Abstract
Abstract We study the parameter dependence of numerical results obtained by the tensor renormalization group. We often observe irregular behavior as the parameters are varied with the method. Using the 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.
Highlights
Tensor renormalization group (TRG) is a promising approach that can solve the sign problem inherent in the Monte-Carlo simulations
We briefly review the TRG method in two-dimensional Ising model presenting a couple of numerical results for later convenience
We present a couple of representative results in the TRG analysis for two-dimensional Ising model on V = 216 × 216 lattice as a preparation of our study explained
Summary
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 [2–13]. It would be difficult to increase Dcut for general lattice theories with multi-dimensional fields so that it should be important to understand and avoid the irregular behavior of the 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.
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