Tensor RG approach to high-temperature fixed point

TL;DR

This paper rigorously analyzes a tensor network RG map for 2D lattice spin systems, proving convergence to the high-temperature fixed point without truncation, advancing understanding of tensor RG maps near criticality.

Contribution

It introduces a non-truncated tensor RG map acting on infinite-dimensional spaces and proves convergence near the high-temperature fixed point, unlike simpler maps that fail to contract.

Findings

The RG map converges to the fixed point tensor when starting close to it.

The simple tensor RG map does not contract due to CDL tensors.

Provides a foundational step towards rigorous tensor network RG analysis near critical points.

Abstract

We study a renormalization group (RG) map for tensor networks that include two-dimensional lattice spin systems such as the Ising model. Numerical studies of such RG maps have been quite successful at reproducing the known critical behavior. In those numerical studies the RG map must be truncated to keep the dimension of the legs of the tensors bounded. Our tensors act on an infinite-dimensional Hilbert space, and our RG map does not involve any truncations. Our RG map has a trivial fixed point which represents the high-temperature fixed point. We prove that if we start with a tensor that is close to this fixed point tensor, then the iterates of the RG map converge in the Hilbert-Schmidt norm to the fixed point tensor. It is important to emphasize that this statement is not true for the simplest tensor network RG map in which one simply contracts four copies of the tensor to define the renormalized tensor. The linearization of this simple RG map about the fixed point is not a contraction due to the presence of so-called CDL tensors. Our work provides a first step towards the important problem of the rigorous study of RG maps for tensor networks in a neighborhood of the critical point.