# Renormalization group flows of Hamiltonians using tensor networks

## TL;DR

This paper develops a tensor network-based renormalization group method for Hamiltonians in two-dimensional classical systems, preserving positivity and enabling interpretation as Hamiltonian flows, with applications to critical phenomena.

## Contribution

It introduces a tensor network renormalization approach that maintains positivity and interprets the flow as Hamiltonian evolution, differing from traditional spin blocking methods.

## Key findings

- Derived algebraic relations for fixed point tensors
- Calculated critical exponents for models
- Benchmarked method on Ising and six-vertex models

## Abstract

A renormalization group flow of Hamiltonians for two-dimensional classical partition functions is constructed using tensor networks. Similar to tensor network renormalization ([G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)], [S. Yang, Z.-C. Gu, and X.-G Wen, Phys. Rev. Lett. 118, 110504 (2017)]) we obtain approximate fixed point tensor networks at criticality. Our formalism however preserves positivity of the tensors at every step and hence yields an interpretation in terms of Hamiltonian flows. We emphasize that the key difference between tensor network approaches and Kadanoff's spin blocking method can be understood in terms of a change of local basis at every decimation step, a property which is crucial to overcome the area law of mutual information. We derive algebraic relations for fixed point tensors, calculate critical exponents, and benchmark our method on the Ising model and the six-vertex model.