be a result of a 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. Therefore, one can conclude that the coarse-graining process erroneously removes the information that is crucial to distinguish the two operators. This results in the unexpected value $\partial_{y}^{2}\sigma$ 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 the result that GILT artificially suppresses defects near the edges and corners of the coarse-grained block, which will be discussed in Sec. IV.8.

We do not directly compare $T_{\sigma_{x,0}}$ with 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 substantially 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.

Figure 12 compares the eigenvectors of the transfer matrix Eq. (30) and the lattice state associated with the placement of a defect at $(r_{0},\theta)$. The lattice state is obtained by 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 in the first tensor. At $\theta\geq\pi$, the defect is in the second tensor.

In Fig. 12, one can see that the eigenvector of the scaling dimension $\Delta\approx 0.125$, which corresponds to $\sigma$, is approximately constant in expansion Eq. (31); the eigenvectors of the scaling dimension $\Delta\approx 1.125$, which correspond to $\partial\sigma$ and $\bar{\partial}\sigma$, have a sine and cosine coefficients in the expansion; similarly, eigenvectors associated with $\Delta\approx 2.125$ have coefficients $1$, $\sin(2\theta)$ and $\cos(2\theta)$, respectively. This observation confirms that, as in lTRG, the eigenvectors of the transfer matrix can also be seen as conformal states.

The use of GILT, as discussed in Ref. [37] and briefly reviewed in Sec. II.4, is critical to ensure the stability of the RG flow. As shown 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. On the contrary, 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 spectrum decomposition (see App. F). The use of MCF furthermore increases stability. As shown in Fig. 13, with GILT and without MCF, the stability of our RG flow is comparable to Ref. [39]. With the introduction of MCF, the stability of our RG flow has improved further.

Figure 14 illustrates the scaling dimensions and central charge extracted from the spectrum of the transfer matrix of the coarse-grained tensors. The method is described in App. E. 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.

At the critical point, but with a finite-size lattice, the extracted scaling dimension is subject to both finite-size and finite-bond-dimension effects, as depicted in Fig. 15. At the beginning of the RG flow, the shift in the scaling dimension $\Delta-\Delta_{\text{CFT}}$, as estimated from the transfer matrix, scales proportionally to $L^{-2}$, where $L$ is the size of the system. This behavior aligns with the expectations for the finite-size effect, as discussed in Ref. [49]. However, the shift estimated from the lTRG method is $L^{-1}$.

During the RG flow, when the system reaches a relatively stable state in the “valley”, the shifts obtained from both the transfer matrix and the lTRG method exhibit a scaling behavior of $L^{0.08}$ and $L^{0}$, respectively. As the system deviates from the RG fixed point, both methods produce shifts that scale as $L^{1.5}$.

In contrast to the findings in Ref. [49], our observations do not reveal the emergence of perturbations such as $\epsilon$ (scaling as $L$) and $\sigma$ (scaling as $L^{3.75}$) at larger length scales due to finite-$\chi$ effects. This discrepancy may be attributed to the difference that Ref. [49] employed the Loop-TNR method, whereas we utilized the HOTRG+GILT scheme.

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 the CDL tensor network, we expect $s^{(odd)}_{i}\approx s^{(even)}_{i}$. Figure 16 shows the oscillation of the 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 the capability of GILT to remove CDL tensors.

The scaling dimensions error obtained from the transfer matrix and from the lTRG method is summarized in Fig. 17. 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).

However, 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 on which the lTRG equation is based, 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 floating-point errors or to the imprecision of the critical temperature we found. Note that due to the finite dimension of the bonds, 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.

Most prior use cases in HOTRG and GILT have 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 insertions $w$, $g$, and $h$, 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:

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

(32)

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. (32) 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. (7) holds if both $T_{1}$ and $T_{2}$ are vacuum tensors. Therefore, 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 the HOTRG coarse grain 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. (9) holds only when the tensors $ABCD$ in the subnetwork Eq. (9) are all vacuum tensors. One may ask whether the Hilbert space that GILT truncates is the subspace corresponding to the defect tensors. This conjecture is supported by the results shown in Fig. 21 showing that 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. On the contrary, 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.

Figure 18 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 artificially suppressed by GILT, as discussed in Sec. IV.8.

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

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. (33) below shows a type of regularized two-point function, where the point-like defect is replaced by a Gaussian profile Eq. (34), as illustrated in Fig. 19. In the log-log graph, Eq. (33) features a flat line, which indicates the smeared tip of the correlation function, followed by a negative slope, which indicates the unsmeared part of the correlation function, which follows the power law Eq. (35),

	$\displaystyle f_{\text{smeared}}(x,y)$	$\displaystyle=\int\int\int\int dx_{1}dy_{1}dx_{2}dy_{2}$		(33)
	$\displaystyle G(x-x_{1},y-y_{1})$	$\displaystyle G(x-x_{2},y-y_{2})P(x_{1}-x_{2},y_{1}-y_{2}),$

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

(34)

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

(35)

In HOTRG, the fusion of two defects occurs when two coarse-grained tensors, each with one defect in its region, are coarse-grained into a larger tensor. As illustrated in Fig. 20, the two point defects $(2,2)$ and $(4,6)$ meets in a $8\times 8$ block at level $l=6$. Naturally, one would expect that the amount of smearing that the two defects suffer is related to the block size of the largest coarse-grained tensor before the two defects meet.

Figure 21 illustrates how the size of the coarse-grained block, $l_{x}\times l_{y}$, just before two defects fuse, regulates the short-distance behavior of the 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 to illustrate the smearing effect.

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

Figures 22 and 23 further demonstrate the trend of the smeared correlation function for a given coarse grain 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. Together, the two trends confirm the smearing picture described in Eq. (33). Note that there are also data scattered between the two trends, as we have already seen in Fig. 21.

The first trend resembles the exact, unsmeared correlation function. In Fig. 21, 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 smearing or that the smearing radius is much smaller than the separation $r$.

The second trend resembles the smeared correlation function, which is a flat line that clipped the tip of the first trend at short distances. The flat line indicates that 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 $r_{s}$. The smearing radius $r_{s}$ is estimated by the intersection of the first and the second trend.

The smearing radius $r_{s}$ increases with the increase of the coarse grain block size ($l_{x}\times l_{y}$) before two defect tenors merge, as shown in Fig. 24 and Fig. 25. However, the smearing radius grows faster than the growth of the block size: $r_{s}\propto l_{x}^{\delta_{\text{LS}}}$, with $\delta_{\text{LS, No GILT}}=1.20$ and $\delta_{\text{LS, GILT}}=1.61$. This indicates that as the coarse-graining iteration proceeds, the smearing radius will catch up with the lattice size and may result in errors in further coarse-graining.

Now, we compare 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. This means that 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 follow more precisely to the fitted line, while the data with GILT have a more significant error with a much lower value. This means that GILT also removes some of the information on the long-range behavior of the defects.

The data analysis in this subsection confirms that the defect is smeared after coarse graining. Without GILT, defects in certain places might not be smeared as much. With GILT, most defects are smeared to 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 defects are being missed from smearing in the non-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 aforementioned special place, but further numerical evidence is required to support this conjecture.

Finally, we provide an explanation for why naive averaging in 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. 23, to obtain the unsmeared two-point function, we require $r>r_{s}$. When points are randomly sampled, it is most likely 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 at the corner or edge of the largest unfused block, which is a very unlikely scenario. Thus, in Fig. 23, we intentionally increase the probability that $r$ is inside the smearing radius, so we can observe the reduction in correlation function due to smearing, and the fitted scaling dimension has a larger error.

However, in Fig. 5, when the points are randomly sampled, most of them satisfy $r>r_{s}$, which guarantees that smearing occurs only on 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.

When GILT is not applied, the dominance of the CDL tensor suggests that the truncated Hilbert space of the coarse-grained tensor mainly captures 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.

In particular, we compare the torus correlation function at $x\rightarrow 0$ and $x\rightarrow L_{x}$. As shown in Fig. 27, the correlator at $x\rightarrow L_{x}$ is 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. 26 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 where GILT is not used and the two defects are localized at the corner of the block.

It suggests that the defect is not 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. 28c,d, the smearing effect is stronger when the two defects are on the midpoint of the opposite edges. Given the finiteness of the dimension of the truncated Hilbert space, it suggests that the defects are 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. 28a,b. Note that there are some inconsistencies in the correlation function when the defects are located in the corners. It suggests that GILT introduces some artificial suppression for corner excitation.