Faster methods for contracting infinite two-dimensional tensor networks

We revisit the corner transfer matrix renormalization group (CTMRG) method of Nishino and Okunishi for contracting two-dimensional (2D) tensor networks and demonstrate that its performance can be substantially improved by determining the tensors using an eigenvalue solver as opposed to the power method used in CTMRG. We also generalize the variational uniform matrix product state (VUMPS) ansatz for diagonalizing 1D quantum Hamiltonians to the case of 2D transfer matrices and discuss similarities with the corner methods. These two new algorithms will be crucial to improving the performance of variational infinite projected entangled pair state (PEPS) methods.

Figure 1

Fixed point equations for the CTMs.

Figure 2

Plots (a) and (b) show the error in the magnetization for the isotropic 2D classical Ising model as a function of computation time at two temperatures near criticality, where (b) is closer to criticality than (a). The network has a bond dimension of $d=2$, and a boundary MPS bond dimension of $\chi =600$ is used. A fully symmetric CTM ansatz is used for CTMRG and the FPCM, and full symmetry is exploited in VUMPS. The speedup of VUMPS and the corner method over CTMRG increases as one gets closer to criticality. Stars indicate the environment tensors have reached a fixed point, and data points beyond those points are numerical fluctuations and were not shown in order to simplify the plot. Plot (c) shows convergence time as a function of inverse temperature above criticality, $\beta /{\beta }_c-1$, for the 2D classical Ising model. For all data points, a boundary MPS bond dimension of $\chi =600$ is used. All data is converged to an error in the magnetization of $<2\times {10}^{-9}$. The inset shows the ratio of the convergence time of CTMRG and VUMPS with respect to the FPCM convergence time (note the logarithmic scale).

Figure 3

Plots of error in magnetization for the isotropic ferromagnetic 2D classical Ising model at $\beta /{\beta }_c-1={10}^{-3}$ with random nonunitary gauge transformations introduced on the horizontal and vertical links, as shown in Eq. (43). This artificially breaks the lattice symmetry in order to test each method on an asymmetric network. Plot (a) shows results for the asymmetric CTMRG algorithm by Corboz et al. in Ref. [51] and plot (b) shows results for the FPCM introduced in this work combined with the CTMRG algorithm used in (a).

Figure 4

(a) Plot of magnetization for the 2D classical XY model, for network bond dimension $d=25$ and boundary MPS bond dimension $\chi =50$. (b) Plot of error in energy (compared to Monte Carlo results) for the 2D quantum Heisenberg model. The network bond dimension is $d=25$ (or PEPS bond dimension $D=\sqrt{d}=5$), and the MPS boundary bond dimension $\chi =100$. (c) Plot of error in the norm (where the “exact” results is taken to be an extrapolation of the norm in the limit of a large environment bond dimension) of the chiral RVB PEPS. The network bond dimension is $d=9$ (or PEPS bond dimension $D=\sqrt{d}=3$), and the boundary MPS bond dimension is $\chi =800$.

Figure 5

Comparison of spectra that need to be inverted to perform the biorthogonalization in different versions of the asymmetric CTMRG algorithm. Note that what is plotted, ${\mathrm{\Sigma }}_{L,i}$, are the square roots of the singular values that are calculated during the biorthogonalization. See the main text for details.