Problems with the definition of renormalized Hamiltonians for momentum-space renormalization transformations

TL;DR

This paper demonstrates that momentum-space renormalization transformations for classical lattice systems can lead to ill-defined Hamiltonians in certain phase diagram regions, highlighting fundamental issues similar to real-space methods.

Contribution

It identifies and analyzes the problems with defining renormalized Hamiltonians in momentum-space renormalization for classical lattice systems.

Findings

Renormalized Hamiltonians are ill-defined in some phase regions.

Similar issues occur in momentum-space and real-space renormalization.

Highlights fundamental limitations of current renormalization approaches.

Abstract

For classical lattice systems with finite (Ising) spins, we show that the implementation of momentum-space renormalization at the level of Hamiltonians runs into the same type of difficulties as found for real-space transformations: Renormalized Hamiltonians are ill-defined in certain regions of the phase diagram.