Exact blocking formulas for spin and gauge models

TL;DR

This paper demonstrates that tensor renormalization group methods enable exact, local blocking formulas for partition functions in 2D spin and gauge models, offering a new analytical approach for lattice theories.

Contribution

It provides the first exact, local blocking formulas for various 2D spin models and 3D gauge theories using tensor renormalization group techniques.

Findings

Exact blocking formulas for 2D O(2), O(3), and 3D Z_2, U(1), SU(2) gauge models.

Tensor RG allows for compact and local coarse-graining expressions.

Potential for generalizations to other models and higher dimensions.

Abstract

Using the example of the two-dimensional (2D) Ising model, we show that in contrast to what can be done in configuration space, the tensor renormalization group (TRG) formulation allows one to write exact, compact, and manifestly local blocking formulas and exact coarse grained expressions for the partition function. We argue that similar results should hold for most models studied by lattice gauge theorists. We provide exact blocking formulas for several 2D spin models (the O(2) and O(3) sigma models and the SU(2) principal chiral model) and for the 3D gauge theories with groups Z_2, U(1) and SU(2). We briefly discuss generalizations to other groups, higher dimensions and practical implementations.