# Scale invariant transfer matrices and Hamiltionians

## TL;DR

This paper introduces the concepts of scale invariant and weakly scale invariant operators in quantum systems across different scales, revealing conditions under which transfer matrices and Hamiltonians exhibit these properties and their implications for spectral parameters.

## Contribution

It defines scale invariance notions for operators in quantum systems and characterizes when transfer matrices and Hamiltonians are scale invariant or weakly so, linking to classical dynamical systems.

## Key findings

- Scale invariance implies spatial inhomogeneity of the spectral parameter.
- Weakly scale invariant transfer matrices can be spatially homogeneous.
- The change of spectral parameter may follow a fractal dynamical system.

## Abstract

Given a direct system of Hilbert spaces $s\mapsto \mathcal H_s$ (with isometric inclusion maps $\iota_s^t:\mathcal H_s\rightarrow \mathcal H_t$ for $s\leq t$) corresponding to quantum systems on scales $s$, we define notions of scale invariant and weakly scale invariant operators. Is some cases of quantum spin chains we find conditions for transfer matrices and nearest neighbour Hamiltonians to be scale invariant or weakly so. Scale invariance forces spatial inhomogeneity of the spectral parameter. But weakly scale invariant transfer matrices may be spatially homogeneous in which case the change of spectral parameter from one scale to another is governed by a classical dynamical system exhibiting fractal behaviour.