Loop Optimization for Tensor Network Renormalization

We introduce a tensor renormalization group scheme for coarse graining a two-dimensional tensor network that can be successfully applied to both classical and quantum systems on and off criticality. The key innovation in our scheme is to deform a 2D tensor network into small loops and then optimize the tensors on each loop. In this way, we remove short-range entanglement at each iteration step and significantly improve the accuracy and stability of the renormalization flow. We demonstrate our algorithm in the classical Ising model and a frustrated 2D quantum model.

Figure 1

Three key steps of the Loop-TNR algorithm. (a) The entanglement filtering step. Projectors are inserted to eliminate local entanglements on the squares labeled with gray circles [see (g) for details]. (c) The loop optimization step. Each of the shaded squares is deformed to a octagon made up of 8 rank-3 tensors with bond dimensions no greater than $\chi$. The best approximation is found by minimizing the cost function in (h). (e) The same coarse graining step as in the standard LN-TNR algorithm. (h) The cost function of the loop optimization can be regarded as the distance between two MPS wave functions. The well-developed variational MPS method is applied to minimize the cost function.

Figure 2

Comparison of the relative errors of the free energy per site computed using LN-TNR and Loop-TNR. Results were obtained on a square lattice with $2^{50}$ spins. (a) Relative error as a function of bond dimension $\chi$ at the critical point. (b) Relative error as a function of temperature for off-critical Ising models.

Figure 3

Comparison of central charge and scaling dimensions for LN-TNR and Loop-TNR at different iteration steps. The red dotted line denotes the central charge, the blue (light gray) solid lines denote the scaling dimensions in the $Z_2$-odd sector, and the black solid lines denote the scaling dimensions in the $Z_2$-even sector. In the $L=2$ ($L=4$) case, a transfer matrix is constructed using two (four) columns of tensors [shown in (g) and (h)]. The central charge and scaling dimensions are determined from the eigenvalues of the transfer matrix [8].

Figure 4

Benchmark of the variational energy of the $D=3$ PEPS proposed in Ref. [64] for the maximally frustrated $J_1-J_2$ antiferromagnetic Heisenberg model on a square lattice (with $J_2=0.5J_1$). Here, we consider a 256 site system with PBC. Because the benchmark energy (dashed line) is an extrapolation for infinite systems, it could be slightly lower than the actual variational energy for 256 sites.