Scaling dimensions from linearized tensor renormalization group transformations

Tensor network renormalization group (TNRG) is a novel numerical technique for 2D and 3D classical statistical models. The approximation accuracy is controlled by an integer called bond dimension χ, roughly corresponding to the number of coupling kept in a conventional RG scheme. The estimation of the free energy of 2D Ising model converges to the exact value when χ increases; TNRG also provides a highly-accurate estimation of the critical temperature of 3D Ising model. However, it is the universal properties, like scaling dimensions of a critical model, that are more interesting and important. In 2D, methods based on conformal field theory (CFT) arguments exist, but are not applicable in 3D.

In our current work, we put the TNRG method into the standard Wilsonian RG framework, and argue that scaling dimensions can be extracted directly from the linearized tensor RG transformation near a critical fixed point. Then, we propose a concrete 2D numerical implementation of this idea and provide benchmark results using the 2D Ising model. Finally, we show the results of the 3D Ising model based on a straightforward generalization of the 2D implementation.