Rotations, Negative Eigenvalues, and Newton Method in Tensor Network Renormalization Group

Abstract

In the tensor network approach to statistical physics, properties of the critical point of a 2D lattice model are encoded by a four-legged tensor which is a fixed point of a renormalization group (RG) map. The traditional way to find the fixed-point tensor consists in iterating the RG map after having tuned the temperature to criticality. Here, we develop a different and more direct technique, which solves the fixed-point equation via the Newton method. This is challenging due to the existence of marginal deformations—linear transformations of the coordinate frame, which parametrize a two-dimensional family of fixed points. We address this challenge by including a 90° rotation into the RG map. This flips the sign of the problematic marginal eigenvalues, rendering the fixed point isolated and accessible via the Newton method. We demonstrate the power of this technique via explicit computations for the 2D Ising and three-state Potts models. Using the Gilt-TNR algorithm at bond dimension χ=30, we find the fixed-point tensors with 10-9 accuracy, much higher than what was previously achieved.