Renormalization of tensor networks using graph-independent local truncations

We introduce an efficient algorithm for reducing bond dimensions in an arbitrary tensor network without changing its geometry. The method is based on a quantitative understanding of local correlations in a network. Together with a tensor network coarse-graining algorithm, it yields a proper renormalization group (RG) flow. Compared to existing methods, the advantages of our algorithm are its low computational cost, simplicity of implementation, and applicability to any network. We benchmark it by evaluating physical observables for the two-dimensional classical Ising model and find accuracy comparable with the best existing tensor network methods. Because of its graph independence, our algorithm is an excellent candidate for implementation of real-space RG in higher dimensions. We discuss some of the details and the remaining challenges in three dimensions. Source code for our algorithm is freely available.

Figure 1

Graphical depiction of the construction of the tensor network associated with a given classical Hamiltonian. To each vertex of the square lattice is associated a configuration variable. We can associate (i) to each edge of the lattice a local Boltzmann weight $W$ which is a function of its neighboring configuration variables. In step (ii) we decompose each $W$ into two matrices $M$ and $M^{\top }$. The contraction (iii) of such four matrices define the initial tensor $A$ of the tensor network.

Figure 2

Single iteration of the TRG algorithm, producing coarse-grained tensor $A^{(s+1)}$ starting from tensor $A^{\left(s\right)}$. In step (i), the $A^{\left(s\right)}$ tensors of the homogeneous network are split via a truncated SVD along two different diagonals. In step (ii), sets of four tensors are contracted together to form $A^{(s+1)}$. This results in a new square lattice, rotated by ${45}^{\circ }$ and with the lattice spacing multiplied by $\sqrt{2}$. The same two steps can be repeated to return the lattice to the original orientation, now with the lattice spacing doubled. The red shading represents short-range correlations of the CDL type, introduced in Fig. 3. Half of the red loops are captured in the contraction of $A^{(s+1)}$, but the rest remain, and become nearest-neighbor correlations of the coarse-grained tensors. This violates the principle of RG that the coarse-grained description of physics should not include UV details. A detailed analysis of how CDL tensors are a fixed point of the TRG transformation, to complement the more schematic picture here, is given in Appendix pp1-s1.

Figure 3

Lattice of $A^{\mathrm{CDL}}$ tensors forms closed loops within plaquettes. Any observable inserted on a leg would be highly correlated with other observables around the same plaquette, but entirely uncorrelated with anything further away. Thus the CDL model is a toy model for purely short-range physics, and having it flow to a trivial RG fixed point is a common test for tensor network RG algorithms. In this paper we often accompany tensor network diagrams with the red shading shown here that schematically represents loops of CDL-type correlations. We use it to illustrate how CDL tensors behave under different algorithms. A more detailed and rigorous analysis of how the algorithms discussed in this paper apply to CDL tensors is given in Appendix pp1.

Figure 4

Single iteration of Gilt-TNR. In step (i), four matrices $R_1^{\prime },\cdots ,R_4^{\prime }$ are inserted between neighboring tensors, using Gilt. These matrices are factorized in step (ii) via an SVD, and the pieces are absorbed into the neighboring tensors in step (iii). This results in a checkerboard network with two kinds of tensors which then undergoes a regular TRG iteration, depicted in steps (iv) and (v). It shows in particular that the TRG iteration restores the homogeneity of the network. As before, the red shading represents short-range correlations which behave like CDL loops. By the end of the coarse-graining step all such correlations have been removed. Note that in step (iii) short-range correlations are only removed from around every other plaquette, since these were the plaquettes that were used as the neighborhood $T$ when creating $R_1^{\prime },\cdots ,R_4^{\prime }$.

Figure 5

Error in free energy of the critical 2D classical Ising model at different bond dimensions, for TRG and for Gilt-TNR. The numbers next to the data points are total running times in minutes, for the simulation consisting of 25 iterations of the algorithm. The exact running times of course depend on hardware and implementation details, but worth noting is the relatively small difference between the TRG and Gilt-TNR algorithms. Even though adding Gilt into the algorithm slows it down a bit, this is more than compensated for in the quality of the results. For the Gilt-TNR results shown here, the parameter $\epsilon$ has been chosen to be $8\times {10}^{-7}$. Note that this is not the optimal choice of $\epsilon$ for this whole range of $\chi$. Instead, one should vary $\epsilon$ as one varies $\chi$, making it smaller as $\chi$ grows. It is only for simplicity of presentation that we have chosen to stick to a single value of $\epsilon$ that performs well over the whole range of $\chi$'s shown.

Figure 6

RG flow of the coarse-grained tensors, illustrated for TRG (top row) and Gilt-TNR (bottom row) for five different temperatures. The horizontal axis is the linear system size, or in other words the number of RG transformations applied. At each system size, the data points are the 60 largest singular values of the coarse-grained tensor, with the same decomposition as that shown in (18). Thus each of the lines follows the development of one of the singular values along the RG flow. These singular values provide a rough, basis independent characterization of the structure of the tensor. Note how, for TRG, the spectrum is different at every temperature, even at the end of RG flow, when a fixed point has been reached. In contrast, for Gilt-TNR, on both sides of the critical point the RG flow ends in a trivial fixed point characteristic of that phase, with either one or two dominant singular values. At the critical point a complex fixed point structure is found that comes from the CFT. This critical fixed point is maintained over several orders of magnitude in linear system size. These results were obtained with $\chi =110$ for both TRG and Gilt-TNR, and $\epsilon =5\times {10}^{-8}$ for Gilt-TNR.

Figure 7

Typical environment spectra of a single leg with respect to a square plaquette in 2D (bottom four spectra) and a cube in 3D (top four spectra), each labeled with the corresponding bond dimension $\chi$. Recall that in each spectrum, the large values correspond to parts of the vector space of the leg that are relevant for physics outside the plaquette or the cube, whereas small values signify contributions relevant only for short-range details. The spectra in 3D can be seen to decay much more slowly, indicating that larger bond dimensions are necessary, before truncations with a small error are possible. The behavior of the spectra as $\chi$ is increased, is also somewhat different in 2D and 3D. In 2D the longer spectra have more values mainly at the bottom end, whereas in 3D new values appear almost throughout the whole spectrum. The example spectra shown here are for the Ising model, from systems that have been coarse grained thrice. Many other choices of system sizes would yield qualitatively similar results, and the same overall difference between 2D and 3D can also be seen with the three-state Potts model.

Figure 8

Top left panel displays the tensor network representation of the quantum state $|\psi \rangle$ on an infinite lattice covering the half-plane. The hatched strip represents an open boundary, where the boundary indices are uncontracted. In steps (i) to (v) we apply one full step of the Gilt-TNR algorithm leaving open indices untouched. Note that in the coarse-graining steps (iv) and (v) we have chosen to use a slightly different, but qualitatively equivalent, coarse-graining procedure, which more resembles 2D HOTRG [56]. A final contraction is performed in step (vi), resulting in a new homogeneous network with an additional strip of tensors. By iterating this procedure we can obtain a representation for $|\psi \rangle$ as a network of the form shown in Fig. 9.

Figure 9

Approximate representation of a ground state of a quantum Hamiltonian $\mathbb{H}$ obtained by iterating the Gilt-TNR algorithm on a network for the Euclidean path integral, while leaving the indices ending at the open boundary untouched.

Figure 10

The TRG algorithm, as it applies to the CDL model. In step (i) the CDL tensors are split with an SVD. In step (ii) the pieces are contracted together, as shown in the inset, to form the new coarse-grained tensors, which are of the CDL form as well.

Figure 11

Gilt-TNR algorithm applied to the CDL model. In steps (i)–(iii) the $R^{\left(n\right)}$ matrices are inserted on all the legs, using every second plaquette as the environment, as explained in Fig. 4. This results in the removal of half of the CDL loops. Note that truncating each of these loops results in a multiplicative scalar factor $Tr\left[M\right]$, which we omit from the figure. In steps (iv) and (v) TRG is applied to the remaining tensors, which disposes of the remaining CDL loops, yielding trivial tensors of bond dimension 1.