We use this term inclusively, not limited to the first TRG algorithm proposed in Ref. [1], but to all RG maps manipulating tensor networks, whether or not they involve disentangling steps or only coarse graining.

Although significant progress was recently reported in Ref. [15].

We are aware of only one previous work [20] which tried to apply the Newton method to locate tensor RG fixed points. Their search failed beyond bond dimension $\chi =8$, for reasons which were not clearly identified at the time but likely due to a combination of not recognizing the importance of eliminating the stress-tensor eigenvalue 1 and working with a tensor RG algorithm without disentangling.

This is the term we propose hoping that it will stick, as there is no generally accepted term.

This is because Hilbert-Schmidt tensors can be contracted along one bond: The result is finite and Hilbert-Schmidt. This implies that the partition function of an $N\times M$ tensor network built of Hilbert-Schmidt tensors is finite [17], while this property is not guaranteed for tensors which are not Hilbert-Schmidt.

Recently, there were claims of exact tensor network RG fixed points constructed in Refs. [32, 33, 34]. We find some aspects of those works questionable. In particular, it has not been verified that the proposed exact fixed-point tensors remain Hilbert-Schmidt when the cutoff is removed. See Ref. [35], where we also provide an argument that tensor RG without disentangling cannot have an exact RG fixed point which is Hilbert-Schmidt and corresponds to a nontrivial CFT.

The critical temperature $T_c$ may be exactly known if the lattice model is exactly solvable, like the nearest-neighbor 2D Ising. In practical calculations at finite $\chi$, the critical temperature shifts a bit from the exact one. To find the shifted critical temperature, one may use the bisection method; see Ref. [37].

As mentioned in Ref. [36], the critical temperature shifts from the exact $t_c=1$ when working at finite bond dimension. To locate the finite-$\chi$ critical temperature, one uses the bisection algorithm: (i) Start from a “bracketing interval” $I=[t_l,t_h]$ such that $t_l$ is in the low-temperature phase and $t_h$ is in the high-temperature phase. (ii) Test the midpoint of the interval $I$—which phase is it in? This is done by running tensor RG for several steps until the tensor starts approaching the high-temperature or lower-temperature fixed points, which can be noticed by monitoring the singular values computed as in Eq. (3.6). (iii) Bisection: Depending on the outcome of (ii), replace the interval $I$ by its lower or upper half $I^{\prime }$ whose end points are in different phases. (iv) Repeat the above three steps until the bracketing interval has length smaller than the required accuracy $\mathrm{\Delta }t$.

This standard terminology may be a bit confusing, since temperature is a microscopic variable and cannot be varied directly around $A_{*}$. What is meant is that the eigenvector arises from the temperature perturbation of the microscopic tensor $A^{(0)}(T_c)$ after many RG steps.

For prior discussions of relations between tensor RG and CFT, see, e.g., Refs. [21, 22, 24, 40]. Aspects related to rotation needed for us were not discussed in those references.

Perturbing by the unit operator $\mathcal{O}=\mathbb{1}$ simply rescales the partition function. In tensor RG, this would translate to rescaling $A$. Since we fixed the normalization of $A$ [Eq. (2.4)], such a perturbation is not present among the $\nabla R$ eigenvectors.

In the standard RG parlance, perturbation by a total derivative is classified as “redundant” [26].

By spatial symmetry, we mean point group symmetry. Lattice translational symmetry is always present and is left implicit. For unequal horizontal and vertical nearest-neighbor couplings $J_x\ne J_y$, spatial symmetry gets reduced to a smaller group. The final spatial symmetry depends on whether the reduction of the spin model to a tensor network is performed rotating by $\pi /4$ not. In this paper, we study the case $J_x\ne J_y$ with $\pi /4$ rotation; see Appendix pp2.

The isotropic TNR scheme from Appendix C in Ref. [28] would preserve rotation symmetry. This complicated map acts on networks built of three different tensors. To our knowledge, it has not been implemented in practice.

If we used a tensor RG algorithm preserving one or both axis reflections, such as Refs. [7, 10, 46, 47], we could project out the second perturbation in Eq. (2.9), but the first one would still be present.

The earliest tensor RG map [1] did change the orientation; however, the point we make—that this eliminates eigenvalues 1—did not play any role in that pioneering paper, which was focused on other issues.

We do not know of any interesting model where operators of spin $\ell =4,8,\dots$ have dimension 2. In reflection positive models, they should have dimension at least $\ell$ by the unitarity bounds.

Note that we use superscripts to number RG iterates (2.5) while subscripts for Newton iterates.

Below, when extracting the CFT spectrum from the linearized RG map, we also consider the Jacobian in the ${\mathbb{Z}}_2$-odd subspace.

On the contrary, if we try to apply this to $R$ instead of $R^{\circ }$, $\nabla R$ has eigenvalues 1, $J$ is not invertible, and the Newton method fails.

In particular, for the bond dimension $\chi =30$ used in our numerical calculations, just storing full $J(A)$ would require around a terabyte of RAM (assuming we use 64-bit floating point numbers). Such memory requirement is far beyond our computational resources.

We thank Clément Delcamp for explaining this point and for passing to us a minimal working version of the original Gilt-TNR code [58] where the described strategy is realized.

To compute the distance between tensors $A^{(n)}$ with $n\le 10$, we applied additional relative discrete gauge fixing (akin to that used in Ref. [21]), as using $I_{+}$ evaluated at the tensor approximating the critical one was not enough to bring tensors close to each other (see Appendix pp4).

Except for a blip around step 11, where a small irregularity is also seen in the singular values (Fig. 4).

For the trajectory originating in the nearest-neighbor Ising model, operator $T\overline{T}$ is known to be fine-tuned to zero [40, 63, 64, 65].

See Remark 3.1. Apart from $\chi =30$, the fixed point $A_{*}^{[30]}$ also depends, of course, on other parameters of Gilt-TNR such as ${\epsilon }_{\mathrm{gilt}}$, but we leave this dependence implicit.

One can also see from Table 3 that the horizontal reflection is broken by a smaller amount than rotations and the vertical reflection. For example, in column 2, we have $A_{1^{+}{1}^{+}{1}^{-}{1}^{-}}\approx A_{1^{-}{1}^{+}{1}^{+}{1}^{-}}$, $A_{1^{-}{1}^{-}{1}^{+}{1}^{+}}\approx A_{1^{+}{1}^{-}{1}^{-}{1}^{+}}$ (horizontal reflections) hold to a much better accuracy than say $A_{1^{+}{1}^{+}{1}^{-}{1}^{-}}\approx A_{1^{-}{1}^{-}{1}^{+}{1}^{+}}$ (vertical reflection). A likely reason for this is given in Ref. [68].

That $Q_1$ and $Q_3$ are chosen before $Q_2$ and $Q_4$ is the likely reason for why the horizontal reflection symmetry is broken weaker than the vertical one [67].

For $\chi =10$, 20, we follow the same procedure as for $\chi =30$. First, we find approximate $t_c$ by bisecting to ${10}^{-10}$ and looking for the longest plateau in singular values as in Fig. 4. We find $t_{\chi =10}=1.0012778632$ (${\epsilon }_{\mathrm{gilt}}={10}^{-4}$) and $t_{\chi =20}=1.0000618664$ (${\epsilon }_{\mathrm{gilt}}=2\times {10}^{-5}$). Then, we produce a plot of Hilbert-Schmidt distances between subsequent gauge-fixed tensors, as in Fig. 5. The approximate fixed-point tensor is defined as $A^{(n)}$ corresponding to the minimum of this plot.

Other tensor RG algorithms involving disentanglers, such as, e.g., TNR [7], also involve iterative optimization of disentanglers, to minimize a cost function expressing a truncation error. It would be interesting to see if those algorithms are, or may be made, continuous or differentiable in a small neighborhood of the critical fixed point. To our knowledge, this has never been tested carefully.

A tensor $A$ is ${\mathbb{Z}}_2$-even or ${\mathbb{Z}}_2$-odd if $A_{ijkl}{s}_i{s}_j{s}_k{s}_l=\pm A_{ijkl}$, where $s_i$, $s_j$, $s_k$, and $s_l$ are the quantum numbers of the four legs and the sign $\pm$ is $+$ for even and $-$ for odd.

Computing $O(10)$ largest eigenvalues and the corresponding eigenvectors of $\nabla R$ by this method requires $O(100)$ calls of the RG map and for $\chi =30$ takes a few minutes on a laptop.

Note that the Jacobian matrix is real but, in general, not symmetric, and so some eigenvalues come in complex-conjugate pairs.

Other works [9, 24] said they could reach an approximate fixed-point tensor by gauge fixing, but without a plot like Fig. 5 the accuracy is hard to know. Reference [59] quantified the rate of convergence to the fixed point by plotting the norm difference of the vectors of singular values for consecutive tensors as a function of the RG step but without performing gauge fixing.

The change in the RG map can be inferred from the linearization expression [Eq. (40) in Ref. [21] ] and is also stated explicitly in Ref. [22] below Eq. (18).

Approximate values of $t_c(a)$ were obtained by the bisection method as explained after Eq. (3.6) with $\mathrm{\Delta }t={10}^{-10}$.

For all values of $a$ in Fig. 8, there is a pair of complex-conjugate eigenvalues in the lower part of the plot. For most values of $a$, the pair is the 9th and 10th eigenvalues, but for a few values of $a$ it is the 8th and 9th. For complex eigenvalues, we plot their absolute value. Since the two eigenvalues in this pair have the same absolute value, the result is an “$\times$” and a “$+$” right on top of each other.

Tensor $A^{(n^{\prime })}$ in this section should not be confused with tensor $A^{(n_{*})}$ in Sec. 3b. They are different tensors because the RG map is different. Also, $n_{*}=23$ in Sec. 3b, which was chosen to minimize $\parallel R(A^{(n)})-A^{(n)}\parallel$, while here we choose $n^{\prime }=23$ simply as a sufficiently large number.

We thank Naoki Kawashima for a discussion.

Sometimes, when the lattice model is translated to a tensor network representation, the spin lattice is rotated by $\pi /4$ with respect to the tensor network, as in Appendix pp2. The lattice model then has somewhat unusual periodic boundary conditions. This microscopic detail does not play a role in our paper.

Because of the lattice rotation by $\pi /4$, the equality requires nonstandard periodic boundary conditions on the Ising side, assuming the usual periodic boundary conditions on the tensor network side.

The critical temperature is given implicitly by $\sinh (2J_y/T_c)\sinh (2J_x/T_c)=1$.

The Arnoldi method requires $O(s)$ evaluations of directional derivatives $\nabla R^{\circ }.v$, each of which takes $O({\chi }^6)$ with our finite difference approximation method.

The linear solver returns one solution and, optionally, the kernel of the matrix, corresponding in our case to vectors that realize symmetries discussed before Eq. (d4).

In practice, this need arises when evaluating the Jacobian eigenvalues around the fixed point in the ${\mathbb{Z}}_2$-odd sector. This requires computing $R(A_{*}+\delta A)$ for small ${\mathbb{Z}}_2$-odd perturbations $\delta A$.

As a matter of fact, this observation also applies to Gilt-TNR used to our work. In fact, we have checked that the fixed point is reached even if the continuous gauge-fixing step is omitted in our algorithm. Nevertheless, we decided to keep the continuous gauge-fixing step, since it is not expensive. Also, having this step prompted us to reflect more deeply why the orthogonal gauge fixing is actually sufficient in our algorithm (Appendix pp4-s3).

By care we mean, e.g., the need to solve the CDL problem. By luck we mean, e.g., that it is hard to a priori exclude such situations as cycles of finite length. For example, discrete dynamical systems are known to generically undergo period-doubling bifurcations.

We believe the statement to the contrary below Eq. (B5b) in Ref. [21] to be incorrect.

We obtain the eigenvector using the Arnoldi method. For this, we approximate $\nabla R.v$ using finite difference formula Eq. (e1) with $h={10}^{-4}$.

While the basis vectors $|0⟩$, $|1⟩$, and $|2⟩$ are complex, one may show that the tensor $A_{3P}(t)$ is real due to $S_3$ symmetry of $A$. The initial tensor is invariant under an $S_3$ action, and our RG map preserves this symmetry.