Irregular parameter dependence of numerical results in tensor renormalization group analysis

We study the parameter dependence of numerical results obtained by the tensor renormalization group. We often observe an irregular behavior as the parameters are varied with the method, which makes it difficult to perform the numerical derivatives in terms of the parameter. With the use of two-dimensional Ising model we explicitly show that the sharp cutoff used in the truncated singular value decomposition causes this unwanted behavior when the level crossing happens between singular values below and above the truncation order as the parameters are varied. We also test a smooth cutoff, instead of the sharp one, as a truncation scheme and discuss its effects.


I. INTRODUCTION
Tensor renormalization group (TRG) is a promising approach that can solve the sign problem inherent in the Monte-Carlo simulations. Since it was proposed in two-dimensional Ising model [1], many studies have been carried out for various models of lattice field theories [2][3][4][5][6][7][8][9][10][11][12][13]. In TRG, the truncated singular value decomposition (SVD) is used to define a coarsegrained tensor, which is given in a manner of sharp cut-off such that the D cut largest singular values and corresponding singular vectors are kept and the others are thrown away. Although the cutoff yields possible systematic errors, it is expected that the result should converge to the correct value as D cut increases.
The results of TRG, however, do not smoothly depend on the parameters in the theory.
They often show irregular behavior at some parameters off the critical point. This behavior is controlled by D cut , but for small D cut it is difficult to obtain the smooth parameter dependence of the results, to which we may apply the numerical derivative with respect to the parameter. We can of course obtain a satisfactory result for a simple model such as twodimensional Ising model taking a sufficiently large D cut to avoid such misbehavior. However, it would be difficult to increase D cut for general lattice theories with multi-dimensional fields so that it should be important to understand and avoid the irregular behavior of the results.
In this paper we investigate the origin of the irregular parameter dependence shown in the TRG results. We present some numerical evidence that it is caused by the level crossing between singular values within and beyond the sharp truncation as the parameters are varied. In this sense the irregular behavior is inevitable for the TRG method with the sharp cutoff. In order to obtain a hint of improving the behavior, we also test other cutoff schemes such as a smooth cutoff.
The rest of this paper is organized as follows. In Sec. II we review the standard TRG method with the sharp cutoff in two-dimensional Ising model with sample numerical results.
The mechanism of irregular behavior is explained in detail with some numerical evidence in Sec. III. We also test other cutoff schemes. Our conclusions are summarized in Sec. IV.

II. TRG IN TWO DIMENSIONAL ISING MODEL
We briefly review the TRG method in two-dimensional Ising model presenting a couple of numerical results for later convenience.

A. Numerical procedures in TRG
We consider a two-dimensional square lattice whose sites are labeled by n = (n 1 , n 2 ) for n 1 , n 2 ∈ Z. The spin variable σ n assigned on the site n takes the discrete values σ n ∈ {1, −1}. Two dimensional Ising model is then defined by the Hamiltonian where i, j denotes possible pairs of the nearest neighbor sites and J is the coupling constant.
The partition function Z = Tr e −βH with the inverse temperature β = 1/T can be expressed as a tensor network form: where for i, j, k, l = −1, 1.
Let us denote the bond dimension of T ijkl as N for the sake of argument. Note that the initial tensor of Eq. (3) is defined with N = 2. We apply the truncated SVD to T ijkl : where T ijkl is treated as a matrix with the column (ij) and row (kl) in Eq. (4) and a matrix with the column (li) and row (jk) in Eq. (5). The above expressions assume the case of N 2 > D cut , while D cut in Eqs. (4) and (5) is replaced by N 2 for N 2 ≤ D cut without any truncation. We apply the decomposition of Eq. (4) to the tensors at even sites defined by mod(n 1 + n 2 ,2)=0 and that of Eq. (5) to ones at odd sites with mod(n 1 + n 2 ,2)=1.
Here U, V, U ′ , V ′ are unitary matrices and λ m and λ ′ m are singular values that are sorted in descending order.
We immediately find that the expression of Eq. (2) is approximated as where Note that the number of tensors decreases because an old tensor is decomposed into two unitary matrices U and V (or U ′ and V ′ ) and then four unitary matrices are assembled into a new tensor.
After repeating the above procedures, the tensor network is finally reduced to a single tensor. Taking the appropriate trace for its indices we obtain the approximate value of Z with D cut . The numerical cost of this algorithm is O(D 6 cut ) which comes from the computations of Eqs. (4) and (5).

B. Numerical examples with TRG analysis
TRG is a powerful tool to study two-dimensional lattice models. Although the exact value is obtained in the D cut → ∞ limit, we can reach a sufficient level of accuracy with moderate number of D cut in practical computations. We present a couple of representative results in the TRG analysis for two-dimensional Ising model on V = 2 16 × 2 16 lattice as a preparation of our study explained in the following section.
The numerical value of the partition function Z is obtained by repeating the renormalization step of TRG with a given value of D cut . Then the Helmholtz free energy F is also obtained with the use of F = −T log(Z). The critical temperature T c is determined from the peak position of the specific heat C V obtained by the numerical derivative of Z with respect to β as C V = −β 2 ∂ 2 ∂β 2 logZ. Figure 1 shows the temperature dependence of the free energy density. The black curve denotes the exact solution given in Ref. [14,15], and the black dotted line denotes the critical temperature. As clearly seen in the figure, the TRG results approach the exact solution as where u (m) ( v (m) ) is the left (right) singular vector corresponding to the m-th singular value IJ is an approximation of the tensor T ijkl for I = (i, j) and J = (i, j). We assume that a set of singular values and corresponding singular vectors (λ, u(λ), v(λ)) smoothly change under a local variation of parameters.
We now consider a case in which the level crossing takes place such that K-th and (K +1)-  Let us consider the following modification for the approximation of tensor at the final step of SVD: The meaning of this approximation is obvious from the definition. T (2) coincides with T (1) before the level crossing. After the level crossing, however, T (2) continues to keep the same sets (λ, u(λ), v(λ)) unlike T (1) . If the irregular behavior is caused by the change of the associated singular vectors, it is expected that the jump at T ref should vanish with the use of T (2) . Figure 5 shows the residues obtained from T (2) , which are drawn by the blue curve. They In order to define another truncation scheme, we introduce a weight function w m to approximate the tensor T ijkl : where U and V are unitary matrices and λ m are singular values sorted in descending order.
Note that Eq. (4) is given by the choice of the weight function, w m = 1 for m ≤ D cut . It can be expected that the crossover effect depends on w m and may become weaker if we employ a smoother cutoff function for w m . Note that the introduction of w m itself does not demand extra computational cost.
As possible choices of cutoff schemes we consider two types of weight functions: (A) "a slanting-cut" given by and (B) "a FDF-cut" inspired by by the Fermi distribution function m . ∆ in w (A) and σ in w (B) are the tunable parameters which basically give the smeared size of cutoff. In this paper we employ ∆ = 3 and σ = 1. Figure 7 shows the relative residues obtained by these two cutoffs, which show smoother temperature dependence compared to those in Fig. 3. It is confirmed that the smooth cutoff scheme is effective to tame the irregular parameter dependence found in the sharp cutoff scheme in the standard TRG method.

IV. SUMMARY
We have discussed the issue of the irregular parameter dependence observed in the TRG results. We have investigated its origin using the two-dimensional Ising model and concluded that the irregular behaviors is caused by the level crossing between the singular values in the sharp cutoff scheme with D cut . When the level crossing occurs between the D cut -th and (D cut + 1)-th singular values, D cut -th singular vector is replaced by the completely different one across the crossover point, though the D cut -th singular value changes continuously as a function of the parameter. Thus the constructed tensor drastically changes and yields the jump of the numerical result at the crossover point.
We have shown that the smooth cutoff improves the irregular behavior of the free energy in two-dimensional Ising model. Further improvements would be important to obtain the precise results in more complicated lattice models or higher dimensional models with the tensor network schemes.